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Let $\mathbb{P}$ be a Borel probability measure on the space of bounded operators on $\mathcal{B}$, equipped with the operator norm topology. By the subadditive ergodic theorem, the limit $$\lim_{n \t …

The set of all $f$-invariant Borel probability measures on $\mathbb{R}$ is $A$-strongly mutually singular. To see this let $L(\mu)$ be the set of all points $x$ such that $\frac{1}{n}\sum_{k=0}^{n-1}\ …

There are very many such measures. In fact, every zero-entropy transformation has a representation as such a measure: Corollary 4.14.3 in Walters' book states that every zero-entropy measure-preservi …

This is intended more as a long comment than an answer. My main experience of nuclearity in this area is that it is a key ingredient in the proofs of correctness (or the proof of the convergence rate) …

Let $T\colon [0,1] \to [0,1]$ be a homeomorphism such that $T(1/n)=1/n$ for all $n \geq 1$ and $T(x)<x$ for all other $x \in (0,1]$. If $\frac{1}{m+1}<x<\frac{1}{m}$ then $T^n(x)$ is monotone decreasi …

As Will shows, the case in which $\mu$ is absolutely continuous with respect to Lebesgue measure and has density bounded away from zero and infinity is constrained in that the Lyapunov exponents of $\ …

The set of Bernoulli measures $B$ is closed, and the closure of $C$ is precisely the set of measures of the form $\int m\,d\mathbb{P}$ where $\mathbb{P}$ is a Borel probability measure on the set of B …

The Jewett-Krieger Theorem states that every ergodic measure-preserving transformation of a standard probability space can be realised as a uniquely ergodic topological dynamical system on a compact m …

The desired result is false for all mixing systems other than a point: Proposition: let $T$ be an invertible totally ergodic transformation of a standard probability space $(X,\mathcal{F},\mu)$. Then …

The discrete-time analogue -- there exists a norm in which $|A_1^nx|\leq |x|$, $|A_2^nx|\leq |x|$ for all $n\geq 1$ and $x \in\mathbb{R}^d$ -- is equivalent to the property that the semigroup generat …

There are two standard methods for pulling back results about decay of corellations from an induced system: Young towers, and the operator renewal theory introduced by Sarig and developed by Gouëzel. …

Just knowing that a transformation $T$ is minimal is no guarantee that $T^n$ is also minimal. For example, let $T_1$ be the non-identity homeomorphism of a two-point metric space and let let $T_2$ be …

May I suggest Michael Lacey's more-or-less definitive paper on this topic, On central limit theorems, modulus of continuity and Diophantine type for irrational rotations (Journal d'Analyse Mathematiqu …

If $X$ is minimal and not a periodic orbit then it cannot contain a periodic orbit and hence in particular cannot contain a Markov shift. A classical construction by Grillenberger shows that one can c …

The classical Birkhoff ergodic theorem considers a map $T$ acting on a probability space $(X,\mathcal{F},\mu)$. This could alternatively be thought of as an action of $\mathbb{N}$ on $(X,\mathcal{F},\ …