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\(C_{0}\)-continuity of the Fröbenius-Perron semigroup. (English) Zbl 1016.37002

Summary: We consider the Fröbenius-Perron semigroup of linear operators associated to a semidynamical system defined in a topological space \(X\) endowed with a finite or a \(\sigma\)-finite regular measure. We prove that if there exists a faithful invariant measure for the semidynamical system, then the Fröbenius-Perron semigroup of linear operators is \(C_{0}\)-continuous in the space \(L_\mu^1(X)\). We also give a geometrical condition which ensures \(C_{0}\)-continuity of the Fröbenius-Perron semigroup of linear operators in the space \(L_\mu^p(X)\) for \(1\leq p<\infty\), as well as in the space \(L_{\text{loc}}^1\).

MSC:

37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations
47D03 Groups and semigroups of linear operators
47D60 \(C\)-semigroups, regularized semigroups
28D10 One-parameter continuous families of measure-preserving transformations
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