A unique ergodicity of minimal symbolic flows with linear block growth. (English) Zbl 0602.28008

Boshernitzan, Michael

J. Anal. Math. 44, 77-96 (1985).

Let (\(\Omega\),T) be a minimal symbolic flow over a finite alphabet. Denote by P(n) the number of different n-blocks which appear in any \(\omega \in \Omega\) (the minimality of the flow makes P(n) independent of the choice of \(\omega \in \Omega\)). Let \({\mathcal I}_ 0\) denote the set of ergodic measures of the flow \((\Omega,T)\) and let \(n({\mathcal I}_ 0)\) be its cardinality. Then the main results of the paper are contained in the following two theorems:
Theorem. If for some integer \(r>1\) we have \(\liminf_{n\to +\infty}(P(n)-rn)=-\infty\) then n(\({\mathcal I}_ 0)<r.\)
Theorem. If \(\limsup_{n\to \infty}(\frac{P(n)}{n})<3,\) then the flow (\(\Omega\),T) is uniquely ergodic.

Reviewer: J.Šiška

MSC:

28D10

One-parameter continuous families of measure-preserving transformations

Keywords:

block growth; unique ergodicity; minimal symbolic flow over a finite alphabet

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