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Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension

Kaenmaki, A and Morris, Ian (2017) Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension Proceedings of the London Mathematical Society, 116 (4). pp. 929-956.

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A fundamental problem in the dimension theory of self-affine sets is the construction of high- dimensional measures which yield sharp lower bounds for the Hausdorff dimension of the set. A natural strategy for the construction of such high-dimensional measures is to investigate measures of maximal Lyapunov dimension; these measures can be alternatively interpreted as equilibrium states of the singular value function introduced by Falconer. Whilst the existence of these equilibrium states has been well-known for some years their structure has remained elusive, particularly in dimensions higher than two. In this article we give a complete description of the equilibrium states of the singular value function in the three-dimensional case, showing in particular that all such equilibrium states must be fully supported. In higher dimensions we also give a new sufficient condition for the uniqueness of these equilibrium states. As a corollary, giving a solution to a folklore open question in dimension three, we prove that for a typical self-affine set in R 3, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Kaenmaki, A
Date : 30 November 2017
Funders : EPSRC
DOI : 10.1112/plms.12089
Copyright Disclaimer : Copyright 2017 London Mathematical Society. This is not the version of record.
Uncontrolled Keywords : affnity dimension, fractal, iterated function system, Lyapunov dimension, self-affine set, thermodynamic formalism.
Depositing User : Melanie Hughes
Date Deposited : 29 Sep 2017 15:07
Last Modified : 11 Dec 2018 11:23

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