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A strongly irreducible affine iterated function system with two invariant measures of maximal dimension

Part of: Classical measure theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  30 October 2020

IAN D. MORRIS  and
CAGRI SERT Open the ORCID record for CAGRI SERT [Opens in a new window]
IAN D. MORRIS
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK (e-mail: i.morris@qmul.ac.uk)
CAGRI SERT*
Affiliation:
Department Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland
*
e-mail: cagri.sert@math.uzh.ch
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Abstract

A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb {R}^{d}$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor, which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of $\mathbb {R}^{d}$ . In this paper we give an example where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.

Keywords

self-affine setiterated function systemequilibrium statenon-conformal repellersubadditive thermodynamic formalism

MSC classification

Primary: 28A80: Fractals
Secondary: 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3417 - 3438
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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