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I found it surprisingly difficult to find a reference for this when I was studying Mane's papers on multiplicative ergodic theorems. My hypothesis was that people working with the Grassmannian in othe …

Let $X=\{0,1\}^{\mathbb{N}}$ with the infinite product topology (which is metrisable). For each $n \geq 1$, define $x_n$ to be the sequence given by $x_i=0$ for $1 \leq i \leq n$, $x_i=1$ for $n+1 \le …

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},\ …

The existence of such an example is prevented by the ergodic decomposition theorem, which asserts that every $T$-invariant measure on a standard probability space $(X,\mathcal{B},m)$ can be expressed …

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 …

This can be found in Walters' book An introduction to ergodic theory where it occurs as Theorem 6.19. The proof is not terribly difficult and is in my opinion quite instructive. I will deal with (3). …

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 …

I believe that Hillel Furstenberg uses the von Neumann ergodic theorem quite frequently in his work on recurrence, which has applications to number theory. For example, in section 3 of the article Poi …

The question of the existence of a nonatomic measure which is jointly invariant under these two maps, and is not equal to Lebesgue measure, is called the Furstenberg $\times 2 \times 3$ problem and is …

A construction of a topologically transitive mixing subshift with a fully supported invariant measure, but no fully supported ergodic measure, is given by Benjamin Weiss in the article "Topological tr …

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 …

As you are perhaps aware, the standard method for investigating decay of correlations of expanding maps is using operator theory, either directly or via an induced dynamical system. The decay of corre …

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 $\ …

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 …