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Non-invertible planar self-affine sets

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior Classical measure theory

Published online by Cambridge University Press:  08 May 2024

ANTTI KÄENMÄKI  and
PETTERI NISSINEN
ANTTI KÄENMÄKI
Affiliation:
Research Unit of Mathematical Sciences, P.O. Box 8000, FI-90014 University of Oulu, Finland. e-mail: antti.kaenmaki@oulu.fi
PETTERI NISSINEN
Affiliation:
Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland. e-mail: pettern@student.uef.fi
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Abstract

We compare the dimension of a non-invertible self-affine set to the dimension of the respective invertible self-affine set. In particular, for generic planar self-affine sets, we show that the dimensions coincide when they are large and differ when they are small. Our study relies on thermodynamic formalism where, for dominated and irreducible matrices, we completely characterise the behaviour of the pressures.

MSC classification

Primary: 28A80: Fractals
Secondary: 37C45: Dimension theory of dynamical systems 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 17
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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