Discontinuities cause essential spectrum

TL;DR

This paper investigates how discontinuities in piecewise monotone interval transformations influence the essential spectrum of associated transfer operators, revealing that non-Markov transformations lead to a large essential spectrum.

Contribution

It introduces a family of Banach spaces that demonstrate the optimal lower bounds on the essential spectral radius for these operators.

Findings

Essential spectrum is large when transformations are non-Markov.

Constructed Banach spaces achieve near-optimal essential spectral radius.

The bounds on the spectral radius are proven to be sharp.

Abstract

We study transfer operators associated to piecewise monotone interval transformations and show that the essential spectrum is large whenever the Banach space bounds $L^\infty$ and the transformation fails to be Markov. Constructing a family of Banach spaces we show that the lower bound on the essential spectral radius is optimal. Indeed, these Banach spaces realise an essential spectral radius as close as desired to the theoretical best possible case.