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On geometrical properties of two-dimensional diffeomorphisms with homoclinic tangencies. (English) Zbl 0885.58062

Summary: Two-dimensional diffeomorphisms with a quadratic tangency of invariant manifolds of a saddle fixed point are considered in the cases where the saddle value \(\sigma\) is either less than 1 or equal to it. A description of the structure of hyperbolic subsets is given. In the case \(\sigma=1\), it is shown that almost all such diffeomorphisms admit the complete description in distinction with the case \(\sigma<1\).

MSC:

37D99 Dynamical systems with hyperbolic behavior
37G99 Local and nonlocal bifurcation theory for dynamical systems
37C75 Stability theory for smooth dynamical systems
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