C0-continuity of the Fröbenius-Perron semigroup

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 σ-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 C0-continuous in the space Lμ 1(X). We also give a geometrical condition which ensures C0-continuity of the Fröbenius-Perron semigroup of linear operators in the space Lμ p(X) for 1≤p<∞, as well as in the space Lloc 1