## $$C_{0}$$-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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