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Geometry and Topology of Escape. I. Epistrophes
Mitchell, K. A. Handley, J. P. Tighe, B. Delos, John B. Knudson, Stephen
Mitchell, K. A.
Handley, J. P.
Tighe, B.
Delos, John B.
Knudson, Stephen
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.
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2003-09-01
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American Institute of Physics
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Physics
DOI
https://doi.org/10.1063/1.1598311