Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies

Abstract

We consider C r (r = 3, . . . , ∞, ω) diffeomorphisms with a generic homoclinic tangency to a hyperbolic periodic point, where this point has at least one complex (non-real) central multiplier, and some explicit conditions are satisfied so that the dynamics near the homoclinic tangency is not effectively one-dimensional. We prove that C 1 -robust heterodimensional cycles of co-index one appear in any generic two-parameter C r -unfolding of such a tangency. These heterodimensional cycles also have C 1 -robust homoclinic tangencies.

Persistence of heterodimensional cycles

“…In support of such claim, we have shown in a series of papers [40,[42][43][44] that heterodimensional cycles emerge due to several types of homoclinic bifurcations. In fact, in the spirit of [37,64,65], one can conjecture that coindex-1 heterodimensional cycles can appear, with very few exceptions, in any homoclinic/heteroclinic bifurcation whose effective dimension allows it, i.e., when the dynamics of the map under consideration are not reduced to a two-dimensional invariant manifold and the map is not sectionally-dissipative (i.e., not area-contracting).…”

“…in [44] where we show the C 1 -robustness of homoclinic tangencies near homoclinic tangencies of effective dimension larger than one.…”

“…Without the contraction of areas, one has coexisting sets of sinks, saddles, and sources [32], and, in the multi-dimensional case, coexisting saddles of different indices [34,56]. In [44], we use the results of the present paper to give conditions under which the saddles of different indices that are born out of a homoclinic tangency get involved into the C 1 -robust heterodimensional dynamics, as in Definition 1.3.…”

“…5 Indeed, Ures [68] discovered that for C 1 surface diffeomorphisms, the thickness of a horseshoe loses continuous dependence on the system, which is a crucial condition for the Newhouse construction; and later Moreira [47] built a more general theory and proved the non-existence of persistent homoclinic tangencies for C 1 surface diffeomorphisms. However, there are several examples of C 1 -persistent homoclinic tangencies in higher dimensions, see [4,15,44,63], which use blenders, or their variants, as supporting structures.…”

“…It should be noted that the blenders obtained in this paper have additional dynamical properties (partial hyperbolicity, the existence of a special Markov partition) which are not included in Definition 1.9. This additional structure is important for the actual construction of blenders and it is also crutial in applications, for example for proving the C 1 -persistence of a certain type of homoclinic tangencies [16,44]. We describe such blenders in Definition A.1, and call them standard.…”