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1 results

  • 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

  • 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.