Discontinuous observables as an obstruction for small essential spectral radius

TL;DR

This paper demonstrates that for many Banach spaces of functions on [0,1], there are inherent lower bounds on the essential spectral radius of transfer operators linked to piecewise smooth expanding maps, especially those allowing discontinuities.

Contribution

It establishes intrinsic lower bounds for the essential spectral radius of transfer operators on broad classes of Banach spaces, including those with discontinuities.

Findings

Lower bounds on the essential spectral radius are intrinsic to certain Banach spaces.

Discontinuities in function spaces do not eliminate these spectral bounds.

Results apply to a wide class of piecewise smooth expanding maps.

Abstract

We show that for a very wide class of Banach spaces of functions on [0,1] there are intrinsic lower bounds for the essential spectral radius of the transfer operator associated to piecewise smooth expanding maps. The class of Banach spaces studied includes any reasonable space which permits discontinuities.