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Strong mixing measures for linear operators and frequent hypercyclicity. (English) Zbl 06117531
The authors show the following theorem for operators T on F-spaces X that satisfy the Frequent Hypercyclic Criterion: If there is a dense subset X 0 of X and a sequence of maps S n :X 0 X such that for every xX 0 the sums n=0 T n x and n=0 S n x converge unconditionally, and such that T n S n x=x and T m S n x=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:
47A16Cyclic vectors, hypercyclic and chaotic operators
28D05Measure-preserving transformations
37A05Measure-preserving transformations
37A25Ergodicity, mixing, rates of mixing