1 results
Dynamical Borel–Cantelli lemma for recurrence theory
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 42 / Issue 6 / June 2022
- Published online by Cambridge University Press:
- 12 April 2021, pp. 1994-2008
- Print publication:
- June 2022
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- Article
- Export citation
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We study the dynamical Borel–Cantelli lemma for recurrence sets in a measure-preserving dynamical system
$(X, \mu , T)$
with a compatible metric d. We prove that under some regularity conditions, the 
$\mu $
-measure of the following set $$\begin{align*}R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\mathbb{N} \} \end{align*}$$obeys a zero–full law according to the convergence or divergence of a certain series, where
$\psi :\mathbb {N}\to \mathbb {R}^+$
. The applications of our main theorem include the Gauss map, 
$\beta $
-transformation and homogeneous self-similar sets.
