Abstract
Axiom A system is an important class in smooth dynamical systems. In the ergodic theory of Anosov diffeomorphisms or of Axiom A attractors of diffeomorphisms, it was shown that there is an invariant Borel probability measure that is characterized by each of the following properties:
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(1)
Pesin’s entropy formula (PEF) holds.
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(2)
SRB property: its conditional measures on unstable manifolds are absolutely continuous with respect to the Lebesgue measures on the corresponding sub-manifolds.
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(3)
Lebesgue a.e. point in an open set is generic with respect to this measure.
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(4)
This measure is approximable by measures that are invariant under suitable stochastic perturbations.
Each one of these properties has been shown to be significant in its own right, but more striking is the fact that they are all equivalent to each one another. Many of these ideas are due to Sinai, Ruelle and Bowen. We refer the reader to [88], [10], [76] and [79].
In this chapter, we study the Axiom A endomorphisms along the line of [72].
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© 2009 Springer-Verlag Berlin Heidelberg
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QUIAN, M., XIE, JS., ZHU, S. (2009). Axiom A Endomorphisms. In: Smooth Ergodic Theory for Endomorphisms. Lecture Notes in Mathematics(), vol 1978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01954-8_4
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DOI: https://doi.org/10.1007/978-3-642-01954-8_4
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-01953-1
Online ISBN: 978-3-642-01954-8
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