Smooth interval maps have symbolic extensions

• 成果类型: Article
• 署名作者: Downarowicz, Tomasz; Maass, Alejandro
• 署名单位: Wroclaw University of Science & Technology; Universidad de Chile; Universidad de Chile
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-008-0172-4
• 发表日期: 2009
• 页码: 617-636
• 关键词: entropy; GROWTH

摘要

We prove that if X denotes the interval or the circle then every transformation T:X -> X of class C (r) , where r > 1 is not necessarily an integer, admits a symbolic extension, i.e., every such transformation is a topological factor of a subshift over a finite alphabet. This is done using the theory of entropy structure. For such transformations we control the entropy structure by providing an upper bound, in terms of Lyapunov exponents, of local entropy in the sense of Newhouse of an ergodic measure nu near an invariant measure mu (the antarctic theorem). This bound allows us to estimate the so-called symbolic extension entropy function on invariant measures (the main theorem), and as a consequence, to estimate the topological symbolic extension entropy; i.e., a number such that there exists a symbolic extension with topological entropy arbitrarily close to that number. This last estimate coincides, in dimension 1, with a conjecture stated by Downarowicz and Newhouse [13, Conjecture 1.2]. The passage from the antarctic theorem to the main theorem is applicable to any topological dynamical system, not only to smooth interval or circle maps.