Exact and K decomposition theorems, with application to Lorentz gases

Vortrag von Prof. Dr. Marco Lenci

Datum: 13.05.26  Zeit: 13.30 - 14.30  Raum: ETH HG G 19.1

Exactness and the K-property (a.k.a. K-mixing) are strong ergodic properties whose significance is that a dynamical system loses all information about its initial conditions, as time flows. These properties were the subject of intense investigation in the second half of the last century, before being somewhat superseded by the stronger Bernoulli property, which was at the time proved for a number of popular systems, including many billiards. But the Bernoulli property can only be formulated for dynamical systems preserving a finite measure, while exactness and the K-property work equally well in infinite-measure contexts, which partly explains a revived interest in them.

I will first present two general theorems on the exact and K decomposition of extensions of dynamical systems. One straightforward application is that, in large generality, a recurrent _periodic_ “Lorentz gas” whose finite-measure factor (Sinai billiard) is hyperbolic is K-mixing. Here the expression “Lorentz gas” is intended in a general sense, including virtually all previously studied 2-dimensional periodic Lorentz gases and d-dimensional periodic Lorentz tubes, as well as many d-dimensional periodic Lorentz slabs. (Joint work with Daniele Galli)

Later, if time permits, I will present a K decomposition theorem specifically tailored to 2-dimensional uniformly hyperbolic billiards in infinite measure. Its main application is the K-property for a general class of planar _aperiodic_ Lorentz gases. (Work in progress with Giovanni Canestrari)