HARMONIC ANALYSIS ON TREES

There is a relationship which has now been studied for a few years between thermodynamic formalism, as in the book of Parry and Pollicott and the theory of unitary representations of certain discrete groups (see for example the work of Boyer and Mayeda). The goal of the minicourse is to explain why such a relation exists, in the case where the group is a finitely generated free group. Then, the constructions rely on geometric properties of trees.

1. Ordered List Item Trees. I will present the standard geometric language on trees. This is mostly a flash course in CAT(-1) geometry.
2. Tree lattices. I will discuss tree lattices and their relationship with graph theory and symbolic dynamics.
3. The Ikehara trace formula. This formula draws a link between the eigenvalues of the discrete Laplace operator and the lengths of closed geodesics on a finite graph. It is a discrete analogue of the Selberg trace formula. I will state it precisely and prove it
4. Thermodynamic formalism and unitary representations. Give a tree lattices, I will exhibit relationships between the thermodynamic formalism on the associated subshift and the theory of its unitary representations.

References:

1. Book of Parry-Pollicott
2. Work of Boyer and Mayeda,

Notes:

Introduction to Selberg's trace formula

Excellent references for the following, very rough, notes are:

• Martin R. Bridson and André Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.
• Adrien Boyer and Dustin Mayeda. Equidistribution, ergodicity and irreducibility associated with Gibbs measures. Comment. Math. Helv., 92(2):349–387, 2017.
• Frédéric Paulin, Mark Pollicott, and Barbara Schapira. Equilibrium states in negative curvature. Asterisque, (373):viii+281, 2015.

Seminário de Manuel Stadlbauer (grupos hiperbólicos) e Carlos Matheus (Fórmula traço de Selberg)