Matrix valued Brownian motion and a paper by Polya

We give a geometric description of the motion of eigenvalues of a Brownian motion with values in some matrix spaces. In the second part we consider a paper by Polya where he introduced a function close to the Riemann zeta function, which satisfies Riemann hypothesis. We show that each of these two functions can be related to Brownian motion on a symmetric space

Phase transitions for suspension flows

This paper is devoted to study thermodynamic formalism for suspension flows defined over countable alphabets. We are mostly interested in the regularity properties of the pressure function. We establish conditions for the pressure function to be real analytic or to exhibit a phase transition. We also construct an example of a potential for which the pressure has countably many phase transitions.Comment: Example 5.2 expanded. Typos corrected. Section 6.1 superced the note "Thermodynamic formalism for the positive geodesic flow on the modular surface" arXiv:1009.462