Invariant measures of full dimension for some expanding maps

Gatzouras, D.; Peres, Yuval

doi:10.1017/s0143385797060987

Cited by 85 publications

(90 citation statements)

References 6 publications

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“…Consider y ∈ X such that π(y) ∈ [2 n 1] and z ∈ X such that π(z) = 2 ∞ . Then (6) and the fact that (S n ϕ)(z) = 0 for all n, and var n (S n ϕ) is not bounded. This is a contradiction.…”

Section: Proof Of (2)(d) and (E)mentioning

confidence: 99%

“…Shin [18] showed that if there is a saturated compensation function G • π (see page 5 for the definition for the factor map π), then for any α > 0 the set of all shift-invariant measures µ on X that maximize h µ (σ X ) + αh πµ (σ Y ) is the set of equilibrium states for the function (α/(α + 1))G • π. A saturated compensation function helps us make some progress on the problem, giving us a systematic way to approach the problem (see pages 5,6, and 6). We will answer questions about SFT-NC carpets for which a saturated compensation function exists (when they are represented in symbolic dynamics, see page 6).…”

Section: Introductionmentioning

confidence: 99%

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Dimensions of compact invariant sets of some expanding maps

Yayama

2009

Ergod. Th. Dynam. Sys.

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Abstract. We study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a "general Sierpinski carpet" was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by using compensation functions to study a general Sierpinski carpet represented by a shift of finite type. We give some conditions under which a general Sierpinski carpet has a unique measure of full Hausdorff dimension and study the properties of the unique measure.

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Section: Proof Of (2)(d) and (E)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dimensions of compact invariant sets of some expanding maps

Yayama

2009

Ergod. Th. Dynam. Sys.

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“…The smoothness C 1+γ was recently relaxed to C 1 [9]. An estimate from above for the Hausdorff dimension of compact invariant sets for differentiable maps has been given by A. Douady and J. Oesterlé [5], and by Ledrappier [11].…”

Section: Introductionmentioning

confidence: 99%

The dimensions of a non-conformal repeller and an average conformal repeller

Ban

Cao

2009

Trans. Amer. Math. Soc.

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Abstract. In this paper, using thermodynamic formalism for the sub-additive potential, upper bounds for the Hausdorff dimension and the box dimension of non-conformal repellers are obtained as the sub-additive Bowen equation. The map f only needs to be C 1 , without additional conditions. We also prove that all the upper bounds for the Hausdorff dimension obtained in earlier papers coincide. This unifies their results. Furthermore we define an average conformal repeller and prove that the dimension of an average conformal repeller equals the unique root of the sub-additive Bowen equation.

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“…[Bow79], [Rue82], [Fal89], [Bar96], [GP97] and our introduction. It seems to be the first time that it is stated in the above generality.…”

Section: Random Julia Sets and Their Dimensionsmentioning

confidence: 99%

On the dimensions of conformal repellers. Randomness and parameter dependency

Rugh

2008

Ann. Math.

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Bowen's formula relates the Hausdorff dimension of a conformal repeller to the zero of a 'pressure' function. We present an elementary, self-contained proof to show that Bowen's formula holds for C 1 conformal repellers. We consider time-dependent conformal repellers obtained as invariant subsets for sequences of conformally expanding maps within a suitable class. We show that Bowen's formula generalizes to such a repeller and that if the sequence is picked at random then the Hausdorff dimension of the repeller almost surely agrees with its upper and lower box dimensions and is given by a natural generalization of Bowen's formula. For a random uniformly hyperbolic Julia set on the Riemann sphere we show that if the family of maps and the probability law depend realanalytically on parameters then so does its almost sure Hausdorff dimension. Random Julia sets and their dimensionsLet (U, d U ) be an open, connected subset of the Riemann sphere avoiding at least three points and equipped with a hyperbolic metric. Let K ⊂ U be a compact subset. We denote by E(K, U ) the space of unramified conformal covering maps f : D f → U with the requirement that the covering domain D f ⊂ K. Denote by Df : D f → R + the conformal derivative of f , see equation (2.4), and by Df = sup f −1 K Df the maximal value of this derivative over the set f −1 K. Let F = (f n ) ⊂ E(K, U ) be a sequence of such maps. The intersectiondefines a uniformly hyperbolic Julia set for the sequence F. Let (Υ, ν) be a probability space and let ω ∈ Υ → f ω ∈ E(K, U ) be a ν-measurable map. Suppose that the elements in the sequence F are picked independently, each according to the law ν. Then J(F) becomes a random 'variable'. Our main objective is to establish the following 696 HANS HENRIK RUGH Theorem 1.1. I. Suppose that E(log Df ω ) < ∞. Then almost surely, the Hausdorff dimension of J(F) is constant and equals its upper and lower box dimensions. The common value is given by a generalization of Bowen's formula.II. Suppose in addition that there is a real parameter t having a complex extension so that: (a) The family of maps (f t,ω ) ω∈Υ depends analytically upon t. (b) The probability measure ν t depends real-analytically on t. (c) Given any local inverse, f −1 t,ω , the log-derivative log Df t,ω •f −1 t,ω is (uniformly in ω ∈ Υ) Lipschitz with respect to t. (d) For each t the condition number Df t,ω · 1/Df t,ω is uniformly bounded in ω ∈ Υ.Then the almost sure Hausdorff dimension obtained in part I depends realanalytically on t. (For a precise definition of the parameter t we refer to Section 6.3, for conditions (a), (c) and (d) see Definition 6.8 and Assumption 6.13, and for (b) see Definition 7.1 and Assumption 7.3. We prove Theorem 1.1 in Section 7). Example 1.2. Let a ∈ C and r ≥ 0 be such that |a| + r < 1 4 . Suppose that c n ∈ C, n ∈ N are i.i.d. random variables uniformly distributed in the closed disk B(a, r) and that N n , n ∈ N are i.i.d. random variables distributed according to a Poisson law of parameter λ ≥ 0. We consider the seque...

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Invariant measures of full dimension for some expanding maps

Cited by 85 publications

References 6 publications

Dimensions of compact invariant sets of some expanding maps

Dimensions of compact invariant sets of some expanding maps

The dimensions of a non-conformal repeller and an average conformal repeller

On the dimensions of conformal repellers. Randomness and parameter dependency

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