Dynamics: a probabilistic and geometric perspective
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.
scientific article; zbMATH DE number 1194163