SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms
type: Item
instance of: scholarly article
title: SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms
zbMATH Open document ID: 1312.37021
author: Zhicong Liu
published in: Discrete and Continuous Dynamical Systems
publication date: 2013-04-24T00:00:00Z
review text: Let \(M\) be a compact Riemannian manifold with dimension \(n\) and \(I\) be the unit interval \([0,1]\). The paper under review the author considers the following assumptions on a \(C^2\) diffeomorphism \(F:M\times I\to M\times I\) with \(F(x,t)=(h(x),f_x(t))\) for \((x,t)\in M\times I\): {\parindent=6mm\begin{itemize}\item{} (\(H_1\)) \(f_x(0)=0\) and \(f_x(1)=1\) for every \(x\in M\); \item{} (\(H_2\)) \(h:M\to M\) is an Anosov diffeomorphism preserving the Lebesgue measure \(\mu\); suppose \(h\) has at least two fixed points \(l\) and \(r\); \item{} (\(H_3\)) \(D_t f_l(0)<1\), \(D_t f_r(1)<1\), \(D_t f_l(1)>1\), \(D_t f_r(0)>1\); and \(f_l (t)<t\), \(f_r(t)>t\) for \(t\in(0,1)\); \item{} (\(H_4\)) \(\int_{M\times \{0\}} \log|D_t f_x(0)|d\mu_0<0\) and \(\int_{M\times \{1\}} \log|D_t f_x(1)|d\mu_1<0\); \item{} (\(H_5\)) \(|D_t f_x|<\lambda_{\min}\), where \(\lambda_{min}>1\) is the minimal expanding rate of the Anosov system \(h\) in the unstable directions; where \(D_t f_x(0)=\frac{d}{dt}f_x(t)|_{t=0}\), \(D_t f_x(1)=\frac{d}{dt}f_x(t)|_{t=1}\), and \(\mu_t\) denotes the Lebesgue measure on \(M\times \{t\}\). \end{itemize}} Using Pesin's theory combined with the general theory of differentiable dynamical systems, the author proves (as consequences of two more general results) that under the above assumptions, the system \(f\) admits exactly two physical measures (SRB measures), \(\mu_0\) and \(\mu_1\). Moreover, the basins of the SRB attractors \((M\times \{0\}, \mu_0)\) and \((M\times \{1\}, \mu_1)\) are topologically dense in the total space \(M\times I\). An illustrative example is also given.
reviewed by: Nicolae Lupa
Mathematics Subject Classification ID: 37C40
Mathematics Subject Classification ID: 37D25
Mathematics Subject Classification ID: 37D30
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 6157266
zbMATH Keywords: SRB attractors
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: Pesin theory
zbMATH Keywords: intermingled basins
zbMATH Keywords: non-hyperbolic diffeomorphisms
MaRDI profile type: Publication
full work available at URL: dcds.2013.33.1545
OpenAlex ID: W2329118240
DOI: 10.3934/DCDS.2013.33.1545
Recommended article: Diffeomorphisms with intermingled attracting basins
Recommended article: SRB measures for partially hyperbolic attractors of local diffeomorphisms
Recommended article: SRB measures of singular hyperbolic attractors
Recommended article: Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps
Recommended article: SRB measures for attractors with continuous invariant splittings
Recommended article: SRB measures for a class of partially hyperbolic attractors in Hilbert spaces
Recommended article: Differentiating potential functions of SRB measures on hyperbolic attractors
Recommended article: SRB measures for higher dimensional singular partially hyperbolic attractors
Recommended article: Basin of attraction of cycles of discretizations of dynamical systems with SRB invariant measures.
Recommended article: Existence and uniqueness of SRB measure on \(C^1\) generic hyperbolic attractors
description: scientific article; zbMATH DE number 6157266