Hyperbolic Basic Set Research Articles

Stable Transitivity of Certain Noncompact Extensions of Hyperbolic Systems

Let ƒ : X → X be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We find several criteria for transitivity of noncompact connected Lie group extensions. As a consequence, we find transitive extensions for any finite-dimensional connected Lie group extension. If, in addition, the group is perfect and has an open set of elements that generate a compact subgroup, we find open sets of stably transitive extensions. In particular, we find stably transitive SL(2, R)-extensions. More generally, we find stably transitive Sp(2n-R)-extensions for all n ≥ 1. For the Euclidean groups SE(n) with n ≥ 4 even, we obtain a new proof of a result of Melbourne and Nicol stating that there is an open and dense set of extensions that are transitive. For groups of the form K × Rn where K is compact, a separation condition is necessary for transitivity. Provided X is a hyperbolic attractor, we show that an open and dense set of extensions satisfying the separation condition are transitive. This generalises a result of Nitica and Pollicott for Rn-extensions.

Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in [formula omitted

We study the dynamics near a symmetric Hopf-zero (also known as saddle-node Hopf or fold-Hopf) bifurcation in a reversible vector field in R 3 , with involutory an reversing symmetry whose fixed point subspace is one-dimensional. We focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddle-foci. We study local perturbations of this degenerate family of heteroclinics within the class of reversible vector fields and establish the generic existence of hyperbolic basic sets (horseshoes), independent of the eigenvalues of the saddle-foci, as well as cascades of bifurcations of periodic, heteroclinic and homoclinic orbits. Finally, we discuss the application of our results to the Michelson system, describing stationary states and travelling waves of the Kuramoto–Sivashinsky PDE.

Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets

We obtain sharp results for the gencricity and stability of transitivity, ergodicity and mixing for compact connected Lie group extensions over a hyperbolic basic set of a C-2 diffeomorphism. In contrast to previous work, our results hold for general hyperbolic basic sets and, are valid in the C-r-topology for all r > 0 (here r need not be an integer and C-1 is replaced by Lipschitz). Moreover, when r >= 2, we show that there is a C-2-open and C-r-dense subset of C-r -extensions that are ergodic. We obtain similar results on stable transitivity for (non-compact) R-m-extensions, thereby generalizing a result of Nitica and Pollicott, and on stable mixing for suspension flows.

A dichotomy for three-dimensional vector fields

We prove that a generic C1 vector field on a closed 3-manifold either has infinitely many sinks or sources or else is singular Axiom A without cycles. Singular Axiom A means that the non-wandering set of the vector field has a decomposition into compact invariant sets, each being either a hyperbolic basic set or a singular hyperbolic attractor (like the Lorenz-like ones) or a singular hyperbolic repeller. An attractor is a transitive set which attracts all nearby future orbits, and a repeller is an attractor for the time-reversed flow. Our result implies that generic C1 vector fields on closed 3-manifolds do exhibit either attractors or repellers.

Stable transitivity of Euclidean group extensions

The topological transitivity of non-compact group extensions of topologically mixing subshifts of finite type has been studied recently by Niţică. We build on these methods, and give the first examples of stably transitive non-compact group extensions of hyperbolic dynamical systems. Our examples include extensions of hyperbolic basic sets by the Euclidean group SE(n) for n even, n\ge4.

Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions

Holder continuous observations on hyperbolic basic sets satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles (approximation by Brownian motion). Using an equivariant version of the Ruelle transfer operator studied by Parry and Pollicott, we obtain similar results for equivariant observations on compact group extensions of hyperbolic basic sets.

Homoclinic tangencies and fractal invariants in arbitrary dimension

For generic families of diffeomorphisms in arbitrary dimension unfolding a homoclinic tangency associated to a hyperbolic basic set, we prove that uniform hyperbolicity prevails in parameter space if and only if the Hausdorff dimension of the basic set is less than 1.

Hölder Continuity of the Holonomy Maps for Hyperbolic Basic Sets II

Hölder Continuity of the Holonomy Maps for Hyperbolic Basic Sets II

Almost flow equivalence for hyperbolic basic sets

Almost flow equivalence for hyperbolic basic sets

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

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.

Structural stability of C1 diffeomorphisms

In this paper we prove that if f is a C 1 diffeomorphism that satisfies Axiom A and the strong transversality condition then it is structurally stable. J. Robbin proved this theorem for C 2 diffeomorphisms. In addition to reducing the amount of differentiability necessary to prove the theorem, we also give a new proof combining the d f metric of Robbin with the stable and unstable manifold proof of D. Anosov. We also prove structural stability in the neighborhood of a single hyperbolic basic set (independent of its being part of a diffeomorphism that satisfies Axiom A and the strong transversality condition). These proofs are adapted to prove the structural stability of C 1 flows in another paper.