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.


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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