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Statistical properties of quadratic polynomials with a neutral fixed point


Avila, Artur; Cheraghi, Davoud (2018). Statistical properties of quadratic polynomials with a neutral fixed point. Journal of the European Mathematical Society, 20(8):2005-2062.

Abstract

We describe the statistical properties of the dynamics of the quadratic polynomials $P_α$$( z )$ =$e^{2παi}$ $z+z^2$ on the complex plane, with $\alpha$ of high type. In particular, we show that these maps are uniquely ergodic on their measure-theoretic attractors, and the unique invariant probability is a physical measure describing the statistical behaviour of typical orbits in the Julia set. This confirms a conjecture of P´erez-Marco on the unique ergodicity of hedgehog dynamics, in this class of maps.

Abstract

We describe the statistical properties of the dynamics of the quadratic polynomials $P_α$$( z )$ =$e^{2παi}$ $z+z^2$ on the complex plane, with $\alpha$ of high type. In particular, we show that these maps are uniquely ergodic on their measure-theoretic attractors, and the unique invariant probability is a physical measure describing the statistical behaviour of typical orbits in the Julia set. This confirms a conjecture of P´erez-Marco on the unique ergodicity of hedgehog dynamics, in this class of maps.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > General Mathematics
Physical Sciences > Applied Mathematics
Uncontrolled Keywords:Applied Mathematics, General Mathematics
Language:English
Date:6 June 2018
Deposited On:17 Jan 2019 08:29
Last Modified:29 Jul 2020 07:58
Publisher:European Mathematical Society
ISSN:1435-9855
OA Status:Green
Publisher DOI:https://doi.org/10.4171/jems/805

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