Let $(X,T,\mu)$ be a measure preserving system, with $\mu$ a probability measure. Let $E \subset X$ of positive measure and $\tau_E$ be the first return time to $E$. Then the Kac Lemma asserts that $$\int_E \tau_E d \mu = 1,$$ that is the average return time to $E$ is $\mu(E)^{-1}$. My question is when is it known that there is some $x \in E$ so that $\tau_E(x) \geq C \mu(E)^{-1}$, where $C$ is some large (fixed) constant? Any pointers to relevant literature would be much appreciated!
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$\begingroup$ Hi. Can you be a bit more precise about "when"? Are you in a fixed dynamical system and are asking which $E$ have some $x$ with $\tau_E(x) \ge C\mu(E)^{-1}$. Or are you asking about properties of a dynamical system that guarantee the existence of some $E$? Note that for any dynamical system, $E=X$ won't have $||\tau_E||_\infty \ge C\mu(E)^{-1}$. $\endgroup$– mathworker21Apr 3, 2020 at 19:16
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$\begingroup$ @mathworker21 Sure. I am interested in theorems of the form: suppose a dynamical system (or measure preserving system) and a set E satisfy a certain property. Then there is an x in E with a large return time. Homomorphisms of the torus are of particular interest to me. $\endgroup$– George ShakanApr 4, 2020 at 2:45
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$\begingroup$ Perhaps it helps having a look at link.springer.com/article/10.1007/s002200100427. The notion of "long return time" seems to be quite close to what you want. $\endgroup$– John BApr 18, 2020 at 3:27
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