Expanding Semiflows on Branched Surfaces and One-Parameter Semigroups of Operators

TL;DR

This paper advances the spectral analysis of expanding semiflows on branched surfaces by examining associated transfer operators as a semigroup, improving understanding of their spectral properties and smoothness without suspension encoding.

Contribution

It introduces new spectral results for transfer operators of semiflows on branched surfaces and enhances the analysis of their smoothness properties without relying on suspension representations.

Findings

Spectral characterization of transfer operators for semiflows.

Relation between the spectrum of transfer operators and their generators.

Improved methods for studying semiflows on branched manifolds.

Abstract

We consider expanding semiflows on branched surfaces. The family of transfer operators associated to the semiflow is a one-parameter semigroup of operators. The transfer operators may also be viewed as an operator-valued function of time and so, in the appropriate norm, we may consider the vector-valued Laplace transform of this function. We obtain a spectral result on these operators and relate this to the spectrum of the generator of this semigroup. Issues of strong continuity of the semigroup are avoided. The main result is the improvement to the machinery associated with studying semiflows as one-parameter semigroups of operators and the study of the smoothness properties of semiflows defined on branched manifolds, without encoding as a suspension semiflow.