Shuhei Hayashi
On the C¹-creation of good periodic orbits
After Pugh developed a C¹ 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 C¹ 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^n_i(x ) → x (i → +∞) for some n₁< n_2< ... , then some finite part between x and f^n_i(x) is closed up. So the Hausdorff distance between { x, ... , f^n_i(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 C¹-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 C¹$-creation of observable periodic orbits becomes possible.