Ergodic Theory and Dynamical Systems

Table of Contents - 11 July 2012 - Volume 32, Issue 04  

Research Article

The packing spectrum for Birkhoff averages on a self-affine repeller

HENRY W. J. REEVEa1

a1 Department of Mathematics, The University of Bristol, University Walk, Clifton, Bristol BS8 1TW, UK (email: henrywjreeve@googlemail.com)

Abstract

We consider the multifractal analysis of Birkhoff averages of continuous potentials on a self-affine Sierpiński sponge. In particular, we give a variational principle for the packing dimension of the level sets. Furthermore, we prove that the packing spectrum is concave and continuous. We give a sufficient condition for the packing spectrum to be real analytic, but show that for general Hölder continuous potentials, this need not be the case. We also give a precise criterion for when the packing spectrum attains the full packing dimension of the repeller. Again, we present an example showing that this is not always the case.

(Received June 22 2010)

(Revised March 15 2011)

(Online publication September 08 2011)