Problem 12
Shub’s entropy conjecture: $h(f) \geq \log |\lambda| $ for diffeos $\Omega $ finite plus hyperbolic Axiom A with cycles.
Shub’s entropy conjecture: $h(f) \geq \log |\lambda| $ for diffeos $\Omega $ finite plus hyperbolic Axiom A with cycles.
Comments
See [gang2010entropy] .
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.