R2 (with smoothness the same as that of the flow) such that Sp(m) = (0,0) E R2 and Sp(> n U) = {0} x [-1, 1] C R2 (we remark that Sp carries the trajectories of f t in U into the lines y = const). A simple closed curve C (the image of an imbedding of a circle in M) is called a contact free cycle or a closed transversal of the flow if its arcs are contact-free segments.

1. 29 The condition of geometric continuity is an analogue of the theorem on continuous dependence on the initial conditions. 22 1. DYNAMICAL SYSTEMS ON SURFACES Let F be a family of disjoint curves on M. For a curve l in F and for points m1i m2 E l denote by mim2 the arc of l between m1 and m2, oriented from m1 to m2 We say that the family F satisfies the condition of geometric continuity if for any > 0, any curve l E F, and any arc m1'm2 C l there is a number 8> 0 such N w that if d(m1, m1) < S, then there exists an arc rn1rn2 of a curve l E F passing through m1 for which rn1rn2 is in the E-neighborhood of m m2 and d(m2, m2)

### Introduction to the Qualitative Theory of Dynamical Systems on Surfaces by G. R. Belitsky, and E. V. Zhuzhoma S. Kh. Aranson

by Jason

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