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3 - Probability (Finite) Invariant Measures: Basic Properties and Existence
from Part I - Ergodic Theory and Geometric Measures
Janina Kotus, Warsaw University of Technology, Mariusz Urbański, University of North Texas
Book:

Meromorphic Dynamics

Published online:

20 April 2023

Print publication:

04 May 2023, pp 102-121
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Summary

This chapter is devoted to the stochastic laws for measurable endomorphisms preserving a probability measure that are finer than the mere Birkhoff Ergodic Theorem. Under appropriate hypotheses, we prove the Law of the Iterated Logarithm. We then describe another powerful method of ergodic theory, namely Young towers, which are also frequently called Kakutani towers. With appropriate assumptions imposed on the first return time, Young's construction yields the exponential decay of correlations, the Central Limit Theorem, and the Law of the Iterated Logarithm follows too.

Random Young towers and quenched limit laws
Part of
- Limit theorems
YAOFENG SU
Journal:

Ergodic Theory and Dynamical Systems / Volume 43 / Issue 3 / March 2023

Published online by Cambridge University Press:

03 February 2022, pp. 971-1003

Print publication:

March 2023
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We obtain quenched almost sure invariance principles (with convergence rates) for random Young towers if the average measure of the tail of return times to the base of random towers decays sufficiently fast. We apply our results to some independent and identically distributed perturbations of some non-uniformly expanding maps. These imply that the random systems under study tend to a Brownian motion under various scalings.

Explicit coupling argument for non-uniformly hyperbolic transformations
Part of
- Dynamical systems with hyperbolic behavior
- Ergodic theory
A. Korepanov, Z. Kosloff, I. Melbourne
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 149 / Issue 1 / February 2019

Published online by Cambridge University Press:

06 July 2018, pp. 101-130

Print publication:

February 2019
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The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. We verify that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map. For non-uniformly expanding maps with a uniformly expanding induced map, we obtain explicit estimates for mixing rates (exponential, stretched exponential, polynomial) that again depend continuously on the constants for the induced map together with data associated with the inducing time. Finally, for non-uniformly hyperbolic transformations, we obtain the corresponding estimates for rates of decay of correlations.