Problem 12

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Shub's entropy conjecture: $h(f) \geq \log |\lambda| $

  1.  for diffeos
  2.  $\Omega $ finite plus hyperbolic
  3.  Axiom A with cycles.

Tags

Tags: 
Hyperbolic
Entropy
Topology

Comments

Submitted by Ledrappier on Mon, 11/14/2016 - 11:40

See [1].

References

  1. [gang2010entropy] Gang L, Viana M, Yang J.  2010.  The entropy conjecture for diffeomorphisms away from tangencies. arXiv preprint arXiv:1012.0514.

Submitted by shub on Mon, 07/10/2017 - 10:04

The conjecture is proven for C^1 Axiom A no cycle diffeomorphisms by Shub and Williams and Ruelle and Sullivan. It has also been proven in general for C^infinity endomorphisms of compact manifolds by Yomdin. For references,see Shub,M., All,Most,Some Differentiable Dynamical Systems, in Proceedings of the ICM,Madrid 2006.

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