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Attractors associated to a family of hyperbolic $p$-adic plane automorphisms

Part of: Algebraic number theory: local and $p$-adic fields Dynamical systems with hyperbolic behavior Topological dynamics Complex dynamical systems Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  20 February 2020

CLAYTON PETSCHE
CLAYTON PETSCHE*
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, OR97331, USA email petschec@math.oregonstate.edu
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Abstract

We consider a certain two-parameter family of automorphisms of the affine plane over a complete, locally compact non-Archimedean field. Each of these automorphisms admits a chaotic attractor on which it is topologically conjugate to a full two-sided shift map, and the attractor supports a unit Borel measure which describes the distribution of the forward orbit of Haar-almost all points in the basin of attraction. We also compute the Hausdorff dimension of the attractor, which is non-integral.

Keywords

p-adic or non-Archimedean dynamical systemsstrange attractorssymbolic dynamicsphysical measuresHausdorff dimension

MSC classification

Primary: 37P20: Non-Archimedean local ground fields
Secondary: 37D45: Strange attractors, chaotic dynamics 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37B10: Symbolic dynamics 37F35: Conformal densities and Hausdorff dimension 11S82: Non-Archimedean dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 6 , June 2021 , pp. 1867 - 1882
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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