Dynamics: a probabilistic and geometric perspective
type: Item
instance of: scholarly article
title: Dynamics: a probabilistic and geometric perspective
zbMATH Open document ID: 0911.58013
author: Marcelo Viana
published in: Documenta Mathematica
publication date: 1998-08-24T00:00:00Z
full work available at URL: https://eudml.org/doc/228700
full work available at URL: Viana.MAN.html
review text: An overview of recent developments and open questions aiming at a global theory of general (non-conservative) dynamical systems. This useful and well-written survey paper consists of 8 sections as follows: 1. Introduction; 2. Setting the scenario; 3. One-dimensional maps; 4. Hénon-like attractors; 5. Homoclinic tangencies; 6. Singular flows; 7. Cycles -- partial hyperbolicity; 8. Ergodic properties of partially hyperbolic systems. In Section 1, the basic notions and notations are given. The results are referred mostly to transformations, since the definitions and results for flows are often similar. All manifolds are smooth, compact, without boundary, and measures are probabilities on the Borel \(\sigma\)-algebra. Section 2 deals with SRB (Sinai-Ruelle-Bowen) measures, stochastic stability and decay of correlations. In Section 3, several results concerning the quadratic map \(f_a(x)= x^2+ a\) and SRB measure are discussed. In Section 4, the statistical properties of non-hyperbolic attractors are studied. In Section 5, the crucial role of fractal dimension in the bifurcation theory is explained. Section 6 deals with the Lorenz-like attractors in dimension 3. In Section 7, \(C^1\)-robustly transitive sets of transformations \(f: M\to M\) are discussed. Finally, Section 8 deals with partially hyperbolic attractors and the Pesin-Sinai construction of Gibbs \(u\)-states.
Mathematics Subject Classification ID: 37Cxx
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37Dxx
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 34F05
zbMATH DE Number: 1194163
zbMATH Keywords: uniformly hyperbolic system
zbMATH Keywords: sensitive dynamics
zbMATH Keywords: nonuniform hyperbolicity
zbMATH Keywords: physical measure
zbMATH Keywords: structural stability
zbMATH Keywords: stochastic stability
zbMATH Keywords: homoclinic orbit
zbMATH Keywords: heteroclinic circle
zbMATH Keywords: partial hyperbolicity
zbMATH Keywords: dominated splitting
zbMATH Keywords: robust transitiveness
zbMATH Keywords: singular hyperbolicity
zbMATH Keywords: singular attractor
reviewed by: A. Klíč
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1194163