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Recent developments in dynamics. (English) Zbl 0844.58001

Chatterji, S. D. (ed.), Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 246-265 (1995).
The author provides a survey of recent developments in dynamics. He asserts that the goal of the theory of dynamical systems is to understand \(\underline {\text{most}}\) of the dynamics of \(\underline {\text{most}}\) systems. “Most” can assume both a topological and a metric significance; topologically, “most” means open and dense or \(G_\delta \) dense, while in a metric sense, “most” means almost every orbit of the system. For smoothly parametrized families of mappings, “most” means almost every value of the parameter.
The author’s survey includes results which bear principally on hyperbolic dynamics and quasiperiodic dynamics. He discusses three approaches to quasi periodic dynamics which have proven useful. These are a function-theoretic approach for KAM-type theories, a variational approach for symplectic systems, and the more geometric renormalization approach. The author also reviews recent developments in hyperbolic dynamics, and suggests the development of a conceptual theory of weakly hyperbolic basis sets which includes features of Henon and Henon-like families of diffeomorphisms.
For the entire collection see [Zbl 0829.00014].

MSC:

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37D99 Dynamical systems with hyperbolic behavior
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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