Pressure, Poincaré series and box dimension of the boundary. (18th June 2021)

• Record Type: Journal Article
• Authors: Iommi, Godofredo; Velozo, Anibal
• Is Part Of: Nonlinearity. Volume 34:Number 6(2021)
• Page Start: 3936
• Page End: 3952
• Publication Date: 2021-06-18
• Subjects: dimension theory -- boundary of hyperbolic space -- uniformly expanding Markov maps; 37A05; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/abfecf
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this note we prove two related results. First, we show that for certain Markov interval maps with infinitely many branches the upper box dimension of the boundary can be read from the pressure of the geometric potential. Secondly, we prove that the box dimension of the set of iterates of a point in ∂ H n with respect to a parabolic subgroup of isometries equals the critical exponent of the Poincaré series of the associated group. This establishes a relationship between the entropy at infinity and dimension theory.