Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2000-09

July 2000

Maximally Chaotic Homeomorphisms of Sigma-Compact Manifolds

Steve Alpern and V.S. Prasad


Let  \mu  be a locally positive Borel measure on a  \sigma-compact  n-manifold  Xn => 2.  We show that there is always a  \mu-preserving homeomorphism of  X  which is maximally chaotic in that it satisfies Devaney's definition of chaos, with the sensitivity constant chosen maximally. Futhermore, maximally chaotic homeomorphisms are compact-open topology dense in the space of all  \mu-preserving homeomorphisms of  X  if and only if  (X,\mu)  has at most one end of infinite measure. (For example, for Lebesgue measure on the plane but not on the strip  X = R x [0,1].) This work extends that of Aarts and Daalderop, Daalderop and Fokkink, Kato et. al., and Alpern, regarding chaotic phenomena on compact manifolds, and that of Besicovitch and Prasad for other dynamical properties on noncompact manifolds.

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