Discontinuous observables as an obstruction for small essential spectral radius

2025/06/09 by Oliver Butterley, Butterley, Oliver, Daniel Smania +1
Mathematics · #37A05 #37A25 #37C30 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.2506.07613

openalex publication_date 2025/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Transfer operators and atomic decomposition
• Transfer operators, atomic decomposition and the Bestiary
• Ruelle spectrum of linear pseudo-Anosov maps
• A spectral gap for transer operators of piecewise expanding maps
• Dynamical determinants and spectrum for hyperbolic diffeomorphisms
• Anisotropic Holder and Sobolev spaces for hyperbolic diffeomorphisms
• Banach spaces adapted to Anosov systems
• Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems

Related

• A footnote on Expanding maps
• Good Banach spaces for piecewise hyperbolic maps via interpolation
• A spectral gap for transer operators of piecewise expanding maps
• An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions
• Local Spectrum, Local Spectral radius,and Growth Conditions