No abstract

We consider a dynamical system given by an area-preserving map on a two-dimensional phase plane and consider a one-dimensional line of initial conditions within this plane. We record the number of iterates it takes a trajectory to escape from a bounded region of the plane as a function along the line of initial conditions, forming an "escape-time plot". For a chaotic system, this plot is in general not a smooth function, but rather has many singularities at which the escape time is infinite; these singularities form a complicated fractal set. In this article we prove the existence of regular repeated sequences, called "epistrophes", which occur at all levels of resolution within the escape-time plot. (The word "epistrophe" comes from rhetoric and means "a repeated ending following a variable beginning".) The epistrophes give the escape-time plot a certain self-similarity, called "epistrophic" self-similarity, which need not imply either strict or asymptotic self-similarity.

We continue our study of the fractal structure of escape-time plots for chaotic maps. In the preceding paper, we showed that the escape-time plot contains regular sequences of successive escape segments, called epistrophes, which converge geometrically upon each endpoint of every escape segment. In the present paper, we use topological techniques to: (1) show that there exists a minimal required set of escape segments within the escape-time plot; (2) develop an algorithm which computes this minimal set; (3) show that the minimal set eventually displays a recursive structure governed by an "Epistrophe Start Rule": a new epistrophe is spawned ∆ = D + 1 iterates after the segment to which it converges, where D is the minimum delay time of the complex. Topological methods and symbolic dynamics have long been valuable tools for describing orbits of dynamical systems. For example, if a particle in the plane scatters from three fixed disks, labeled A,B, and C, its orbit can be characterized by a sequence of symbols, such as ...ABA*BCBCA..., giving the sequence of collisions with the disks. The asterisk gives the location of the particle at the present time; as time goes by the asterisk takes one step to the right. In this paper, we describe a new kind of symbolic dynamics, in which the symbol sequence describes the structure of a curve in the plane. The relevant curve is not the trajectory of a particle, but rather an ensemble of initial points in phase space -the line of initial conditions. This line winds around "holes" in the plane in a manner described by the symbol sequence. The dynamical map applied to the line induces a map on the symbol sequence, which is more complicated than a simple shift. We use this symbolic dynamics to derive properties of the epistrophes introduced in the preceding paper. In particular, we use it to obtain a "minimal set of escape segments" and an "epistrophe start rule."