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AN INEQUALITY FOR THE MATRIX PRESSURE FUNCTION AND APPLICATIONS

Morris, ID (2016) AN INEQUALITY FOR THE MATRIX PRESSURE FUNCTION AND APPLICATIONS Advances in Mathematics, 302 (Oct). pp. 280-308.

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Abstract

We prove an a priori lower bound for the pressure, or p-norm joint spectral radius, of a measure on the set of d × d real matrices which parallels a result of J. Bochi for the joint spectral radius. We apply this lower bound to give new proofs of the continuity of the affinity dimension of a selfaffine set and of the continuity of the singular-value pressure for invertible matrices, both of which had been previously established by D.-J. Feng and P. Shmerkin using multiplicative ergodic theory and the subadditive variational principle. Unlike the previous proof, our lower bound yields algorithms to rigorously compute the pressure, singular value pressure and affinity dimension of a finite set of matrices to within an a priori prescribed accuracy in finitely many computational steps. We additionally deduce a related inequality for the singular value pressure for measures on the set of 2 × 2 real matrices, give a precise characterisation of the discontinuities of the singular value pressure function for two-dimensional matrices, and prove a general theorem relating the zero-temperature limit of the matrix pressure to the joint spectral radius.

Item Type: Article
Subjects : Mathematics
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
AuthorsEmailORCID
Morris, IDUNSPECIFIEDUNSPECIFIED
Date : 1 August 2016
Identification Number : 10.1016/j.aim.2016.07.025
Copyright Disclaimer : © 2016. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Depositing User : Symplectic Elements
Date Deposited : 29 Jul 2016 11:30
Last Modified : 25 Nov 2016 16:40
URI: http://epubs.surrey.ac.uk/id/eprint/811564

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