Introduction to Ergodic Theory

@inproceedings{Walters1977IntroductionTE,
  title={Introduction to Ergodic Theory},
  author={Peter Walters},
  year={1977}
}
Ergodic theory concerns with the study of the long-time behavior of a dynamical system. An interesting result known as Birkhoff’s ergodic theorem states that under certain conditions, the time average exists and is equal to the space average. The applications of ergodic theory are the main concern of this note. We will introduce fundamental concepts in ergodic theory, Birkhoff’s ergodic theorem and its consequences. 

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  • P. Walters
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1996
Abstract We give some topological ergodic theorems inspired by the Wiener-Wintner ergodic theorem. These theorems are used to give results for uniquely ergodic transformations and to study unique

Generators of Probability Dynamical Systems

TLDR
The concept of entropy will be extended to the countable partitions and the ergodic properties of probability dynamical systems are investigated and a version of Kolmogorov-Sinai theorem concerning the entropy of a probability dynamicals system is given.
...

References

Uniform distribution of sequences

( 1 ) {xn}z= Xn--Z_I Zin-Ztn-I is uniformly distributed mod 1, i.e., if ( 2 ) lim (1/N)A(x, N, {xn}z)-x (0x<_ 1), where A(x, N, {Xn)) denotes the number of indices n, l<=n<=N such that {x} is less