Random perturbations of codimension one homoclinic tangencies in dimension 3

Details Random perturbations of codimension one homoclinic tangencies in dimension 3 Random perturbations of codimension one homoclinic tangencies in dimension 3

2002-04-22

arxiv.org/abs/math/0204266v1

Dynamical Systems, An International Journal, Vol. 18, No. 1 (2003), 35-55.

Mathematics

Dynamical Systems

22 pages; 5 figures

Scientific paper

Adding small random parametric noise to an arc of diffeomophisms of a manifold of dimension 3, generically unfolding a codimension one quadratic homoclinic tangency q associated to a sectionally dissipative saddle fixed point p, we obtain not more than a finite number of physical probability measures, whose ergodic basins cover the orbits which are recurrent to a neighborhood of the tangency point $q$. This result is in contrast to the extension of Newhouse's phenomenon of coexistence of infinitely many sinks obtained by Palis and Viana in this setting. There is a similar result for the simpler bidimensional case whose proof relies on geometric arguments. We now extend the arguments to cover three dimensional manifolds.

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