We introduce and study skew product Smale endomorphisms over finitely irreducible shifts with countable alphabets. This case is different from the one with finite alphabets and we develop new methods. In the conformal context we prove that almost all conditional measures of equilibrium states of summable Hölder continuous potentials are exact dimensional, and their dimension is equal to the ratio of (global) entropy and Lyapunov exponent. We show that the exact dimensionality of conditional measures on fibers implies global exact dimensionality of the original measure. We then study equilibrium states for skew products over expanding Markov-Rényi transformations, and settle the question of exact dimensionality of such measures. We apply our results to skew products over the continued fractions transformation. This allows us to extend and improve the Doeblin-Lenstra Conjecture on Diophantine approximation coefficients, to a larger class of measures and irrational numbers.MSC 2010: 37D35, 37A35, 37C45, 37A45, 46G10, 60B05. Kewords: Skew product endomorphisms, equilibrium measures, exact dimensional measures, dimensions, stable sets, entropy, natural extensions, continued fractions.Contents 1 EUGEN MIHAILESCU AND MARIUSZ URBAŃSKI Proof. By Theorem 4.12 there exists ν ∈ M σ (E A ) such that ν •π −1 = µ. The other theorems listed immediately above give:We have the following two definitions. Definition 4.16. The measure µ ∈ M T (J) is called an equilibrium state of the continuous potential ψ :Ŷ → R, if ψ dµ > −∞ and h µ (T) + J ψ dµ = P T (ψ).Definition 4.17. The potential ψ : J → R is called summable if e∈E exp sup(ψ| [e] T ) < ∞.Observation 4.18. ψ : J → R is summable if and only if ψ •π : E A → R is summable.Definition 4.19. We call a continuous skew product Smale endomorphism T :Ŷ →Ŷ Hölder, if the projectionπ : E A → J is Hölder continuous.We now establish an important property of Hölder skew product Smale endomorphisms of compact type, and then will describe a general construction of such endomorphisms.Theorem 4.20. If T : J → J is Hölder skew product Smale endomorphism of compact type and ψ : J → R is a Hölder summable potential, then ψ admits a unique equilibrium state, denoted by µ ψ . In addition µ ψ = µ ψ•π •π −1 , where µ ψ•π is the unique equilibrium state of ψ •π : E A → R with respect to σ :Proof. ψ •π : E A → R is a summable Hölder continuous potential, so it has a unique equilibrium state µ ψ•π by Theorem 2.6. By Observation 4.14 and Observation 4.4 we haveWe have to show that, if µ is an equilibrium measure of ψ, then µ = µ ψ•π •π −1 . In this case, from Theorem 4.12, we get µ = ν •π −1 for some ν ∈ M σ (E A ). Then by Observation 4.14,Hence, ν is an equilibrium state of ψ •π : E A → R and ν = µ ψ•π (see Theorem 2.6). Now we provide the promised construction of Hölder Smale skew product endomorphisms. Start with (Y, d), a complete bounded metric space, and assume given for every ω ∈ E + A a continuous closed injective map T ω : Y → Y , satisfying the following conditions (4.13) d(T ω (y 2 ), T ω (y 1 ...
Abstract. We study the case of an Axiom A holomorphic non-degenerate (hence non-invertible) map f : P 2 C → P 2 C, where P 2 C stands for the complex projective space of dimension 2. Let Λ denote a basic set for f of unstable index 1, and x an arbitrary point of Λ; we denote by δ s (x) the Hausdorff dimension of W s r (x) ∩ Λ, where r is some fixed positive number and W s r (x) is the local stable manifold at x of size r; δ s (x) is called the stable dimension at x. Mihailescu and Urbański introduced a notion of inverse topological pressure, denoted by P − , which takes into consideration preimages of points. Manning and McCluskey studied the case of hyperbolic diffeomorphisms on real surfaces and give formulas for Hausdorff dimension. Our non-invertible situation is different here since the local unstable manifolds are not uniquely determined by their base point, instead they depend in general on whole prehistories of the base points. Hence our methods are different and are based on using a sequence of inverse pressures for the iterates of f , in order to give upper and lower estimates of the stable dimension. We obtain an estimate of the oscillation of the stable dimension on Λ. When each point x from Λ has the same number d ′ of preimages in Λ, then we show that δ s (x) is independent of x; in fact δ s (x) is shown to be equal in this case with the unique zero of the map t → P(tφ s − log d ′ ). We also prove the Lipschitz continuity of the stable vector spaces over Λ; this proof is again different than the one for diffeomorphisms (however, the unstable distribution is not always Lipschitz for conformal non-invertible maps). In the end we include the corresponding results for a real conformal setting.
Abstract. We study invariant measures for random countable (finite or infinite) conformal iterated function systems (IFS) with arbitrary overlaps. We do not assume any type of separation condition. We prove, under a mild assumption of finite entropy, the dimensional exactness of the projections of invariant measures from the shift space, and we give a formula for their dimension, in the context of random infinite conformal iterated function systems with overlaps. There exist many differences between our case and the finite deterministic case studied in [7], and we introduce new methods specific to the infinite and random case. We apply our results towards a problem related to a conjecture of Lyons about random continued fractions ([10]), and show that for Lebesgue-almost all parameters λ > 0, the invariant measure ν λ is exact dimensional. The finite IFS determining these continued fractions is not hyperbolic, but we can associate to it a random infinite IFS of contractions which have overlaps. We study then also other large classes of random countable iterated function systems with overlaps, namely: a) several types of random iterated function systems related to Kahane-Salem sets; and b) randomized infinite IFS in the plane which have uniformly bounded number of disc overlaps. For all the above classes, we find lower and upper estimates for the pointwise (and Hausdorff, packing) dimensions of the invariant measures.
A unique feature of smooth hyperbolic non-invertible maps is that of having different unstable directions corresponding to different prehistories of the same point. In this paper we construct a new class of examples of non-invertible hyperbolic skew products with thick fibers for which we prove that there exist uncountably many points in the locally maximal invariant set (actually a Cantor set in each fiber), having different unstable directions corresponding to different prehistories; also we estimate the angle between such unstable directions. We discuss then the Hausdorff dimension of the fibers of for these maps by employing the thickness of Cantor sets, the inverse pressure, and also by use of continuous bounds for the preimage counting function. We prove that in certain examples, there are uncountably many points in with two preimages belonging to , as well as uncountably many points having only one preimage in . In the end we give examples which, also from the point of view of Hausdorff dimension, are far from being homeomorphisms on , as well as far from being constant-to-1 maps on .
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