Does positive topological entropy imply horseshoes?

Question

I am aware that positive topological entropy implies chaos in the sense of Li-Yorke. I want to get an idea of what consequences positive topological entropy has for the the presence (or lack of) chaotic dynamics in the following alternative formulation of "chaotic dynamics", specifically:

If a given invertible dynamical system $\Phi$ on a phase space $M$ has positive topological entropy, does this imply that it has a Smale horseshoe structure in at least a part of the phase space?

By a horseshoe, I mean an invariant (Cantor) set on which $\Phi$ has dynamics which is topologically conjugate to a shift map $\sigma$ on $N$ symbols.

Conversely, does zero topological entropy imply that horseshoes cannot exist?

Answer 1

The answer is yes if $\Phi$ is a $C^{1+\alpha}$ diffeomorphism, due to work of Anatole Katok in the context of smooth ergodic theory, in his beautiful paper Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137-173.

To my best knowledge, it is unknown whether this part of smooth ergodic theory goes over to $C^1$ diffeomorphisms, this would require a very careful inspection of every detail. I would say that it has some possibilities.

On the other hand, the answer to your last question is obviously positive: horseshoes carry positive topological entropy, due to the topological conjugacy to the full shift.

PS: By the way: do you really know that your first sentence holds for maps other than maps of the interval? I presume not, but otherwise I would be glad to know.