## Goals

The project aims to develop new tools in ergodic theory and dynamical
systems, and explore applications to problems related
to mathematical physics, geometry and arithmetics. The
first general objective is to advance large deviation
theory for non-compact dynamical systems. We plan to
deduce new subexponential large deviation bounds for
Gibbs measures on the countable Markov shift and
explore how these results are linked to Manneville-Pomeau dynamics describing
intermittence in the theory of turbulent flows,
dynamical properties of the Gauss map, which is deeply
connected to Diophantine approximation, and
homogeneous dynamics such as the Teichmüller flow on
translation surfaces. The second general objective is
to investigate Host-type measure rigidity theory for
toral automorphisms and homogeneous dynamics. This
topic relates to currently ongoing research on measure
classification theorems, which have been influential
in several applications such Diophantine approximation
and quantum ergodicity. The main tools used in these
works are additive combinatorics, thermodynamical
formalism, and the spectral/ergodic theory of the scenery flow.