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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 …

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 …

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 …

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 …

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 …

Page 108 of Peter Walters' classic textbook An Introduction to Ergodic Theory suggests the reference "An uncountable family of $K$-automorphisms" by Donald Ornstein and Paul Shields [Advances in Mathe …

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 …

Let $Y$ be the support of $\mu$ and suppose that $\mu$ is not ergodic. Then there exists a $T$-invariant measurable set $A\subset Y$ such that $0<\mu(A)<1$. The measures $\mu_1$, $\mu_2$ defined by $$ …

Let $\phi_t$ denote the flow map at time $t$. Suppose that $\mu$ and $\nu$ agree on every invariant Borel set,and let $f \colon X \to \mathbb{R}$ be continuous. Define $\overline{f}:=\liminf_{t \to \i …

I would not be surprised if small $\sigma^2$ does indeed imply small d-bar distance to Bernoulli, but I think that the converse is false. Rather, processes with arbitrarily small d-bar distance to Ber …

Let $T \colon X \to X$ be a minimal transformation of a compact metric space which is not uniquely ergodic, let $\mu$ be a non-ergodic $T$-invariant measure on $X$, and let $A$ be a set with nonempty …

There is a book by Denker, Grillenberger and Sigmund which deals extensively with this topic: they prove a whole range of theorems which construct subshifts whose invariant measures have specified pro …

My recollection is that in one of the last chapters of the book by Denker-Grillenberger-Sigmund, there is a theorem which says something like: given $n$ ergodic measure-preserving transformations with …

Let $T_1 \colon X_1 \to X_1$ be an Anosov diffeomorphism and let $T_2 \colon X_2 \to X_2$ be a uniquely ergodic expansive homeomorphism which is not a periodic orbit. Let $T \colon X_1 \times X_2 \to …

The Poincaré recurrence theorem is sometimes useful because of the way it translates into recurrence in metric spaces. For example, a corollary of the Poincaré theorem is that for a measure-preserving …