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, \inv_b°\inv_a] Certain theorems on a.e. periodicity of one-parameter families of homeomorphisms of the circle with singularities of break type (jump in derivative) are proven. They find application to the following dynamical system suggested by V.I. Arnold. Consider a convex closed set S. Suppose that its boundary ¶S is a curve of length 1. An involution \inv_a of this curve with respect to the line ( direction) a is defined as follows. Consider all translations of line a. Since the set S is convex, any translation of a meets ¶S at none, one or two points. In the last case we exchange these two points. For any two directions a and b the convolution \inv_b°\inv_a of corresponding involutions defines an orientation-preserving homeomorphism of the boundary ¶S. Suppose that ¶S is smooth everywhere except at one point C, where we have a corner (see Figure 1). Let us also choose a and b passing through the point C.

Suppose now that ¶S is parameterized by the arc length parameter: x=x(l), y=y(l), where l Î [0,1), so that 0 corresponds to a singular point C. Suppose that one of the following assumptions holds: i) x(l),y(l) Î C³([0,1]) and all tangent lines to ¶S have non-degenerous tangency, ii) x(l),y(l) Î C^¥([0,1]) and there are no flat points.

Also assume that the curve (x(l),y(l)) has finite one-sided curvatures which are k₊ > 0 at l=0 and k_- > 0 at l=1. Notice that the last condition is obviously satisfied in the case i). The number of (break-type) singularities for the homeomorphism \inv_b\inv_a obtained depends on the choice of a and b.

Theorem. Directions a and b are defined by angles z₁ an z₂ Î [0,p) with two tangent lines at point C resp. The Lebesgues measure of the set

Z={(z1,z2) Î [0,p)×[0,p) | \invb°\inva has no periodic trajectories} is 0.

This theorem is a corollary of a more general result on one-parameter families of homeomorphisms of the circle with two break-type singularities. Similar result holds for several breaks if one break is ``larger'' in a sense than the others.

Footnotes:

¹ Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 3G3

E-mail: dkhmelev(at)math dot toronto dot edu