## Institute for Mathematical Sciences

## Preprint ims97-4

** Andre de Carvalho**
* Pruning fronts and the formation of horseshoes*

Abstract: Let f:E -> E be a homeomorphism of the plane E. We define open sets P, called {\em pruning fronts} after the work of
Cvitanovi\'c, 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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