Persistence of Heterodimensional Cycles

Abstract

A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least C 2 , we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family always create robust heterodimensional dynamics, i.e., chain-transitive sets which contain coexisting orbits with different numbers of positive Lyapunov exponents and persist for an open set of parameter values. In particular, we solve the so-called C r -stabilization problem for the coindex-1 heterodimensional cycles in any regularity class r = 2, . . . , ∞, ω. The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders.

Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies

“…The heterodimensional cycles obtained in each case of Theorem 1 can be made C 1 -robust by a C r -small perturbation, using a recently established C r -stabilization theory for heterodimensional cycles in [19]. The stabilization process is essentially based on the emergence of Bonatti-Díaz blenders discovered in [8].…”

“…By definition, the existence of either of them prevents a system to be uniformly hyperbolic. Moreover, they bifurcate to be robust (for homoclinic tangencies, see [23,25], for heterodimensional cycles, see [8,9] for C 1 case, and [19] for C r case). They exhibit behaviors which are not seizable among uniformly hyperbolic systems such as presence of zero Lyapunov exponents [15], super-exponential growth of number of periodic points [2,3,5,16], coexistence of infinitely many sinks [12,14,22].…”

“…, ∞, ω) diffeomorphism, heterodimensional cycles can be obtained by arbitrarily C r -small perturbations from generic homoclinic tangencies whose local dynamics do not admit low-dimensional reductions, namely, those associated to a hyperbolic periodic point with complex central multipliers (see Theorem 1). Using the results in [19], the found heterodimensional cycles can be made C 1 -robust and such that the two hyperbolic sets involved are blenders (hyperbolic sets on which non-transverse intersections are robust, see Definition 2). With a further application of the results in [10], we show that one of the blenders also has a C 1 -robust homoclinic tangency (see Corollary 1).…”

Homoclinic tangencies leading to robust heterodimensional cycles

“…Currently, there are versions (as the ones here) where the central dimensions are greater than one. Applications of blenders of central dimension one include: robustly transitive dynamics [19], robust heterodimensional cycles [18,33], robust homoclinic tangencies [20], stable ergodicity [46], and construction of nonhyperbolic ergodic measures [14], among others. Blenders with larger central dimensions were introduced in [37,8] to study instability problems in symplectic dynamics, in [6] to obtain robust heterodimensional cycles of large coindex, and in [7,2] to get robust tangencies of large codimension.…”

“…(i.e., the absolute value of the difference of the u-indices of the hyperbolic sets in the cycle) one case, see [18] for the C 1 case and [33] for the C r case, r 2. The case of heterodimensional cycles of higher coindex is widely open.…”