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+[[ebsd2021:participantsquint|Comments and questions of participants]]
+</WRAP>
 HARMONIC ANALYSIS ON TREES
-. Trees. I will present the standard geometric language on trees. This is mostly a flash course in CAT(-1) geometry.
+There is a relationship which has now been studied for a few years between
-. Tree lattices. I will discuss tree lattices and their relationship with graph theory and symbolic dynamics.
+thermodynamic formalism, as in the book of Parry and Pollicott and the theory
-. 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.
+of unitary representations of certain discrete groups (see for example the work
-. 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.
+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.
+  - Ordered List Item Trees. I will present the standard geometric language on trees. This is mostly a flash course in CAT(-1) geometry.
+  - Tree lattices. I will discuss tree lattices and their relationship with graph theory and symbolic dynamics.
+  - 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
+  - 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:
+  - [[http://www.numdam.org/issue/AST_1990__187-188__1_0.pdf|Book of Parry-Pollicott]]
+  - [[https://webusers.imj-prg.fr/~adrien.boyer/IrredGibbsr.pdf|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.
+[[https://drive.google.com/file/d/1wOXY1__0PjVpcNzxMWNCFzjlxSqFg-dx/view|Seminário de Manuel Stadlbauer (grupos hiperbólicos) e Carlos Matheus (Fórmula traço de Selberg)]]
+<WRAP center round box 60%>
+[[ebsd2021:Bourdon|Bourdon metric]]
+</WRAP>
+<WRAP center round box 60%>
+[[ebsd2021:Gibbscocycle|Gibbs cocycle]]
+</WRAP>
+<WRAP center round box 60%>
+[[ebsd2021:Patterson|Patterson-Sullivan density]]
+</WRAP>
+<WRAP center round box 60%>
+[[ebsd2021:Gibbsstates|Gibbs states]]
+</WRAP>
 ~~DISCUSSION~~