STATIONARY MEASURES FOR PROJECTIVE TRANSFORMATIONS ...

STATIONARY MEASURES FOR PROJECTIVE TRANSFORMATIONS: THE BLACKWELL AND FURSTENBERG MEASURES ´ ANY, ´ B. BAR M. POLLICOTT AND K. SIMON

Abstract. In this paper we study the Blackwell and Furstenberg measures, which play an important role in information theory and the study of Lyapunov exponents. For the Blackwell measure we determine parameter domains of singularity and give upper bounds for the Hausdorff dimension. For the Furstenberg measure, we establish absolute continuity for some parameter values. Our method is to analyse linear fractional iterated function schemes which are contracting on average, have no separation properties and, in the case of the Blackwell measure, have place dependent probabilities. In such a general setting, even an effective upper bound on the dimension of the measure is difficult to achieve.

1. Invariant measures for projective transformations In this note we will be interested in the problem of understanding invariant measures for an interesting class of iterated function schemes of the interval. Such problems have been extensively studied in the case of affine iterated function schemes, and there have been a number of partial results in more general settings. However, in this note we will consider the interesting case of linear fractional transformations, whose significance is that they arise naturally from projective transformations. To avoid too much abstraction, we will concentrate on two cases which have historically been particularly important. Namely, the Blackwell measure and the Furstenberg measure. The fundamental question we want to address in each case is when these measures are absolutely continuous or singular. The Blackwell measure, at least in the particular case of binary symmetric channels, is an invariant measure for linear fractional transformations with a certain class of rational weights. This measure was introduced by Blackwell in 1957 and plays a central role in understanding the entropy rate, and other important characteristics, of fundamental models in information theory. We show that for a suitable range of parameter values this measure is necessarily singular. In order to study 1

2

´ ANY, ´ B. BAR M. POLLICOTT AND K. SIMON

the regions upon which stationary measures are singular (in particular in the case of the Blackwell measures), we will employ some recent results established by Jaroszewska and Rams [6]. Another important invariant measure for linear fractional iterated function schemes, arising this time in the study of Lyapunov exponents for matrices, is the Furstenberg measure. This too arises from linear fractional transformations, but this time with constant weights. In this case we can show the more positive result that for almost all parameters in a certain range of values the Furstenberg measure is absolutely continuous. In section 2 we introduce the general framework common to both classes of measures and in section 3 we recall a useful result of JaroszewskaRams. In section 4 we specialise to the case of the Blackwell measure, and in section 5 we concentrate on a particular class of examples corresponding to certain binary symmetric channels with noise. In section 6, we give new criteria for the singularity of the Blackwell measure, and illustrate this with a simple example. In section 7, we turn to the Furstenberg measure, and in section 8 we concentrate on a particular class of examples corresponding to a pair of positive 2 × 2 matrices. We give criteria for the absolute continuity of the Furstenberg measure, and again illustrate this with a simple example. 2. The general framework Consider an Iterated Function Scheme F which consists of the functions {Si }m i=1 defined on a separable metric space (X, %) with the (possibly) m P non-constant probabilities {pi (x)}m , i.e., pi (x) ≡ 1. Let µ be an i=1 i=1

invariant measure for F. That is µ is a stationary measure for the discrete time Markov chain with transition probability function (t.p.f.) m X P (x, E) = pi (x) · 1Si−1 (E) (x). i=1

In this case, for the linear operator T : C(X) → C(X) on continuous functions C(X) defined by m X (1) (T f )(x) := pi (x) · f (Si (x)), i=1

the probability measure µ satisfies (2)

µ = T ∗ µ.

STATIONARY MEASURES FOR PROJECTIVE TRANSFORMATIONS

3

In other words Z Z X m pi (x) · f (Si (x))dµ(x) (3) f (x)dµ(x) = i=1

and, in particular, for a Borel set E ⊂ X we have m Z X pi (x)dµ(x). (4) µ(E) = i=1

Si−1 (E)

One way to prove that a measure is singular is to verify that its Hausdorff dimension is less than one. by definition the Hausdorff dimension of a measure µ is (5)

dimH (µ) := inf {dimH (Z) : µ(Z c ) = 0} ,

where Z c is the complement of the set Z. A very recent theorem of Jaroszewska-Rams [6] gives upper bound under the most general conditions on the Hausdorff dimension of a measure which is invariant for a contracting of average IFS with place dependent probabilities. 3. The Jaroszewska-Rams Theorem In order to study the regions upon which stationary measures are singular (in the case of the Blackwell measures), we will employ some recent general results established by Jaroszewska and Rams [6]. Let M be the set of invariant measures. It is also assumed that µ is ergodic. This means that µ is an extremal point of M. To get a more dynamic characterization Jaroszewska and Rams introduced the skew product map S acting on X × Σ (where Σ = {1, . . . , m}N ) defined by S(x, i) := (Si1 (x), σi). Let µ ∈ M be an invariant measure for the Iterated Function Scheme F and let g : X × Σ → R be continuous function then we define a Borel measure µ e on X × Σ by Z Z Z g(x, i)de µ(x, i) := g(x, i)dpx (i)dµ(x), where px (·) is a probability measure on Σ defined by (6) px ([i1 , . . . , in ]) := pi|n (x) := pi1 (x) · pi2 (Si1 (x)) · · · pin (Sin−1 ...i1 (x)). Then µ e is a S-invariant measure. Theorem 1 (Jaroszewska, Rams). The measure µ is ergodic with respect to the Iterated Function Scheme F if and only if µ e is ergodic w.r.t. (X × Σ, F).


4

In what follows we use the notation that if i = (i1 , . . . , in ) then we ← − write i := (in , . . . , i1 ). Definition 2 (Entropy and Lyapunov exponent defined in [6]). For an |i| = N let n o − (x), S← − (y) < δ, ∀k = 1, . . . , N BN (x, i, δ) := y ∈ X : % S← i|k i|k and hN (µ, δ) := −

Z

X

X |i|=N

pi (x) ·

inf y∈BN (x,i,δ)

log pi (y)dµ(x).

We then define hN (µ) := lim hN (µ, δ) δ→0

and

1 · hN (µ). N To define the Lyapunov exponent we need to introduce: Z X − (x), S← − (y) % S← i i pi (x) · sup log dµ(x), λN (µ, δ) := %(x, y) y6=x,y∈BN (x,i,δ) X h(µ) := lim

N →∞

|i|=N

and λN (µ) := lim λN (µ, δ). δ→0

Finally, the Lyapunov exponent is defined by 1 λN (µ). λ(µ) := lim N →∞ N The following result will be particularly useful in the sequel. Theorem 3 (Jaroszewska, Rams). Let µ be an ergodic measure with respect to the Iterated Function Scheme F. Assume that λ(µ) < 0, i.e., the Iterated Function Scheme F is contracting on average, then we have h(µ) (7) dimH (µ) ≤ . −λ(µ) In the next section we will consider natural examples of contracting on average IFS with place dependent probabilities and no separation properties. Such examples arise quite naturally in the study of certain practical problems. Indeed, these appear to be the first significant examples of such systems for which dimension theoretical properties are treated. Moreover, in these examples we can make precise numerical estimates on h(µ)/λ(µ).


5

4. The Blackwell measure In this section we recall the general construction of the Blackwell measure [1]. Consider a stationary ergodic Markov process X := {xn }∞ n=−∞ with states i = 1, . . . , B, with transition matrix M = [mij ]B ij=1 and associated probability measure P. Then the entropy H(X) is given by (8)

H(X) = −

B X ij=1

P {xn = i} · m(i, j) · log m(i, j).

Assume that A < B, then for any surjective map Φ {1, . . . , B} → {1, . . . , A} we can consider the ergodic stationary process Y := {yn = Φ(xn )}∞ n=−∞ with states i = 1, . . . , A.

Blackwell [1] expressed the the entropy for Y as follows: Z X A (9) H(Y ) = − ra (w) log ra (w)dQ(w), a=1

where the measure Q is called the Blackwell measure and Q can be characterized as the invariant measure of an Iterated Function Scheme below. To do so we first define ra (w) :=

B X

X

i=1

{j: Φ(j)=a}

wi · m(i, j)

for a ∈ {1, · · · , A}. Note that (r1 (w), . . . , rA (w)) is a probability vector for all ) ( B X w ∈ W := w ∈ RB : wi ≥ 0, wi = 1 . i=1

To define the Blackwell measure Q we introduce the functions fa (w) :=

B X j=1

ej · δa (Φ(j)) ·

B X i=1

wi · m(i, j)/ra (w),

where ej is the j-th coordinate unit vector. Note that f : W → Wa , where for a = 1, . . . , A we define Wa := {w ∈ W : Φ(j) = a} . Now we are ready to define Q as the invariant measure of the Iterated Function Scheme:


6

(10)

{f1 (w), . . . , fA (w)} with probabilities {r1 (w), . . . , , rA (w)} .

That is for a Borel set E ⊂ W we have Z A X ra (w)dQ(w). (11) Q(E) = a=1

fa−1 (E)

Alternatively, for a continuous function F : W → R we have Z A Z X F (w)dQ(w) = ra (w) · F (fa (w))dQ(w). (12) a=1

Clearly we have suppQ ⊂

(13)

A [

Wa .

a=1

5. A class of examples of the Blackwell measure We recall the following particular example of a Blackwell measure, from [3, Example 4.1]. We set B = 4, A = 2 and let Φ : {1, 2, 3, 4} → {1, 2} be given by Φ(1) = Φ(4) = 1 and Φ(2) = Φ(3) = 2. In this case we have ( W :=

4

w = (w1 , . . . , w4 ) ∈ R : wi ≥ 0 and

4 X

) wi = 1 ,

i=1

and W1 := {w ∈ W : w = (w1 , 0, 0, w4 )} and W2 = {w ∈ W : w = (0, w2 , w3 , 0)} . In this case the support of the Blackwell measure supp(Q) = W1 ∩ W2 . Theorem 4. The Blackwell measure is singular in the region marked on Figure 1. Moreover, for every parameter value ε, p the Hausdorff dimension of the Blackwell measure is smaller than or equal to the entropy over Lyapunov exponent as defined in Section 3. This ratio in the above theorem can be numerically computed with high precision. Figure 1 shows some level curves of this ratio on the parameter domain (ε, p).


dimH (Q) ≤ 0.5

p

0,35 0,3 0,25 0,2 0,15 0,1 0,05 0

7

dimH (Q) ∈ [0.9, 1]

dimH (Q) ∈ [0.5, 0.7] dimH (Q) ∈ [0.7, 0.9]

dimH (Q) = 1

0,2

0,3

0,4

0,5

0,7

0,6

0,8

e

Figure 1. Estimates on where the Blackwell measure is singular (via higher iterates).

We can then associate the matrix  Π00 (1 − ε) Π00 ε  Π00 (1 − ε) Π00 ε M :=   Π10 (1 − ε) Π10 ε Π10 (1 − ε) Π10 ε Furthermore, let  Π00 (1 − ε)  Π00 (1 − ε) M1 :=   Π10 (1 − ε) Π10 (1 − ε)

0 0 0 0

0 0 0 0

Π01 (1 − ε) Π01 (1 − ε) Π11 (1 − ε) Π11 (1 − ε)

 Π01 ε Π01 ε  . Π11 ε  Π11 ε

  Π01 ε 0   Π01 ε  0 and M2 :=   0 Π11 ε  Π11 ε 0

Π00 ε Π00 ε Π10 ε Π10 ε

Π01 (1 − ε) Π01 (1 − ε) Π11 (1 − ε) Π11 (1 − ε)

Then for a = 1, 2 we have that the weight functions are (14)

ra (w) = w · Ma · 1 =: kw.Ma k1 and fa (w) := w · Ma /ra (w).

More precisely, we can write (15) r1 (w) = (w1 +w2 )·[Π00 (1 − ε) + Π01 · ε]+(w3 +w4 )·[Π10 (1 − ε) + Π11 · ε]

 0 0  . 0  0


8

and (16) r2 (w) = (w1 +w2 )·[Π00 ε + Π01 · (1 − ε)]+(w3 +w4 )·[Π10 ε + Π11 · (1 − ε)] . We can parameterize subsets W1 , W2 ⊂ W by maps wi : [0, 1] → Wi (i = 1, 2) defined by w1 (x) := (x, 0, 0, 1 − x) and w2 (x) := (0, x, 1 − x, 0),

x ∈ [0, 1].

Using the above weights r1 (w), r2 (w) we define the place dependent probabilities pa : [0, 1] → (0, 1) for a = 1, 2 as follows: (17) p1 (x) := r1 (wa (x)) = x·[Π00 (1 − ε) + Π01 · ε]+(1−x)·[Π10 (1 − ε) + Π11 ε] , and (18) p2 (x) := r2 (wa (x)) = x·[Π00 ε + Π01 (1 − ε)]+(1−x)·[Π10 ε + Π11 (1 − ε)] . It is easy to see that p1 (x) + p2 (x) ≡ 1 for all x and so px (i) (defined as in (6)) is a probability measure on Σ for every x ∈ [0, 1]. Finally, we define the Iterated Function Scheme {S1 , S2 } acting on [0, 1] by (19)

S1 (x) := [x · Π00 · (1 − ε) + (1 − x) · Π10 · (1 − ε)] /p1 (x)

and S2 (x) := [x · Π00 · ε + (1 − x) · Π10 · ε] /p2 (x).

A simple calculation shows that in the particular case where Π00 = Π11 = p and Π10 = Π01 = 1 − p then we have S1 (x) =

x (2 p − 2 p ε + ε − 1) + 1 − p − ε + p ε x (−1 − 4 p ε + 2 p + 2 ε) + (1 − p) (1 − ε) + p ε | {z } =:p1 (x)

(20) S2 (x) =

x (2 p ε − ε) + ε − p ε . x (1 + 4 p ε − 2 p − 2 ε) + p + ε − 2 p ε | {z } =:p2 (x)

We know that the Blackwell measure Q is supported by W1 ∪ W2 . Let us write Qa := Q|Wa for a = 1, 2. This leads naturally to the definition of the Borel measures m1 , m2 and m on the interval [0, 1] defined by (21) ma (E) := Qa (wa (E)) for a = 1, 2 and m(E) := m1 (E) + m2 (E). Using the definition (14) of fa for a = 1, 2, we write fa,b := fa |Wb . Then fa,b : Wb → Wa and the following diagram commutes:


(22)

Sa

[0, 1] wb

Wb

/

[0, 1] /

fa,b

9

wa

Wa

This shows the reason that we introduced the IFS {S1 , S2 }. Now we prove that our IFS is eventually contracting. Lemma 5. There exists an n such that for every (i1 , . . . , in ) ∈ {1, 2}n and for every x ∈ [0, 1] we have 0 Si ...i (x) < 1. (23) 1

n

Proof. Using the formulaes in (20) by an immediate calculation we obtain that the compact set J := S1 ([0, 1]) ∪ S2 ([0, 1]) is contained in the open interval (0, 1). Let Jek := wk (J) ⊂ Wk for k = 1, 2. Then Jek is a compact subset of the interior of Wk . Let L := max {|Si0 (x)| : x ∈ [0, 1] and i = 1, 2} . Then by [3, Proposition 2.2] we can choose an n such that for every (i1 , . . . , in−1 ) ∈ {1, 2}n−1 the map fi1 ...in−1 |Je1 ∪Je2 is a contraction with ratio less than 1/2L. By the chain rule, for an arbitrary x ∈ [0, 1] we have (24)

Si01 ...in (x) = Si01 ...in−1 (Sin (x)) · Si0n (x).

The commutative diagram (22) yields that Si1 ...in−1 (y) = wi−1 ◦ fi1 ◦ wi2 ◦ · · · wi−1 ◦ fin−2 ◦ win−1 wi−1 ◦ fin−1 ◦ win 1 n−2 n−1 = wi−1 ◦ fi1 ...in−1 ◦ win (y). 1

√ Since both w1 (x) and w2 (x) are similarities with ratio 2, the map Si1 ...in−1 |J is a contraction with ration less than 1/2L. Thus we obtain that for every (i1 , . . . , in ) ∈ {1, 2}n and x ∈ [0, 1], 1 |Si01 ...in (x)| < . 2

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Using (21) and (22) we obtain from the change S of variable formula that for a Borel set B ⊂ [0, 1] and E = w1 (B) w2 (B) Z r1 (w)dQ(w) m1 (B) = Q1 (E) = f1−1 (E) Z Z (25) r1 (w)dQ2 (w) r1 (w)dQ1 (w) + = −1 (E) f1,2

−1 (E) f1,1

Z

Z = S1−1 (B)

p1 (x)dm1 (x) +

S1−1 (B)

p1 (x)dm2 (x)

Z = S1−1 (B)

p1 (x)dm(x).

In the same way we obtain that Z Z (26) m2 (B) = r2 (w)dQ(w) = f2−1 (E)

S2−1 (B)

p2 (x)dm(x).

Adding the last two formulae together we obtain that Z Z (27) m(B) = p1 (x)dm(x) + p2 (x)dm(x). S1−1 (B)

S2−1 (B)

Thus it follows that for every n we have that supp(m) ⊂ ∪i1 ···in Si1 ···in [0, 1]. Therefore, if max |S10 (x)| + max |S20 (x)| < 1 then m ⊥ leb. It is immediate from the definition of the Hausdorff dimension that dimH (Q) = dimH (m), so it is enough to estimate the latter one. Let us call S the IFS which consists of the functions {S1 (x), S2 (x)}, chosen with the place dependent probabilities {p1 (x), p2 (x)}, respectively. It follows from (5) that S is eventually contracting. By (27) the measure m is an invariant measure for the IFS S. We fix an N which satisfies Lemma 5. Then the IFS N Sb := {Si } i∈{1,2}

with probabilities pi (x) := pi1 (x) · pi2 (Si1 (x)) · · · pin (Sin−1 ...i1 (x)). is strictly contracting. We write Tb for the N -th iterate of the operator T : C 1 ([0, 1]) → C 1 ([0, 1]) defined by the formulae (1). That is for an f ∈ C 1 ([0, 1]) we have X (Tbf )(x) = pi (x) · f (Si (x)). i∈{1,2}N


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Using that the IFS Sb is contracting, a standard and easy calculation shows that there exists a constant C > 0 such that for any f ∈ C 1 ([a, b]) k(Tbn f )0 k∞ ≤ Ckf k∞ + kf 0 k∞ .

(28)

The following lemma summarizes two of the important properties of the measure m. Lemma 6. The measure m is the unique invariant measure for the IFS S. Moreover, it is necessarily of pure type (i.e., it is either singular or absolutely continuous with respect to the Lebesgue measure). Proof. Let f be a C 1 function, then we claim that T n f converges to a constant. Observe that it follows from (28) that the sequence of C 1 [0, 1] functions {Tbn f }∞ n=1 is an equicontinuous family. Thus by the AzelaAscoli theorem we can deduce that there is a convergent subsequence {T ni f }∞ i=1 to a continuous function f , say. Furthermore, inf f (x) ≤ inf T f (x) ≤ inf T 2 f (x) ≤ · · · ≤ inf f (x). x

x

x

x

and so for every n we have inf x f (x) = inf x T n f (x). Moreover, since for any x the set {Si1 · · · Sin (x) : i1 , · · · , in , n ≥ 1} is dense, we can deduce that f (x) ≡ inf x f (x), i.e., the limit is a constant c(f ), say. Furthermore, T n f → c(f ) as n → +∞ and if T∗ µ = µ is an invariant measure then µ(f ) = µ(T n f ) → c(f ). In particular, since c(f ) is independent of µ we deduce that there is a unique T -invariant measure. The uniqueness of the T -invariant measure µ leads easily to the conclusion that µ is of pure type. If we write µ = µac + µsing for the unique lebesgue decomposition into absolutely continuous and singular parts then we observe that T∗ µac = µac and T∗ µsing = µsing . However, by the uniqueness of the stationary measure we can deduce that one of these vanishes. R Remark 7. For any g ∈ C 1 ([0, 1]) we have that T n g → gdµ(x) uniformly (at an exponential rate). In practice the rate of convergence might be difficult to compute. However, some rigorous estimates can come either using the Birkhoff cone method [10] or cycle expansions [2]. It is easy to see that in this case the entropy and Lyapunov exponents in Definition 2 have the simpler forms: Z X 1 pi (x) · log pi (x)dm(x) (29) h(m) = lim − · n→∞ n |i|=n


12

and 1 λ(m) = lim · n→∞ n

(30)

Z X |i|=n

0

− (x)dm(x). pi (x) · log S← i

Lemma 8. We can write Z (31) h(m) = − p1 (x) · log p1 (x) + p2 (x) log p2 (x)dm(x) Z 0 0 (32) λ(m) = p1 (x) · log S1 (x) + p2 (x) · log S2 (x)dm(x). 0

Proof. Let τa (x) be either pi (x) or Si (x). For an i = (i1 , . . . , in ) ∈ {1, 2}n we write τi (x) := τi1 (x) · τi2 (Si1 (x)) · · · τin (Sin−1 ...i1 (x)). Then what we need to prove it is that (33) Z Z X 1 pi (x)·log τi (x)dm(x) = p1 (x) log τ1 (x)+p2 (x) log τ2 (x)dm(x). lim n→∞ n |i|=n

Let F (x) := p1 (x) · log τ1 (x) + p2 (x) · log τ2 (x) and then Z (34)

F (x) =

log τi1 (x)dpx (i).

Observe that for every 1 ≤ k ≤ n and x ∈ [0, 1] we have (35)

X |i|=n

pi (x) · log τik (Sik−1 ...i1 (x)) =

Z log τik (Sik−1 ...i1 (x))dpx (i).

We can define (36)

g(x, i) := log τi1 (x)

and then using the previous three identities, the fact that m e is Sinvariant in the third step and the notation dm(x, e i) = dpx (i)dm(x) we


obtain that Z Z F (x)dm(x) = Z (37) = Z = Z = Z =

13

Z g(x, i)dpx (i)dm(x) g(x, i)dm(x, e i) g(Sik−1 ...i1 (x), σ k i)dm(x, e i) Z log τik (Sik−1 ...i1 (x))dpx (i)dm(x) X |i|=n

pi (x) · log τik (Sik−1 ...i1 (x))dm(x).

Adding these equalities for 1 ≤ k ≤ n we obtain that Z Z X (38) F (x)dm(x) = pi (x) · log τi (i). |i|=n

Which completes the proof of the lemma.

It follows from the above lemma that Z Z (39) h(m) = h(x)dm(x) and λ(m) = `(x)dm(x), where (40)

h(x) := − (p1 (x) log p1 (x) + p2 (x) log p2 (x)) , = ϕ(p1 (x)) + ϕ(p2 (x)), 0

0

`(x) = p1 (x) log |S1 (x)| + p2 (x) log |S2 (x)|,

where ϕ(x) = −x · log x. An easy calculation now shows the following: Lemma 9. We can write that 0

S1 (x) = (41)

0

S2 (x) =

∆ p21 (x)

, and

∆ , p22 (x)

where (42)

∆ := 2εp − 2ε2 p − ε + ε2 = −ε(1 − ε)(1 − 2p).

This then gives the following corollary: Corollary 10. (1) For p < 1/2 both S1 (x) and S2 (x) are monotone decreasing.


14

(2) For p > 1/2 both S1 (x) and S2 (x) are monotone increasing. (3) For p = 1/2 both of the functions S1 (x) and S2 (x) are constant. So, in what follows we always assume that 1 (43) p 6= . 2 Another immediate consequence of Lemma 9 is the following corollary. Corollary 11. `(x) = log |∆| + 2h(x). This in turn immediately implies the following corollary. Corollary 12. We have that h(m) (44) ≤a≤1 −λ(m) if and only if Z a (45) h(x)dm(x) ≤ − · log |∆|. 1 + 2a 6. Singularity of the Blackwell measure We are now in a position to study ranges of parameter values for which the Blackwell measure is singular. It follows from (40) that we get a better understanding of the function h(x) by analyzing the potentials p1 (x), p2 (x). To further simplify the calculations we introduce (46)

q := ε + p − 2εp.

Since 0 < ε, p < 1 the range of q is: 0 < q < 21 . Using (20) by some elementary calculations we obtain that: Lemma 13. We can write (47)

p1 (x) = (2q − 1)x + 1 − q, p2 (x) = (1 − 2q)x + q.

In this way p1 (x) is always monotone decreasing and p2 (x) is always monotone increasing. In particular we have that p1 (0) = p2 (1) = 1 − q, p2 (0) = p1 (1) = q. Moreover, for every x ∈ [0, 1] we have the identity (48)

S1 (x) + S2 (1 − x) = 1.

Finally, the function h(x) satisfies:


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(a): h(x) = h(1 − x) for all x ∈ [0, 1], (b): h00 (x) < 0 for all x ∈ [0, 1], (c): h(x) attains its maximum at x = 12 , (d): h(0) = h(1) is the value of the minimum of the function h(x). Properties (c) are (d) are immediate consequences of properties (a) and (b), which themselves can be verified by direct calculation. The fact that our system is in a sense a contracting on average system follows from Lemmas 9 and 13: Definition 14. We say the a smooth real function f defined on [0, 1] is a reversed cup function if f satisfies the prosperities (a)-(d) above. Lemma 15. Let f (x) be a reversed cup function. Then the function (T f )(x) is also a reversed cup function, where T is the transfer operator (defined in (1)). Proof. Using (41) and the chain rule we obtain that (T f )0 (x) = (2q − 1) · f (S1 (x)) + (1 − 2q) · f (S2 (x)) + f 0 (S1 (x)) · S10 (x) · p1 (x) | {z } ∆/p1 (x)

0

+ f (S2 (x)) ·

S20 (x) |

· p (x) {z 2 }

∆/p2 (x)

Similarly, using that ∆, S10 (x), S20 (x) < 0 we get (T f )00 (x) = (2q − 1) · f 0 (S1 (x)) · S10 (x) + (1 − 2q) · f 0 (S2 (x)) · S20 (x) ∆ ∆ + f 00 (S1 (x)) · S10 (x) · + f 00 (S2 (x)) · S20 (x) · p1 (x) p2 (x) 0 −2 + f (S1 (x)) · (−1) · ∆ · p1 (x)(2q − 1) | {z } S10 (x)

+ f 0 (S2 (x)) · (−1) · ∆ · p−2 2 (x)(1 − 2q) | {z } S20 (x)

= f 00 (S1 (x)) · S10 (x) · < 0.

∆ ∆ + f 00 (S2 (x)) · S20 (x) · p1 (x) p2 (x)

16


In this way we have checked that property (b) in Lemma 49 holds. To see that property (a) holds as well we need to observe that

(49)

(T f )(x) = p1 (x)f (S1 (x)) + p2 (x)f (S2 (x)) = p1 (x)f (S1 (x)) + p1 (1 − x)f (1 − S1 (1 − x)) | {z } S2 (x)

= p1 (x)f (S1 (x)) + p1 (1 − x)f (S1 (1 − x)) = (T f )(1 − x). This completes the proof.

This leads to the next result. Proposition 16. For every n we have

(50)

(T n h)(0) ≤ h(m) ≤ (T n h)(1/2).

Proof. Since T ∗ m = m it follows from Lemma 8 that

Z h(m) =

Z h(x)dm(x) =

(T n h)(x)dm(x).

Using Lemma 13 and Lemma 15 we obtain that (T n h)(x) is a reversed cup function. This immediately implies the assertion of the Proposition. Example 17. Let us consider a concrete example: p = 0.2 and = 0.3. By the preceding we can sandwich it in the nested intervals [(T n h)(1/2)], (T n h)(0)]. In the following table we present these values for n = 1, · · · , 15.


17

n

(T n h)(1/2)

n

(T n h)(0)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.6931471805599453094 0.6885320764504560426 0.6873399079169119219 0.6870204663672780007 0.6869351099501216909 0.6869122847651288517 0.6869061819804526502 0.6869045502238403004 0.6869041139291720920 0.6869039972737272948 0.6869039660826768519 0.6869039577428888054 0.6869039555130163633 0.6869039549167984387 0.6869039547573831256 0.6869039547147590431

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.6640641265641080113 0.6807022640392729670 0.6852455729100437792 0.6864604838094890865 0.6867853827159174941 0.6868722510733396336 0.6868954778673486178 0.6869016881839058585 0.6869033486838228601 0.6869037926642685923 0.6869039113746924475 0.6869039431151996655 0.6869039516019000626 0.6869039538710535796 0.6869039544777743832 0.6869039546399979308

Thus we can get bounds on the ration h/χ of the blackwell measure in terms of the parameter values (p, ). Figure 2 shows where this bound lies below 1, and thus the measure is singular. Remark 18. Unfortunately, there don’t appear to be any techniques available to show that the Blackwell measure is absolutely continuous on particular domains. 7. The Furstenberg measure We now turn to the complementary problem of understanding the nature of the Furstenberg measure. We begin with the basic definition. The Furstenberg measure is associated to a finite set of matrices A1 , · · · .Ak ∈ GL(d, R) chosen randomly with respect to a given iid Bernoulli measure associated to (p1 , · · · , pk ), say, and is used to characterize the lyapunov exponents. 7.1. The Furstenberg measure. Definition 19. The Lyapunov exponent λ is given by the limit 1 X (51) λ := lim pi1 · · · pin log ||Ai1 · · · Ain || n→+∞ n i ,··· ,i 1

n

18


Figure 2. The level sets of the entropy minus the Lyapunov

exponent can be better estimated using iterates of the operator T to estimate the entropy (which is expressed as an integral with respect to the Blackwell mesure).

By a famous result of Kesten and Furstenberg one has the following pointwise estimate with repesct to the Bernoulli measure µ = + (p1 , · · · , pk )Z : Theorem 20 (Furstenberg-Kesten). For a.e. (µ) i ∈ Σ one has 1 log ||Ai1 · · · Ain ||. n→+∞ n

λ = lim

Thus the Lyapunov exponent λ estimates the rate of growth of the norm ||Ai1 Ai2 Ai3 · · · Ain ||


19

for a typical random product of matrices Ai1 , Ai2 , Ai3 , . . . Ain , with respect to the Euclidean norm k · k on Rd , say. We now turn to the associated Furstenberg measure, which is defined on projective space P Rd−1 . Let (p1 , · · · , pk ) be a probability vector, + and let µ = (p1 , · · · , pk )Z be the associated Bernoulli measure on the + space of sequences Σ = {1, · · · , k}Z . More precisely, consider the real projective sphere P Rd−1 = (Rd − {(0, 0)})/ ∼,

where v ∼ w if there exists β > 0 such that βv = w. The usual linear action Aj : Rd → Rd allows one to define a projective action Aj : P Rd−1 → P Rd−1 by Aj x , Aj (x1 , · · · , xd ) = qP d 2 (A x) j i i=1 for j = 1, · · · , k. Standing assumption: Henceforth, we shall always assume that the matrices are all positive (i.e., each of the entries of each of the matrices A1 , · · · , Ak are strictly positive). In particular, we can then deduce that the induced action on projective space is a contraction, with respect to a suitable metric. More precisely, by the well known Birkhoff cone theorem, we know that the associated actions contract with respect to the Hilbert-Birkhoff metric on projective space. This implies (c.f. [4]) that there is a unique probability measure ν on P Rd satisfying (52)

ν(E) =

k X

pj ν(A−1 j (E)).

j=1

Actually we obtain this measure as (53)

ν(E) = Π∗ (µ) where Π(i) := lim Ai1 · · · Ain (w), n→∞

d

for an arbitrary w ∈ P R . One of the reasons that this measure ν is important is as follows Theorem 21 (Furstenberg [5]). The Standing hypothesis implies that Z X k (54) λ= pj · log kAj (x)kdν(x). P Rd

j=1


20

Therefore the measure ν is called Furstenberg measure. 8. A class of examples of the Furstenberg measure In this section we consider the particular case that d = 2 and p1 = a b p2 = 12 . We can associate to a 2 × 2 matrix A = , say, a linear c d fractional transformation on the unit interval [0, 1] denoted by g : x 7→

ax + b(1 − x) . (a + c)x + (b + d)(1 − x)

In particular, this corresponds to the projective action for a particular choice of chart. Furthermore, if we assume that d = 2 we can associated to two such matrices a0 b 0 a1 b 1 A0 = and A1 = c0 d0 c1 d 1 the corresponding orientation preserving maps g0 , g1 : [0, 1] → [0, 1] are denoted by g0 : x 7→

a0 x + b0 (1 − x) a1 x + b1 (1 − x) and g1 : x 7→ (a0 + c0 )x + (b0 + d0 )(1 − x) (a1 + c1 )x + (b1 + d1 )(1 − x)

The hypothesis on the positivity of the matrices makes it natural to make the following simplifying assumptions. (1) the maps are contracting, i.e., ||g00 ||∞ , ||g10 ||∞ < 1 (2) the maps g0 , g1 have distinct fixed points g0 (x1 ) = x0 and g1 (x1 ) = x1 . We can assume without loss of generality that x0 < x1 then the limit set of the iterated function scheme lies in the interval [x0 , x1 ] and so after a simple affine change of coordinate we can assume without loss generality that x0 = 0 and x1 = 1. In particular, in these coordinates we can assume that α0 x d1 x + (1 − α1 )(1 − x) and g1 : x 7→ , x + d0 (1 − x) d1 x + (1 − x) The functions g0 and g1 depend on altogether four parameters from 4 which we form the vector t := (α0 , α1 , d0 , d1 ) ∈ (R+ ) . As in the general case here we also write ν = νt for the Furstenberg measure. That is ν is the unique probability measure on the interval [0, 1] satisfying 1 1 (55) ν(E) = · ν g0−1 (E) + · ν g1−1 (E) , 2 2 g0 : x 7→


21

for all Borel sets E ⊂ [0, 1]. We know that for πt : Σ → [0, 1],

πt (i) := lim gi0 ◦ · · · ◦ gin (0), n→∞

we have ν = νt = (πt )∗ µ, N Bernoulli measure on Σ. where Σ = {0, 1} and µ = 12 , 12 Next, we can associate to the Furstenberg measure ν its entropy h(ν) and its Lyapunov exponent λ(ν), as before. By Birkhoff Ergodic Theorem we have Z (56) λ(ν) := log gi00 (σi)dµ(i). N

The following follows as a special case of the Jaroszewska-Rams theorem, Theorem 3. Theorem 22. We have dimH (νt ) ≤

(57)

h(νt ) . −λ(νt )

In particular, for those parameters t for which h(νt ) < −λ(νt ) the measure νt is singular. 8.1. The absolute continuity of the Furstenberg measure. We want to find sufficient conditions for the Furstenberg measure to be absolutely continuous. We will find a parameter set U of positive Leb4 measure such that for almost all parameters t from U the Furstenberg measure νt is absolute continuous. To achieve this we apply the theory introduced by Simon, Solomyak and Urbanski c.f. [11] and [12] which says roughly speaking that if the (so called) transversality condition holds on a parameter domain then for Lebesgue almost parameters from this domain the measure νt is absolute continuous whenever the ratio of entropy/Lyapunov exponent is greater than 1. Throughout this section we use the notation for i = 0, 1 we write bi := 1 − i. and

−1 di := min di , d−1 , d := max d , d . i i i i In the rest of the paper, the parameter t will be chosen from the open set U ⊂ (R+ )4 defined as follows: Definition 23. U := U1 ∩ U2 ∩ U3 , where U1 , U2 , U3 ⊂ (R ) are the set of parameters t ∈ (R+ )4 for which the conditions in points (1), (2) and (3) below hold respectively. + 4


22

(1): The functions g0 , g1 are monotone increasing strict contractions and g0 ([0, 1)) ∪ g1 ((0, 1]) = [0, 1] = [fix(g0 ), fix(g1 )],

where f ix(g0 ), f ix(g1 ) are the unique fixed points of g0 , g1 , respectively. This is equivalent to (58)

0 < αi < di for i = 0, 1 and α0 + α1 > 1 (2): gii ([0, 1]) ∩ gbi ([0, 1]) = ∅ holds for i=0,1. Equivalently, both of the following two inequalities hold:

(59)

(g00 (1) =) and

(60)

(g11 (0) =)

α02 < 1 − α1 (= g1 (0)) α0 + d0 (1 − α0 )

d1 (1 − α1 ) + (1 − α1 )α1 > α0 (= g0 (1)). d1 (1 − α1 ) + α1

(3): We write T 0 (x) = x is the identity, finally T 1 (x) := 1 − x. We assume that either for i = 0 or for i = 1 : αbi · dbi · T i ◦ gbi−1 ◦ (T i )−1 (αi ) α · α · d · d 0 1 0 1 + (61) 1 then νt is absolute continuous with relog λ spect to the Lebesgue measure.

There are very few results about the absolute continuity of measures which are invariant with respect to a non-linear IFS. The only method available, up to our knowledge, is the theory developed in the papers [11] and [12]. In order to apply it we need to require that the maps are strict contractions (condition (1) of Definition 23) and that the transversality condition holds (mostly condition (3)). Even in the linear case (e.g. Bernoulli convolution measures) we can apply the transversality method to prove absolute continuity, only if the contractions are ”small”. This is corresponds to our condition (2). In


23

Definition 23 the first two conditions were stated first with a rather geometric description and then we expressed them in terms of formulas. The equivalence of these formulations are immediate. The only thing we need to observe to verify this is: α0 d0 α1 d1 (62) g00 (x) = and g10 (x) = 2 (x + d0 (1 − x)) (d1 x + (1 − x))2 implies that (63)

αi di ≤ gi0 (x) ≤ αi di for i = 0, 1.

In the particular case that d0 = d1 = 1 then the maps g0 , g1 are linear with slope α0 , α1 respectively. Next we define the transversality condition and then we prove that conditions (1), (2) and (3) of Definition 23 imply that the transversality condition holds. Definition 25 (Transversality). Let d0 , d1 ∈ R+ be arbitrary fixed. We say that the transversality condition holds on an open set of V = Vd0 ,d1 ⊂ [0, 1]2 of parameters (α0 , α1 ) if there exists C > 0 such that for every sequence i = (i0 , i1 , . . . ), j = (j0 , j1 , . . . ) ∈ Σ with i0 6= j0 we have: Leb2 {(α0 , α1 ) ∈ V : t = (α0 , α1 , d0 , d1 ), |πt (i) − πt (j)| < r} ≤ Cr

for all r > 0.

It is easy to see (c.f. [11] and [12]) that in order to establish transversality we only need to do a computation involving the gradients: Lemma 26. Fix d0 , d1 ∈ R+ . Let V ⊂ [0, 1]2 be an open set. Assume that for all (α0 , α1 ) ∈ V for t = (α0 , α1 , d0 , d1 ) and i, j ∈ Σ with πt (i) = πt (j) and i0 6= j0 we have (64)

where

||∇α0 ,α1 πt (i) − ∇α0 ,α1 πt (j)|| > 0,

∂ ∂ πt (i), πt (i) . ∇α0 ,α1 πt (i) := ∂α0 ∂α1 Then the transversality condition holds on V .

Now we can prove the main result of this section: Proof of Theorem 24. Using [12, Theorem 7.2], to verify our theorem the only thing left to do is to prove that the conditions (1), (2) and (3) imply that (64) holds. First we observe that the roles of the functions g0 and g1 are symmetric. Namely, for α1 x d0 x + (1 − α0 )(1 − x) (65) γ0 (x) = and γ1 (x) = . x + d1 (1 − x) d0 x + (1 − x)


24

we have (66)

g0 (x) = T 1 ◦ γ1 ◦ (T 1 )−1 (x) and g1 (x) = T 1 ◦ γ0 ◦ (T 1 )−1 (x).

That is, we obtain the graph of the function gi by reflecting the graph of γbi to the center of the unit square. First we assume that condition (3) holds with i = 0. That is now we assume that α1 d1 g1−1 (α0 ) α0 α1 d0 d1 + < 1. α0 1 − α 0 d0 To see that this implies (64) first we verify that ∂ 1 πt (k) ≤ (68) ∀k ∈ Σ we have . ∂α0 1 − α0 d0 Namely, we observe that for every y ∈ [0, 1] we have 1 ∂ g0 (y) = · g0 (y). (69) ∂α0 α0 (67)

If ∂α∂ 0 πt (k) 6= 0 then there is an n ≥ 0 such that the first zero in k = (k0 , k1 , . . . ) is kn . Then for z = πt (σ n+1 k) we have 1 ∂ ∂ n 0 0 n+1 πt (k) = (g1 ) (z) · g0 (g0 (z)) + g0 (z) · πt (σ k) ∂α0 α0 ∂α0 ∂ ≤ 1 + α 0 · d0 · πt (σ n+1 k), ∂α0 where in the first line we used (69) and in the second line we used the facts that g1 is a contraction, maxx g0 (x) = α0 and (63). We obtain that (68) holds by induction. Next we fix arbitrary i, j ∈ Σ with πt (i) = πt (j) and i0 6= j0 . Without loss of generality we may assume that i0 = 0 j0 = 1. Then by condition (2) we have i1 = 1 and j1 = 0. To verify (64) now we prove that ∂ ∂ (70) (πt (i)) − (πt (j)) > 0. ∂α0 ∂α0 We define the real functions h1 (α0 ) := πt (σ 2 i) and h2 (α0 ) := πt (σ 2 j). We apply the chain rule, (63) and (69) twice to get ∂ πt (i) πt (i) (πt (i)) = + g00 (g1 (h1 (α0 ))) · g10 (h1 (α0 )) · h01 (α0 ) ≥ ∂α0 α0 α0 and ∂ g0 (h2 (α0 )) 0 0 0 (πt (j)) = g1 (g0 (h2 (α0 ))) · + g0 (h2 (α0 )) · h2 (α0 ) ∂α0 α0 ≤ α1 d1 ·

g1−1 (πt (i)) α0 d0 · α1 d1 + , α0 1 − α 0 d0


25

where in the last line we used that πt (i) = πt (j). Thus ∂ (71) (πt (i) − πt (j)) > f (z), ∂α0 where f : [1 − α1 , α0 ] → R is defined as

g −1 (z) α0 d0 · α1 d1 z − α1 d1 · 1 − . α0 α0 1 − α 0 d0 By (63) the function f is strictly decreasing and by (67) we have that f (z) ≥ f (α0 ) > 0 holds for all 1 − α1 ≤ z ≤ α0 . Hence, (70) holds. Now we turn to the case when (61) holds with i = 1. Using the first part of (66) this means that f (z) :=

α0 d0 γ1−1 (α1 ) α0 α1 d0 d1 + < 1. α1 1 − α 1 d1 Below we use that the Conditions (1) and (2) of Definition 23 remain unchanged if we interchange (α0 , d0 ) and (α1 , d1 ), so we can consider et = (α1 , α0 , d1 , d0 ) instead of t := (α0 , α1 , d0 , d1 ). By definition the natural projection in this case will be (72)

(73) πet (i) = =

lim γi1 ◦ γi2 ◦ · · · ◦ γin (0)

n→∞

lim T 1 (gbi1 ◦ gbi2 ◦ · · · ◦ gbin ((T 1 )−1 0)) = T 1 πt (bi) .

n→∞

Note that assumption (72) for the IFS {γ1 , γ2 } and et is exactly the same as the assumption (67) for the IFS {g0 , g1 } and t. So, we can apply the previous argument to show that whenever πet (i) = πet (j) (which is equivalent to πt (bi) = πt (bj) ) with i1 = 0, j1 = 1 then ∂ ∂ ∂ ∂ (πt (bi)) − (πt (bj)) = (πet (i)) − (πe(j)) > 0. (74) − ∂α1 ∂α1 ∂α1 ∂α1 t This completes the proof of the Theorem.

9. Examples of absolute continuity of the Furstenberg measure We can restrict attention to considering matrices of the particular form α0 0 d1 1 − α 1 A0 = and A1 = 1 − α0 d0 0 α1

where 0 < 1 − α1 < α0 < 1 and d0 , d1 ∈ R+ , since, as we have seen, this can be achieved by an affine change of coordinates and since scaled matrices have the same projective actions. This gives a four dimensional parameter space.

26


If we assume for simplicity that d0 = d1 = 1 then the functions g0 , g1 are linear: g0 (x) = α0 x, g1 (x) = α1 x + 1 − α1 . The absolute continuity of the invariant measure for this linear case was investigated by Ngai, Wang [8]. (C.f. [8, Figure 1] for the domain of absolute continuity.) While in [8] only the linear case was considered, our calculations also work in the case of (d0 , d1 ) which are sufficiently close to the linear case d0 = d1 = 1 . The conditions (1) and (2) of Definition 23 can be summarized in the linear case d0 = d1 = 1 as follows: α02 < 1 − α1 < α0 < 1 − α12 .

(75)

Moreover, in the linear case, condition (3) of Definition 23 is as follows: (76) either

α0 α1 α1 + α0 − 1 α0 α1 α0 + α1 − 1 + < 1 or + < 1. α0 1 − α0 α1 1 − α1

In particular, solutions include α0 = 17/128 and α1 = 7/8 and thus we could consider a full measure set of matrices in a sufficiently small (four or eight dimensional) neighborhood of A0 =

17/128 0 111/128 1

1 1/8 and A1 = 0 7/8

We can also consider the set of parameters (α0 , α1 ) satisfying the above inequalities and for a.e. (α0 , α1 ) in this set we have that the Furstenberg measure is absolutely continuous. For the Furstenberg measure, with probability vector (p, 1 − p), the entropy of the measure ν = (p, 1 − p)Z+ is simply h(ν) = −p log p − (1 − p) log(1 − p). In particular, when p = 21 this becomes log 2. However, the Lyapunov exponent λ(ν) of the Furstenberg measure cannot be written in a closed form and can only be numerically estimated. We recall that by definition λ(ν) = limn→∞ λn , where λn :=

1 2n

X i0 ,...,in−1

log kAi1 · · · Ain k.

The following table gives the values of these approximations. (It is also possible to get estimates on λ using cycle expansions.)


Figure 3. d0 = d1 = 1. In the green region dimension is equal

to entropy over Lyapunov exponent. In the yellow region this remains true almost surely. In the red region we have absolute continuity. In the striped region the absolute continuity was proved by Ngai Wang.

n

λn

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.287024757051608011 0.258808591315294742 0.245624281188549106 0.239666895599546522 0.237089539695933433 0.236022533561996868 0.235598268433514587 0.235435765643042602 0.235375877561900943 0.235354815592289776 0.235347897218792846 0.235345887479272859 0.235345459418261633 0.235345473898739984 0.235345571492518357

27


28

h(ν) In particular, in this case we see that λ(ν) = 2 · 94523. The value needed to be at least 1 to be consistent with the absolute continuity of (nearby) measures. A non-linear example: We can also consider the case d0 = d11 = d 6= 1. In this case g0 , g1 are fractional linear functions. Unfortunately, the Lyapunov exponent of ν = ( 12 , 12 )Z+ is complicated to calculate. However, we can approximate it. By using the definition of Lyapunov exponent, we have Z λν = − log gi00 (πt (σi))dν(i) = Σ Z 1 − log d − log α0 α1 + 2 log (πt (i) + d(1 − πt (i))) dν(i) 2 Σ

Since log (x + d(1 − x)) is monotone increasing or decreasing, depending on d, we can approximate the integral of it. For d < 1 by the following way Z X 1 log (gi (0) + d(1 − gi (0))) ≤ log (πt (i) + d(1 − πt (i))) dν(i) ≤ 2n Σ |i|=n

X 1 log (gi (1) + d(1 − gi (1))) 2n

|i|=n

We can see an approximation of the set of (α0 , α1 ), where the Furstenberg measure is absolutely continuous for d = 0.9, on Figure 4. The absolute continuity domain include for example α0 = 9/20 and α1 = 3/5 and d = d0 = 1/d1 = 9/10 and thus we could consider a full measure set of matrices in a sufficiently small (four or eight dimensional) neighborhood of 9 10 2 0 A0 = 20 and A1 = 9 53 . 11 9 0 5 20 10 h(ν) > 1. So the measure ν corresponding to the parameters where λ(ν) which belong into this small neighborhood is almost surely absolute continuous.

10. Appendix Here we briefly summarize the results of the papers [11] and [12] used in this note. We are given the family Φt = {φt1 , . . . , φtm }t∈U of hyperbolic IFS on the a compact interval X ⊂ R. (Hyperbolic means that there exist 0 < c1 < c2 < 1 such that for the C 2 maps φtk we have c1
0.

Then n o (1) For Lebd almost all t ∈ U , dimH νt = min h(µ) , 1 ; −λt (2) νt Leb for Lebd almost all t ∈ {t ∈ U : −h(µ)/λt > 1} We start with a Lemma from [11]. For a set F ⊂ Rd let Nr (F ) be the minimal number of balls needed to cover the set F . Lemma 28. (see [11, Lemma 7.3]) Let U ⊂ Rd be as above. Suppose that f is a C 1 real-valued function defined in a neighborhood of the closure of U such that for some i ∈ {1, . . . , d} there exists an η > 0 satisfying (77)

t ∈ U, |f (t)| ≤ η =⇒

∂f (t) ≥ η. ∂ti

Then there exists C = C(η) such that (78)

Nr ({t ∈ U : |f (t)| ≤ r}) ≤ c · r1−d , ∀r > 0.

For every (i, j) ∈ Σ2 we may apply this lemma for f = fi,j (t) since (H) implies that (77) holds. Write ηi,j and Ci,j for the corresponding constants. For compactness η := min(i,j)∈Σ2 ηi,j > 0 and C := max(i,j)∈Σ2 C(i,j) < ∞. So, ∀(i, j) ∈ Σ2 :

Nr ({t ∈ U : |fi,j | ≤ r}) ≤ C · r1−d

The last statement is called the strong transversality condition (c.f. [11, p. 454]) which clearly implies that the transversality condition holds: There exists a e c > 0 such that (79)

∀r > 0 : Lebd {t ∈ U : |fi,j | ≤ r} ≤ e c·r

Then we can apply [12, Theorem 7.2] which immediately yields the assertion of our theorem. References [1] D. Blackwell, The entropy of functions of finite-state Markov chains, Trans. First Prague Conf. Information Theory, Statistical Decision Functions, Random Processes, (1957), 13-20. [2] P. Cvitanovic, Chaos classical and quamtum http://chaosbook.org/index.html [3] G. Han, B. Marcus, Analyticity of entropy rate of hidden Markov chains, IEEE Trans. Inform. Theory 52 (2006), no. 12, 5251–5266. [4] J. E. Hutchinson, Fractals and self-similarity, Indiana University Mathematics Journal 30, (1981), 713-747.


31

[5] H. Furstenberg, Non-commuting random products, Transactions of the American Mathematical Society 108 (1963), 377-428. [6] J. Jaroszewska, M. Rams, On the Hausdorff dimension of invariant measure of weakly contracting on average measurable Iterated Function Scheme, Journal of Stat. Phys. to appear [7] J. Marklof, Y. Tourigny and L. Wolowski, Explicit invariant measures for products of random matrices, Transactions of the AMS 360 (2008), 33913427. [8] Sz.M. Ngai, Y. Wang, Self-similar measures associated with ifs with nonuniform contraction ratios Asian J. Math. 9 (2005), 227-244. [9] Y. Peres, Domain of analytic continuation for the top Lyapunov exponent, Ann. Inst. H. Poincar Probab. Statist. 28 no.1 (1992), 131–148. [10] E. Seneta, Non-negative matrices and Markov chains, Springer, 1981. [11] K. Simon, Solomyak and M. Urbanski, Hausdorff dimension of limit sets for parabolic IFS with overlaps, Pacific J. Math. 201 (2001), no. 2, 441–478 [12] K. Simon, Solomyak and M. Urbanski, Invariant measures for parabolic IFS with overlaps and random continued fractions, Trans. Amer. Math. Soc. 353 (2001), no. 12, 5145–5164