A gentle introduction to anisotropic Banach spaces

TL;DR

This paper introduces anisotropic Banach spaces and demonstrates their application to hyperbolic dynamical systems through a concrete example, making the complex theory more accessible and establishing a new connection with expanding maps.

Contribution

It provides an accessible introduction to anisotropic Banach spaces and presents an original result linking dissipative Baker's transformations with expanding maps.

Findings

Established a continuous family of transfer operators on a single Banach space

Connected dissipative Baker's transformations with expanding maps

Enhanced understanding of anisotropic Banach spaces in hyperbolic dynamics

Abstract

The use of anisotropic Banach spaces has provided a wealth of new results in the study of hyperbolic dynamical systems in recent years, yet their application to specific systems is often technical and difficult to access. The purpose of this note is to provide an introduction to the use of these spaces in the study of hyperbolic maps and to highlight the important elements and how they work together. This is done via a concrete example of a family of dissipative Baker's transformations. Along the way, we prove an original result connecting such transformations with expanding maps via a continuous family of transfer operators acting on a single Banach space.