Strange triangular maps of the square

BULL. AUSTRAL. MATH. SOC.

VOL. 51 (1995)

54H20, 58F13, 26A18

[395-415]

STRANGE TRIANGULAR MAPS OF THE SQUARE G.L. FORTI, L. PAGANONI AND J. SMITAL We show that continuous triangular maps of the square I7, F : (z ,y) •-» {f{x)i g(x, y)), exhibit phenomena impossible in the one—dimensional case. In particular: (1) A triangular map F with zero topological entropy can have a minimal set containing an interval {a} X / , and can have recurrent points that are not uniformly recurrent; this solves two problems by S.F. Kolyada. (2) In the class of mappings satisfying Per(.F') = F i x ^ ) , there are nonchaotic maps with positive sequence topological entropy and chaotic maps with zero sequence topological entropy. 1. INTRODUCTION

Let / = [0,1] be the unit interval. By triangular map we always mean a continuous map F : I2 -> I2 of the form F(x,y) = (f(x),g(x,y)) = (f(x),g,(y)) . The map / of the corresponding dynamical system is called the base for F; and gx is a map from the layer Ix = {x} x I to I. Recall that recent results by Kolyada and Sharkovsky show that the dynamical systems generated by triangular maps are essentially more complex than one-dimensional systems, regardless of the fact that for both the same version of Sharkovsky's theorem [14] on the coexistence of cycles is valid [9]; for other references, see [1] and the survey paper [10]. In the present paper we first give a solution to the following two problems from [10]: P\: Can a minimal set of a triangular map contain an interval? P2 : Is the condition h(F) = 0 equivalent to the property that any recurrent point of F is uniformly recurrent, in the class of triangular maps of the square? (Here h(F) denotes the topological entropy of F.) In Section 2 we provide examples showing that the answer to Pi is positive and to P2 negative. (See Theorem 1 below.) In Section 3 we exhibit other examples of strange triangular maps showing that some conditions characterising chaos in the sense of Li and Yorke in the one-dimensional Received 14th June, 1994 Work supported by M.U.R.S.T. Research funds (40%) and C.N.R. Most of the work on this paper was done while the third author was a Visiting Professor at the University of Milan. The support of this institution is gratefully acknowledged. Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/95 395

SA2.00+0.00.

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[2]

case are not equivalent even in the class of triangular maps satisfying Per (F) — Fix (F); it turns out that such maps can have positive sequence topological entropy without being chaotic (Theorem 2). Throughout our paper we use standard terminology as, for example, in [10] (see also [2] or [15]). Some basic notion and results are stated in the text. 2. SOLUTION OF KOLYADA'S PROBLEMS

THEOREM 1 . properties: (i) (ii) (iii)

There are triangular maps Fi,F2 : I2 -* I2 with the following

Both JFI , F2 have zero topological entropy. F\ has a minimal set M containing an interval of the form {a} x I. Fz has a recurrent point which is not uniformly recurrent.

REMARK 1. Recall that M is a minimal set for F if, for any z G M, the w-limit set wp{z) of z is the whole of M. Note that the projection of any minimal set for a triangular map F onto the x-axis is a minimal set for the base map / , and hence, a Cantor set or a cycle [15]. However, the second case implies h(F) > 0, as can be easily verified (see also [9]), so our minimal set M is contained in Q x I, where Q is a minimal w-limit set of the Feigenbaum's type. PROOF: The proof of our theorem has three parts. First we introduce a general construction of triangular maps F with zero topological entropy, and then in the next two parts, by specifying parameters, we prove (ii) and (iii). PART I. GENERAL CONSTRUCTION. Let / : / — » / be a continuous map of type 2°° having a unique infinite w-limit set Q such that / | Q is one-to-one. In this case Q is homeomorphic to the space {0,1}^ of sequences of two symbols equipped with a metric p of pointwise convergence defined, for example, by p(a,j3) = max{l/t ; a(i) ^ /?(*)} for any distinct a = {a(i)} and /3 = {(3(i)} in {0,1}^. Moreover, / acts on Q as the adding machine, that is, for a G {0,1}^, f(a) — a +1000 • • • where the addition is modulo 2 from left to right; for example, /(101100• • •) = 011100• • • , /(11100•• •) = 00010 • • • , et cetera. For details the reader is referred to [3] where the representation of Q is described; the concept of the adding machine was indicated in [13]. Clearly u x f( ) = Q f° r a n 3 r " £ Q | hence Q is a minimal set for / . Our general map F with free parameters will be monotonic on any layer Ix. This will imply h(F) = 0. Indeed, we have (see [9]) Sup{h(F,Ix)

; x £ / } + h{f) > h{F),

where h(F,Ix) denotes the topological entropy of the map F : I2 —» I2 with respect to the compact subset Ix, that is, the entropy h(F,Ix) is computed only from trajectories

[3]

Triangular maps

397

starting from / „ . But since Fx is monotonic on Ix for any t , we clearly have h(F, Ix) = 0, and of course, h(f) = 0 since / is of type 2°°. Thus, h(F) = 0. To have F monotonic on all layers, it suffices to define its restriction to the set Q X I, monotonic on all layers, and then extend it continuously and monotonically to the whole of I2. So we shall define F on Q x / . As already mentioned, we may assume without loss of generality that Q is the space {0, \}N with the metric p as above. For any Jfe ^ 1 and any a € {0,1} 2 , let 2 "(fc +

Thus we have constructed, under conditions (1) and (2) in particular, a triangular map F : I2 —»I2 of zero topological entropy satisfying the identities (4)-(7). Now we specify parameters to get F\ and F2. PART II. PROOF OF (ii). Let {r*}£li be a sequence of rational numbers from (0,1), dense in / and such that

(8)

Km^ r # f = 1. 1/2**

It is easy to construct such a sequence. Put 6k+i = 1 — fk+i P g i v e n b y ( l i m [Fi,F2, ••• ,Fk])uF°°. \k—»oo

I

PROOF OF THEOREM 2: We apply Lemma 1. Let F = [Fi,F2,---] where Fi = T(Ki,Li,Mi,6i,l/4) is defined on the rectangle [A 2 '^ 1 / 4 ),!] X I, h W j = 0 and Ki,Li,Mi are suitable compact sets. Then by (i) of Lemma 1, F is in T\. Denote Jfc(O) = 0 and k(i) = 2s(«i, 1/4) + • • • + 2a(£,-, 1/4). Next let {fflj} be a decreasing sequence of points with ai = 1 and limOj = A. For all i > 0 with 2''1 ^ i < 2> define Mj = [o,-,l] and 5,- = 1/2^'. Since limM; = (A,l], by (iii) of Lemma 1, the trajectory of any point (x,y) € I2 eventually is in oo

(J (F*(Mj X 7)) for some i. Hence this trajectory is eventually constant in the second fc=o variable and consequently, F is nonchaotic. points, namely Let Sj C [a;-+i,a,-) be a set of 22

We write the indexes / in base 2: they are all strings of length 2 J - 1 of digits 0 and 1. For all i with 2 J - 1 ^ i < 2 ; , define Kt as the set of points of Sj having zero at the (i + 1 - 2'~ 1 )-th place and define Li = Sj \ K{. We now show that h^iF) > 0 for a suitable sequence A. By Lemma 1 for any distinct a, 6 £ Sj there exists i = i(a, b) satisfying 2 7 " 1 ^ i < 27 and 1/2 independently of 1/1,3/2, where s(i) = s(6i,l/4) (see Lemma 1). Define

Then, if S(A>m,F,e) is an (A,m,.F,e)-span with the minimal possible number of points, for e = 1/4 and m = 1? — 1 we have

and so

lim lim sup — log #S(A, e-»0 jn-.+oo m

m, F, e) ^ lim sup —z—- log 22 j-»+oo

& ~ 1

1 9lo«2>0 2

D

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G.L. Forti, L. Paganoni and J. Smital

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REMARK 6. It is possible to modify the function F of Theorem 2 in order to get a function F' G T\ which still has positive sequence topological entropy but satisfies conditions (i), (iv) and (v) of Theorem A. To do this, we take a new decreasing sequence {a,} with lim a,j — fj, > A and we construct the sets Ki, Li as in the proof of Theorem 2. Then we define F' = [F[,F^---] £ Tx where F[ = T(JJT!,£J,0,^,1/4), K[ = Ki U [(3fi + A)/4, p] and L\ = U U [(/x + A)/4, (2fj, + A)/4]. Obviously if we don't look for a function in T\ but only for a triangular map, a much simplier example is G(x,y) — (Xx,g(y))

where g(y) is the tent map.

PROOF OF THEOREM 3: Instead of a map from T\ we shall construct a topologically conjugated map F defined on [0,oo] x / (where oo stands for +oo) by F(x,y) = (x 4- l,gx{y)) if x < oo, a.ndF(oo,y) — (oo,y). Moreover we may assume that [0,oo]x I is equipped with the metric |(a:i,yi) — (*2,y2)| = rnaxj AZl — A*2 ,\yi — y2\ >, which makes [0, oo] x / homeomorphic to I2 with the usual metric. This will simplify our notation and, on the other hand, will not change the properties considered, including the sequence topological entropy, since they are invariant with respect to homeomorphic transformations, see [7]. For any integer k > 1 define continuous maps fiki^k '• I —* I by lik{t) = max{0, t - I/A;}, Then

vk{t) = min{2 = (vi,v2) and let m be the greatest integer with m ^ |«i - «! |. (In the sequel we call this m the iterative distance between u and v.) Then, for every n there is jfe(t) such that

\F»{u)-F»(v)\ + 1?±±±, and so, by (27), \Fn(u) - Fn(v)\ - » 0 a s n - » o o . All that remains to prove is (v). Put n(0) = 0, n(i) — 2(fc(l) + Jfe(2) H for i > 0, and choose the sequence {&(i)}j>o of (27) so that (31)

k(i)

divides

k(i +1)

and

—> 0

1- k(i))

monotonically.

nl(i + 1)

Let A — {t(i)}^. 1 be an increasing sequence of positive integers. Fix m ^ 1 and consider the elements U, 1 ^ i ^ m. For each i there exists a unique r such that n(r) ^U < n(r + 1). Denote by ri 0 and choose i — ie such that 10/k(i + 1) < e. Put p := k(i + 1). Since in the definition (19), Ax(F) doesn't change by removing a finite number of elements from A, we may assume without loss of generality that

and A*1 < e. To prove (v) we need an (A,m,F, e)-span T with sufficiently small cardinality. Since A*1 < e, for any u,v 6 [0, oo] x / and a ^ t\ we have (33)

|*"(u)-.F'(t>)|