Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-98-09

May 1998

Periodic, Locally Linear, and Chaotic Approximations to Volume Preserving Homeomorphisms

Steve Alpern


We modify a combinatorial idea of Peter Lax, to uniformly approximate any volume preserving homeomorphism  h  of the closed unit cube  Inn >= 2,  by another one  h'  such that either: (i)  h'  rigidly permutes an infinite family of dyadic cubes of total volume one, or (ii)  h'  is chaotic in the sense of Devaney. The first result (i) generalizes a similar result of Oxtoby and Ulam, yielding an approximation which is periodic and locally linear almost everywhere. The second result (ii) is the first combinatorial proof of the existence of chaotic manifold homeomorphisms, a result obtained independently, using different methods, by J. Aarts and F. Daalderop. Our constructions may also be applied on compact connected manifolds of dimension  n >= 2.

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