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Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.

3 votes
0 answers
53 views

On the relative growth rates of occupancy times in ergodic theory

Let $(X, \mathcal{F}, \mu)$ be a general measure space, and let $T: X \to X$ be a measure-preserving transformation on $X$. Assume that $T$ is ergodic and satisfies the property that, for any set $A \ …
2 votes
1 answer
195 views

Deriving an inequality for the integral of maximum indicator functions under measure-preserv...

Let's denote the measure space by $(X, \mathcal{B}, \mu)$ and the measure-preserving transformation by $T: X \to X$. Let $A \in \mathcal{B}$ be a measurable set with $0 < \mu(A) < \infty$. Let $0 < t …
1 vote
1 answer
112 views

Ensuring the measure condition $\mu(E) = \lambda$ in a lemma: need some clarification regard...

I was studying a lemma from my notes on ergodic theory and encountered a difficulty. The lemma states: Let $(X, \mathcal{B}, \mu)$ be an infinite non-atomic measure space, and let $T$ be an ergodic no …
3 votes
2 answers
250 views

Existence of a positive measurable set with disjoint preimage under iterated transformation

Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal …