t-Entropy formulae for concrete classes of transfer operators

t-Entropy is a principal object of the spectral theory of operators, generated by dynamical systems, namely, weighted shift operators and transfer operators. In essence t-entropy is the Fenchel – Legendre transform of the spectral potential of an operator in question and derivation of explicit formulae for its calculation is a rather nontrivial problem. In the article explicit formulae for t-entropy for two the most exploited in applications classes of transfer operators are obtained. Namely, we consider transfer operators generated by reversible mappings (i. e. weighted shift operators) and transfer operators generated by local homeomorphisms (i. e. Perron – Frobenius operators). In the first case t-entropy is computed by means of integrals with respect to invariant measures, while in the second case it is computed in terms of integrals with respect to invariant measures and Kolmogorov – Sinai entropy.

TAUBERIAN THEOREMS AND SPECTRAL THEORY IN TOPOLOGICAL VECTOR SPACES

We investigate the spectral theory of integrable actions of a locally compact abelian group on a locally convex vector space. We start with an analysis of various spectral subspaces induced by the action of the group. This is applied to analyse the spectral theory of operators on the space, generated by measures on the group. We apply these results to derive general Tauberian theorems that apply to arbitrary locally compact abelian groups acting on a large class of locally convex vector spaces, which includes Fréchet spaces. We show how these theorems simplify the derivation of Mean Ergodic theorems.