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It is true that a system is ergodic if (and only if) every measurable function constant along trajectories is globally constant almost everywhere. For indicator functions this is just the usual defini …

For a nice introductory discussion to the maximal ergodic inequality, see [1]. In particular, inequality (5) there is Wiener's maximal ergodic theorem. See also Lemma 15.3 in [2]. A more advanced a …

The property does not hold if the measure $\mu$ is atomic, so we assume that the given standard probability space $(X, \mathcal{B}, \mu)$ is nonatomic, whence it is isomorphic to he unit interval …

There is a simple reduction of the Birkhoff ergodic Theorem for $L^1$ functions to the bounded case using Kakutani-Rokhlin towers, that I learned from H. Furstenberg and B. Weiss decades ago. We use …

In the most standard examples, the set of invariant ergodic measures has the cardinality of the continuum. For example, take $X=\{0,1\}^{\mathbb Z}$ equipped with the product, topology, the Borel $\si …

Let $\{b_n\}_{n \ge 1}$ be a sequence of independent random variables taking the values $\pm 1$ with probability $1/2$ each. Define $\{a_n\}_{n \ge 1}$ by $a_n=b_{n-1} b_{n-2}$ if $\; n \equiv 0 \mod …

Yes. Let $\nu$ be a probability measure on $[0,1]$. Let $T$ be the left shift on a sequence space $X:=[0,1]^{\mathbb N}$ equipped with the product $\sigma$-field and the product measure $\mu:=\nu^{\ma …

Indeed $D(x)$ is empty for almost all $x$. It suffices to show that for all integers $b,\ell>0$, the set $$ D_b(x,\ell) = \left\{ a \in [b,b+1) \colon \liminf_{n \to \infty} \frac{ \# \{0 \leq k < n \ …

A sufficient condition: Commuting Markov operators will have a common invariant measure by https://en.wikipedia.org/wiki/Markov%E2%80%93Kakutani_fixed-point_theorem . See https://projecteuclid.org/do …

For any $p>0$ the sequence of fractional parts $x_n=\{n^p\alpha\}$ cannot be random-like in the sense defined in the appendix. The case of integer $p$ was already discussed in the comment by Goldster …

The extension you want was proved in the 1990s by Bergelson and Leibman. See [1] and also further developments in [2]. [1] Bergelson, Vitaly, and Alexander Leibman. "Polynomial extensions of van der W …

False for p infinite. True for finite. See e.g. the book by Krengel, Ergodic theorems. Other sources (that also go further) are [1, Sec. I.2.1] or [2, Theorem 8.8]. [1] T. Eisner, Stability of opera …

Uniform convergence holds when $R_n$ is at most a power of $n$. Using the tail of a Poisson variable, you can easily infer that $P(X_t>r) \le r^{-C\log \log r}$ for some $C$ that depends on the maxi …