chaotic behavior of interval maps and total ... - Semantic Scholar

International Journal of Bifurcation and Chaos, Vol. 14, No. 7 (2004) 2161–2186 c World Scientific Publishing Company

CHAOTIC BEHAVIOR OF INTERVAL MAPS AND TOTAL VARIATIONS OF ITERATES GOONG CHEN∗† and TINGWEN HUANG‡ Department of Mathematics, Texas A&M University, College Station, TX 77843, USA ∗ [email protected] ‡ [email protected] YU HUANG§ Department of Mathematics, Zhongshan University, Guangzhou, Guangdong 510275, P. R. China [email protected] Received August 2, 2002; Revised June 5, 2003 Interval maps reveal precious information about the chaotic behavior of general nonlinear systems. If an interval map f : I → I is chaotic, then its iterates f n will display heightened oscillatory behavior or profiles as n → ∞. This manifestation is quite intuitive and is, here in this paper, studied analytically in terms of the total variations of f n on subintervals. There are four distinctive cases of the growth of total variations of f n as n → ∞: (i) the total variations of f n on I remain bounded; (ii) they grow unbounded, but not exponentially with respect to n; (iii) they grow with an exponential rate with respect to n; (iv) they grow unbounded on every subinterval of I. We study in detail these four cases in relations to the well-known notions such as sensitive dependence on initial data, topological entropy, homoclinic orbits, nonwandering sets, etc. This paper is divided into three parts. There are eight main theorems, which show that when the oscillatory profiles of the graphs of f n are more extreme, the more complex is the behavior of the system. Keywords: Total variations; chaos; sensitive dependence; topological entropy; periodic points.

1. Introduction The study of chaotic phenomena in deterministic dynamical systems is a focus of attention in nonlinear science and mathematics. But what is meant rigorously by the term “chaos”? Up to now, there appears no universally accepted definition. To quote

Brown and Chua: “At present chaos is a philosophical term, not a rigorous mathematical term. It may be a subjective notion illustrating the present day limitations of the human intellect or it may describe an intrinsic property of nature such as the “randomness” of the sequence of prime number.

†

Supported in part by Texas A&M University Interdisciplinary Research Initiative IRI 99-22, a TITF initiative, and a DARPA grant. § Supported in part by the National Natural Science Foundation of P. R. China and the Natural Science Foundation of Guangdong Province. 2161

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Moreover, chaos may be undecidable in the sense of Godel in that no matter what definition is given for chaos, there are some examples of chaos which cannot be proven to be chaotic from the definition” [Brown & Chua, 1996]. This notwithstanding, several viable mathematical definitions of chaos exist, each reflecting its own background. The term “chaos” was first coined by Li and Yorke [1975] for a map on a compact interval. Following the work of Li and Yorke, Zhou [1997] gave a definition of chaos for a topological dynamical system on a general metric space as follows. Definition 1.1. A continuous map f on a compact

metric space (X, d) is said to be chaotic on an invariant set X0 in the sense of Li -Yorke provided that there is an uncountable set S ⊂ X 0 , such that

(i) lim supn→∞ d(f n (x), f n (y)) > 0 ∀ x, y ∈ S, x 6= y , (ii) lim inf n→∞ d(f n (x), f n (y)) = 0 ∀ x, y ∈ S .

Another explicit definition of chaos belongs to [Devaney, 1989]. Definition 1.2. Let X be a metric space with metric d(·, ·), and let f : X → X be continuous. We say that f is chaotic on X if

(i) f is topologically transitive on X, i.e. for every pair of nonempty open sets U and V of X, there exists an n ∈ N+ such that f n (U ) ∩ V 6= ∅; (ii) the set of all periodic points of f is dense in X; (iii) f has sensitive dependence1 on initial data, i.e. there exists a δ > 0 such that for every x0 ∈ X and for every open set U containing x0 , there exists a y ∈ U and one n ∈ N+ such that d(f n (y), f n (x0 )) > δ. In Devaney’s definition, condition (i) means that a chaotic system is indecomposable, i.e. the system cannot be decomposed into the sum of two subsystems. Condition (ii) implies that all systems with no periodic point are not chaotic, and condition (iii) says that the system is unpredictable. Which means that a small change of initial data can cause an unavoidable error after many iterations. However, these three conditions are not logically independent. Banks et al. [1992] proved that conditions (i) and (ii) imply condition (iii). Condition (ii) says that all 1

minimal systems cannot be chaotic (a minimal system has no periodic points). This is not reasonable. Thus, [Robinson, 1999] gave a refined definition (see also [Zhou, 1997]) as follows. Definition 1.3. A continuous map on a metric

space (X, d) is said to be chaotic provided that (i) f is topologically transitive, and (ii) f has sensitive dependence on initial data. There are other ways of quantitative measurement of the complex or chaotic nature of the dynamics. They are the Liapunov exponents, various concepts of (fractal) dimensions including the box dimension and the Hausdorff dimension, and topological entropy, as we briefly comment on them as follows. (a) The definition of the Liapunov exponents goes back to the dissertation of Liapunov in 1892 [Liapunov, 1907]. Half a century later Cesari [1959] and Hartman [1964] gave the definition of the Liapunov exponents for time-dependent linear differential equations. For more details, see [Katok & Hasselblatt, 1995; Mane, 1987; Ruelle, 1989; Walters, 1982]. If a system has a positive Liapunov exponent, we say it is chaotic. This definition is perhaps the most computable (in an approximation sense) on a computer. (b) The box dimension and the Hausdorff dimension are two important notions in fractal geometry. See [Edgar, 1990] and [Falconer, 1990] for a detailed discussion. Defined in a constructive way, the box dimension seems to serve a reasonable quantitative measure of chaos [Robinson, 1999]. (c) Topological entropy was first introduced by Adler, Konheim and McAndrew in [Adler et al., 1965] for a compact dynamical system. Later Bowen in 1970 gave a new, but equivalent, definition for a uniformly continuous map on a (not necessarily compact) metric space [Bowen, 1970, 1971]. Topological entropy is an invariant under topological conjugacy. It is indisputable that a system is complex if it has positive topological entropy. We have given in the above six alternative definitions of chaos, each having its own significance. The definition of chaos in terms of topo-

In this condition (iii), the constant δ will henceforth be called a sensitivity constant of f .

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logical entropy is excellent from a mathematical perspective but somehow is not very computable. The notions of chaos in the sense of Li–Yorke and Robinson are very agreeable theoretically and have been widely accepted by mathematicians. The definitions of chaos in terms of Liapunov exponents and (fractal) dimension are more computable on a computer and thus have been widely used by the physics and engineering communities. As we stated earlier, for a topological dynamical system on a general (compact or noncompact) metric space, there are always certain examples which are chaotic in one sense but not in the other. For one-dimensional dynamical systems, nevertheless, things are simpler and the characterization of chaos in terms of periodic orbits, homoclinic orbits, topological entropy, etc., has been widely studied by many mathematicians. See, for example [Block & Coppel, 1992; Katok & Hasselblatt, 1995; Zhou, 1997], and the references therein. For example, we point out that for f ∈ C 0 (I), the space of all continuous maps on a compact interval I, f is chaotic on the nonwandering set Ω(f ) in the sense of Li–Yorke is equivalent to that f has a positive topological entropy [Zhou, 1997]. Throughout this paper, let I be a compact interval and f be a continuous map from I into itself. The nth iterate of f is denoted as f n for n = 1, 2, . . . , while f 0 means the identity map x. In this paper, we characterize the chaotic behavior of f by means of the growth rate of the total variation VI (f n ) of iterates f n of f on the interval I. This paper is divided into three parts. In Part I, we study the growth of the total variations of f n on any subintervals of I in relation to the sensitive dependence on initial data of f . The following results are obtained. If f has sensitive dependence on initial data (according to condition (iii) in Definition 1.2), then limn→∞ VJ (f n ) = ∞ for every closed subinterval J of I. Conversely, if limn→∞ VJ (f n ) = ∞ holds for every closed subinterval J of I and f is piecewise monotone, that is, f has finitely many extremal points on I (cf. Definition 5.1 for TV-chaoticity), then f has sensitive dependence on initial data. We also give an example to show that if f is not piecewise monotone, then f may not have sensitive dependence on initial data even if lim n→∞ VJ (f n ) = ∞ holds for every closed subinterval J of I. Let D ⊂ I be a closed invariant set of f . We call f weakly chaotic on D in the sense of total variation (or WTV-chaotic on D in short) if

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f is piecewise monotone and limn→∞ VD (f n ) = ∞ (cf. Definition 5.2). In Part II, we consider TV-chaotic and WTVchaotic cases in relations to topological entropy and the distribution of the periodic points of f . More precisely, we obtain the following results. If f is TV-chaotic, then f has positive topological entropy. But the converse is not true. If f has positive entropy (equivalently, if f has a periodic point whose period is not a power of 2 or f has a homoclinic point), then limn→∞ VP (f ) (f n ) = ∞, where P (f ) is the set of all periodic points of f . Conversely, if limn→∞ VΩ(f ) (f n ) = ∞, then f has periodic points of period 2k for any k = 1, 2, . . . . Here Ω(f ) is the nonwandering set of f . In Part III, the relations between the growth rates of the total variations of f n and the distribution of the periodic points of f are studied. We obtain the following classification. If f has a periodic point whose period is not a power of 2, then the growth rate of the total variation of f n is exponential as n → ∞. The converse is also true provided that f is piecewise monotone. If f has a periodic point of period 4, then limn→∞ VI (f n ) = ∞. Furthermore, if f has two distinct fixed points and a periodic point of period 2, then limn→∞ VI (f n ) = ∞ also holds. However, in the last two cases, the growth rate to infinity is not exponential with respect to n. Thus we can measure the complexity or the chaotic nature of f by means of the growth rates of the total variations. This is quite intuitive and appears quite agreeable as far as intuition and possible physics and engineering applications are concerned. More importantly, it has enabled us to define spatiotemporal chaos for some one-dimensional partial differential equation systems; for some applications in this direction, see [Chen et al., 2001; Chen et al., 2002; Huang, 2003]. As a matter of fact, this is the main motivation of this paper.

Part I: Sensitive Dependence on Initial Data 2. The Main Results In this Part I, we will probe some fundamental properties of chaos for interval maps. Throughout, we denote N∗ = {1, 2, 3, . . .}. A typical model for explaining chaos is the quadratic map f µ = µx(1 − x), µ > 0, on the unit interval I = [0, 1]. Let 0 < µ ≤ 4 so that fµ : I → I. Then as µ is increasing, a

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Fig. 2. The profile of fµ100 on the interval [0, 1], where fµ is the quadratic map with µ = 3.6. Note that there are many oscillations in the graph.

Fig. 1. The profile of fµ100 on the interval [0, 1], where fµ is the quadratic map with µ = 3.2. Note the nearly piecewise constant feature of the graph.

period-doubling cascade emerges. Let µ 0 be the parameter value such that the period-doubling cascade has been completed. Thus, for each µ: 0 < µ < µ0 , lim d(fµn (x0 ), P2 (fµ )) = 0, for all x0 ∈ [0, 1],

n→∞

(1)

where P2 (fµ ) = the set of all periodic points of fµ of periods 2k , for any k ∈ N∗ , and the distance, d(x, S), between a point x and a set S is defined by d(x, S) = inf{|x − y||y ∈ S}.

(2)

From (1), we can easily deduce the following: “There exists a subsequence {nj |j = 1, 2, . . . , } n ⊆ N∗ such that limj→∞ fµ j (·) is a step function taking values in P (fµ )” . (3) Graphically, this is illustrated in Fig. 1. Next, let µ be in the post period-doubling era, i.e. µ > µ0 . Then the period-doubling route is complete and chaos begins after µ0 . The interval I now

contains many periodic points as well as nonperiodic points. Therefore (3) will no longer be valid for any subsequence {nj |j ∈ N∗ } in N∗ . Graphically, this is reflected in the fact that the profiles of fµn on I contain many “ripples” or oscillations and, as n → ∞, their numbers tend to infinity. We illustrate this in Fig. 2. As the number of oscillations of fµn on I grows, the total variations of fµn on I increase with n and, finally, may become unbounded. (In contrast, this appears uncertain if (3) holds instead.) This phenomenon, quite intuitive and obvious, yet does not seem to have been carefully studied. It has attracted our close attention when we studied wave propagation on an interval (see [Chen et al. 2001]). We were able to establish that when the composite nonlinear boundary reflection map therein is chaotic, the total variations of snapshots of the gradient (wx (·, t), wt (·, t)) tend to infinity as t → ∞ for the solution w(x, t) of such a wave equation on I. We will use VJ (g) to denote the total variation of a function g on an interval J; see [Hewitt & Stromberg, 1965, p. 266], for example. In this part, we wish to use the unbounded growth of V J (f n ), as n → ∞, of an interval map f on subintervals J of I to characterize the chaotic behavior of f . Our conclusion shows that this property is strongly connected to the sensitive dependence on initial data and infinite multi-periodicity of the map f . Let us state the main theorems of this Part below. Main Theorem 1. Let I be a finite closed interval of R and let f : I → I be continuous. Assume

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that f has sensitive dependence on initial data on I (according to condition (iii) in Definition 1.2). Then limn→∞ VJ (f n ) = ∞ for every closed subinterval J of I.

Let x ∈ int(Ii ) ≡ (xi−1 , xi ), the interior of Ii . Then there is a y ∈ int(Ii ) and an Ni ∈ Z+ such that

A straightforward converse of Theorem 1 will not be true, as Example 4.1 in Sec. 4 serves as a good contradiction. Some additional, but natural, assumption on f is needed, as the following theorem shows.

This implies

Main Theorem 2. Let I be a finite closed interval of R and let f : I → I be continuous with finitely many extremal points2 on I. If limn→∞ VJ (f n ) = ∞ as n → ∞ holds for every closed subinterval J of I, then f has sensitive dependence on initial data on I.

In the main body of this Part I, we devote Secs. 3 and 4, respectively, to the proofs of Main Theorems 1 and 2. See Parts II and III, of this paper for further results pertinent to topological entropy, periodic points and growth rates of total variations of chaotic maps.

3. Proof of Main Theorem 1: Sensitive Dependence on Initial Data on an Interval Implies Unbounded Growth of Total Variations of Iterates on Every Subinterval For any interval J = [c, d], let |J| = d−c, the length of J. Throughout the rest of this paper, we denote I = [a, b] as the interval whereupon f is defined. Lemma 3.1. Assume that f : I → I is continuous and f has sensitive dependence on initial data on I. Let J = [c, d] be an arbitrary subinterval of I, with |J| ≥ δ, where δ is a sensitivity constant of f . Then there is an A : 0 < A ≤ δ/2, independent of J, such that

|f n (J)| ≥ A,

for all

n ∈ N∗ .

(4)

Proof. Let N = [2(b − a)/δ] + 1, where for a pos-

itive r ∈ R, [r] denotes the usual integral part of the number r. Divide I into N equal-length subintervals Ii , i = 1, . . . , N , with Ii = [xi−1 , xi ], x0 = a and xN = b, |Ii | = (b − a)/N . Then |Ii | ≤ δ/2 holds for each i.

2

|f Ni (x) − f Ni (y)| ≥ δ . |f Ni (Ii )| ≥ δ .

(5)

For i = 1, 2, . . . , N , let ai = min{|f j (Ii )||j = 0, 1, 2, . . . , Ni } ,

(6)

from which we define A = min{ai |i = 1, 2, . . . , N };

A > 0.

(7)

Then A ≤ δ/2 because |Ii | ≤ δ/2 for each i. Now let J = [c, d] satisfy |J| = d − c ≥ δ. Then there is at least an interval Ijo , 1 ≤ j0 ≤ N , such that J ⊇ Ij0 . Thus, f k (J) ⊇ f k (Ij0 ),

for

k = 0, 1, 2, . . . .

(8)

We are ready to establish (4). We divide the discussion into the following cases: (i) 0 ≤ n ≤ Nj0 . Then |f n (J)| ≥ |f n (Ij0 )| ≥ aj0 ≥ A, by (6)–(8). So (4) holds. (ii) n > Nj0 . If 0 < n − Nj0 ≤ min{Ni |i = 1, 2, . . . , N }, then because of (5) and (8), f Nj0 (J) ⊇ f Nj0 (Ij0 ) and f Nj0 (Ij0 ) being with length at least δ then further contains at least one subinterval I j1 , we obtain f n−Nj0 (f Nj0 (J)) ⊇ f n−Nj0 (f Nj0 (Ij0 ) ⊇ f n−Nj0 (Ij1 ) .

(9)

But n − Nj0 ≤ min{Ni |i = 1, 2, . . . , N }. By (6), (7) and (9), we have |f n (J)| = |f n−Nj0 (f Nj0 (J))| ≥ f n−Nj0 (Ij1 )|| ≥ A . The above restriction that 0 < n − Nj0 ≤ min{Ni |i = 1, 2, . . . , N } can actually be relaxed to 0 < n−Nj0 ≤ Nj1 , if Ij1 is the subinterval satisfying (9), by (6) and (7). One can then extend the above argument inductively to any n = Nj0 + Nj1 + · · · + Njk + Rk , where Rk ∈ N∗ ∪ {0}, 0 ≤ Rk ≤ Njk+1 , and where f n (J) = f Rk (f Nj0 +···Njk (J)) ⊇ f Rk (f Njk (Ijk )) ⊇ f Rk (Ijk+1 )

(10)

For a continuous map f of a real variable, a (local) extremal point of f is either a local maximum or a local minimum of f .

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is satisfied for a sequence of intervals I j0 , Ij1 , . . . , Ijk and Ijk+1 . From (5)–(7) and (10), we have proved (4).  Equipped with Lemma 4, we can proceed to give the following. Let J = [c, d] be any subinterval of I and let M > 0 be a sufficiently large number. We want to prove that Proof of Main Theorem 1.

“there exists N (M ) ∈ N∗ such that VJ (f n ) ≥ M for all n ≥ N (M )” . (11) First, divide J into N subintervals, with N = [M/A] + 1, where A satisfies Lemma 3.1. Thus J = J1 ∪ J2 ∪ · · · ∪ JN , with Jk = [xk−1 , xk ]; xk = c+k((d−c)/N ), k = 1, 2, . . . , N . By the sensitive dependence of f on I, for any k = 1, 2, . . . , N , there exists an Nk such that |f Nk (Jk )| ≥ δ .

(12)

Since f Nk (Jk ) is a connected interval, by (12) we can apply Lemma 3.1 and obtain |f n (Jk )| = |f n−Nk (f Nk (Jk ))| ≥ A, if n ≥ Nk ,

or, more generally, by D(S) = sup{d(x, y)|x, y ∈ S} if a metric d is used in lieu of the absolute value. Also, recall the definition of a perfect set [Hewitt & Stromberg, 1965, p. 70]. A perfect set S is a closed set with no isolated points. Thus, for every point x ∈ S and every open neighborhood U of x, there exists at least a y ∈ U ∩ S such that y 6= x. For example, every Cantor set is perfect. We have the following. Proposition 3.2. Let X and Y be sequentially

compact complete metric spaces and let J be a perfect set in X. Then (1) J has infinite cardinality, except the trivial cases that J is a singleton or the empty set; (2) Assume that f : J → Y is continuous and f is sensitively dependent on initial data on J. Then f (J) is also a perfect set in Y . Proof. Part (1) is easy, so let us prove only part (2).

We first show that f (J) does not have isolated points. If y0 ∈ f (J) were an isolated point of f (J), then there is an x0 ∈ J such that f (x0 ) = y0 . Since x0 is an accumulation point of J, there exists a sequence {xn |n = 1, 2, . . . , } ⊆ J such that dX (·, ·) ≡ the metric in X .

lim dX (xn , x0 ) = 0;

n→∞

for k = 1, 2, . . . , N . Now take N (M ) = max{N1 , . . . , NN }. Then for n ≥ N (M ), n

VJ (f ) =

VJk (f ) ≥

N X

A = NA > M .

k=1

≥

N X

N X

n

k=1

n

|f (Jk )|

k=1

The proof of (11) and, therefore, of Main Theorem 1 is complete.  Very often, for an interval map f : I → I, f has sensitive dependence on initial data only on a “strange” (or hyperbolic) attractor I˜ in I, rather than on the entire interval I. Then Main Theorem 1 is not directly applicable. We need the following modifications. Let I˜ be a bounded closed set in R and let f : ˜ ˜ define I → I˜ be continuous. For any subset S ⊆ I, its diameter D(S) by D(S) = sup{|x − y||x, y ∈ S} ;

(13) By the continuity of f , lim dY (f (xn ), f (x0 )) = lim dY (f (xn ), y0 ) = 0;

n→∞

n→∞

dY (·, ·) ≡ the metric in Y.

(14)

However, y0 is an isolated point. Therefore (14) implies that “there exists an N0 ∈ N∗ , sufficiently large, such that f (xn ) = y0 , for all n ≥ N0 .” (15) But (15) can be seen to violate the sensitive dependence if we choose the elements xn in the sequence of (14) as follows. Let Un (x0 ) be a nested sequence of relatively open sets in J of the point x 0 , such that U1 (x0 ) ⊃ U2 (x0 ) ⊃ · · · ⊃ Un (x0 ) ⊃ · · · , lim D(Un (x0 )) = 0 . n→∞

Because f has sensitive dependence on initial data on J, in each Un (x0 ), there exists an

Chaotic Behavior of Interval Maps and Total Variations of Iterates

x ˜n ∈ Un (x0 ), x ˜n 6= x0 , such that

dX (f N (˜ xn ), f N (x0 )) > δ, for some N ∈ N∗ ,

(16)

where δ is a sensitivity constant of f . Then (13) and (14) hold with x ˜n replacing xn therein. However, we see that (15) contradicts (16) after x n in (15) is replaced by x ˜n . Therefore f (J) does not contain isolated points. Next, we show that f (J) is closed in Y . Let y 0 be an accumulation point of f (J). Therefore, there exists a sequence {xn |n = 1, 2, . . . , } ⊆ J such that lim f (xn ) = y0 .

(17)

n→∞

Since X is sequentially compact, the sequence has a subsequence {xnk |k = 1, 2, . . .} such that lim xnk = x0 ∈ J,

k→∞

for some point x0 .

(18)

lim f (xnk ) = f (x0 ) ∈ f (J) .

(19)

k→∞

From (17) and (19), we obtain y0 = f (x0 ) ∈ f (J). Hence f (J) is closed. Therefore f (J) is a perfect set in Y .  Corollary 3.3. Assume that I˜ is a perfect set in R

and f : I˜ → I˜ is continuous and has sensitive dependence on initial data on I˜ with sensitivity constant δ > 0. Let J be a perfect subset of I˜ with D(J) ≥ δ and J = I˜ ∩ [min J, max J]. Then there is an A : 0 < A ≤ δ/2, independent of J, such that for all n ∈ N∗ .

(20)

Proof. Let us adapt the proof of Lemma 3.1. Let

˜ N = [2(b − a)/δ] + 1. a = min I˜ and b = max I, Define I˜i = I˜ ∩ Ii ; Ii = [xi−1 , xi ] , with x0 = a, xN = b, xi = a + i



By the sensitive dependence of f , again as in (5) we obtain, similarly, D(f Ni1 (I˜i1 )) ≥ δ, for some

for each N i1 ∈ N ∗ .

i 1 ∈ I1 ,

(21)

For each i1 ∈ I1 , let ai1 = min{D(f j (I˜i1 ))|j = 0, 1, 2, . . . , Ni1 } , (22) and define A = min{ai1 |i1 ∈ I1 };

A > 0.

(23)

Then A ≤ δ/2. Now let J be a perfect set with D(J) ≥ δ and J = I˜ ∩ [min J, max J]. Then because D(J) = max J − min J > δ, and J 6= ∅, we have J ⊇ I˜¯i1 for some ¯i1 ∈ I1 . Thus, similarly as in (8), we have (24) f k (J) ⊇ f k (I˜¯i ), for k = 0, 1, 2, . . . . 1

By the continuity of f on J, we have

D(f n (J)) ≥ A,

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,

for i = 1, 2, . . . , N . In general, I˜i may be empty for some i. If I˜i is nonempty, then it may not be perfect because I˜i may contain the endpoints of the interval I i that somehow become isolated points of I˜i . In that case, we modify I˜i by excluding any such isolated points from I˜i . Therefore, we can now separate {I˜i |i = 1, 2, . . . , N } into two groups: {I˜i0 |i0 ∈ I0 ⊆ {1, 2, . . . , N }, I˜i0 = ∅, the empty set} and 2, . . . , N }, I˜i1 6= ∅, I˜i1 is perfect}, {I˜i1 |i1 ∈ I1 ⊆ {1, S ˜ and we have I = i1 ∈I1 I˜i1 .

Again, we divide the discussion into the following cases: (i) 0 ≤ n ≤ N¯i1 . Then D(f n (J)) ≥ D(f k (I˜¯i1 )) ≥ a¯i1 ≥ A, by (21)–(24). (ii) n > N¯i1 . If 0 < n − N¯i1 ≤ min{Ni1 |ii ∈ I1 }, then because of (21) and (24), f N¯i1 (J) ⊇ f N¯i1 (I˜¯i1 ), and f N¯i1 (I˜¯i1 ) being perfect by Proposition 3.2, with diameter at least δ and containing [min f N¯i1 (I¯i1 ), max f N¯i1 (I¯i1 )] ∩ f N¯i1 (I˜¯i1 ), then further contains at least one I˜¯i2 with ¯i2 ∈ I1 . We obtain f n−N¯i1 (f N¯i1 (J)) ⊇ f n−N¯i1 (f N¯i1 (I˜˜i1 )) ⊇ f n−N¯i1 (I˜¯i ) . 2

The rest of the procedures can be carried out similarly as in (9)–(10); we thus omit the details here. Therefore (20) is established.  For a function g defined on a closed subset S of R, we define the total variation of g on S to be the following: (N X VS (g) = sup |g(xk ) − g(xk−1 )| P (S) k=1

x0 , x1 , . . . , xN ∈ P (S) ,  x0 < x 1 < · · · < x N

(25)

where P (S) represents a partition x 0 < x1 < · · · < xN of S, with xj ∈ S for each j = 0, 1, . . . , N and

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the supremum in (25) is taken over all partitions of S. Now we can establish the following refined version of Main Theorem 1. Theorem 3.4. Let I˜ be a perfect set in R and let f : I˜ → I˜ be continuous with sensitive dependence ˜ Let J be a perfect subset of I˜ on initial data on I. with D(J) > 0 and J = I˜ ∩ [min J, max J]. Then lim VJ (f n ) = ∞ .

(26)

n→∞

We refer back to the proof of Main Theorem 1. As before, we want to show that (11) is true. Because J is a perfect set and D(J) > 0, by Proposition 3.2(a) we know that J contains infinitely many points. Let S = {x1 , x2 , . . . , xN } be any finite subset of J with N distinct points, where N = [M/A] + 1, with A given by (20). For x k ∈ S, choose a small interval Jk centered at xk such that J˜k ≡ J ∩ Jk = I˜ ∩ Jk is a perfect set and J˜k1 , J˜k2 overlap at most at their “endpoints” which are defined as max J˜ki , min J˜ki , i = 1, 2, for each distinct pair J˜k1 and J˜k2 . Then by the sensitive dependence of f , for each k = 1, 2, . . . , N , there exists an N k such that D(f Nk (J˜k )) ≥ δ, for k = 1, 2, . . . , N .

[min f Nk (Jk ), max f Nk (Jk )]. So Corollary 3.3 is applicable and we obtain D(f n (J˜k )) = D(f n−Nk (f Nk (J˜k ))) ≥ A ,

if n ≥ Nk , for k = 1, 2, . . . , N .

Now take N (M ) = max{N1 , . . . , NN }. Then for n ≥ N (M ), n

VJ (f ) ≥

Proof.

By

the

continuity

of

f Nk ,

f Nk (Jk ) = I˜ ∩

≥

N X

k=1

N X

n

VJ˜k (f ) ≥

N X

D(f n (J˜k ))

k=1

A = NA > M .

k=1

The proof of (11) and, therefore, of Theorem 3.4 is complete. 

4. A Counterexample. The Proof of Main Theorem 2 First, we show that the direct converse of Main Theorem 1 is not true. Example 4.1. A continuous map f : I → I on an interval I satisfying limn→∞ VJ (f n ) = ∞ for every subinterval J, but f is not sensitively dependent on initial data on I. Define f : I ≡ [0, 1] → I by

 0, x = 0 ;        2  1 2n2 + 2n + 1 1  2n + 2n + 1 x −  , ≤ x ≤ ;   2n + 1 (n + 1)2 (n + 1)2 2n2 (n + 1)2 f (x) = 2 + 2n + 1  2 + 2n + 1   2n 2n 2n2 + 2n + 1 2n2 + 2n + 1 1   − x − + , ≤ x ≤ 2;   2 2 2 2 2 2  2n + 1 2n (n + 1) 2n (n + 1) 2n (n + 1) n     n = 1, 2, 3, . . . .

See Fig. 3. Note that on each subinterval In ≡ [1/(n + 1)2 , 2 1/n ], VIn (f ) = 2 · ((2n2 + 2n + 1)/2n2 (n + 1)2 ) = 2n2 + 2n + 1/n2 (n + 1)2 . Therefore VI (f ) =

∞ X

n=1

VIn (f ) =

∞ X 2n2 + 2n + 1

n=1

n2 (n + 1)2

< ∞ . (28)

The midpoint of the interval In , (2n2 + 2n + 1)/ 2n2 (n + 1)2 , is a fixed point of f . But f vanishes at the endpoints of In , for each n = 1, 2, . . . . We list other properties of f below. (P1) For any a ∈ (0, 1), V[0,a]

(f n )

= ∞ for any

(27)

n = 2, 3, 4, . . . Proof of (P1).

First, we show that

Vh0, 1 i (f ) ≥ j2

1 , j+1

for j = 1, 2, . . . .

(29)

We partition the interval [0, 1/j 2 ] by the points x0 = 0, x2 =

x1 =

1 ,..., (2j)2

1 , [2(j + 1)]2 xj+2 =

1 . j2

Chaotic Behavior of Interval Maps and Total Variations of Iterates

2169

we have Vh0, 1 i (f 2 ) = ∞ . j2

It is also clear that we have Vh0, 1 i (f n ) = ∞

for

n = 3, 4, 5, . . . .

j2

For any a ∈ (0, 1), there is a j0 ∈ N∗ such that 1/j02 < a. Thus V[0,a] (f n ) ≥ V

0,

1 j2 0

 (f n )

= ∞,

and (30) has been proved.

for

n = 2, 3, 4, . . . ,



(P2) For any a, b ∈ [0, 1], a < b, we have Fig. 3.

lim V[a,b] (f n ) = ∞ .

The graph of the function f (x) defined by (27).

n→∞

Proof of (P2).

0,

1 j2

i (f )

≥

j+1 X i=0

≥2

V[xi ,xi+1 ] (f ) ≥ 2

2j X i=j

1 ≥2 (i + 1)2

2j X 2i2 + 2i + 1 i=j

2j X i=j

2i2 (i + 1)2

1 (i + 1)(i + 2)

 2j  X 1 1 1 − . ≥ =2 i+1 i+2 j+1 i=j

Next, we prove that

Since

We divide the discussion into several

cases.

Then Vh

(31)

Vh f



Because Vh

0, 12 j

0,

i (f 2 )

P∞

1 ,1 (j+1)2 j 2

j=N0

i (f 2 )

j∈N .

for any

1 1 , 2 2 (j + 1) j Vh

1 j2

=∞

∗



i (f 2 )



1 ⊇ 0, (j + 1)2 ≥

1 , j+1

(30)



,

by (29).

1/j is divergent, ≥

N1 X

≥

N1 X

i=j

i=j

Vh

1 ,1 (i+1)2 i2

i (f 2 )

1 → ∞ as N1 → ∞ , i+1

(i) Let A1 = {0} ∪ {1/n2 |n ∈ N∗ }. We claim that if [a, b] ∩ A1 6= ∅, then lim V[a,b] (f n ) = ∞,

n→∞

for

n = 2, 3, . . . .

(32)

This is easy to establish, because if x 0 ∈ [a, b]∩ A1 , then f (x0 ) = 0. Therefore the range of f ([a, b]) contains [0, δ] for some δ > 0 and we can apply (P1) to obtain (32). (ii) Let A2 = {(2n2 + 2n + 1)/2n2 (n + 1)2 |n ∈ N∗ }. We claim that if [a, b] ∩ A2 6= ∅, then lim V[a,b] (f n ) = ∞,

n→∞

for

n = 2, 3, . . . .

(33)

To show (33), let a0 ∈ [a, b] ∩ A2 , a0 ≡ (2n20 + 2n0 + 1)/2n20 (n0 + 1)2 , for some n0 ∈ N∗ . Assume that for some b0 > a0 , I0 ≡ [a0 , b0 ] ⊆ [a, b]. (The case [b0 , a0 ] ⊆ [a, b] can be treated in the same way.) If I0 3 1/n20 , then part (i) above is applicable, and (33) is proven. If I0 63 1/n2 or if   2 1 2n0 + 2n0 + 1 , , I0 ( 2n20 (n0 + 1)2 n20 then it is easy to show that there is an N 0 ∈ N∗ such that 1 f N0 (b0 ) ≤ (n0 + 1)2 Therefore, by (32) we again have (33).

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(iii) Let A ≡ A1 ∪ A2 . By parts (i) and (ii), if I0 ∩ A 6= ∅, then (31) holds. Or, if there exists an n0 such that f n0 (I) ∩ A 6= ∅, then (31) is again valid. Therefore, we assume that I 0 = [a, b] is a subinterval such that f n (I0 ) ∩ A = ∅ for all n ∈ N∗ . We choose an N0 ∈ N∗ such that   N0 1 5 . > 3 b−a Then by the piecewise linearity of (27), the absolute value of the slope on each subinterval   2 2n + 2n + 1 1 , 2n2 (n + 1)2 n2 or   1 2n2 + 2n + 1 , (n + 1)2 2n2 (n + 1)2 is no less than 5/3, so we have 1 ≥ |f N (a) − f N (b)|

2n21 + 2n1 + 1 N −1 |f (a) − f N −1 (b)| 2n21 (n1 + 1)2

=

(for some n1 ∈ N∗ ) 5 ≥ |f N −1 (a) − f N −1 (b)| 3  2 5 ≥ |f N −2 (a) − f N −2 (b)| 3  N 5 ≥ |a − b| > 1 , 3 a contradiction. Therefore, (P2) has been proven.



(P3) The map f is not sensitively dependent on initial data on [0, 1]. Consider the point x0 = 0. For any δ > 0, let U = [0, δ/2], a (relatively) open neighborhood of x0 . Then a simple graphical analysis from Fig. 3 shows that f n (U ) ⊆ [0, δ/2], for any n ∈ N∗ . Therefore f : [0, 1] → [0, 1] does not have sensitive dependence on initial data.  Proof of (P3).

Properties (P2) and (P3) have furnished a counterexample to the converse of Main Theorem 1. A major reason why the direct converse of Main Theorem 1 fails, as manifested by Example 4.1, is

that the map f in (27) itself “has too many extremal points”. To state a “good” converse to Main Theorem 1, we therefore need to add an extra assumption that f has only finitely many extremal points on I in the statement of Main Theorem 2. For convenience, we label the basic assumptions on f below. [H1 ] On the interval I = [a, b], f : I → I is continuous, such that for any closed subinterval J ⊆ I, limn→∞ VJ (f n ) = ∞. [H2 ] f : I → I is continuous, and f has finitely many extremal points on I. If f satisfies [H2 ], we also say that f is piecewise monotone. We now proceed to prove Main Theorem 2, step by step, below. Proposition 4.1. Assume [H1 ]. Then

(i) f (x) 6≡ c on any subinterval J of I, for any constant c; (ii) f (x) 6≡ x on any subinterval J of I; (iii) Assume also [H2 ]. Let J be a subinterval of I whereupon f is monotone. Then there exists at most one point x ¯ ∈ J such that f (¯ x) = x ¯. Consequently, f has at most finitely many fixed points on I. (iv) Let J be a subinterval of I and x0 ∈ J satisfies f (x0 ) = x0 . If f is increasing on J, then f (x) > x for all x > x0 , x ∈ J, and f (x) < x for all x < x0 , x ∈ J. This property also holds if J is an interval with x0 either as its left or right endpoint. (v) If f satisfies both [H1 ] and [H2 ] on I, then so does f n for any n ∈ N∗ . Proof. Part (i) is obvious. Consider part (ii). If f (x) ≡ x on J, then

VJ (f n ) = |J|

for every

n.

This violates [H1 ]. Now consider (iii). Let us first assume that f is monotonically decreasing on J. Define F (x) = f (x) − x. Then F is also decreasing on J. If there were two points x ¯1 and x ¯2 ; x ¯1 , x ¯2 ∈ J, x ¯1 6= x ¯2 , such that f (¯ xi ) = x ¯i , i = 1, 2, then F (¯ x1 ) = F (¯ x2 ) and therefore F (x) ≡ 0 on a subinterval of J, implying f (x) ≡ x on J, contradicting part (ii). If f is monotonically increasing on J and there exist two fixed points x ¯1, x ¯2 ∈ J, x ¯1 < x ¯2 , then f is monotonically increasing on J0 ≡ [¯ x1 , x ¯2 ], and f n is also increasing on J0 , such that f n (J0 ) = J0

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Chaotic Behavior of Interval Maps and Total Variations of Iterates

for every n ∈ N∗ . Thus VJ0 (f n ) = |J0 | 9 ∞ as n → ∞, contradicting [H1 ]. Therefore, we have established (iii). Further, consider (iv). If there exists x 1 ∈ J such that x1 > x0 and f (x1 ) ≤ x1 , then f (x1 ) < x1 because f (x1 ) = x1 is ruled out by (iii). Consider J1 ≡ [x0 , x1 ], if x0 < x1 . Then f is increasing on J1 , so are f n for any n ∈ N∗ , such that f n (J1 ) ⊆ J1 . Hence VJ1 (f n ) ≤ |J1 |, violating [H1 ]. The case that x1 < x0 and f (x1 ) ≥ x1 similarly also leads to a contradiction. Finally, part (v) is obvious. Its proof is omitted.  We see that Proposition 4.1 (iv) is actually a hyperbolicity result, i.e. if [H 1 ] holds, and if f is increasing and differentiable at a fixed point x0 , then f 0 (x0 ) > 1. (One can actually prove that under [H1 ] if f is monotone and differentiable in a neighborhood of x0 , then |f 0 (x0 )| > 1. The proof is easily done by using f 2 instead of f .) However, throughout this paper we never needed the differentiability assumption on f . Remark 4.1.

Lemma 4.2. Assume [H1 ] and [H2 ]. Let x ¯0 be a

fixed point of f on I and U be a small open neighborhood of x ¯ 0 in I. Then there exists a δ0 > 0 such that for any x ∈ U \{¯ x0 }, there exists an Nx ∈ N∗ , Nx depending on x, such that ¯ 0 | > δ0 . |f Nx (x) − x

(34)

Proof. By [H2 ], we have two possibilities: (i) f is

monotone on U = [¯ x0 − δ, x ¯0 + δ] for some sufficiently small δ > 0; (ii) x ¯ 0 is an extremal point of f . First, consider case (i) when f is increasing on U . Since [H1 ] is assumed, Proposition 4.1 (iv) gives f (¯ x0 − δ) < x ¯0 − δ,

f (¯ x0 + δ) > x ¯0 + δ .

Thus, we can find x1 ∈ (¯ x0 −δ, x ¯0 ), x2 ∈ (¯ x0 , x ¯0 +δ), such that f (x1 ) = x ¯0 − δ,

f (x2 ) = x ¯0 + δ .

(35)

Define δ0 = min{¯ x 0 − x1 , x2 − x ¯0 }. We now show that (34) is true. Assume the contrary that (34) fails for some x ˆ ∈ (¯ x0 − δ, x ¯0 ) ∪ (¯ x0 , x ¯0 + δ). Then |f n (ˆ x) − x ¯ 0 | ≤ δ0 ,

for all

n ∈ N∗ .

(36)

We consider the case x ˆ>x ¯ 0 . (The case x ˆ x ¯0 and f (¯ x0 + δ) < x ¯0 . Thus we can find x1 ∈ (¯ x0 − δ, x ¯0 ) and x2 ∈ (¯ x0 , x ¯0 + δ) such that f (x1 ) = x ¯0 + δ,

f (x2 ) = x ¯0 − δ .

Let δ0 = min{x2 − x ¯0 , x ¯0 − x1 }. If (34) were not true for this δ0 , then there is an x ˆ ∈ (¯ x 0 − δ0 , x ¯0 ) ∪ (¯ x0 , x ¯0 + δ0 ) such that |f n (ˆ x) − x ¯ 0 | < δ0 ,

for all

n ∈ N∗ .

(37)

We may assume that x ˆ > x ¯ 0 . (The case x ˆ < x ¯0 can be treated similarly.) Since f 2 is increasing on [¯ x0 , x ˆ] and by (37) and f (¯ x0 ) = x ¯0 , we have f 2n ([¯ x0 , x ˆ]) ⊆ [¯ x0 , x ¯0 + δ0 ] for all n ∈ N∗ . Therefore V[¯x0 , xˆ] (f 2n ) ≤ δ0 ,

for any

n ∈ N∗ ,

contradicting [H1 ]. So case (i) is fine for (35). Now, we proceed to treat case (ii), i.e. x ¯ 0 , as a fixed point of f , is also an extremal point of f . Note that it is also possible that x ¯ 0 = a or x ¯0 = b, i.e. x ¯0 is a boundary extremal point. Let us divide the discussion into the following four subcases: (1) x ¯0 = a; (2) x ¯0 = b; (3) x ¯0 ∈ (a, b) is a (relative maximum; and (4) x ¯ 0 ∈ (a, b) is a relative minimum. Subcase (1) implies that x ¯ 0 = a, as a fixed point, must be a local minimum. Let x ˜1 = min{˜ x|˜ x is an extremal point, x ˜>x ¯0} .

Then by Proposition 4.1 (iv), we have f (˜ x1) > x ˜1 . Then there exists an x ˆ 1 ∈ (¯ x0 , x ˜1 ) such that f (ˆ x1 ) = x ˜1 . Define δ0 = x ˆ1 − x ¯0 . Then since f is increasing on [¯ x0 , x ˜1 ], the case can be treated as in case (i) earlier. Subcase (2) is a mirror image of subcase (1) and can be treated in the same way. So let us treat subcase (3). Let x ˜1 = max{˜ x|˜ x is an extremal point, x ˜x ¯ 0 }.

Then f is increasing on [˜ x1 , x ¯0 ] and decreasing on [¯ x0 , x ˜2 ]. By Proposition 4.1 (iv), we have f (˜ x1 ) < x ˜1 . Therefore, there exists an x ˆ 1 ∈ (˜ x1 , x¯0 ) such that f (ˆ x1 ) = x ˜1 . If f (˜ x2 ) < x ˜1 , then there is an x ˆ2 = (¯ x0 , x ˜2 ) such that f (ˆ x2 ) = x ˜1 . In this case, we set δ0 = min{¯ x0 − x ˆ1 , x ˆ2 − x ¯0 }. If f (˜ x2 ) ≥ x ˜1 then we set δ0 = x ¯0 − x ˆ1 . The remaining arguments go the same way as in (i) earlier.

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G. Chen et al.

Subcase (4) can be treated in the same way as Subcase (3).  Lemma 4.3. Let x ˆ ∈ I and A = {x0 , . . . , xk } ⊆ I

satisfy

(i) limn→∞ d(f n (ˆ x), A) = 0; (ii) for any xi ∈ A, i = 0, 1, . . . , k, there exists a (i) subsequence nj , j ∈ N∗ such that lim f

(i)

nj

j→∞

(ˆ x) = xi .

Then f k (xi ) = xi , for i = 0, 1, . . . , k. First, we prove that f (xi ) ∈ A, for i = 0, 1, . . . , k. If this were not true, then there is an i0 : 0 ≤ i0 ≤ k, such that yˆ = f (xi0 ) ∈ / A. Define δ = d(ˆ y , A) > 0. Since f is continuous, for each i: 0 ≤ i ≤ k, there is a δi , 0 < δi < δ/2, such that δ |f (x) − f (xi )| < , for any 2 (38) x ∈ (xi − δi , xi + δi ) ∩ I , Proof.

(If xi happens to be a or b, the endpoints of I, then we simply modify the interval in (38) to [x i , xi + δi ] or [xi − δi , xi ] accordingly.) Define δ¯ = min |{δi |i = ˆ ∈ N∗ 0, 1, . . . , k}. By assumption (i), there is an N such that δ ˆ. (39) d(f n (ˆ x), A) < , for any n > N 2 By assumption (ii), for xi0 there is a sufficiently ˆ and large Nj0 ∈ N∗ such that Nj0 > N δ |f Nj0 +1 (ˆ x) − f (xi0 )| < , or 2 (40) δ Nj0 +1 |f (ˆ x) − yˆ| < . 2 By (38) and (40), we have x) − yˆ| y − xi0 | − |f Nj0 +1 (ˆ |f Nj0 +1 (ˆ x) − xi0 | ≥ |ˆ δ δ ≥ . 2 2

|f N0 +n (ˆ x)

− x[n]k0 | < min



 δ0 ¯ ,δ ; 3

(41)

[n]k0 ≡ n(modk0 ) . Since xk0 ∈ A, there is an N1 > N0 such that |f N1 (ˆ x) − xk0 | < δ 0 /3 .

(42)

But xk0 ∈ A\O(x0 ), |xk0 −x[N1 ]k | ≥ d(x0 , O(x0 )) = 0 δ1 . By 41, we have |f N1 (ˆ x) − xk0 | ≥ |xk0 − x[N1 −N0 ]k0 | − |x[N1 −N0 ]k − f N1 (ˆ x)| ≥ δ0 −

δ , 2 which contradicts (39). Hence we have proved that f (A) ⊆ A. Since O(x0 ) ≡ {f n (x0 )|n ≥ 0} ⊆ A, the orbit of x0 must be finite so we let d(f Nj0 +1 (ˆ x), A) ≥

n ≥ 0} .

δ0 3

=

2δ 0 , 3

contradicting (42). Therefore k0 = k + 1. The proof is complete.  Lemma 4.4. Let J be any subinterval of I. Then

there exists an infinite sequence {n j ∈ N|j = 1, 2, . . .}, nj → ∞, such that f nj (J) contains at least an extremal point of f for all n j . Proof. If f is not monotone on J, take n 1 = 0. Then

f n1 (J) = J contains an extremal point of f . If f is monotone on J, then because f cannot be constant on J, f must be either strictly increasing or strictly decreasing on J. Assume first that f is strictly increasing. Then there exists some m1 ≥ 2 such that f m1 is not monotone on J because otherwise VJ (f n ) ≤ b − a

implying

k0 ≡ min{n|f n (x0 ) = x0 ,

By induction, we further have

0

0 ≤ i ≤ k.

≥δ−

Obviously, k0 ≤ k + 1. We now prove that k0 = k + 1. Relabel the indices of the elements of A so that O(x0 ) = {x0 , x1 , . . . , xk0 −1 }. Suppose k0 < k + 1. Define δ 0 = d(xk0 , O(x0 )). Then by condition (ii) there is an N0 ∈ N∗ such that  0  δ ¯ N0 , δ ; cf. δ¯ in (39) . |f (ˆ x) − x0 | < min 3

for all

n = 1, 2, . . . ,

a contradiction. This implies that f is not monotone on f m1 −1 (J) and, therefore, f m1 −1 (J) has an extremal point of f . We then choose n1 = m1 − 1 ≥ 1 in this case. (If instead f is strictly decreasing on J, then the proof is similar.) From [H1 ], f n1 (J) does not collapse to a single point by Proposition 4.1. Choose a subinterval J 1 of f n1 (J) where f is monotone on J1 . Using the above

Chaotic Behavior of Interval Maps and Total Variations of Iterates

arguments again, we have some m2 ≥ 2 such that f m2 is not monotone on J1 . Therefore f m2 −1 (J1 ) contains an extremal point of f . But J 1 ⊆ f n1 (J), and so f m2 −1 (J1 ) ⊆ f n1 +m2 −1 (J) contains an extremal point of f . Define n2 = n1 + m2 − 1. This process can be continued indefinitely. The proof is complete.  Lemma 4.5. Assume [H1 ] and [H2 ]. Let x ˜0 be an

extremal point of f . Then there is a δ > 0 such that for any (relatively) open neighborhood U of ˆ ∈ N∗ such that x ˜0 , there is an x ˆ ∈ U and an N ˆ ˆ N N |f (ˆ x) − f (˜ x0 )| ≥ δ.

Consider Case (2a). Let y0 ∈ W but y0 ∈ / E, and let δ0 = (1/2)d(y0 , E). By Lemma 4.4, for U there is an N1 ∈ N∗ and a sequence {nj } such that f nj (U ) contains at least an extremal point of f for all nj ≥ N1 . Since y0 ∈ W , there is an nk > N1 such that |f nk (˜ x0 ) − y0 | < δ0 /3. Let x ˜˜j ∈ E be such that x ˜˜j ∈ f nk (U ), and let x ˆ ∈ U be such that f nk (ˆ x) = x ˜˜j . Then |f nk (˜ x0 ) − f nk (ˆ x)| ≥ |y0 − f nk (ˆ x)| − |y0 − f nk (˜ x0 )| ≥ d(y0 , E) − |y0 − f nk (˜ x0 )|

δ0 1 ≥ d(y0 , E) − 2 3 δ0 δ0 δ0 − = . = 2 3 6

Proof. Let E = {˜ x0 , x ˜1 , . . . , x ˜k } be the set of all

extremal points of f . We may note that by Proposition 4.1 (iv) and [H2 ], we have a, b ∈ E. Consider the orbit of x ˜0 : O(˜ x0 ) = {f n (˜ x0 )|n = 1, 2, . . .}. There are two possibilities. There are n1 , n2 : n1 > n2 ≥ 0, such that n2 x ); 0 0 ) = f (˜

Case 1.

x f n1 (˜

x0 ) x0 ) 6= f n2 (˜ For any n1 , n2 ∈ N∗ , f n1 (˜ if n1 = 6 n2 .

Case 2.

x0 ). For Consider Case 1 first. Let y0 = f n2 (˜ any interval U , f n2 (U ) is also an interval because f n2 is continuous. This interval f n2 (U ) can never degenerate into a point by Proposition 4.1 (i). Set F (x) = f n1 −n2 (x). Then F also satisfies [H1 ] and ˆ1 ∈ U sat[H2 ]. Pick y1 ∈ f n2 (U ) but y1 6= y0 . Let x n 2 x1 ) = y1 . Then because y0 is a fixed point isfy f (ˆ of F , by Lemma 4.2 there are a δ > 0 (independent of y1 ) and an N ∈ N∗ (dependent on y1 ) such that |F N (y1 ) − F N (y0 )| = |F N (y1 ) − y0 | ≥ δ ,

or

|f

|f

N (n1 −n2 )

N (n1 −n2 )+n2 )

(y1 ) − f

(ˆ x1 ) − f

N (n1 −n2 )

N (n1 −n2 )+n2

(y0 )| ≥ δ ,

(˜ x0 )| ≥ δ .

Therefore, Lemma 4.5 holds for Case 1. Next, consider Case 2. Define W = ω + (˜ x0 )

(i.e. the ω + limit set of x ˜0 )

= {y ∈ R | there is a subsequence nj ,

j = 1, 2, . . . , such that lim f nj (˜ x0 ) = y}. j→∞

There are two subcases: Case 2a. Case 2b.

W 6⊆ E;

W ⊆ E.

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ˆ = nk . Then we have Set δ = δ0 /6 and N ˆ

ˆ

|f N (˜ x0 ) − f N (ˆ x)| ≥ δ , so Lemma 4.5 holds true. Now consider Case (2b). We divide this into two further subcases: Case (2bi).

For all n ∈ N∗ , f n (U ) ∩ W = ∅;

There is an n0 ∈ N∗ such that 6 ∅. f n0 (U ) ∩ W =

Case (2bii).

Consider Case (2bi). Since E is finite by [H 2 ] and by Lemma 4.4, there is an x ˜˜i ∈ E and a ∗ subsequence {ni ∈ N |i = 1, 2, . . .} such that ˜˜i . Since f n (U ) ∩ W = f ni (U ) always contains x ∅, x ˜˜i ∈ / W . Let δ = 1/2d(˜ x˜i , W ) > 0. Since limn→∞ d(f n (˜ x0 ), W ) = 0, there is a j0 sufficiently large such that 1 x˜i , W ), d(f nj (˜ x0 ), W ) < d(˜ 2

for all

j ≥ j0 .

˜˜i , Now, choose N = nj0 > N1 . Since f nj0 (U ) 3 x nj 0 there is an x ˆ ∈ U such that f (ˆ x) = x ˜˜i . Therefore |f nj0 (˜ x0 ) − f nj0 (ˆ x)| = |f nj0 (˜ x0 ) − x ˜˜i |

≥ d(˜ x˜i , W ) − d(f nj0 (˜ x0 ), W ) 1 x˜i , W ) = δ. ≥ d(˜ 2

Hence Lemma 4.5 holds for Case (2bi). Finally, consider Case (2bii). Since f n0 (U ) ∩ x) = W 6= ∅, there is an x ˆ ∈ U such that f n0 (ˆ x ˜˜j , for some x ˜˜j ∈ W ⊆ E. Pick a point y0 ∈ ˆˆ) = ˆˆ ∈ U be such that f n0 (x x)}. Let x f n0 (U )\{f n0 (ˆ

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y0 . Since W ⊆ E and E is finite, W is finite and has, say, k1 elements. By Lemma 4.3, we have f k1 (x∗ ) = x∗ for all x∗ ∈ W . Define F (x) = f k1 (x). Then each x∗ ∈ W is a fixed point of F , and F satisfies [H1 ] and [H2 ] as well, by Proposition 4.1(v). By Lemma 4.2, there exists a 2δ > 0 and N1 (depending on x ˜˜j ) such that |F N1 (y0 ) − F N1 (˜ x˜j )| = |F N1 (y0 ) − x ˜˜j | ≥ 2δ .

Let N = N1 k1 + n0 . Then ˆˆ) − f N (ˆ |f N (x x)| = |F N1 (y0 ) − F N1 (˜ x˜ )| ≥ 2δ . j

Hence, by an application of the triangle inequality, we have or

either |f N (˜ x0 ) − f N (ˆ x)| > δ , ˆˆ)| > δ . |f (˜ x0 ) − f (x N

N

Therefore, Lemma 4.5 holds for Case (2bii). The proof is complete.  Let E = {˜ x0 , x ˜1 , . . . , x ˜k } be the set of all extremal points of f . By Lemma 4.5, for any interval U 3 x ˜ i , there is a δi > 0 (independent of U ) such that there is an x ˆ i ∈ U \{xi } and Ni (dependent on x ˆi ) satisfying

Proof of Main Theorem 2.

|f Ni (ˆ xi ) − f Ni (˜ xi )| > δi ,

i = 0, 1, 2, . . . , k . (43)

Define 2δ ≡ min{δi |i = 0, 1, . . . , k}. For any x ∈ I and any interval U 3 x, by Lemma 4.4, for some 0 N 0 ∈ N∗ , f N (U ) contains an extremal point, say 0 0 x ˜˜j , i.e. x ˜˜j ∈ f N (U ). Since f N (U ) is an interval with positive length, by (43) and Lemma 4.5 there 0 is an x ˆ ∈ f N (U ) and an N˜j such that |f N˜j (ˆ x) − f N˜j (˜ x˜j )| ≥ δ˜j ≥ 2δ . Now, let N = N˜j + N 0 , y1 , y2 ∈ U satisfy 0 0 f N (y1 ) = x ˆ, f N (y2 ) = x ˜˜j . We have |f N (y1 ) − f N (y2 )| = |f N˜j (ˆ x) − f N˜j (˜ x˜j )| ≥ 2δ .

Therefore, for any x ∈ U , by an application of the triangle inequality, we have either |f N (y1 ) − f N (x)| ≥ δ or |f N (y2 ) − f N (x)| ≥ δ . The sensitive dependence of f on initial data has been proven. 

5. Further Results and Comments Huang [2002] has further proved the following result. Theorem 5.1. Assume the same conditions as

Main Theorem 2. Then the map f has periodic points of period 2k for all k ∈ N∗ . T. Huang’s proof in [Huang, 2002] was carried out by a thorough classical analysis of the function behavior of f under the given assumptions. It motivated Juang and Shieh [2001] to further establish the following. Theorem 5.2 [Juang & Shieh, 2001].

Let f satisfy

[H1 ] and [H2 ]. Then (i) htop (f ), i.e. the topological entropy of f on I, is positive; (ii) f has a periodic point whose period is not a power of 2; (iii) VI (f n ) grows exponentially as n → ∞. In particular, part (ii) of Theorem 5.2 has surpassed Theorem 5.1. For this reason, we will not include the proof of Theorem 5.1 here. Instead, in Part II, we will address Theorem 5.2 and several closely related results. In this Part I, we have shown that if an interval map f : I → I has properties [H1 ] and [H2 ], then it has one of the distinctive features of chaos: sensitive dependence on initial data (and the possession of infinitely many periodic points). Intuitively also, [H1 ] means that the number of oscillations f n tends to infinity as n → ∞. This motivates us to give the following alternative definition of chaos. Definition 5.1. A continuous map f : I → I on an

interval I = [a, b] is said to be chaotic in the sense of total variations, or TV-chaotic, if f satisfies [H 1 ] and [H2 ]. A significant difficulty associated with Definition 5.1 is that, in general, it is not an easy task to directly establish that VJ (f n ) → ∞ as n → ∞ for every subinterval J of I. We may make a compromised attempt by giving the following instead. Definition 5.2. A continuous map f : I → I on an interval I = [a, b] is said to be weakly chaotic on a closed invariant set D of I in the sense of total variations, or WTV-chaotic on D, if f satisfies [H 2 ] and limn→∞ VD (f n ) = ∞.

Chaotic Behavior of Interval Maps and Total Variations of Iterates

In particular, two major results in [Chen et al., 2001] have shown that the following are true. Theorem 5.3 [Chen et al., 2001, Theorem 3.2 and Corollary 3.3]. Let I be a closed interval and f : I → I be continuous satisfying [H2 ]. If f has a periodic orbit of prime period m · 2k , where m is odd and k = 0, 1, 2, . . . , then f is WTV-chaotic on I. Theorem 5.4 [Chen et al., 2001, in the proof of Theorem 3.5]. Let I be a closed interval and f : I → I be continuous satisfying [H2 ]. Assume, say x ¯0 = 0, is a fixed point of f. If this fixed point has a homoclinic orbit, then f is WTV-chaotic on I.

If an interval map is TV-chaotic, then it is WTV-chaotic. But the converse is not true in general. Actually, in Part III, we will show that the rate of growth of VI (f n ) as n → ∞ is important, because if the rate of growth is not exponential (even though VI (f n ) → ∞ as n → ∞ is satisfied), then f may not necessarily even be chaotic in the usual sense according to Main Theorems 7 and 8 in Part III.

Part II: Positive Topological Entropy and WTV-Chaoticity 6. The Main Results We continue the study on chaos and total variation of an interval map f : I → I, where I = [a, b], from Part I. The definition of topological entropy, h top f , of a continuous map f : X → X defined on a metric space X can be found in [Kotak & Hasselblatt, 1995] and [Robinson, 1999], for example. When X = I = [a, b] and f is piecewise monotone, it is known [Kotak & Hasselblatt, 1995, Corollary 15.2.14] that 1 (44) htop (f ) = lim log VI (f n ) . n→∞ n Here, let us adopt (44) as our definition of topological entropy. In [Juang & Shieh, 2001], by using the Smale horseshoe theory, Juang and Shieh have proved Theorem 5.2. This result is interesting. It brings the relationship between chaos and total variations of iterates all the more closer than those already obtained in Part I. In this Part II, we first give an alternative proof of Theorem 5.2 using unstable manifolds and homoclinic orbits. The proof presented in Sec. 7 here is

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shorter than that in [Juang & Shieh, 2001], which constitutes about 20 manuscript pages. We then note that the converse of Theorem 5.2 is not true; see Remark 7.1, i.e. positive topological entropy does not necessarily imply TV-chaoticity. Actually, for a piecewise monotone interval map f , htop (f ) > 0 is equivalent to f having a homoclinic orbit; see Lemma 7.4 in Sec. 7. In Sec. 8, we further refine Theorems 5.3 and 5.4 and obtain the following. If f : I → I is continuous and has a homoclinic point, then

Main Theorem 3.

lim VP (f ) (f n ) = ∞ .

n→∞

(45)

Here P (f ) denotes the set of all periodic points of f , P (f ) is its closure, and VS (g) denotes the total variation of g on a closed bounded set S (see 25). Let Ω(f ) denote the set of all nonwandering points of f (see Sec. 7). Since P (f ) ⊆ Ω(f ), from Theorem 3, we have the following consequence. Corollary 6.1. If f : I → I is continuous and has

a homoclinic point, then

lim VΩ(f ) (f n ) = ∞ .

n→∞

(46)

From the viewpoint of dynamical system theory, it is well known that the dynamical behavior of f n on its nonwandering set can completely determine the dynamical behavior of the entire system. Thus, Main Theorem 3 not only sharpens Theorems 5.3 and 5.4, but also has its own significance. The last theorem of this Part II is a partial converse of Corollary 6.1. It is also a generalization of Theorem 5.1 in Part I. Main Theorem 4.

If f : I → I is continuous and

lim VΩ(f ) (f n ) = ∞ ,

n→∞

(47)

then f has periodic points of period 2 k for any k = 1, 2, . . . . We remark that we do not need the piecewise monotonicity of f in Main Theorem 4.

7. Proof of Theorem 5.2 We need some preliminary definitions and results. Let C 0 (I, I) denote the set of all continuous maps of I into itself. As usual, let f ∈ C 0 (I, I),

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F (f ) and P (f ) denote, respectively, the set of all fixed points and the set of all periodic points of f on I. A point x ∈ I is said to be a nonwandering point of f if for every neighborhood V (x) of x and any positive integer N , there is an integer n > N , such that f n (V (x)) ∩ V (x) 6= ∅. Denote by Ω(f ) the set of all nonwandering points of f . Let p ∈ P (f ). The unstable manifold W u (p, f ) is defined as follows: x ∈ W u (p, f ) if for every neighborhood V of p, x ∈ f n (V ) for some n ∈ N∗ . Lemma 7.1. Let p ∈ F (f ). Then we have

(i) f (W u (p, f )) = W u (p, f ); (ii) W u (p, f n ) = W u (p, f ), integer n.

for

any

positive

Proof. It follows directly from the definition.



Let f ∈ C 0 (I, I). A point x ∈ I is said to be a homoclinic point of f if there is a periodic point p of f such that the following holds (see [Block, 1978], for example): (i) x 6= p; (ii) x ∈ W u (p, f n ) where n is the period of p; and (iii) f nm (x) = p for some positive integer m. The following three lemmas may be found in [Block, 1978].

that (i) implies (iii) is a theorem due to Bowen and Franks [1976], and the proof that (iii) implies (i) can be found in [Zhou, 1988, Theorem 7.3]. Let f ∈ C 0 (I, I) and P (f ) denote the closure of P (f ). Set I − P (f ) =

∞ [

(ai , bi ) .

(48)

i=1

We say the set P (f ) has structure of the first kind if at least one of the two endpoints of every interval (ai , bi ) in the above decomposition belongs to P (f ) ∪ {a, b}. Otherwise, we say P (f ) has structure of the second kind [Zhou, 1988, Definition 5.1]. Lemma 7.5 [Zhou, 1988, Corollary 5.13.1 and

Lemma 5.14]. Suppose that f ∈ C 0 (I, I) has no homoclinic points. Then the following holds: (i) If the two endpoints of (ai , bi ) in 48 are contained in P (f ) − P (f ), then lim |f n ([ai , bi ])| = 0 .

n→∞

(Here, as in Part I, | · | denotes the length.) (ii) If P (f ) has structure of the first kind, then n

lim |W u (pni , f 2 i )| = 0 ,

i→∞

Lemma 7.2. Let f ∈ C 0 (I, I). Then f has a

for any periodic points sequence {p ni }. Here each pni has period 2ni with ni tending increasingly to infinity as i → ∞.

Lemma 7.3. Let f ∈ C 0 (I, I). Assume that p1

By Lemma 7.5, we have the following.

homoclinic point if and only if for some positive integer n, there is a fixed point p of f n and a point z ∈ W u (P, f n ) with z 6= p and f n (z) = p.

and p2 are fixed points of f with p1 < p2 and that there are no fixed points of f in the interval (p 1 , p2 ). Then (i) (p1 , (p1 , (ii) (p1 , (p1 ,

p2 ) ⊂ W u (p1 , f ) if f (x) > x for all x ∈ p2 ); p2 ) ⊂ W u (p2 , f ) if f (x) < x for all x ∈ p2 ).

Lemma 7.4. Let f ∈ C 0 (I, I). The following are

equivalent:

(i) f has a periodic point whose period is not a power of 2 ; (ii) f has a homoclinic point; (iii) htop (f ) > 0. We remark that the equivalence between (i) and (ii) follows from [Block, 1978, Theorem A]. The fact

Corollary 7.6. If f is a TV-chaotic map and has

no homoclinic points, then P (f ) has structure of the first kind. Proof. Assume the contrary. Then there is an in-

terval (ai , bi ) in (48) satisfying (i) in Lemma 7.5, i.e. lim |f n ([ai , bi ])| = 0 .

n→∞

This contradicts the sensitivity of f in Main Theorem 2 in Part I.  Let p be a fixed point of f and let P = (p, f (p)) = (p, p). Let Br (P) be a disk in R2 with center at P and radius r. Set (see [Juang & Shieh, 2001])

Chaotic Behavior of Interval Maps and Total Variations of Iterates

Ip,r = {(x, y) ∈ R2 : p > y > x} ∩ Br (P)

IIp,r = {(x, y) ∈ R2 : y < x < p} ∩ Br (P)

IIIp,r = {(x, y) ∈ R2 : x > y > p} ∩ Br (P)

IVp,r = {(x, y) ∈ R2 : y > x > p} ∩ Br (P)

(49)

Vp,r = {(x, y) ∈ R2 : x < p, y > p} ∩ Br (P)

V Ip,r = {(x, y) ∈ R2 : x > p, y < p} ∩ Br (P) . If f is decreasing on a neighborhood of p. That is, there exists r > 0 such that the graph of f passes by the point (p, f (p)) from Vp,r to V Ip,r , then p is said to be a fixed point of type A for f . If the graph of f passes by (p, f (p)) from II p,r to IVp,r , then p is called a fixed point of type B for f . From Lemma 7.3, we have the following. Lemma 7.7. If p is a fixed point of type B for f, then p is an interior point of W u (p, f ).

If f is a TV-chaotic map and p is a fixed point of f , then neither the graph of f enters the point (p, f (p)) via way of region Ip,r , nor it leaves the point (p, f (p)) by way of region IIIp,r for some r > 0. Thus, f (a+) > a and f (b−) < b. So, f has at least one fixed point p of type A for f and p is an interior point in I. Next, we consider f 2 . The graph of f 2 enters the point (p, f 2 (p)) by IIp,r and leaves it by IVp,r for some r > 0. That is, f 2 is increasing in some neighborhood of p. Since f is TV-chaotic, so is f 2 . By the same argument above, we can conclude that f 2 has at least two points of type A for f 2 , one is in (a, p), another in (p, b). Repeating above procedure, we have (also see [Juang & Shieh, 2001, Lemma 2.5]) the following. Lemma

7.8. Suppose

that f

is TV-chaotic.

Then (i) If p is a fixed point of type A for f, then p is a fixed point of type B for f 2 ; (ii) f has an interior period-2n point which is of n type A for f 2 for any positive integer n; (iii) Suppose p is an interior period 2 n point which n is of type A for f 2 . Set pk = f k (p), 1 ≤ k < 2n , with p0 = p. Then pk is also of type A for n f 2 and any 0 ≤ k < 2n . Proof of Theorem 5.2. Since (i) implies (iii) in Theorem 5.2 by [Kotak & Hasselblatt, 1995, Corollary 15.1.14] and Lemma 7.4, it suffices to prove that TV-chaoticity implies the existence of a homoclinic point.

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Suppose that f ∈ C 0 (I, I) is TV-chaotic and has no homoclinic points. From Corollary 7.6, P (f ) must have structure of the first kind. By (ii) in Lemmas 7.5 and 7.8, for any δ > 0, there exists an integer K and a periodic point p of f with period K 2K such that p is a fixed point of type A for f 2 and K

|W u (pi , f 2 )| < δ ,

f i (p),

(50)

2K .

where pi = 0≤i< On the other hand, again by Lemma 7.8, p i is K also a fixed point of type A for f 2 and so is a K+1 fixed point of type B for f 2 . Lemma 7.7 implies K+1 that pi is an interior point of W u (pi , f 2 ). Thus K u 2 pi is an interior point of W (pi , f ) from (ii) in Lemma 7.1. Therefore, by the continuity of f , there exists a neighorhood V of p such that K

f i (V ) ⊂ W u (pi , f 2 ), ∀ i = 0, 1, . . . , 2K − 1 . (51) From (50), (51) and (i) in Lemma 7.1, we obtain |f n (V )|
3), by Lemma 8.1, x11 is a periodic point of g with period m + 2. We assume x11 , x12 , x13 , x14 in I1 has the following order:

(54)

Thus, for a periodic point with odd period m, we have a Stefan cycle (54) [Robinson, 1999]. Conversely, we also have the following.

(55)

x1 < x13 < x11 < x12 < x14 < x2 .

(59)

From (57), we have I1 ⊂⊂ g 4 ([x11 , x12 ]),

I1 ⊂⊂ g m ([x1 , x11 ]) , (60)

and all the endpoints of the above intervals are periodic points of g. Thus, VI1 ∩P (g) (g m ) ≥ V[x1 ,x11 ]∩P (g) (g m ) + V[x11 , x12 ]∩P (g) (g m ) ≥ (x2 − x1 ) + (x2 − x1 )

= 2(x2 − x1 ) .

(61)

Now we have obtained a periodic point x 11 of g with odd period m + 2. Using the same argument above, we can again obtain a periodic point x 111 of g with odd period m + 4 such that [x11 , x12 ] ⊂⊂ g 4 ([x111 , x112 ]) , and [x11 , x12 ] ⊂⊂ g m+2 ([x11 , x111 ]) , where x112 = g(x111 ). All the endpoints of the subintervals above belong to P (g). Thus, VI1 ∩P (g) (g m+2+4 ) ≥ V[x1 ,x11 ]∩P (g) (g m+6 ) + V[x111 ,x112 ]∩P (g) (g m+6 )

Chaotic Behavior of Interval Maps and Total Variations of Iterates

+ V[x11 ,x111 ]∩P (g) (g m+6 )

Proof. For (i), since all the periodic points of f have

≥ (x2 − x1 ) + (x2 − x1 ) + (x2 − x1 ) ≥ 3(x2 − x1 ) . This process can be continued indefinitely. In general, we can obtain by induction that VI1 ∩P (g) (g m+2k+4k ) ≥ (k + 2)(x2 − x1 ) .

(62)

So we have

finite periods, there exists a positive integer N such that P (f ) = Fix(f N ). Here Fix(g) denotes the set of all fixed points of g. On the other hand, since f N is continuous on the closed interval I, the set of all fixed points of f N is closed. Thus (i) holds because P (f ) is the union of all Fix(f N ). (ii) follows from the Sarkovskii Theorem.  Proof of Theorem 4.

lim V (g k→∞ I1 ∩P (g)

m+6k

2179

) = ∞.

From 62 and the fact that I1 ⊂ g(I1 ), it is easy to see that lim VI1 ∩P (g) (g n ) = ∞ .

lim VΩ(f ) (f n ) = ∞ ,

n→∞

k

lim VI1 ∩P (f ) (f n2 ) = ∞ ,

P (f ) = P (f ) . From Lemma 8.2 and the finiteness of the cardinality of P P (f ), there exists a positive N such that Ω(f ) = P (f ) = Fix(f N ) .

n→∞

and hence

(63)

but there exists a positive integer k such that 2k ∈ / P P (f ). From (ii) of Lemma 8.3, we have that P P (f ) is a finite set. Thus, by (i) of Lemma 8.3,

n→∞

This completes the proof of our claim 55. Recall that P (g) ⊂ P (f ), so we have proved that

Suppose that

Therefore lim VP (f ) (f

n2k

n→∞

) = ∞.

The remaining part of the proof is just the same as the proof of [Chen et al., 2001, Corollary 3.1]. We omit it.  We need two more lemmas in the following in order to prove Main Theorem 4. The first, Lemma 8.2, provides information about the relationship between Ω(f ) and P (f ); its proof can be found in [Nitecki, 1982] and [Zhou, 1988]. Lemma 8.2. Let f ∈

C 0 (I,

I). Then Ω(f ) = P (f )

if and only if P (f ) = P (f ). Lemma 8.3. Assume that f ∈ C 0 (I, I) is piece-

wise monotone.

(i) If all the periodic points of f have finite periods, then P (f ) is closed. (ii) Let P P (f ) denote the set of positive integers that are periods belonging to periodic point of f . If P P (f ) contains an infinite number of integers, then {2k : k = 0, 1, 2, . . .} ⊂ P P (f ) .

That is, f has periodic points with periods 2 k for k = 0, 1, 2, . . . .

lim VΩ(f ) (f kN ) = lim VFix(f N ) (f kN )

k→∞

k→∞

= VFix(f N ) (f N ) ≤ b − a . This contradicts (63). The proof of Main Theorem 4 is complete. 

Part III: Periodic Orbits and Exponential Growth of Total Variations 9. The Main Results Let I = [a, b] and C 0 (I, I) be the space of all continuous maps from I into itself. As in Parts I and II, we continue the investigation of total variations and chaos. In this Part, we consider the growth rate of the total variations of f n as n → ∞ in relation to the distribution of the periods. We have obtained the following main results. Let f ∈ C 0 (I, I). Suppose that f has a periodic point whose period is not a power of 2. Then the growth rate for the total variation of f n on I is exponential as n → ∞. Main Theorem 5.

The converse of Main Theorem 5 is given as follows.

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Let f ∈ C 0 (I, I). Suppose f is piecewise monotone. If the growth rate of the total variation of f n on I is exponential as n → ∞, then f has a periodic point whose period is not a power of 2.

Main Theorem 6.

Thus, by combining Main Theorems 5 and 6 with the known results in Parts I and II we have the following. Corollary 9.1. Let f ∈ C 0 (I, I). Suppose f is

piecewise monotone. Then the following conditions are equivalent:

(i) f has a periodic point whose period is not a power of 2; (ii) f has a homoclinic point; (iii) f has positive topological entropy; (iv) The growth rate for the total variation of f n on I is exponential. Let f ∈ C 0 (I, I). Suppose f has a periodic point whose period is 4. Then Main Theorem 7.

lim VI (f n ) = ∞ .

n→∞

(64)

By Sarkovskii’s Theorem, if f has a 4-periodic point and has no other periodic points whose periods are greater than 4, then f has only 2-periodic points and fixed points besides the 4-periodic points. Thus, the dynamics of such f is expected to be simple. However, from Main Theorem 7, such mappings can still have total variations V I (f n ) growing unbounded as n → ∞. A surprise, we feel. A natural question here is what happens if f only has a periodic point with period 2. Do we still have VI (f n ) → ∞ as n → ∞? In general, the answer is no. For example, let f (x) = −x on x ∈ I = [−1, 1]. Then it is easy to see that, for any positive integer n, VI (f n ) = 2 . However, the following holds. Let f ∈ C 0 (I, I). Suppose that f has two distinct fixed points and a periodic point with period 2. Then

on I = [0, 1] satisfies the conditions of Main Theorem 8 when µ ≥ 3 (see, for example, [Davaney, 1989, p. 33]). Although the results given in Main Theorems 7 and 8 imply that the number of oscillations of f n on I grows as n becomes large, in general these oscillations happen only near repelling fixed points or the repelling periodic points. Remark 9.2.

In the rest of this Part III, we first give some preliminary definitions and results which are necessary in the proofs of our main theorems. Then in Sec. 11, the main theorems will be proven.

10. Definitions and Existing Results Let f ∈ C 0 (I, I) and P = = 0, 1, 2, . . .} be a periodic orbit of f with period m, where m is a power of 2 and m ≥ 2. We say P is simple if for any subset {q1 , . . . , qn } of P where n divides m and n ≥ 2, and any positive integer r which divides m such that {q1 , . . . , qn } is a periodic orbit of f r with q1 < q2 < · · · < qn , we have

Definition 10.1 [Block, 1979].

{f n (x)|n

f r ({q1 , . . . , qn/2 }) = {qn/2+1 , . . . , qn } .

Let us expound the definition above. Let P = {p, f (p), . . . , f 2

lim VI (f ) = ∞ .

n→∞

Remark 9.1.

The quadratic family fµ (x) = µx(1 − x) ,

(p)}

(65)

be a period orbit of f with period 2n for some p ∈ I. n It is easy to see that {p, f 2 (p), . . . , f 2 −2 (p)} and n {f (p), f 3 (p), . . . , f 2 −1 (p)} are two different peri2 odic orbits of f with the same period 2n−1 . When n − 1 > 1, the above two periodic orbits can again be decomposed into four periodic orbits of f 4 with each being the same period 2n−2 . Continuing the above process, we can decompose P into 2 n−1 difn−1 ferent periodic orbits of f 2 , each having the same period 2. On the other hand, the periodic orbit P has another representation by the ordering P = {p1 < p2 < · · · < p2n } .

Main Theorem 8.

n

n −1

(66)

Denote P1 = {p1 < p2 < · · · < p2n−1 } , P2 = {p2n−1 +1 < · · · < p2n } . From Definition 10, P is simple if and only if the decomposition P = P1 ∪ P2 satisfies the following conditions:

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Chaotic Behavior of Interval Maps and Total Variations of Iterates

(1) f (P1 ) = P2 and f (P2 ) = P1 . That is, P1 and P2 are two periodic orbits of f 2 with period 2n−1 ; (2) P1 and P2 can be decomposed by the same treatment as above with P1 and P2 individually, in lieu of P ; (3) The process above can be continued until P is decomposed into 2n−1 different periodic orbits, each has the same period 2. That n−1 n−1 is, f 2 (p1 ) = p2 , f 2 (p2 ) = p1 , . . . , n−1 n−1 f 2 (p2n −1 ) = p2n , f 2 (p2n ) = p2n −1 . Let f ∈ 0 C (I, I). Then f has a periodic orbit whose period is not a power of 2 if and only if f has a periodic orbit with period a power of 2 which is not simple.

Lemma 10.1 [Block, 1979, Theorem 5].

Let P be a periodic orbit of f containing at least two points. Let Pmin (f ) and Pmax (f ) denote, respectively, the smallest and largest elements of P . Denote

(ii) f does not have a fixed point in (p 1 , p2 ) or (p3 , p4 ); (iii) f 2 has a fixed point in (p1 , p2 ) and in (p3 , p4 ), respectively.

11. Proofs of Main Theorems Lemma 11.1. Given two positive integers a 1 and a2 , define a sequence {an } by

an = an−1 + an−2 ,

(70)

Proof. Since a1 ≥ 1 and a2 ≥ 1, we have

and

√ !3−2 1+ 5 . 2

Assuming (70) is true for n ≤ k. Then

D(f ) = {x ∈ I : f (x) < x} , and let PU (f ) (PD (f )) denote the largest (resp. the smallest) element of P ∩ U (f ) (resp. P ∩ D(f )). The following lemmas follow from [Stefan, 1977]. Lemma 10.2. Let f ∈ C 0 (I, I) and let P be

a periodic orbit of f . If f has a fixed point between Pmin (f ) and PU (f ) (resp. between PD (f ) and Pmax (f )), then f has periodic orbits of every period.

Lemma 10.3. Let f ∈ C 0 (I, I) and let P be a

periodic orbit of f . If PD (f ) < PU (f ), then f has periodic orbits of every period.

Let f ∈ C 0 (I, I) and let P = {p1 , p2 , p3 , p4 } be a periodic orbit of period 4 and p1 < p2 < p3 < p4 . If f does not have any periodic orbit whose period is not a power of 2, then, by Lemmas 10.1–10.3, we have Remark 10.1.

f ({p1 , p2 }) = {p3 , p4 } ,

(69)

Then, for every n ∈ N∗ , we have √ !n−2 1+ 5 . an ≥ 2

a3 = a 1 + a 2 ≥ 2 ≥

U (f ) = {x ∈ I : f (x) > x}

n ≥ 3.

(67)

ak+1 = ak + ak−1 √ !k−3 √ !k−2 1+ 5 1+ 5 + ≥ 2 2 # √ ! √ !k−3 " 1+ 5 1+ 5 +1 = 2 2 √ !k−1 1+ 5 = . 2 By induction, (70) holds for every n ∈ N ∗ .



Lemma 11.2 [Kotak & Hasselblatt, 1995, Proposi-

tion 15.2.12].

Let f ∈ C 0 (I, I). Then

htop (f ) ≤ lim inf n→∞

1 log VI (f n ) . n

(71)

Here htop (f ) is the topological entropy of f on I.

and f 2 (p1 ) = p2 , f 2 (p2 ) = p1 , f 2 (p3 ) = p4 , f 2 (p4 ) = p3 . Moreover, the following holds: (i) f has a fixed point in (p2 , p3 );

Lemma 11.3 [Kotak & Hasselblatt, 1995, Corol-

(68)

lary 15.2.14]. Let f ∈ C 0 (I, I). Suppose that f is piecewise monotone. Then 1 log VI (f n ) = htop (f ) . n→∞ n lim

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Thus

11.1. Proofs of Main Theorems 5 and 6 Main Theorem 6 follows from Lemma 11.3 and Lemma 7.4 in Part II. Also, Main Theorem 5 can be obtained from Lemmas 7.4 and 11.2. Here, we will give instead an elementary proof of Main Theorem 5 directly from the hypotheses of the theorem instead of using Lemma 7.4. As a byproduct, we will deduce another proof of the implication from (i) to (iii) in Lemma 7.4. See Remark 11.1. Assume that f has a periodic orbit whose period is not a power of 2. By Sarkovskii’s Theorem, there exists an r = 2 k for some k ∈ N∗ , such that f r has a periodic orbit with period 3. Denote g = f r and let {x1 , x2 , x3 } denote a periodic orbit of g with period 3 with x 2 = g(x1 ), x3 = g(x2 ) and x1 = g(x3 ). Without loss of generality, assume that

VI1 (g) ≥ VI11 (g) + VI12 (g) ≥ |I1 | + |I2 | ≥ 2δ0 = a1 δ0 .

Again, by (78), we have

I1

% &

I1 I2

Proof of Main Theorem 5.

x2 < x 1 < x 3 .

(72)

% & &

I1 I2 I1

So we can find two subintervals in I11 , overlapping at most endpoints, such that they cover I 1 and I2 by g 2 , respectively. Thus we have VI1 (g 2 ) ≥ |I1 | + |I2 | + |I1 | ≥ 3δ0 = a2 δ0 ,

and (77) is proved. For n = 3, we have the following covering relations:

Denote I1 = [x2 , x1 ], I2 = [x1 , x3 ] ,

(73)

δ0 = min{|I1 |, |I2 |} .

(74)

We first claim that VI1 (g n ) ≥

√ !n 1+ 5 δ0 , 2

(75)

N∗ .

for every n ∈ In order to prove our claim, by Lemma 11.1, it suffices to prove n

∗

VI1 (g ) ≥ an δ0 , ∀ n ∈ N ,

(76)

with √ 1+ 5 , and a2 = 3 > a1 = 2 > 2

√ !2 1+ 5 . 2 (77)

We prove (76) by induction. From (72) and (73), we have I1 → I 1 ∪ I 2 ,

I2 → I1 .

(78)

So we can find two subintervals I11 and I12 of I1 , overlapping at most endpoints, such that g(I11 ) = I1 , and g(I12 ) = I2 .

I1

% &

I1 I2

% & &

I1 I2 I1

% & & % &

I1 I2 I1 I1 I2

From the above diagram, similar to the last two steps, we conclude that VI1 (g 3 ) ≥ 5δ0 = (a1 + a2 )δ0 .

Assume that we can find an (= an−1 + an−2 ) subintervals of I1 , overlapping at most endpoints, such that each of them covers I1 or I2 by g n , and the number of subintervals of I1 which cover I1 or I2 , respectively, by g n−1 and g n−2 are an−1 and an−2 . Thus VI1 (g n ) ≥ (an−1 + an−2 )δ0 .

(79)

VI1 (g n+1 ) ≥ (an + an−1 )δ0 = an+1 δ0 .

(80)

Since I1 covers I1 and I2 , and I2 covers I1 by the map g, an−1 subintervals of I1 and an−2 subintervals of I2 cover a total number of an−1 + an−2 subintervals of I1 and an−1 subintervals of I2 by map g. Thus, we can find (an−1 + an−2 ) + an−1 = an + an−1 subintervals of I1 , overlapping at most endpoints, such that each of them covers I1 or I2 by g n+1 . Therefore,

Chaotic Behavior of Interval Maps and Total Variations of Iterates

This proves (76) by induction, so does our claim (75). We have proven that, for every n ∈ N∗ , √ !n 1 + 5 k δ0 , (81) VI1 (f n·2 ) ≥ 2

4 with p1 < p2 < p3 < p4 . From Remark 10.1, we have f ({p1 , p2 }) = {p3 , p4 } ,

for any n ∈ N∗ . In fact, for any l with 0 < l < 2k , we can find a subinterval I0 in I, such that I0 covers I1 by f l (see e.g. [Chen et al., p. 32]). Thus, by the same argument as above, we can deduce √ !n 1 + 5 k k δ0 . VI (f n·2 +l ) ≥ VI0 (f n·2 +l ) ≥ 2 That is, (82) holds. Proof of Main Theorem 5 is complete.  Let f ∈ C 0 (I, I). If f is piecewise monotone and has a periodic orbit of period m · 2k where m is odd for some k ∈ N∗ , then, by Lemma 11.3 and (82), we have √ 1+ 5 −(k+1) htop (f ) ≥ 2 log . 2

f 2 (p1 ) = p2 , f 2 (p2 ) = p1 ,

11.2. Proofs of Main Theorems 7 and 8 C 0 (I,

Let f ∈ I). Suppose f has a periodic orbit whose period is 4. If f has a periodic orbit whose prime period is not a power of 2, then from Main Theorem 5, the growth rate of the total variation of f n on I is exponential as n → ∞. Thus, in this case,

Proof of Main Theorem 7.

lim VI (f n ) = ∞ .

n→∞

So Main Theorem 7 holds. Now assume that f does not have any periodic orbit whose period is not a power of 2. Let P = {p1 , p2 , p3 , p4 } be a periodic orbit of f of period

(84)

f 2 (p3 ) = p4 , f 2 (p4 ) = p3 ,

and there is a fixed point of f in (p2 , p3 ), but no fixed point of f exists in (p1 , p2 ) and (p3 , p4 ). Denote by p0 the fixed point of f in (p2 , p3 ). Thus p1 < p 2 < p 0 < p 3 < p 4 .

(85)

Let I1 = [p2 , p0 ], I2 = [p1 , p2 ] , and δ0 = min{|I1 |, |I2 |} . From (84), we see that I1 covers I1 ∪ I2 and I2 covers I2 by f 2 . Let g = f 2 . We have the following covering diagram: I % 1 I % 1 & I2 I1 & I2 & I2

Remark 11.1.

This offers a quantitative proof that (i) implies (iii) in Lemma 7.4.

(83)

f ({p3 , p4 }) = {p1 , p2 }

k

since g = f 2 . To complete our proof of Theorem 5, we need only prove that, for any given l with 0 < l < 2k , √ !n 1+ 5 n·2k +l VI (f )≥ δ0 , (82) 2

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Thus, by using the same argument as in the proof of Main Theorem 5, we conclude that VI1 (f 2n ) ≥ (n + 1)δ0 .

(86)

Let I0 = [p0 , p4 ]. From (83), I0 covers I1 by f . So, by the same reasoning, we have VI0 (f 2n+1 ) ≥ (n + 1)δ0 . Therefore VI (f n ) ≥ [n/2]δ , for any n ∈ N∗ , where [r] is the integral part of r for r ∈ R. So, we have lim VI (f n ) = ∞ .

n→∞

The proof of Main Theorem 7 is complete.



Let x0 and x1 denote the two distinct fixed points of f such that x 0 < x1 , and let P = {p1 , p2 } denote a periodic orbit of f with period 2 on I, where p1 < p2 . There are three Proof of Main Theorem 8.

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possibilities about the distributions of x 0 , x1 and P on I : Case 1

p 1 < x0 < x1 < p2 ;

(87)

Case 2

x 0 < p1 < p2 ;

(88)

Case 3

p 1 < p2 < x1 .

We only need to deal with Cases 1 and 2, since Case 3 can be done similarly as Case 1. Let us consider Case 1 where (87) holds. Denote I1 = [p1 , x0 ], I2 = [x0 , x1 ], I3 = [x1 , p2 ] . Since f (x0 ) = x0 ,

f (x1 ) = x1

(89)

f (p1 ) = p2 ,

f (p2 ) = p1 ,

(90)

from (87)–(90), I1 covers I2 ∪I3 by f , I2 covers I2 itself and I3 covers I1 ∪I2 . Thus we have the following covering diagram:

I1

% &

I3 I2

% & &

I2 I1 I2

Therefore, by the same argument as in the proof of Main Theorem 5, we have VI1 (f n ) ≥ (n + 1)δ0 ,

(91)

where δ0 = min{|I1 |, |I2 |, |I3 |} . Next, consider Case 2 where 87 holds. This time, we define I1 = [x0 , p1 ],

Then I1 covers I1 ∪ I2 by f and I2 covers I2 itself by f . Thus we have the following covering diagram

I1

% &

I2

% & &

I1 I2 I2

Therefore VI1 (f n ) ≥ (n + 1)δ0 , where δ0 = min{|I1 |, |I2 |} .

lim VI (f n ) = ∞ ,

n→∞

for both Cases 1 and 2. The proof of Main Theorem 8 is complete.  We mention here the work of [Milnor & Thurston, 1988], who set up an effective way of describing the qualitative behavior of the successive iterates of a piecewise monotone mapping. For a continuous piecewise monotone interval map f on I let I1 , . . . , Il denote the subintervals of I on which f is alternately strictly increasing or strictly decreasing. Each such maximal interval where f is monotone is called a lap of f . We denote by l(f ), the lap number, the number of laps f . As an application of the Milnor–Thurston Theorem which computes the Artin–Mazur zeta function in terms of the kneading determinant, it is known that f has only finitely many distinct periods (all necessarily powers of two by Sarkovskii’s Theorem) if and only if the sequence of the lap numbers l(f n ) is bounded by a polynomial function of n. This implies that the total variation VI (f n ) of f n on I grows at a speed at n most polynomially. Combining our results, we obtain that if a continuous piecewise monotone mapping f on I has only finitely many distinct periods and has a period four, then VI (f n ) grows unbounded and at most polynomially as n goes to infinity. Some further relations between the exponential growth of VI (f n ) for f ∈ C 0 (I) and the topological entropy may be deduced from theorems in [Rothschild, 1971; Misiurewicz & Szlenk, 1980; Katok & Hasselblatt, 1995]. Remark 11.2.

12. Conclusion

I2 = [p1 , p2 ] .

I1

From (91) and (92), we have

(92)

Summing up the results obtained above, we have the following relations of implication. Assume here that f is piecewise monotone, though this condition may not be needed in some of the implications. Then Chaos in the sense of Devaney ⇒ sensitive dependence on initial data ⇔ TV-chaos ⇒ exponential growth of VI (f n ) with respect to n as n → ∞ ⇒ positive topological entropy ⇔ existence of a periodic point of a period being not a power of 2 ⇔ existence of a homoclinic point ⇔ chaos in the sense of Li–Yorke ⇒ WTV-chaos on P (f ) ⇒ WTVchaos on Ω(f ) ⇒ the existence of periodic points of periods 2k for any k = 1, 2, . . . ⇒ the existence of a periodic point of period 4 ⇒ WTV-chaos on I.

Chaotic Behavior of Interval Maps and Total Variations of Iterates

These results have convinced us that for a more extreme oscillatory pattern of the graphs of f n , the more complicated is the behaviour of the system. Thus the growth rates of total variations of f n are a valuable quantitative and qualitative measure for the level of complexity of one-dimensional dynamical systems. On the other hand, this notion is directly applicable to the study of chaos in partial differential equations. See, for example [Chen et al., 2001; Huang, 2003]. Nevertheless, the above results hold only for one-dimensional dynamical systems. Similar generalization to multidimensional dynamical systems is a very important and challenging topic. We will consider this in forthcoming papers.

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