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Strong mixing measures for linear operators and frequent hypercyclicity. (English) Zbl 1288.47009

The authors show the following theorem for operators \(T\) on \(F\)-spaces \(X\) that satisfy the Frequent Hypercyclic Criterion: Theorem. If there is a dense subset \(X_0\) of \(X\) and a sequence of maps \(S_n:X_0\to X\) such that, for every \(x\in X_0\), the sums \(\sum_{n=0}^\infty T^nx \) and \(\sum_{n=0}^\infty S_nx\) converge unconditionally, and such that \(T^nS_nx=x\) and \(T^mS_nx=S_{n-m}x\) for \(n>m\), then there is a \(T\)-invariant strongly mixing Borel probability measure on \(X\) with full support. An application to backward shift operators is given.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
28D05 Measure-preserving transformations
37A05 Dynamical aspects of measure-preserving transformations
37A25 Ergodicity, mixing, rates of mixing
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