Symbolic extensions and smooth dynamical systems

• 成果类型: Article
• 署名作者: Downarowicz, T; Newhouse, S
• 署名单位: Wroclaw University of Science & Technology; Michigan State University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-004-0413-0
• 发表日期: 2005
• 页码: 453-499
• 关键词: volume growth; entropy

摘要

Let f : X --> X be a homeomorphism of the compact metric space X. A symbolic extension of ( f, X) is a subshift on a finite alphabet ( g, Y) which has f as a topological factor. We show that a generic C-1 nonhyperbolic (i.e., non-Anosov) area preserving diffeomorphism of a compact surface has no symbolic extensions. For r > 1, we exhibit a residual subset R of an open set U of C-r diffeomorphisms of a compact surface such that if f is an element of R, then any possible symbolic extension has topological entropy strictly larger than that of f. These results complement the known fact that any C-infinity diffeomorphism has symbolic extensions with the same entropy. We also show that C-r generically on surfaces, homoclinic closures which contain tangencies of stable and unstable manifolds have Hausdorff dimension two.