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Suppose we have a compact metric space $X$, a continuous map $f \colon X \to X$, and Borel probability measures $\mu$ and $\nu$ on $X$ such that the set $$ X_{f,\mu} := \left\{ x \in X \, : \, \frac{1 …

Informal description. Suppose I have a dynamical system $f$ defined on the product of a compact space $X$ representing the state space of an "experimentally visible" variable and a compact space $Y$ r …

Okay, I've seen that the proof of an affirmative answer to both questions in the general case (with $\nu=\mu \otimes \lambda$) is not very hard: Since $X \times Y$ is compact, we can let $(k_n)$ be a …

Preliminaries. (See [1] for further details.) Let $M$ be a compact connected $C^\infty$ Riemannian manifold. We say that a list $\sigma_1,\ldots,\sigma_n$ ($n \in \mathbb{N}$) of $C^\infty$ vector fie …

The answer to my question is yes!! [The assumption that $(X,\mathcal{X})$ is a standard Borel space is not needed.] A beautiful proof of a modified version of the statement has been provided by user65 …

THE QUESTION Let $(X,\mathcal{X})$ be a standard Borel space, $T \colon X \to X$ a measurable map, and $\mu$ a $T$-mixing probability measure. Is it necessarily the case that for all $A \in \mathcal …

Do you not also want that $\mathbb{P}_Y$ is $\phi$-invariant? In any case, yes there are extremely many continuous-time continuous-path real-valued stochastic processes whose law is ergodic under the …

It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it wo …

Let $\Omega$ and $M$ be compact $C^\infty$ manifolds, let $\theta \colon \Omega \to \Omega$ be a $C^\infty$ diffeomorphism, and let $\Theta \colon \Omega \times M \to \Omega \times M$ be a $C^\infty$ …

Let $X$ be a Polish space, and $\mu$ a locally finite measure. Take any $p \in \{0\} \cup [1,\infty)$. We will say that a linear operator $T \colon L^p(\mu) \to L^p(\mu)$ has property $(\ast)$ if ever …

Does there exist a set $F$ of monotone continuous functions $f \colon [0,1] \to [0,1]$ with the following properties? For each $f \in F$ there exists $x \in [0,1]$ such that $f(x)=1$. There exist $0< …

Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a diffeomorphism, and let $\mu$ be a probability measure on $M$ with compact support. As stated in the questio …

Let $X$ be a set equipped with some structure (e.g. topological space, measurable space, probability space, etc.). We say that two endomorphisms $f,g \colon X \to X$ are conjugate to each other if the …

Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a continuous map, and let $\mu$ be a probability measure on $M$ with compact support. Definition. The …

Setting: Let $M$ be a compact connected $C^\infty$ Riemannian manifold of dimension $D \geq 2$, with $\lambda$ the normalised Riemannian volume measure. Write $T_{\neq 0}M \subset TM$ for the non-ze …