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# A little topological counterpart of Birkhoff's ergodic theorem

Author(s): Reinhard Winkler

Journal: Uniform Distribution Theory
ISSN 1336-913X

Volume: 5;
Issue: 1;
Start page: 157;
Date: 2010;

Keywords: Birkhoff’s ergodic theorem | Baire category | topological dynamics | distribution of sequences

ABSTRACT
For a compact metric space $X$ and a continuous transformation $T: X \to X$ with at least one transitive and recurrent orbit, there is a set $M_0(T)$ of $T$-invariant probability measures on $X$ such that for a comeager set of starting points the set of limit measures is exactly $M_0(T)$.