Stable and unstable manifolds of diffeomorphisms with positive entropy
by
Naoya Sumi
Department of Mathematics, Tokyo Metropolitan University

We study chaotic properties of diffeomorphisms with positive entropy. Notions of chaos have been given by Li and Yorke [4], Devaney, and others. It is known that if a continuous map of interval has positive entropy, then it is chaotic according to the definition of Li and Yorke (cf. [1]).

In [3] Katok proved the following: let f be a C^1+\epsilon-diffeomorphism of a closed surface M. If the topological entropy of f is positive, then there exists a hyperbolic set \Gamma such that the restriction of f into \Gamma is topologically conjugate to a subshift of finite type with positive entropy. This implies that f is chaotic in the sense of Li-Yorke. However, Katok's theorem does not hold for the high dimensional case.

In this talk we show the following:

Theorem Let f be a C²-diffeomorphism of a closed C^\infty-manifold M. If the topological entropy of f is positive, then f is chaotic in the sense of Li-Yorke.

Concerning the chaos in the sence of Li-Yorke, Kato introduced the notion of " * -chaos" ([2]). To obtain Theorem A we need the following theorem.

Theorem Let f be a C²-diffeomorphism of a closed C^\infty-manifold M and let \mu be an f-invariant ergodic Borel probability measure on M. If the metric entropy of \mu is positive, then for \mu-a.e. x in M the following hold:

(a) [`(W<sup>s</sup>(x))] is a perfect * -chaotic set, and
(b) [`(W^u(x))] contains a perfect * -chaotic set.

Here W^s(x) and W^u(x) are defined by

Ws(x)={ y in M : limsup n --> \infty  1/n logd(fn(x), fn(y)) < 0 } and, Wu(x)={ y in M : limsup n --> \infty  1/n logd(f-n(x), f-n(y)) < 0 }.

respectively. W^s(x) is called a stable manifold and W^u(x) is called an unstable manifold ().

References

1. L.S. Block and W.A. Coppel, Dynamics in one dimension, Lecture Notes in Math., Springer-Verlag, 1513, 2151-2157.
2. H. Kato, On scrambled sets and a theorem of Kuratowski on independent sets, Proc. Amer. Math. Soc., 126 (1998), 2151-2157.
3. A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. I.H.E.S., 51 (1980), 137-174.
4. T.Y. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.
5. Y.B. Pesin, Families of invariant manifolds corresponding to non-zero characteristic exponents, Math. of the USSR, Izvestija, 10 (1978), 1261-1305.

Date received: July 2, 1999