| 概要 | In their pioneer works, Sinai, Bowen, and Ruelle gave a complete
description of the thermodynamic formalism of uniformly hyperbolic
diffeomorphisms and Holder continuous potentials. In this talk,
I'll report on recent progress in real and complex dimension 1,
where a complete picture is emerging. For simplicity the talk will be
restricted to geometric potentials and the quadratic family, but most
results apply in greater generality. First goal is to describe the
(non-)existence of equilibrium states, their statistical properties,
and the real analytic properties of the geometric pressure function.
The second goal is to describe phase transitions: The phenomenon of
lack of real analyticity. After classifying and describing the
mechanisms that produce phase transitions, the focus will on the
various surprising phenomena that occur at criticality, some of which
illustrate the universality of the quadratic family. |