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This seems like a really basic question, and yet I haven't managed to find the answer! Let $(X,\Sigma,\mu,T)$ be a measure-preserving dynamical system. Does there necessarily exist at least one seque …

Let $\Omega$ be a Polish space equipped with a Borel probability measure $\mathbb{P}$, and let $\theta \colon \Omega \to \Omega$ be a measurably invertible map that is mixing with respect to $\mathbb{ …

Hello all. Suppose $X$ is a Polish space, $\mu$ is a Borel probability measure on $X$, and $T:X \to X$ is a continuous $\mu$-preserving map which is not ergodic. Does there necessarily exist a Borel …

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< …

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 …

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 $C^\infty$ …

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 …

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 …

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 …

Consider the differential equation on $\mathbb{R}^2$ whose polar-coordinate representation is given by $$ \begin{array}{r c l} \dot{r} & = & r(\alpha - r^2) \\ \dot{\theta} & = & -sin(\theta).\end{arr …

The claim is true. (As proved here, it is quite unique to sinusoidal vector fields. The same reference also mentions, in its introduction, existing applications of equation (1) with $g$ taking the par …

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 …

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 …

This is a question in the theory of random dynamical systems. Let $(X,d)$ be a compact metric space, let $(I,\mathcal{I},\nu)$ be a probability space, and let $(f_\alpha)_{\alpha \in I}$ be an $I$-ind …