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The Mathematics of Time pp 1–82Cite as

Differentiable Dynamical Systems

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Abstract

This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M. An action is a homomorphism G→Diff(M) such that the induced map G×M→M is differentiable. Here Diff(M) is the group of all diffeomorphisms of M and a diffeo- morphism is a differentiable map with a differentiable inverse. Everything will be discussed here from the C ∞ or C r point of view. All manifolds maps, etc. will be differentiable (C r, 1 ≦ r ≦ ∞) unless stated otherwise.

The preparation of this paper was supported by the National Science Foundation under grant GN-530 to the American Mathematical Society and partially supported by NSF grant GP-5798.

These footnotes and reference are in no way meant to be complete; I will give some of the more important results which bear directly on Differentiable dynamical systems (DDS) and have appeared since.

Let me first note some of the more comprehensive generel reference. Especially pertinent is the book by Shub, 1978, which gives a very good development of some of the main results of differentiable dynamical systems. There are also dood accounts by Nitecki, 1971 and Palis-Melo, 1978. More elementary texts which give an ordinary differential equations background are Arnold, 1973 and Hirsch-Smale, 1974.

Then there are three survey articles by Shub; Shub, 1974, 1976 and in Peixoto, 1973, which give a brief account of recent research. Bowen’s monograph, Bowen, 1978, is also of this nature. See also Chapter 7 in Abraham-Marsden, 1978, Markus, 1971, and “Fifty Problems by Palis-Pugh in Manning 1975.

Finally, one should mention various proceedings of conferences which comtain research and survey articles on dynamical systems. These include Chern-Dmale, 1970, or simply C-S, Manning, 1975, Markley et al., 1978, Nitecki-Robinson, 1980, and Peixoto, 1973.

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    Steve Smale

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Smale, S. (1980). Differentiable Dynamical Systems. In: The Mathematics of Time. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8101-3_1

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