Fractal Weyl law for skew extensions of expanding maps
type: Item
instance of: scholarly article
title: Fractal Weyl law for skew extensions of expanding maps
zbMATH Open document ID: 1291.37030
DOI: 10.1088/0951-7715/25/6/1671
published in: Nonlinearity
publication date: 2012-07-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1112.5109
Mathematics Subject Classification ID: 37C30
Mathematics Subject Classification ID: 81Q20
Mathematics Subject Classification ID: 37D30
zbMATH DE Number: 6052374
zbMATH Keywords: partially hyperbolic dynamical systems
zbMATH Keywords: transfer operators
zbMATH Keywords: decay of correlations
zbMATH Keywords: semiclassical analysis
author: Jean-François Arnoldi
MaRDI profile type: Publication
OpenAlex ID: W2091162453
arXiv ID: 1112.5109
review text: A natural class of partially hyperbolic dynamical systems can be constructed as follows. Let \(E:\mathbb{S}^1\rightarrow\mathbb{S}^1\) be a smooth expanding map, let \(G\) be a compact Lie group with normalized Haar measure \(m\), and let \(\tau:\mathbb{S}^1\rightarrow G\) be a smooth map. One can define NEWLINE\[NEWLINE\forall(x,g)\in\mathbb{S}^1\times G,\;\hat{E}_{\tau}(x,g):=(E(x),\tau(x)g).NEWLINE\]NEWLINE This extension of the map \(E\) has a \(1\)-dimensional expanding direction, and a neutral direction corresponding to \(G\). It provides one of the simplest models of partially hyperbolic dynamical systems, and one can try to understand the ergodic properties of this map, such as decay of correlations. For instance, it is known that the smooth invariant SRB measures associated to \(E\) is mixing, and it is natural to ask if mixing still holds for the extension with respect to the invariant measure \(\mu\times m\) (at least under some dynamical conditions). Some quantitative results on these questions were first obtained by \textit{D. Dolgopyat} [Isr. J. Math. 130, 157--205 (2002; Zbl 1005.37005)], and many improvements have been obtained since this article.NEWLINENEWLINEThe article under review deals with this question in the case where \(G\) is the simplest non-abelian Lie group, i.e., \(G=\mathrm{SU}(2)\). The case \(G=\mathrm{U}(1)\) was treated in [\textit{F. Faure}, Nonlinearity 24, No. 5, 1473--1498 (2011; Zbl 1233.37018)]. The main objective of the article is to describe spectral properties of the transfer operator \(\mathcal{F}_{\tau}\) associated to the map \(\hat{E}_{\tau}\) using semiclassical techniques. For that purpose, the author uses harmonic analysis to decompose the operator \(\mathcal{F}_{\tau}\) into a sum of operators \(\hat{F}_{j}\) acting on simpler spaces \(L^2(\mathbb{S}^1)\times \mathcal{D}_j\), indexed by the irreducible finite-dimensional unitary representations of \(G\). Then, he studies the spectral properties of the operators \(\…
reviewed by: Gabriel Rivière
description: scientific article; zbMATH DE number 6052374