## Institute for Mathematical Sciences

## Preprint ims94-18

** N. I. Chernov & S. Troubetzkoy**
* Measures with Infinite Lyapunov Exponents for the Periodic Lorentz Gas*

Abstract: In \cite{Ch91a} it was shown that the billiard ball map for the periodic Lorentz gas has infinite topological entropy. In this
article we study the set of points with infinite Lyapunov
exponents. Using the cell structure developed in
\cite{BSC90,Ku} we construct an ergodic invariant probability
measure with infinite topological entropy supported on this
set. Since the topological entropy is infinite this is a
measure of maximal entropy. From the construction it is clear
that there many such measures can coexist on a single component
of topological transitivity. We also construct an ergodic
invariant probability measure with finite entropy which is
supported on this set showing that infinite exponents do not
necessarily lead to infinite entropy.

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