Shuhei Hayashi

On the C^{1}-creation of good periodic orbits

After Pugh developed a C^{1} perturbation technique to
create periodic orbits in the 60's, Mane proved the ergodic closing
lemma in the 80's in order
to solve the C^{1} Stability Conjecture for diffeomorphisms, which is a reformulation of Pugh's perturbation from an ergodic viewpoint.
In Pugh's closing lemma, if x is a recurrent point; i.e.,
f^{ni}(x )
→ x (i → +∞) for some n_{1}< n_2< ... , then some finite part between x and f^{ni}(x) is closed up. So the Hausdorff distance between { x, ... , f^{ni}(x)}
and the created periodic orbit may not be small, while Mane's ergodic
closing lemma realizes the small Hausdorff distance. However, only
having the small Hausdorff distance is not enough to approximate the
Lyapunov exponents at x by those at the created periodic points.
Recently, such approximation became possible by an extended version of
the ergodic closing lemma. Following these achievements, we consider
the C^{1}-creation of good periodic orbits from a numerical
viewpoint. This is motivated by a paper of Gambaudo and Tresser
appeared in 1983. They pointed out that some hyperbolic attracting
periodic orbits (which are thought of as trivial observable attractors
in the theory of dynamical systems) are not necessarily observable in
numerical experimentation when their domains of regular attraction are
too small to be observed by computers. We introduce a mathematical
concept of observability of finite sets in consideration of numerical
procedure and try to find out conditions under which the C^{1}$-creation of observable periodic orbits becomes possible.