Ergodic Theory of One-Dimensional Mappings

Contemporary Mathematics, 2020

This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure or finitely many such measures (finitely ergodic homeomorphisms). Since every Cantor dynamical system (X, T) can be realized as a Vershik map acting on the path space of a Bratteli diagram, we use combinatorial methods developed in symbolic dynamics and Bratteli diagrams during the last decade to study the simplex of invariant measures.

Journal of Applied Mathematics and Stochastic Analysis, 2001

LetTλ(x)=cos(λarccosx),−1≤x≤1, whereλ>1is not an integer. For a certain set ofλ's which are irrational, the density of the unique absolutely continuous measure invariant underTλis determined exactly. This is accomplished by showing thatTλis differentially conjugate to a piecewise linear Markov map whose unique invariant density can be computed as the unique left eigenvector of a matrix.

Nonlinearity, 2016

This dissertation consists of two parts. In the first part, we consider a piecewise expanding unimodal map (PEUM) f : [0, 1] → [0, 1] with µ = ρdx the (unique) SRB measure associated to it and we show that ρ has a Taylor expansion in the Whitney sense. Moreover, we prove that the set of points where ρ is not differentiable is uncountable and has Hausdorff dimension equal to zero. In the second part, we consider a family f t : [0, 1] → [0, 1] of PEUMs with µ t the correspoding SRB measure and we present a new proof of [3] when considering the observables in C 1 [0, 1]. That is, Γ(t) = ∫ ϕdµ t is differentiable at t = 0, with ϕ ∈ C 1 [0, 1], when assuming J(c) = ∑ ∞ k=0 v(f k (c)) Df k (f (c)) is zero. Furthermore, we show that in fact Γ(t) is never differentiable when J(c) is not zero and we give the exact modulus of continuity.

2004

We consider a transformation of a normalized measure space such that the image of any point is a finite set. We call such transformation $m$-transformation. In this case the orbit of any point looks like a tree. In the study of $m$-transformations we are interested in the properties of the trees. An $m$-transformation generates a stochastic kernel and a new measure. Using these objects, we introduce analogies of some main concept of ergodic theory: ergodicity, Koopman and Frobenius-Perron operators etc. We prove ergodic theorems and consider examples. We also indicate possible applications to fractal geometry and give a generalization of our construction. Some results which have analogies in the classical ergodic theory we are proved using standard methods. Other results have no analogies.

Computers & Mathematics with Applications, 1995

Let r [0,1]-, [0,1] be the map defined by ~-(x)-2x(mod 1) and let A denote Lebesgue measure on [0, 1], which is the unique absolutely continuous T-invariant measure. We construct sequences of transformations {Tn} such that ~'n-* T uniformly as n-* oo, but the sequence {~n} of associated absolutely continuous rn-invariant measures does not converge to A, not even weakly. Indeed, we prove that {~n} converges to a measure singular with respect to A. Furthermore, we characterize this singular measure in terms of the approximating transformations. We also show that any T-ergodic invariant measure can be realized as the weak limit of a sequence of absolutely continuous invariant measures associated with appropriate approximating transformations.

arXiv (Cornell University), 2019

We show in this work that the upper and the lower generalized fractal dimensions $D^{\pm}_{\mu}(q)$, for each $q\in\mathbb{R}$, of an ergodic measure associated with an invertible bi-Lipschitz transformation over a Polish metric space are equal, respectively, to its packing and Hausdorff dimensions. This is particularly true for hyperbolic ergodic measures associated with $C^{1+\alpha}$-diffeomorphisms of smooth compact Riemannian manifolds, from which follows an extension of Young's Theorem (Young, L., S. Dimension, entropy and Lyapunov exponents. Ergodic Theory and Dynamical Systems, 2(1):109-124, 1982). Analogous results are obtained for expanding systems. Furthermore, for expansive homeomorphisms (like $C^1$-Axiom A systems), we show that the set of invariant measures with zero correlation dimension, under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each $s\ge 0$, $D^{+}_{\mu}(s)$ is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric.

arXiv: Dynamical Systems, 2018

For topologically mixing locally expanding semigroup actions generated by a finite collection of $C^{1+\alpha}$ conformal local diffeomorphisms, we provide a countable Markov partition satisfying the finite images and the finite cycle properties. We show that they admit inducing schemes and describe the tower constructions associated with them. An important feature of these towers is that their induced maps are equivalent to a subshift of countable type. Through the investigating the ergodic properties of induced map, we prove the existence of a liftable absolutely continuous stationary measure for the original action. We then establish a thermodynamic formalism for induced map and deduce the unicity of eqiliberium state.

We prove that there is a hyperbolic transcendental entire map that generates a class of potentials which is different from the ones studied by Mayer and Urba\'nski (2010). Moreover, a metric on a non-compact subset of the full shift with a countable alphabet is given, which is not necessarily compatible with the natural shift metric. This subset encodes the dynamics of a subset of the Julia set of hyperbolic transcendental maps, which is non-compact and non-Markov, so we study the existence of a conformal and an invariant probability measure of the shift on this subset and a class of weakly H\"older continuous potentials.

Canadian Journal of Mathematics, 1982

1. Introduction. Let be a probability space with standard. Let T be a bimeasurable one-to-one map of Ω onto itself. Let U: Ω → Ω be another measurable transformation whose orbits are contained in the T-orbits; that is, where Z denotes the set of integers. (This is equivalent to saying that there is a measurable mapping L: Ω → Z such that U(ω) = T L(ω) (ω), ω ∈ Ω.) Such pairs (T, U) arise quite naturally in ergodic theory and information theory. (For example, in ergodic theory, one can see such pairs in the study of the full group of a transformation [1]; in information theory, such a pair can be associated with the input and output of a variable-length source encoder [2] [3].) Neveu [4] obtained necessary and sufficient conditions for U to be measure-preserving if T is measure-preserving. However, if U fails to be measure-preserving, U might still possess many of the features of measure-preserving transformations.

The aim of this paper is to establish some results in connection with the chaotic behaviours of the forward shift map   on the generalised one-sided symbol space

2009

In this work we obtain a new criterion to establish ergodicity and non-uniform hyperbolicity of smooth measures of diffeomorphisms. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of non-zero Lyapunov exponents in some contexts. In the partial hyperbolicity context, we obtain that stably ergodic diffeomorphisms are C^1-dense among volume preserving partially hyperbolic diffeomorphisms with two-dimensional center bundle. This is motivated by a well known conjecture of C. Pugh and M. Shub.

2016

In this paper we address the existence and ergodicity of non-hyperbolic attracting sets for a certain class of smooth endomorphisms on the solid torus. Such systems allow a formulation as a skew product system defined by planar diffeomorphisms have contraction on average which forced by any expanding circle map. These attractors are invariant graphs of upper semicontinuous maps which support exactly one SRB measure. In our approach, these skew product systems arising from iterated function systems generated by a finitely many weak contractive diffeomorphisms. Under some conditions including negative fiber Lyapunov exponents, we prove the existence of unique non-hyperbolic attracting invariant graphs for these systems which attract positive orbits of almost all initial points. Also, we prove that these systems are Bernoulli and therefore they are mixing. Moreover, these properties remain true under small perturbations in the space of endomorphisms on the solid torus.

Journal of Pure & Applied Sciences, 2023

This paper presents a detailed investigation of absolutely continuous invariant measures (ACIMs) for piecewise expanding chaotic transformations in ℝ , with particular attention paid to the case where the derivative has summable oscillations. ACIMs are important objects in the study of dynamical systems, as they provide a way to understand the long-term behavior of trajectories and the statistical properties of the system. The paper covers a range of important topics related to ACIMs, including the boundedness condition, distortion condition, localization condition, and Schmitt's condition. It also discusses the Perron-Frobenius operator, which plays a critical role in the existence and properties of ACIMs. The main result of the paper is the proof that the Perron-Frobenius operator is constrictive, which implies the existence of a finite number of ergodic ACIMs that satisfy Schmitt's condition and a condition dependent on the defining partition. This finding has significant implications for the understanding of complex systems and the advancement of research in this field. The paper also discusses the relationship between ACIMs and dynamical systems, highlighting the role of ACIMs in ergodic theory. Overall, this paper provides a valuable reference for researchers interested in the study of ACIMs and their significance in the analysis of dynamical systems and ergodic theory.

2014

Let T)(x) cos(,arccosx), 1 _< x _< 1, where , > 1 is not an integer. For a certain set of ,’s which are irrational, the density of the unique absolutely continuous measure invariant under T,X is determined exactly. This is accomplished by showing that T,X is differentially conjugate to a piecewise linear Markov map whose unique invariant density can be computed as the unique left eigenvector of a matrix.

Proceedings of the American Mathematical Society, 1976

Powers of strongly ergodic transformations need not be strongly ergodic.