 98114 Simanyi N., Szasz D.
 Cylindric Billiards and Transitive Lie group Actions
(63K, AMSTeX)
Mar 2, 98

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Abstract. A conjecture is formulated and discussed which provides
necessary and sufficient condition for the ergodicity
of cylindric billiards (this conjecture improves a previous one
of the second author). This condition requires that the action of a
Liesubgroup $\Cal G$ of the orthogonal group $O(d)$ ($d$ being the dimension
of the billiard in question) be transitive on the unit sphere $S^{d1}$. If
$C_1, \dots, C_k$ are the cylindric scatterers of the billiard, then $\Cal G$
is generated by the embedded Liesubgroups $\Cal G_i$ of $O(d)$, where
$\Cal G_i$ consists of all orthogonal transformations of $\Bbb R^d$
that leave the points of the generator subspace of $C_i$ fixed
($1 \le i \le k$). In this paper we
can prove the necessity of our conjecture and we also formulate some
notions related to transitivity. For hard ball systems, we can also show
that the transitivity holds in general: for arbitrary number $N\ge 2$
of balls,
arbitrary masses $m_1, \dots, m_N$ and in arbitrary dimension $\nu \ge 2$.
This result implies that our conjecture is stronger than the BoltzmannSinai
ergodic hypothesis for hard ball systems. As a byremark, we can give a
somewhat surprising characterization of the positive subspace of the
second fundamental form for the evolution of special orthogonal manifold
(wavefront), namely for the parallel beam of light. Thus we obtain
a new characterization of sufficiency of an orbit segment.
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