The complexity function has been widely studied in the past decades, a survey is provided in. The topological entropy of the subshift is nothing but the exponential growth rate of p X. In the case of a subshift, we focus our attention on the complexity function, defined as the function p X : N → N that counts the number of non-empty cylinders of length n. One could consider a finer notion than entropy and study how it restricts the dimension group of a Cantor minimal system. In particular, entropy does not impose any restriction on the dimension group of a Cantor minimal system. In Giordano, Putnam and Skau showed that two Cantor minimal systems are strongly orbit equivalent if and only if they have homeomorphic dimension groups with unit (we refer to Section 2 for definitions). A fundamental object that appears in this context is the dimension group of a dynamical system, an algebraic concept imported from the study of C ∗-algebras. So one could ask, what are the natural restrictions that arise from being strongly orbit equivalent. All these results make clear that entropy poses no restriction whatsoever to being strongly orbit equivalent. Later, in a series of papers, Sugisaki generalized this result, showing among other things that for any c ∈ any Cantor minimal system is strongly orbit equivalent to a system whose topological entropy equals c. This was dramatically disproved by Boyle and Handelman, who showed that any Cantor minimal system (a system where the phase space is a Cantor space) is strongly orbit equivalent to a zero entropy Cantor system. When the strong orbit equivalence property was introduced in topological dynamics, it was not clear its relationship with entropy, in particular, it was asked if entropy was preserved under strong orbit equivalence. They are strongly orbit equivalent if they are orbit equivalent and the cocyle function is discontinuous at most in one point (see Section 2.7 for precise definitions). Two minimal topological dynamical systems are orbit equivalent if there is a homeomorphism between the phase spaces sending orbits to orbits.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |