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Let $\mathbf X := (X, \mathcal S, \mu, T)$ be an ergodic measure preserving system with finite measure such that for every increasing sequence $\{n_k\}$ of natural numbers with positive lower density, …

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 $\mu$. As …

Let $\mathbf X := (X, \mathcal S, \mu)$ be a probability space without atoms. We say two measure preserving transformations $T$ and $F$ on $\mathbf X$ are $\delta$-close, for $\delta > 0$, if $ \int_X …

Let $\mathbf X := (X, \mathcal F, \mu)$ be a standard probability space. For an ergodic measure preserving transformation $T$, we define the ergodic robustness $\mathcal R(T)$ of $T$ as follows: For $ …

Note: This is a concrete case of the following question: Are almost all measure-preserving flows on compact manifolds ergodic? Let $M$ be a Riemannian manifold with its natural Riemannian measure, and …

Let $(\Omega, \mathcal F, \mu)$ be a standard probability space. Question: For each $f \in L^\infty (\Omega)$, does there exist an ergodic measure preserving transformation $T: \Omega \to \Omega$ such …

Problem set up: Let $\mathbf X := (X, \mathcal A, \mu)$ be a standard probability space. We say that a measure preserving transformation $T$ on $\mathbf X$ is $\varepsilon$-almost mixing if for every …

Let $(X, T, \mathcal F, \mu)$ be an ergodic measure preserving system with finite measure. Suppose $T$ is uniformly recurrent, in the following sense: For every $A \in \mathcal F$, there exists an $M …

Let $(X, T, \mathcal F, \mu)$ be a nonatomic standard probability space equipped with a measure preserving transformation $T$. We say $T$ is uniformly weak mixing if for every $\varepsilon > 0$, there …

Let $\mathbf X := (X, \mathcal S, \mu, T)$ be an ergodic measure preserving system with finite measure such that for every increasing sequence $\{n_k\}$ of natural numbers whose natural density exists …

In Furstenberg's proof of the multiple recurrence theorem in ergodic theory, one makes use of the concept of compact and weak mixing extensions of a measure preserving system. The following definition …

Let $(X, T, \mu)$ be an ergodic measure preserving system with finite measure, and $f \in L^{\infty} (X)$. Define the maximal orbit deviation function $D_f: X \to \mathbb R$ by $$D_f := \sup_{n, m \ge …

Problem set up: Let $\mathbf X := (X, \mathcal A, \mu)$ be a standard probability space. We say that a measure preserving transformation $T$ on $\mathbf X$ is $\varepsilon$-almost weakly mixing if for …

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, measure preserving and uniformly transitive in the sense tha …

Consider the random circle rotation $x \to x + Z \text{ mod 1}$ on $([0, 1], \text{Lebesgue})$ where at each rotation, $Z$ is uniformly distributed on $[0, 1]$ and independent of previous rotations. I …