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By M. Aizenman (Chief Editor)

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4. 3. Stability of Generalized Two-Periodic Trajectories Let Q be a billiard table on M(r, β). For each v ∈ V let t (v) be the corresponding past semitrajectory in Q. Consider the curvature evolution of an infinitesimal beam along t (v). Starting with B(φ −k · v, χ ) for arbitrary χ , we obtain after k steps forward the infinitesimal beam B(v, χ (k) ), χ (k) = φ k · χ . Eqs. (9)–(11) and (12) describe the action of the billiard map on the curvature of infinitesimal beams. Assuming k to be infinity, we obtain a formal continued fraction corresponding to the semitrajectory t (v): c(v) = χ (∞) = a0 + b0 a−1 + b−1 a−2 · · · .

Math. Phys. : On absolutely focusing mirrors. Lecture Notes in Math. Vol. 1514, 1991, pp. : Commun. Math. Phys. : Commun. Math. Phys. : Phys. Rep. : Commun. Math. Phys. : Russian Mathem. : Commun. Math. Phys. : J. Math. Phys. : J. Math. Phys. : Sinai billiard in potential field (constraction of stable and unstable fibers). Coll. Math. Soc. J. : Sinai billiard in potential field (absolute continuity). In Proc. 3rd Pann. Symp. J. Mogyorody, I. Vincze, W. ). : Geometry 2, Encycl. of Math. Sc. Vol.

Let α(l) be the angle, which dγ dl makes with x-axis, then dx = cos α; dl dy = sin α; dl dα = −κ. dl (28) We introduce also an auxiliary variable δ, such that βx = sin δ. For β > 0 there are two different particle trajectories connecting the points γ (l0 ) and γ (l1 ) (resp. s corresponding to these points), see Fig. 12. Below, we consider the trajectory which lies in the lower halfplane. Then, the results for trajectory in the upper halfplane are obtained by the change of the sign of β to the opposite.

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Communications in Mathematical Physics - Volume 217 by M. Aizenman (Chief Editor)


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