**RESEARCH IN PAIRS**

**Mixing and Mixing Rates for a Perturbed Infinite Measure Geodesic Flow**

*Mélange et vitesse de mélange pour des flots géodésiques perturbés en mesure infinie***2-13 July 2018**

Lorentz gases are central examples in infinite ergodic theory. Of interest is both the situation where the scatterers form a doubly periodic array or a periodic tube, and perturbed gases where periodicity of scatters is broken in a localised region. A slightly simpler example is that of a perturbed geodesic flow on the Z cover, where the periodic structure is altered by removing one site/point. The geodesic flow on the Z cover can be represented as a suspension flow over the Z-extension of a probability measure preserving system with a regularly varying roof function. Mixing for such Z-extensions is well-understood, but rates of mixing, as well as mixing and rates of mixing for perturbed flows, remains a major challenge. In recent years we have developed a programme for understanding mixing and rates of mixing for general classes of infinite measure preserving semiflows and flows. We expect that our theory is well suited for addressing such questions for the perturbed geodesic flow and perturbed Lorentz gas. The main aim of the project is to develop the theory as needed so that it applies to the perturbed geodesic flows. This is a first step for understanding the more difficult Lorentz gas setting.

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