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

Navas, Andrés; Plaza, Sergio

Int. J. Math. Math. Sci. 31, No. 5, 307-319 (2002).

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

Keywords:

Fröbenius-Perron semigroup; linear operators; semidynamical system; invariant measure; \(C_0\)-continuity