Ahlfors regularity, extensions by Schatten ideals and a geometric fundamental class of Smale space C*-algebras using dynamical partitions of unity

Gerontogiannis, Dimitrios Michail D. (2021) Ahlfors regularity, extensions by Schatten ideals and a geometric fundamental class of Smale space C*-algebras using dynamical partitions of unity. PhD thesis, University of Glasgow.

Abstract

In this thesis we study the Ahlfors regularity of Bowen's measure on Smale spaces and analyse Smale space C*-algebras in the framework of Connes' noncommutative geometry using smooth extensions by Schatten ideals and summable Fredholm modules.

Bowen's construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry's measure which is Ahlfors regular. By extending Bowen's construction we create a tool for transferring, up to topological conjugacy, the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff, box-counting and Assouad dimensions of a large class of Smale spaces.

In the noncommutative setting, given a Smale space, our generalised Markov partitions yield dynamical partitions of unity which produce explicit θ-summable Fredholm modules that represent a fundamental K-homology class for the Spanier-Whitehead duality of the stable and unstable Ruelle algebras of the Smale space. Therefore, we obtain an exhaustive description of the K-homology classes of Ruelle algebras in terms of Fredholm modules constructed by Markov partitions. Our method involves the construction of dynamical metrics on Smale space groupoids that give rise to smooth (holomorphically stable and dense) *-subalgebras of Smale space C*-algebras. In particular, for every such smooth subalgebra of a Ruelle algebra, we show that every extension class in the BDF-theory group of the Ruelle algebra can be represented by an extension that on the smooth subalgebra reduces to an algebraic extension by a Schatten p-ideal. The value p is related to the dimension of the underlying Smale space. This provides a new approach to the noncommutative dimension theory of Smale spaces.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Keywords: Smale spaces, Markov partitions, Ahlfors regularity, fractal dimensions, C*-algebras, groupoid metrics, smooth algebras, KK-theory, Spanier-Whitehead duality, K-homology, extensions by Schatten ideals, Fredholm modules.
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Whittaker, Dr. Michael F. and Zacharias, Professor Joachim
• Date of Award: 2021
• Depositing User: Mr Dimitrios Michail Gerontogiannis
• Unique ID: glathesis:2021-82082
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 25 Mar 2021 08:46
• Last Modified: 25 Mar 2021 08:54
• Thesis DOI: 10.5525/gla.thesis.82082
• URI: https://theses.gla.ac.uk/id/eprint/82082