Wild hyperbolic sets, yet no chance for the coexistence of infinitely many KLUS-simple Newhouse attracting sets

Abstract

The phenomenon of the coexistence of infinitely many sinks for two dimensional dissipative diffeomorphisms is a result due to Newhouse [Ne1, Ne2]. In fact, for each parameter value at which a homoclinic tangency is formed nondegenerately, there exist intervals in the parameter space containing dense sets of parameter values for which there are infinitely many coexisting sinks (Robinson [R]). The structure of the sinks constructed by Newhouse is limited. “Simple” Newhouse parameter values are values at which there are infinitely many sinks having some special well defined property concerning the structure. A result due to Tedeschini-Lalli and Yorke [TY] says that the Lebesgue measure of the set of simple Newhouse parameter values is zero when the tangencies are due to the standard “affine” horseshoe map. It is argued in [TY] and [PR] that a more general derivation of this measure zero result would be desirable. The main result of this paper is that the Lebesgue measure of the set of KLUS-simple parameter values (including the simple Newhouse parameter values) is zero for saddle hyperbolic basic sets forming tangencies.