Tag Archives: entropy

Katok’s entropy conjecture

In dynamics there are several conjectures related to entropy. The first is the famous Shub Entropy Conjecture:

Let f:M\to M be a differentiable map (or a diffeomorphism), s(f_*) be the spectral radius of the linear map f_{*}:\oplus H_k(M,\mathbb{R})\to \oplus H_k(M,\mathbb{R}). Then
h_{\mathrm{top}}(f)\ge s(f_*).
Moveover find characteristics of maps satisfying h_{\mathrm{top}}(f)= s(f_*) in each homotopy class.

I am not sure whether the following special problem is still open (it is proved if E^u is orientable):

If f:M\to M is an Anosov diffeomorphism, then h_{\mathrm{top}}(f)= s(f_*).

There is another entropy conjecture due to Katok. In fact Katok proved that the topological entropy and the Liouville entropy of a geodesic flow on a negatively curved surface agree only if the curvature is constant, that is, the metric is locally symmetric. He conjectured that the same holds in any dimension. One of the best result in this direction is

Besson, Courtois and Gallot, the topological entropy of geodesic flow is minimized only for locally symmetric metrics.

Katok’s closing property for Pesin blocks

First let us list some interesting results of Katok (Lyapunov exponents, entropy and periodic orbbits for diffeomorphisms 1980).

Let f:M\to M be a C^r diffeo for some r>1.
I. If there exists a nonatomic ergodic hyperbolic measure \mu then h_{\mathrm{top}}(f)>0.
II. Generally \limsup_{n\to\infty}\frac{\log |P_n(f)|}{n}\ge h_\mu(f) for each hyperbolic measure \mu.

He started with that for a hyperbolic ergodic measure \mu, the Pesin set has full \mu-measure. Katok first proved that a recurrent point with respect to the Pesin block is closable and the closed orbit is hyperbolic whose invariant manifolds are of uniform size (depending on the level of block).

Then if the measure is continuous, he picked two points x_i in the support of \mu close to each other and then two recurrent points y_i\in\Lambda_k that are arbitrary close to x_i and found a periodic point p_i for each one. The new points are so close to the recurrent point d(y_i,p_i)\le d(x_1,x_2)/100 that

1. these two are distinct and
2. they are still close: d(p_1,p_2)\le 2d(x_1,x_2).

Since their invariant manifolds have uniform size (with respect to level of Pesin block) and the p_i‘s can be really close, they intersect transversally. This guarantees the existence of horseshoe. So the map has positive topological entropy (although the metric entropy of \mu might be zero).

For the later claim he reduced to the case that \mu is ergodic. Then pick a subset K_n of Pesin blocks with cardinal growth approximating the metric entropy that is d^n_f-separated and recurrent (closer than the separate constant) for some time during [n,(1+\epsilon)n]. This involves a counting definition of metric entropy given in the first section. Then for each point x\in K_n we find a periodic point p_x closing x. Again these periodic points are distinct (and hyperbolic) with period in [n,(1+\epsilon)n]. Thus we have
\sum_{j=n}^{(1+\epsilon)n}|P_j(f)|\ge |K_n|. So the growth of hyperbolic periodic points dominates the metric entropy of any (ergodic and hence invariant) hyperbolic measure.