Analysis on Graphs and Fractals

Aspects of self-similar graphs

D Guido, Univ. Roma 'Tor Vergata', Italy

Abstract

With a suitable notion of self-similar graph, it is possible to construct a trace on the C* - algebra generated by geometric operators acting on l^2 vectors on the graph. Three aspects will be discussed:

• the study of geometric invariants on the graph.
• the study of the Ihara zeta function for the graph.
• the study of thermodynamical properties of the graph.

The talk is based on the following works:

1. Cipriani, Fabio; Guido, Daniele; Isola, Tommaso 'A C*-algebra of geometric operators on self-similar CW-complexes. Novikov-Shubin and L2-Betti numbers', preprint math.OA/0607603
2. Guido D., Isola T., Lapidus M.L. 'A trace on fractal graphs and the Ihara zeta functio', preprint math.OA/0608060.
3. Fidaleo, F., Guido, D., Isola, T. 'Bose Einstein condensation on graphs by examples', in preparation.