An Alternative Approach to Generalised BV and the Application to Expanding Interval Maps

TL;DR

This paper develops a new family of Banach spaces tailored for analyzing weighted transfer operators of piecewise-smooth interval maps, especially with complex partitions and less regular weights, providing spectral bounds.

Contribution

It introduces novel Banach spaces suitable for studying transfer operators with less regular weights and complex partitions, expanding analytical tools in dynamical systems.

Findings

Established upper bounds for spectral radius.

Established upper bounds for essential spectral radius.

Applicable to maps with countable partitions and piecewise Hölder weights.

Abstract

We introduce a family of Banach spaces of measures, each containing the set of measures with density of bounded variation. These spaces are suitable for the study of weighted transfer operators of piecewise-smooth maps of the interval where the weighting used in the transfer operator is not better than piecewise H\"older continuous and the partition on which the map is continuous may possess a countable number of elements. For such weighted transfer operators we give upper bounds for both the spectral radius and for the essential spectral radius.