Pruning fronts and the formation of horseshoes

Let f:E -> E be a homeomorphism of the plane E. We define open sets P, called {\em pruning fronts} after the work of Cvitanović, for which it is possible to construct an isotopy H: E x [0,1] -> E with open support contained in the union of f^{n}(P), such that H(*,0)=f(*) and H(*,1)=f_P(*), where f_P is a homeomorphism under which every point of P is wandering. Applying this construction with f being Smale's horseshoe, it is possible to obtain an uncountable family of homeomorphisms, depending on infinitely many parameters, going from trivial to chaotic dynamic behaviour. This family is a 2-dimensional analog of a 1-dimensional universal family.

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