Maximally chaotic homeomorphisms of sigma-compact manifolds
✍ Scribed by Steve Alpern; V.S. Prasad
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 100 KB
- Volume
- 105
- Category
- Article
- ISSN
- 0166-8641
No coin nor oath required. For personal study only.
✦ Synopsis
Let µ be a locally positive Borel measure on a σ -compact n-manifold X, n 2. We show that there is always a µ-preserving homeomorphism of X which is maximally chaotic in that it satisfies Devaney's definition of chaos, with the sensitivity constant chosen maximally. Furthermore, maximally chaotic homeomorphisms are compact-open topology dense in the space of all µpreserving homeomorphisms of X if and only if (X, µ) has at most one end of infinite measure. (For example, for Lebesgue measure λ on X = R 2 , but not for λ on the strip X = R × [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.