Inverse limits of expanding Thurston maps and pseudo-Anosov homeomorphisms of the 3-sphere

Abstract: The inverse limit and natural extension constructions extend a non-invertible dynamical system "naturally" to an invertible one. Inverse limits were probably first used in dynamics by Bob Williams, in the late 1960's, in his study of expanding 1-dimensional attractors. Since then, several researchers have studied inverse limits in dynamics. In complex dynamics, and rational maps in particular, this started to be done systematically in the 1990's, with Dennis Sullivan's work on renormalization and with the work of Meiyu Su and Minsky-Lyubich on Riemann surface laminations associated to rational maps.

In this talk we discuss inverse limits of expanding Thurston maps with a certain symmetry. Typical examples are real rational maps (i.e., holomorphic maps that can be written with real coefficients). Inverse limit spaces are almost always very complicated topological spaces. We show how a mild collapse on the inverse limit spaces under consideration yields the 3-sphere and the natural extension then becomes a "nice" 3-sphere homeomorphism. More precisely, the statement we discuss is that, starting from our expanding Thurston map of the 2-sphere, there is a mild collapse which turns its inverse limit space into the 3-sphere and its natural extension into a "pseudo-Anosov" homeomorphism. This means that on the 3-sphere there are two singular invariant foliations; these foliations are transverse to each other, are transversely measured and the holonomies preserve these measures; the homeomorphism preserves the foliations and contracts one and expands the other transverse measure.

This talk is closely related to the talk "Inverse limits of tent maps and sphere homeomorphisms" given by Toby Hall on June 9th. There generalized pseudo-Anosov maps of the 2-sphere were constructed from inverse limits of interval endomorphisms. Thus our construction can be viewed as a higher dimensional analog.

This is joint work (in progress) with Daniel Meyer.

analysis of PDEscomplex variablesdynamical systemsmetric geometry

Audience: researchers in the topic

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