Dimension estimates for invariant measures of contracting-on-average iterated function

systems

Michał Rams ^{∗}

Institute of Mathematics, Polish Academy of Sciences ul. Śniadeckich 8, 00-950 Warszawa, Poland e-mail rams@impan.gov.pl

Abstract

We estimate from above and below the dimension of invariant measure for contracting-on-average iterated function systems in ℝ^{d}.

1 Introduction

The goal of this paper is to give dimension estimates for invariant measures of contracting-on-average self-conformal iterated function systems with pointdepending probabilities.

The motivation for this work comes from the recent results of Szarek and Myjak [S], [MS] and also Nicol, Sidorov and Broomhead [NSB] and Fan, Simon and Toth [FST] who gave bounds for the Hausdorff dimension of the invariant measure for some contracting-on-average iterated function systems.

Lately, Johansson and Öberg computed the precise value of the Hausdorff dimension for bounded (not necessarily contracting) iterated function systems on the line satisfying the open set condition, [JO]. Similar questions come from other branches of mathematics, compare [Wa].

There are two kinds of results on the dimension of an invariant measure for (contracting) iterating function systems. The upper bound is easy to obtain, depending on assumptions it takes form of Moran formula (for similitudes) or Bowen (pressure) formula (for smooth conformal maps). For systems with probabilities we get Eckmann-Ruelle formula, i.e. ratio of entropy to the Lyapunov exponent. To get the lower bound for an iterated function system one typically needs a version of the open set condition (like in [LN], [R]).

∗supported by Polish KBN Grant No 2P0 3A 034 25

1991 Mathematics Subject Classification: Primary 28A78,34D45

We are working here with systems that aren’t everywhere contracting. We need to assume some contractiveness-on-average condition to guarantee existence and uniqueness of the invariant measure, as was proved in [BDEG].

We prove the Eckmann-Ruelle formula for the upper bound on the Hausdorff dimension of the invariant measure. It is the same as the bound in [JO] and [FST], but we work on different class of systems, in particular our method works for measures with unbounded supports. As for the lower bound, we can only obtain it under some (bit stronger) version of OSC.

The rest of the paper is divided as follows. In the second section we state some known results on the iterated function systems and formulate our theorem. In the third section we prove some properties of contracting-on-average iterated function systems that we will need in the future. As the results of this section might be of independent interest, they are formulated in more general situation, i.e. for iterated function system acting on a Polish space and satisfying Dini condition. In the fourth section we finish our preparation by formulating some geometric lemmas about typical behaviour of orbits of IFS. Finally, in the fifth section we present the proof of Theorem 2.1.

2 Introduction to iterated function systems

For informations on dimension theory and properties of contracting iterated function systems we refer the reader to standard textbooks like [F]. The existence and most important properties of invariant measures for contractingon-average iterated function systems are to be found in [BDEG] and [E].

Consider an iterated function system consisting of k C^{1+α} maps h_{1}, …, h_{k} : ℝ^{d} → ℝ^{d} and a probability vector field p⃗(x) = (p_{1}(x), …, p_{k}(x)) defined on ℝ^{d}. For a map h we denote h′(x) = ||Dh(x)||. We assume h′_{i} are uniformly bounded away from 0 and ∞. We also assume that p_{i} are C^{α} and uniformly bounded away from 0. Hence, log p_{i} and log h′_{i} are C^{α}.

The system acts as follows: on every iteration we randomly choose i ∈ {1, …, k} according to the vector p⃗(x) (depending on the point x ∈ ℝ^{d} we are in) and then apply the map h_{i}. If all the maps h_{i} are contracting, there exists a unique probability measure solving equation

ν = F (ν) := ∑_{i} h_{i∗}(p_{i} · ν),  (2.1)

it is called invariant measure for the iterated function system. Moreover, the invariant measure satisfies

\overline{dim}_{H}(ν) ≤ \frac{∫ ∑_{i} p_{i}(x) log p_{i}(x)dν(x)}{∫ ∑_{i} p_{i}(x) log |h′_{i}(x)|dν(x)}  (2.2)

where the upper Hausdorff dimension of the measure is defined as the infimum of the Hausdorff dimension over sets of full measure. Let us remind here that the lower Hausdorff dimension \underline{dim}_{H} is defined as the infimum of the Hausdorff dimensions of sets of positive measure and if both are equal, the common value is called Hausdorff dimension of the measure and denoted dim_{H}.

Denote Σ = {1, …, k}^{ℕ}. Given a sequence ω ∈ Σ, we will denote by ω_{n} the n-th element and by ω^{n} the first n elements of ω. We will also denote ω^{−n} = ω_{n}ω_{n−1} … ω_{1}. The n-th level cylinder C_{ω^{n}} is the set of all sequences from Σ that begin with ω^{n}. Let σ, σ_{i} be the left shift map and the k right shift maps on Σ, i.e.

σω = ω_{2}ω_{3} …
σ_{i}ω = iω_{1}ω_{2} …

Define h_{ω^{n}} = h_{ω_{n}} ∘ … ∘ h_{ω_{1}}. Let

p_{ω^{n}}(x) = ∏_{i=1}^{n} p_{ω_{i}}(h_{ω^{i−1}}(x))

and let p_{x} be the unique probability measure satisfying

p_{x}(C_{ω^{n}}) = p_{ω^{n}}(x)

The limit

π(ω) = lim_{n→∞} h_{ω^{−n}}(x_{0})  (2.3)

exists for all ω ∈ Σ and does not depend on x_{0} ∈ ℝ^{d}. The invariant measure ν can be written as

ν = π_{∗}(μ_{−})  (2.4)

where μ_{−} is some σ-invariant probability measure on Σ.

In this paper we assume only that the IFS is contracting-on-average, ie

∑_{1}^{k} p_{j}(x) log \frac{d(h_{j}(x), h_{j}(y))}{d(x, y)} < −b < 0  (2.5)

uniformly in x and y. The existence and uniqueness of the invariant measure ν for such systems was established in [BDEG]. In addition, it was proved that the measure is attractive, i.e. that F^{n}(ν̃) → ν for all probability measures ν̃.

To get a reasonable lower bound on the Hausdorff dimension of the invariant measure we will need stronger geometric assumptions. Namely, we will assume h_{i} are conformal maps (we will call it the conformal case). The conformality assumption is void for d = 1 but very strong in higher dimension: it implies analyticity in ℝ^{2} and only Möbius transformations are conformal in ℝ^{d}, d ≥ 3 (Liouville Thm).

In conformal case, if IFS is contracting and if in addition there exists an open set U such that h_{i}(U) ⊂ U and h_{i}(U) ∩ h_{j}(U) = ∅ for i ≠ j (it is called open set condition, OSC), we have equality in (2.2) ([F]).

Our main result is that the results on the Hausdorff dimension of invariant measures for contracting-on-average iterated function systems are almost the same as for contracting ones. The one difference is that instead of OSC we have to use a bit stronger geometrical conditions.

If the IFS satisfies OSC for some open set U and if (for some R_{1})

dist(h_{i}(U), h_{j}(U)) ≥ R_{1} > 0,  (2.6)

the system satisfies strong open set condition (SOSC). If (in addition to OSC) U satisfies (for some R_{2} and R_{3})

∀_{r<R_{2}}∀_{x∈U} vol(B_{r}(x) ∩ U) ≥ R_{3}r^{d},  (2.7)

the system satisfies regular open set condition (ROSC).

Denote the right hand side of (2.2) by s.

Theorem 2.1. Let (ℝ^{d}, h_{i}, p_{i}) be an iterated function system. Assume h_{i} are C^{α} and with bounded log h′_{i}, p_{i} are C^{α} and bounded away from 0. Assume that (2.5) is satisfied. Denote the invariant measure by ν. Then

\overline{dim}_{H}(ν) ≤ s

If in addition all h_{i} are conformal and either SOSC or ROSC are satisfied,

dim_{H}(ν) = s

3 Coding map and ergodicity

Our main goal in this section is to prove the following proposition:

Proposition 3.1. Let X be a Polish space and fix x_{0} ∈ X. Let (X, h_{i}, p_{i}) be an iterated function system satisfying (2.5), with p_{i} satisfying Dini condition and bounded away from 0 and with h_{i} Lipschitz. Let ν be the invariant measure of this system. Then there exists a σ-invariant probabilistic measure μ_{−} on Σ such that (2.3) is well defined μ_{−}-almost everywhere and (2.4) holds.

Similar result was proved by Werner [We], but he used a stronger assumption instead of (2.5).

We start with some notations. Let

μ_{+}(dω) = ∫_{X} p_{x}(dω)dν(x)

It is a probability σ-invariant measure on Σ. We denote Σ̃ = {1, …, k}^{ℤ} and let σ̃ be the left shift on Σ̃, i.e. (σ̃ω)_{i} = ω_{i+1}. The space (Σ̃, σ̃) has a natural equivalence with (Σ × Σ, S), where

S(τ, ω) = (σ_{ω_{1}}τ, σω)

On Σ̃ there exists a unique σ̃-invariant measure μ such that μ_{+} is the marginal measure of μ with respect to the positive coordinates:

μ_{+}(·) = ∫ μ(dτ, ·)

The measure μ is called natural extension of μ_{+}.

We define the measure μ_{−} as the marginal measure of μ with respect to the nonpositive coordinates, i.e.

μ_{−}(·) = ∫ μ(·, dω)

The explicit formula for μ_{−} can be written as

μ_{−}(C_{ω^{n}}) = μ_{+}(C_{ω^{−n}})  (3.1)

Let π be given by (2.3) (wherever it is defined). In order to prove (2.4) we need only to prove that π is well defined for μ_{−}-almost all ω ∈ Σ. Indeed, assume it is so. The measure π_{∗}(μ_{−}) is then a probability measure on X. Moreover,

h_{i}(π(ω)) = π(σ_{i}ω)

hence π_{∗}(μ_{−}) is invariant under the action of the IFS. By uniqueness of the invariant measure, it must be equal to ν and (2.4) holds.

We will need some estimations on the ergodic sums of integrable functions over typical trajectories of the IFS. The ergodic theorem proved in [E] gives us some estimations, but we need something stronger. The following lemma is a large deviation result for typical trajectories:

Lemma 3.2. Choose arbitrary bounded A ⊂ X. Then there exists L(A) such that for all x ∈ X, n > 0 and K > 0

p_{x}(ω ∈ Σ; diam(h_{ω^{n}}(A)) ≤ K exp(−nb)) > 1 − \frac{L(A)(1 + log^{+}(d(x, x_{0})))n}{(log K)^{2}}

Proof. For j ∈ {1, …, k} and y ∈ X let

H̃(y, j) = sup_{z∈X} log \frac{d(h_{j}(y), h_{j}(z))}{d(y, z)}

and

H(y, j) = H̃(y, j) − b − ∑_{i=1}^{k} p_{i}(y)H̃(y, i)

By (2.5), H(y, j) > H̃(y, j) for all y and j. In addition, as H̃ is uniformly bounded, so is H.

Define a sequence of random variables of fixed expected value b:

H_{n}(ω) = H(h_{ω^{n−1}}(x), ω_{n})

where ω ∈ (Σ, p_{x}). By boundedness of H_{n}, we have a bound on its variation as well:

var H_{n} < v

The random variables H_{n} and ∑_{j=1}^{n−1} H_{j} are uncorrelated (but not necessarily independent). Hence

var(∑_{j=1}^{n} H_{j}) < vn

and by Tchebysheff inequality

p_{x}(ω; ∑_{j=1}^{n} H_{j}(ω) > −nb + log K) < \frac{v}{(log K)^{2}}  (3.2)

Now, we can estimate

diam(h_{ω^{n}}(A)) ≤ 2 sup_{y∈A} d(h_{ω^{n}}(x), h_{ω^{n}}(y)) ≤ 2 exp(∑_{j=1}^{n} H_{j}(ω))(d(x, x_{0})+d(x_{0}, A)+diam A)

(here we use the inequality H̃ < H and (2.5)). The assertion follows from (3.2).

Lemma 3.3.

∫_{X} log^{+}(d(x, x_{0}))dν(x) ≤ M(x_{0}) < ∞

Proof. Let ν̃ be any probability measure on X with bounded support. Let

M_{0} = max_{i} d(x_{0}, h_{i}(x_{0}))

We have for all x ∈ X

d(x_{0}, h_{i}(x)) ≤ M_{0} + d(h_{i}(x_{0}), h_{i}(x))

By (2.5) and Lipschitz property of h_{i} we conclude that there exists M_{1} > 0 such that for d(x_{0}, x) > M_{1}

∑ p_{i}(x) log^{+}(d(x_{0}, h_{i}(x))) ≤ (1 − b/2) log(d(x_{0}, x))  (3.3)

Let

M_{2} = sup_{d(x,x_{0})<M_{1}} max_{i} d(h_{i}(x), x_{0})

We divide ν̃ into two parts: ν̃_{1}, supported on a ball B_{M_{1}}(x_{0}) and ν̃_{2}, supported on the outside. Clearly,

F (ν̃) = F (ν̃_{1}) + F (ν̃_{2})

We have

∫_{X} log^{+}(d(x, x_{0}))F (ν̃_{1})(dx) ≤ log(M_{2})

and (by (3.3))

∫_{X} log^{+}(d(x, x_{0}))F (ν̃_{2})(dx) ≤ (1 − b/2) ∫_{X} log^{+}(d(x, x_{0}))dν̃(x)

Hence,

∫_{X} log^{+}(d(x, x_{0}))F (ν̃)(dx) ≤ log(M_{2}) + (1 − b/2) ∫_{X} log^{+}(d(x, x_{0}))dν̃(x)

The sequence 1/n ∑_{i=1}^{n} F^{i}(ν̃) converges weakly to ν while the upper limit of ∫_{X} log^{+}(d(x, x_{0}))F^{n}(ν̃)(dx) is not greater than 2 log(M_{2})/b.

Let h be the supremum of h′_{i}(x) over all i and all x. Let

A = {x_{0}, h_{1}(x_{0}), …, h_{k}(x_{0})}

Choose two positive constants b′′ < b′ < b and let δ > 1 be such that

b′ − (δ − 1) log h > b′′δ  (3.4)

Let n_{j} = ⌊δ^{j}⌋ and

Z_{j} = {ω ∈ Σ; diam(h_{ω^{−n_{j}}} (A)) < exp(−n_{j}b′)}

Similarly, let

Z̃_{j} = {ω ∈ Σ; diam(h_{ω^{n_{j}}} (A)) < exp(−n_{j}b′)}

Both Z_{j} and Z̃_{j} are disjoint unions of n_{j}-th level cylinders, hence by (3.1)

μ_{−}(Z_{j}) = μ_{+}(Z̃_{j})

By Lemma 3.2,

p_{x}(Z̃_{j}) ≥ 1 − \frac{L(A)(1 + log^{+}(d(x, x_{0})))}{(b − b′)^{2}n_{j}}

Using Lemma 3.3 we get

μ_{+}(Z̃_{j}) = ∫_{X} p_{x}(Z̃_{j})dν(x) ≥ 1 − \frac{L(A)(1 + M(x_{0}))}{(b − b′)^{2}n_{j}}

Hence, the series ∑(1 − μ_{−}(Z_{j})) = ∑(1 − μ_{+}(Z̃_{j})) is convergent. Denote Z = lim inf Z_{j}. By Borel-Cantelli Lemma

μ_{−}(Z) = 1

As

diam(h_{j} ∘ h_{ω^{n}}(A)) ≤ h diam(h_{ω^{n}}(A)),

(3.4) implies that for ω ∈ ⋂_{j≥m} Z_{j} and n > n_{m}

diam(h_{ω^{−n}}(A)) < exp(−nb′′)

In particular, for all ω ∈ Z the sequence diam(h_{ω^{−n}}(A)) is summable. We can estimate

d(h_{ω^{−(n+1)}}(x_{0}), h_{ω^{−n}}(x_{0})) = d(h_{ω^{−n}}(h_{ω_{n+1}}(x_{0})), h_{ω^{−n}}(x_{0})) ≤ diam(h_{ω^{−n}}(A))

hence h_{ω^{−n}}(x_{0}) form a Cauchy sequence. We proved thus that the limit in (2.3) exists for all ω ∈ Z, i.e. for a set of full measure μ_{−}. As stated above, (2.4) follows and we are done with Proposition 3.1.

Consider the space X × Σ with measure

dν̄(x, ω) = p_{x}(dω)dν(x)

This measure is invariant under the map

g(x, ω) = (h_{ω_{1}}(x), σω)

By Elton [E], Lemma 1, this map is ergodic. An immediate corollary of Elton’s Lemma and Proposition 3.1 is the following Proposition 3.4. Under assumptions of Proposition 3.1, the map (Σ̃, σ̃, μ) is ergodic.

Proof. Denote the ergodic average for any f ∈ L^{1}(Σ̃, μ) by f̄. To prove the assertion we need to prove that f̄ exists and is equal to μ(f) μ-almost everywhere.

Note first that (X×Σ, g, ν̄) is a factor of (Σ×Σ, S, μ) under the projection (π, Id). Hence, the assertion is true for functions of the form f(τ, ω) = f(ω), i.e. depending only on the future. As the ergodic average doesn’t depend on the finite number of initial summands, same is true for functions of the form f(τ, ω) = f(τ^{n}, ω) for any n. Those functions are dense in L^{1}(σ×Σ, μ), hence the assertion follows for all functions by standard density argument.

We remind that as (Σ̃, σ̃) is bijective, the map (Σ̃, σ̃^{−1}, μ) is also ergodic.

4 Ergodic properties of typical orbits

Let us define the Lyapunov exponent and (minus) entropy of the system:

λ = ∫ ∑_{j=1}^{k} p_{j}(x) log |h′_{j}(x)|dν(x),

η = ∫ ∑_{j=1}^{k} p_{j}(x) log p_{j}(x)dν(x)

By (2.5) and ergodicity of (ℝ^{d} × Σ, g), λ < 0.

For any z_{1}, …, z_{k} ∈ L^{1}(ℝ^{d}, ν) we denote

z = ∫_{ℝ^{d}} ∑_{j=1}^{k} p_{j}(x)z_{j}(x)dν(x)

and

z_{ω^{n}}(x) = ∑_{i=1}^{n} z_{ω_{i}}(h_{ω^{i−1}}(x)).

The following two lemmas are direct consequences of the Birkhoff Ergodic Theorem; the first by application to the map (Σ̃, σ̃, μ) and the function f(τ, ω) = z_{ω_{1}}(π(τ)), the second is obtained for the same f but for map (Σ̃, σ̃^{−1}, μ). Note that

μ(f) = ν̄(f ∘ (π^{−1}, Id)) = ∫_{ℝ^{d}} ∑_{j=1}^{k} z_{j}(x)p_{j}(x)dν(x) = z

Lemma 4.1. For any ε > 0, μ_{+}-almost every ω ∈ Σ and ν-almost all x ∈ ℝ^{d} one can find a constant K > 0 such that for all n

K^{−1} exp(n(1 + ε)z) ≤ exp(z_{ω^{n}}(x)) ≤ K exp(n(1 − ε)z).

Lemma 4.2. For any ε > 0 and μ_{−}-almost every ω ∈ Σ there exists K > 0 such that for all n

K^{−1} exp(n(1 + ε)z) ≤ exp(z_{ω^{−n}} ∘ π(σ^{n}ω)) ≤ K exp(n(1 − ε)z).

The following is a standard bounded distortion lemma.

Lemma 4.3. For every ε > 0 and K > 0 there exists L(K) such that if for some x ∈ ℝ^{d} and ω ∈ Σ and for all n > 0

K^{−1} exp(n(1 + ε)λ) ≤ h′_{ω^{n}}(x) ≤ K exp(n(1 − ε)λ)  (4.1)

then

∑_{m=0}^{n} diam(h_{ω^{m}}(B_{L(K)}(x))) ≤ 1

Proof. Choose some L. We have

diam h_{ω^{n}}(B_{L}(x)) ≤ L · sup_{y∈B_{L}(x)} h′_{ω^{n}}(y)  (4.2)

As log |h′_{i}| is C^{α}, for all y ∈ B_{L}(x)

|h′_{ω^{n}}(y) − h′_{ω^{n}}(x)| ≤ c ∑_{i=0}^{n−1}(d(h_{ω^{i}}(x), h_{ω^{i}}(y)))^{α}h′_{ω^{n}}(x)  (4.3)

Denote

d_{0} = 2L

and

d_{m} = 2LK exp(m(1 − ε)λ)(1 + c ∑_{j=0}^{m−1} d_{j}^{α})

By (4.2) and (4.3) we have

d_{m} ≥ diam(h_{ω^{m}}(B_{L}(x)))

Consider the series

M(L) = ∑_{i=0}^{∞}(L(1 + c)K exp(i(1 − ε)λ))^{α}

For L small enough (depending on K) M(L) is smaller than 1. For such L for all n

∑_{m=0}^{n} d_{m}^{α} ≤ M(L)

It implies that

∑_{m=0}^{n} diam(h_{ω^{m}}(B_{L(K)}(x))) ≤ 2LK(1 + cM(L)) ∑_{m=0}^{n} exp(m(1 − ε)λ) ≤

≤ \frac{2LK(1 + cM(L))}{e^{−λ} − 1}

and the right hand side is arbitrarily small (independent of n) for L sufficiently small.

Corollary 4.4. Under assumptions of Lemma 4.3, for any C^{α} functions z_{i} the difference z_{ω^{n}}(x) − z_{ω^{n}}(y) is uniformly bounded for all y ∈ B_{L(K)}(x).

The following bounded distortion lemma implies that both (2.6) and(2.7) are preserved under iterations (up to rescaling).

Lemma 4.5. Fix x ∈ ℝ^{d}, R > 0 and ω^{n}. Assume that for some K > 0 and ε > 0 and for all m ≤ n

K^{−1} exp(m(1 + ε)λ) ≤ h′_{(σ^{n−m}ω)^{m}}(h_{ω^{m}}(x)) ≤ K exp(m(1 − ε)λ)

Then there exists l(K, R) such that for all r ≤ l(K) exp(n(1 + 2ε)λ) we have

∑_{m=0}^{n} diam(h_{(σ^{n−m}ω)^{m}}^{−1}(B_{r}(h_{ω^{n}}(x)))) ≤ R

Proof. Choose some l and let

r = l exp(n(1 + 2ε)λ)

Like in the proof of Lemma 4.3, we can estimate

diam(h_{(σ^{n−m}ω)^{m}}^{−1}(B_{r}(h_{ω^{n}}(x)))) ≤ d_{m}

where

d_{0} = 2r

and

d_{m} = 2rK exp(−m(1 + ε)λ)(1 + c ∑_{j=0}^{m−1} d_{j}^{α})

Consider the series

M(l) = ∑_{j=0}^{∞}(2lK(1 + c) exp(jελ))^{α}

For l small enough (depending on K),

M(l) < 1

and

∑_{m=0}^{n−1} d_{m}^{α} < M(l)

Hence,

∑_{m=0}^{n} diam(h_{(σ^{n−m}ω)^{m}}^{−1}(B_{r}(h_{ω^{n}}(x)))) ≤ 2lK(1 + M(l))e^{n(1+2ε)λ} ∑_{m=0}^{n} e^{−m(1+ε)λ} ≤

≤ \frac{2e^{−λ}lK(1 + c)}{e^{−(1+ε)λ} − 1}

and for l small enough the right hand side is smaller than any constant, independently of n.

Corollary 4.6. Under assumptions of Lemma 4.5, for any C^{α} functions z_{i} the difference z_{ω^{n}}(x) − z_{ω^{n}}(y) is uniformly bounded for all y ∈ h_{ω^{n}}^{−1}(B_{r}(x)).

Before we begin the proof of Theorem 2.1, let us show an example of iterated function system satisfying the regular open set condition. The maps are not similitudes (it is impossible to have OSC for similitudes that are not all contracting) but their derivatives are constant in U. Outside U it may well happen that all the maps are expanding at some point, hence (2.5) is satisfied only on U. The system is thus not necessarily contracting-on-average. However, Theorem 2.1 still holds.

Example 4.7. Let B_{n} = (10^{n}, 3 · 10^{n}) for n ≥ 0. Let

h_{1}(x) = \frac{1}{20}x + 15 · 10^{n−2}

for x ∈ B_{n}, n > 0,

h_{1}(x) = \frac{1}{20}x + 1

for x ≤ 3,

h_{2}(x) = 5x + 5 · 10^{n}

for x ∈ B_{n}, n > 0,

h_{2}(x) = 5x + 5

for x ≤ 3. We extend maps h_{1}, h_{2} to be homeomorphisms of ℝ. The system satisfies the regular open set condition for U = ⋃ B_{n} (regularity is satisfied because all the components of U have diameters bounded away from 0). The system is contracting-on-average (on U only) for

p_{1}(x) = p_{1} = 1 − p_{2} > \frac{log(50/7)}{log(500/17)}

For negative x the contracting map h_{1} is stronger than the dilatating map h_{2} (in addition it is applied more often for typical ω) and the fixed point for h_{2} is smaller than the fixed point for h_{1}. Similarly, for positive x both h_{1} and h_{2} are equally strong (former moves B_{i} into B_{i−1}, latter into B_{i+1}) but the former happens more often for typical ω. Hence, wherever we start, the typical trajectory lands inside B_{0} after finite number of iterations. The invariant measure (if it exists) is thus supported on U, hence independent from the way we define h_{1} and h_{2} outside U.

Note that we may define h_{1} and h_{2} in such a way that the system satisfies (2.5) on whole ℝ. Hence, the invariant measure indeed exists and is unique. Its Hausdorff dimension equals

dim_{H}(ν) = \frac{p_{1} log p_{1} + p_{2} log p_{2}}{−2p_{1} log 2 − (p_{1} − p_{2}) log 5}.  (4.4)

More detailed study shows that the necessary and sufficient condition for existence of invariant measure is p_{1} > 1/2, even though the iterated function system doesn’t satisfy (2.5) in this case. The invariant measure is always unique and its Hausdorff dimension is still given by (4.4).

5 Proof of Theorem 2.1

5.1 Upper bound

Fix small ε > 0. Choose some big K > 0 and n > 0.

Let Σ(n, K, ε)(x) be the set of sequences ω ∈ Σ for which

K^{−1} exp(n(1 + ε)η) ≤ p_{ω^{n}}(x) ≤ K exp(n(1 − ε)η)

and

K^{−1} exp(n(1 + ε)λ) ≤ |h′_{ω^{n}}(x)| ≤ K exp(n(1 − ε)λ).  (5.1)

By Lemma 4.1, μ(Σ(n, K, ε)(x)) is (for sufficiently big K, independently of n) greater than (1 − ε) for all x except some set A(n, K, ε) of ν-measure smaller than ε. In addition, ν([−K, K]^{d}) > 1 − ε for K big enough.

Let us assume K is sufficiently big that all the above is satisfied. Note that (as ∑_{ω^{n}} p_{ω^{n}}(x) = 1) ω^{n} can take at most K exp(−n(1 + ε)η) values inside each Σ(n, K, ε)(x). Given ω^{n}, let B(ω^{n}, K, ε) denote the set of x for which ω^{n} = τ^{n} for some τ ∈ Σ(n, K, ε)(x). Denote

ν_{n} = ∑_{ω^{n}} h_{ω^{n}∗}(p_{ω^{n}} · χ_{B(ω^{n},K,ε)∩[−K,K]^{d}} · ν)

As

ν = ∑ h_{ω^{n}∗}(p_{ω^{n}}ν),

we have ν_{n} ≤ ν.

At the same time, for any x ∈ [−K, K]^{d} \ A(n, K, ε) we have

p_{x}(ω; x ∈ B(ω^{n}, K, ε)) ≥ 1 − ε

Hence,

|ν_{n}| ≥ ν([−K, K]^{d}\A(n, K, ε))· inf_{x∈[−K,K]^{d}\A(n,K,ε)} p_{x}(ω; x ∈ B(ω^{n}, K, ε)) ≥ 1−3ε.

Let (E_{i}^{ε,n}) be a family of sets of diameter L(K), covering [−K, K]^{d} \ A(n, K, ε). This family may be chosen to have at most (2dK/L(K))^{d} elements. Let us choose a point x_{i}^{ε,n} inside each E_{i}^{ε,n} \ A(n, K, ε) whenever this set is nonempty. Let (F_{j}^{ε,n}) be a family of sets of the form h_{ω^{n}}(E_{i}^{ε,n}) for ω ∈ Σ(n, K, ε)(x_{i}^{ε,n}). This family has at most 2^{d}d^{d}K^{d+1}L(K)^{−d} exp(−n(1 + ε)η) elements. By Lemma 4.3, Corollary 4.4 and (5.1) each of sets F_{j}^{ε,n} has diameter not greater than K_{0}L(K)K exp(n(1 − ε)λ).

As the support of ν_{n} is contained in ⋃_{ω^{n}} h_{ω^{n}}([−K, K]^{d} ∩ B(ω^{n}, K, ε)), it is also contained in ⋃ F_{j}^{ε,n}. The sum

∑(diam F_{j}^{ε,n})^{s(1+3ε)} ≤ 2^{d}d^{d}K_{0}^{s(1+3ε)}K^{d+1+s(1+3ε)}L(K)^{s(1+3ε)−d} exp(nηε(1−3ε))

is arbitrarily small for big n. Let us choose n for which this sum is smaller than ε and denote Z_{ε} = ⋃ F_{j}^{ε,n}.

We can now repeat the procedure for a smaller ε_{m} = 2^{−m}ε. The set Z = lim sup Z_{ε_{m}} has full ν-measure and we have constructed a family of its covers (G^{M}) = ⋃_{m>M} ⋃ F_{j}^{ε_{m}} such that

∑_{G_{l}^{M} ∈G^{M}}(diam G_{l}^{M})^{s(1+3ε)} ≤ 1.

Hence,

dim_{H}(Z) ≤ s(1 + 3ε)

and ε can be arbitrarily small.

5.2 Lower bound

We claim that if we can find a family of sets Σ_{K} ∈ Σ such that μ(Σ_{K}) ↗ 1 and the measures ν_{K} = π_{∗}(μ_{|Σ_{K}}) have lower Hausdorff dimension not smaller than s_{0} then the lower Hausdorff dimension of ν will be not smaller than s_{0}.

Suppose it is not true, i.e. there exists a set X ⊂ ℝ^{d} such that ν(X) > 0 and dim_{H}(X) < s_{0}. For some K, ν(X) + μ(Σ_{K}) > 1. Hence, the set Y = π^{−1}(X) ∩ Σ_{K} has positive measure μ. It follows that π(Y) has positive measure ν_{K} and its Hausdorff dimension is smaller than s_{0} – a contradiction.

Thus, we need only to prove the existence of such families of sets Σ_{K} (or equivalently, of measures ν_{K}) of lower Hausdorff dimension arbitrarily close to s.

Fix ε > 0, K > 0. Let Σ_{K} be the set of ω ∈ Σ for which for all n

K^{−1} exp(−n(1 − ε)λ) ≤ |(h_{ω^{−n}}^{−1})′ ∘ π(ω)| ≤ K exp(−n(1 + ε)λ)  (5.2)

and

K^{−1} exp(n(1 + ε)η) ≤ p_{ω^{−n}} ∘ π(σ^{n}ω) ≤ K exp(n(1 − ε)η).  (5.3)

For K big enough μ_{−}(Σ_{K}) is arbitrarily close to 1 by Lemma 4.2. Let x = π(ω) for some ω ∈ Σ_{K} and

r_{n} = K^{−1}l(K, R) exp(n(1 + 2ε)λ),

where R = R_{1} in SOSC case or R = R_{2} in ROSC case. The ball B_{r_{n}}(x) may in general intersect many of the sets h_{τ^{n}}(U). Let T be the set of all τ^{n} for which ν_{K}(B_{r_{n}}(x) ∩ h_{τ^{n}}(U)) > 0.

By Lemma 4.5, Corollary 4.6 and (5.3) we have

p_{τ^{n}}(h_{τ^{n}}^{−1}(y)) ≤ K_{0}K exp(n(1 − ε)η)

for ν_{K}-almost all y ∈ B_{r_{n}}(x) ∩ h_{τ^{n}}(U). Hence,

ν_{K}(h_{τ^{n}}(U) ∩ B_{r_{n}}(x)) ≤ ν(h_{τ^{n}}(U) ∩ B_{r_{n}}(x)) ≤ K exp(n(1 − ε)η)

for all τ^{n} ∈ T.

We claim that T has uniformly bounded (independently of n) number of elements. Assume first that the strong open set condition is satisfied. Choose any point y ∈ U and any τ^{n} ≠ ω^{n}. Let m = min{j ≥ 1; ω_{j} ≠ τ_{j}} − 1. By SOSC,

d(h_{(σ^{m}ω)^{−(n−m)}}(h_{ω^{−n}}^{−1}(x)), h_{(σ^{m}τ)^{−(n−m)}}(y)) > R_{1}

Hence, by Lemma 4.5

h_{τ^{−n}}(y) ∉ B_{r_{n}}(x)

and thus τ^{−n} ∉ T. Hence, T has just one element ω^{−n}.

Consider now the regular open set condition situation. Choose τ^{n} ∈ T and ν_{K}-typical y ∈ h_{τ^{n}}(U) ∩ B_{r_{n}}(x). By Lemma 4.5 and Corollary 4.6,

B_{K_{0}r}(h_{τ^{n}}^{−1}(y)) ⊂ h_{τ^{n}}^{−1}(B_{r_{n}}(x)) ⊂ B_{r}(h_{τ^{n}}^{−1}(y))

for some r < R_{2}. By ROSC

vol(U ∩ h_{τ^{n}}^{−1}(B_{r_{n}}(x))) ≥ R_{3}K_{0}^{d} vol(h_{τ^{n}}^{−1}(B_{r_{n}}(x)))

and (by Corollary 4.6 again)

vol(h_{τ^{n}}(U) ∩ B_{r_{n}}(x)) ≥ K_{1} vol(B_{r_{n}}(x))

As h_{τ^{n}}(U) are pairwise disjoint, T can have at most K_{1}^{−1} elements. Hence, if either SOSC or ROSC are satisfied,

ν_{K}(B_{r_{n}}(x)) ≤ K_{2} exp(n(1 − ε)η)

with K_{2} depending on K but not on x or n. Hence,

lim sup_{r→0} \frac{ν_{K}(B_{r}(x))}{(2r)^{s(1−4ε)}} = 0  (5.4)

We only assumed above (5.2) and (5.3), so (5.4) is satisfied ν_{K}-almost everywhere, which (by Frostman Lemma) implies that the lower Hausdorff dimension of ν_{K} is not smaller than s(1 − 4ε).

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