99-139 J. C. Tatjer

Three dimensional dissipative diffeomorphisms with homoclinic tangencies (757K, postscript) May 4, 99

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Abstract. Given a two-parameter family of three-dimensional diffeomorphism $\{ f_{a,b}\}_{a,b},$ with a dissipative (but not sectionally dissipative) saddle fixed point, we prove that the existence of a certain type of homoclinic tangency of the invariant manifolds, implies that there exists a return map $f^n_{a,b},$ near the homoclinic orbit, for values of the parameter near such tangency and for $n$ greater enough, such that after a change of variables and reparametrization depending on $n,$ this return map tends to a simple quadratic map. This implies the existence of strange attractors and infinitely many sinks as in other known cases. Moreover, there appear attracting invariant circles, implying the existence of quasiperiodic behaviour near the homoclinic tangency.

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