# Questions tagged [ergodic-theory]

Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.

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### Ergodic theorem on limit of periodic transformations?

Suppose $(X,\mu)$ is a probability space, and $T_n, n \in \mathbb N$, is a sequence of periodic measure preserving transformations. For $x \in X$ and $f : X \to \mathbb R$, let $\mathrm{avg}_{f,n}(x)$...
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### (Exponential) Mixing property for Gauss map - going from cylinders to intervals

I'm trying to understand the proof of a mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step. The Gauss map $T$, ...
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### Are these topological sequence entropy definition equivalent?

I am working on Möbius disjointness for models of topological dynamic systems. In that purpose, I try to understand the notion of topological entropy. We know, for a t.d.s $(X,T)$ that it is defined ...
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### Rate of convergence of sojourn times of Markov chains

Let $(X_0,X_1,\dots)$ be a time-homogeneous Markov chain with finite state space $\Omega$. Assume that $(X_0,X_1,\dots)$ is irreducible and aperiodic and let $\pi$ be its stationary distribution. By ...
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### Implications for a simple deterministic chaos definition

Among many others, one definition of deterministic chaos terms "chaotic" a classical dynamical system that satisfies the following three properties: sensitive dependence to initial ...
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### positive of the largest Lyapunov exponent

Let $\alpha\in \mathbb{R} / \mathbb{Q}$, A(x)=\left(\begin{array}{ll} \frac{1}{{\lambda}^2}-2 \cos 2\pi x -1& 2\lambda \cos 2\pi x-\frac{1}{{\lambda}} \\ \frac{1}{{\lambda}} &...
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### Is there a condition for a subshift of finite type to be uniquely ergodic?

Are SFTs uniquely ergodic in general, or is there a known necessary and sufficient condition for them to be uniquely ergodic?
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### Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic?

It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure. Marc Peigné [1] constructed examples of geometrically infinite hyperbolic ...
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### Uniformity of convergence in the pointwise ergodic theorem

Definitions and some motivation: Let $X$ be a compact metric space, and $T$ a uniquely ergodic measure preserving transformation on $X$, with associated invariant ergodic probability measure $\mu$. ...