Exponential Decay of Correlations for Families of Suspension Flows over Non-Compact Spaces

The rate of decay of correlations for a mixing dynamical system is a topic of wide interest in Ergodic Theory, as well as an indicator of the chaotic nature of the system in terms of the unpredictability of its long-term behavior. In the PhD research project, we consider two different families of mixing dynamical systems, and we show that they exhibit an exponential rate of decay of correlations, exploiting a well-established method described in an article by D. Dolgopyat. The first family is a generalization of a well-known system, the geodesic flow on the modular surface, for which exponential mixing had already been established, but with a proof of algebraic nature. The first part of the thesis, therefore, provides an alternative proof, which we obtain with classical techniques from the Theory of Dynamical Systems and Ergodic Theory. The second family of systems which we study concerns suspension flows over Markov subshifts with an infinite alphabet. Under suitable properties outlined in the research project, we establish exponential decay of correlation. Both families act on non-compact phase spaces, while many similar results in the literature concern compact spaces.