Abstract
This chapter is an introduction to hyperbolic dynamics. We first introduce the notion of a hyperbolic set. In particular, we describe the Smale horseshoe and some of its modifications. We also establish the continuity of the stable and unstable spaces on the base point. We then consider the characterization of a hyperbolic set in terms of invariant families of cones. In particular, this allows us to describe some stability properties of hyperbolic sets under sufficiently small perturbations. The pre-requisites from the theory of smooth manifolds are fully recalled in Sect. 5.1.
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Notes
- 1.
Theorem (See for example [43]) If f:A→ℝ is a continuous function in a closed subset A⊂X of a normal space (that is, a space such that any two disjoint closed sets have disjoint open neighborhoods), then there exists a continuous function g:X→ℝ such that g|A=f.
References
Munkres, J.: Topology: A First Course. Prentice-Hall, New York (1975)
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Barreira, L., Valls, C. (2013). Hyperbolic Dynamics I. In: Dynamical Systems. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-4835-7_5
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DOI: https://doi.org/10.1007/978-1-4471-4835-7_5
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4834-0
Online ISBN: 978-1-4471-4835-7
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