College of Science & Engineering

Dynamical zeta functions and the irreversible Ledrappier system

Supervisor: Dr Christopher Bruce

School: Mathematics and Statistics

Description:

Dynamical systems of algebraic origin form an important class of topological dynamical systems. Indeed, the theory of algebraic actions of groups is rich, and there are deep connections with other areas of mathematics, such as number theory, commutative algebra, and operator algebras, see [S] and [KL]. A fundamental invariant for algebraic Z^d-actions is their dynamical zeta function as defined by Lind [L]. Such functions are dynamical analogues of the classical Riemann zeta function; they encode certain combinatorial information about the dynamical system in a compact form where analytical techniques can be employed. Computing zeta functions for algebraic Z^d-actions is in general a very difficult and important problem, see, e.g., [MW]. 

The theory of irreversible dynamical systems of algebraic origin is much less well-developed than that of the reversible systems studied in [L], [S], and [KL]. However, recently, it was discovered that the irreversible systems exhibit new and intriguing phenomena in connection to the theory of C*-algebras and groupoids, see [BL1]. These recent developments lead to a new notion of dynamical zeta function for a certain class of algebraic N^d-actions. Such systems do not fit into the framework from [L]. 

This 10-week project focuses on the dynamical zeta functions attached to algebraic N^2-actions. The student will start by carefully examining the zeta function attached to a particular algebraic N^2-action called the irreversible Ledrappier system. The dynamical zeta function for the classical (reversible) Ledrappier system is known to be interesting and was studied by Lind in [L]. This starting point will give the student the opportunity to familiarise themselves with the basics of algebraic N^2-actions, and will allow me to assess the student’s background and capabilities. The specific questions that we will tackle will be adjusted to match the student’s background and interests.  

 

[BL1] C. Bruce and X. Li, Algebraic actions I. C*-algebras and groupoids. Preprint, https://arxiv.org/abs/2209.05823.  

[KL] D. Kerr and H. Li, Ergodic theory. Independence and dichotomies. Springer Monographs in Mathematics. Springer, Cham, 2016. 

[L] D. A. Lind, A zeta function for Z^d-actions. Ergodic theory of Z^d-actions, London Math. Soc. Lecture Note Ser., 228, Cambridge Univ. Press, Cambridge, 1996. 

[MW] R. Miles and T. Ward, The dynamical zeta function for commuting automorphisms of zero-dimensional groups. Ergodic Theory Dynam. Systems 38 (2018), no. 4, 1564-1587. 

[S] K. Schmidt, Dynamical systems of algebraic origin. Progress in Mathematics, 128. Birkhäuser Verlag, Basel, 1995.