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3 - Probability (Finite) Invariant Measures: Basic Properties and Existence

from Part I - Ergodic Theory and Geometric Measures

Published online by Cambridge University Press:  20 April 2023

Janina Kotus  and
Mariusz Urbański
Janina Kotus
Affiliation:
Warsaw University of Technology
Mariusz Urbański
Affiliation:
University of North Texas
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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.

Keywords

Stochastic lawsdecay of correlationsYoung towersRokhlin natural extension
Type
Chapter
Information
Meromorphic Dynamics
Abstract Ergodic Theory, Geometry, Graph Directed Markov Systems, and Conformal Measures
 , pp. 102 - 121
Publisher: Cambridge University Press
Print publication year: 2023

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