Hyperbolic Sets Research Articles

The Jang Equation and the Positive Mass Theorem in the Asymptotically Hyperbolic Setting

We solve the Jang equation with respect to asymptotically hyperbolic “hyperboloidal” initial data. The results are applied to give a non-spinor proof of the positive mass theorem in the asymptotically hyperbolic setting. This work focuses on the case when the spatial dimension is equal to three.

Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra II

AbstractLet $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let $\Lambda _0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of $g_0$ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let $L_{g,\Lambda ,f}$ (respectively $M_{g,\Lambda ,f}$ ) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively highest) values of f along the geodesics in the hyperbolic continuation $\Lambda $ of $\Lambda _0$ . We prove that for generic choices of g and f, the Hausdorff dimensions of the sets $L_{g,\Lambda , f}\cap (-\infty , t)$ vary continuously with $t\in \mathbb {R}$ and, moreover, $M_{g,\Lambda , f}\cap (-\infty , t)$ has the same Hausdorff dimension as $L_{g,\Lambda , f}\cap (-\infty , t)$ for all $t\in \mathbb {R}$ .

Blenders near polynomial product maps of $\mathbb{C}^2$

In this paper we show that if p is a polynomial of degree d\geq2 possessing a neutral periodic point then a product map of the form (z,w)\mapsto(p(z),q(w)) can be approximated by polynomial skew products (z,w)\mapsto(\tilde{p}(z,w),q(w)) possessing special dynamical objects called blenders. Moreover, these objects can be chosen to be of two types: repelling or saddle. As a consequence, such a product map belongs to the closure of the interior of two different sets: the bifurcation locus of the space of holomorphic endomorphisms of degree d of \mathbb{P}^2 and the set of endomorphisms having an attracting set of non-empty interior. Similar techniques also give the first example of an attractor with non-empty interior or of a saddle hyperbolic set which is robustly contained in the small Julia set and whose unstable manifolds are all dense in \mathbb{P}^2 . In an independent part, we use perturbations of Hénon maps to obtain examples of attracting sets with repelling points and also of quasi-attractors which are not attracting sets.

Generalized Lorenz Canonical Form Revisited

This paper completes the description of the generalized Lorenz system (GLS) and hyperbolic generalized Lorenz system (HGLS) along with their canonical forms (GLCF, HGLCF), mostly presented earlier, by deriving explicit state transformation formulas to prove the equivalence between GLS and GLCF, as well as between HGLS and HGLCF. Consequently, complete formulations of the generalized Lorenz canonical systems and forms, and their hyperbolic settings, are obtained and presented. Only potentially chaotic systems are classified, which significantly helps clarify the respective canonical forms. To do so, some tools for systems to exclude chaotic behavior are developed, which are interesting in their own right for general dynamical systems theory. The new insight may inspire future investigations of generalized and canonical formulations of some other types of chaotic systems.

Invariance of entropy for maps isotopic to Anosov * *CHV was partially supported by Proyecto Fondecyt 1171427.

We prove the topological entropy remains constant inside the class of partially hyperbolic diffeomorphisms of with simple central bundle (that is, when it decomposes into one dimensional sub-bundles with controlled geometry) and such that their induced action on is hyperbolic. In absence of the simplicity condition we construct a robustly transitive counter-example.

Dynamical Systems and Topology of Magnetic Fields in Conducting Medium

We discuss application of contemporary methods of the theory of dynamical systems with regular and chaotic hyperbolic dynamics to investigation of topological structure of magnetic fields in conducting media. For substantial classes of magnetic fields, we consider well-known physical models allowing us to reduce investigation of such fields to study of vector fields and Morse–Smale diffeomorphisms as well as diffeomorphisms with nontrivial basic sets satisfying the A axiom introduced by Smale. For the point–charge magnetic field model, we consider the problem of the separator playing an important role in the reconnection processes and investigate relations between its singularities. We consider the class of magnetic fields in the solar corona and solve the problem of topological equivalency of fields in this class. We develop a topological modification of the Zeldovich funicular model of the nondissipative cinematic dynamo, constructing a hyperbolic diffeomorphism with chaotic dynamics that is conservative in the neighborhood of its transitive invariant set.

New examples of stably ergodic diffeomorphisms in dimension 3

We prove that in the isotopy class of any volume preserving partially hyperbolic diffeomorphism in a three-dimensional manifold, there is a non-partially hyperbolic stably ergodic diffeomorphism. In particular, we provide new examples of stably ergodic diffeomorphisms in three-dimensional manifolds with respect to a smooth volume measure.

The existence of semi-horseshoes for partially hyperbolic diffeomorphisms

A topological dynamical system is said to have a semi-horseshoe if there exists a subsystem of its iterates which is topologically semi-conjugate to the full shift. It is shown that several classes of dynamical systems, including partially hyperbolic diffeomorphisms and automorphisms of compact metric abelian groups with positive topological entropy, have semi-horseshoes.

C1 density of stable ergodicity

We prove a C1 version of a conjecture by Pugh and Shub: among partially hyperbolic volume-preserving Cr diffeomorphisms, r>1, the stably ergodic ones are C1-dense.To establish these results, we develop new perturbation tools for the C1 topology: linearization of horseshoes while preserving entropy, and creation of “superblenders” from hyperbolic sets with large entropy.

A central limit theorem for periodic orbits of hyperbolic flows

We consider a counting problem in the setting of hyperbolic dynamics. Let be a weak-mixing hyperbolic flow. We count the proportion of prime periodic orbits of , with length less than T, that satisfy an averaging condition related to a Hölder continuous function . We show, assuming an approximability condition on ϕ, that as , we obtain a central limit theorem. The proof uses transfer operator estimates due to Dolgopyat to provide the bounds on complex functions that we need to carry out our analysis. We can then use contour integration to obtain the asymptotic behaviour which gives the central limit theorem.

Hyers–Ulam stability for hyperbolic random dynamics

We prove that small nonlinear perturbations of random linear dynamics admitting a tempered exponential dichotomy have a random version of the shadowing property. As a consequence, if the exponential dichotomy is uniform, we get that the random linear dynamics is Hyers-Ulam stable. Moreover, we apply our results to study the conservation of Lyapunov exponents of the random linear dynamics subjected to nonlinear perturbations.

Centralizers of partially hyperbolic diffeomorphisms in dimension 3

In this note we describe centralizers of volume preserving partially hyperbolic diffeomorphisms which are homotopic to identity on Seifert fibered and hyperbolic 3-manifolds. Our proof follows the strategy of Damjanovic, Wilkinson and Xu [ 10 ] who recently classified the centralizer for perturbations of time- \begin{document}$ 1 $\end{document} maps of geodesic flows in negative curvature. We strongly rely on recent classification results in dimension 3 established in [ 5 , 6 ].

On local structure of one-dimensional basic sets of non-reversible A-endomorphisms of surfaces

Recently the authors of the article discovered a meaningful class of non-reversible endomorphisms on a two-dimensional torus. A remarkable property of these endomorphisms is that their non-wandering sets contain nontrivial one-dimensional strictly invariant hyperbolic basic sets (in the terminology of S. Smale and F. Pshetitsky) which have the uniqueness of an unstable one-dimensional bundle. It was proved that nontrivial (other than periodic isolated orbits) invariant sets can only be repellers. Note that this is not the case for reversible endomorphisms (diffeomorphisms). In the present paper, it is proved that one-dimensional expanding uniquely hyperbolic and strictly invariant one-dimensional expanding attractors and one-dimensional contracting repellers of non-reversible A-endomorphisms of closed orientable surfaces have the local structure of the product of an interval by a zero-dimensional closed set (finite or Cantor). This result contrasts with the existence of one-dimensional fractal repellers arising in complex dynamics on the Riemannian sphere and not possessing the properties of the existence of a single one-dimensional unstable bundle.

Homoclinic tangency and variation of entropy

In this paper we study the e¤ect of a homoclinic tangency in the variation of the topological entropy. We prove that a diffeomorphism with a homoclinic tangency associated to a basic hyperbolic set with maximal entropy is a point of entropy variation in the C^{\infty} -topology. We also prove results about variation of entropy in other topologies and when the tangency does not correspond to a basic set with maximal entropy. We also show an example of discontinuity of the entropy among C^{\infty} diffeomorphisms of three dimensional manifolds.

Quasiconformality and hyperbolic skew

AbstractWe prove that if $f:\mathbb{B}^n \to \mathbb{B}^n$ , for n ≥ 2, is a homeomorphism with bounded skew over all equilateral hyperbolic triangles, then f is in fact quasiconformal. Conversely, we show that if $f:\mathbb{B}^n \to \mathbb{B}^n$ is quasiconformal then f is η-quasisymmetric in the hyperbolic metric, where η depends only on n and K. We obtain the same result for hyperbolic n-manifolds. Analogous results in $\mathbb{R}^n$ , and metric spaces that behave like $\mathbb{R}^n$ , are known, but as far as we are aware, these are the first such results in the hyperbolic setting, which is the natural metric to use on $\mathbb{B}^n$ .

Accessibility of derived-from-Anosov systems

This paper shows any non-Anosov partially hyperbolic diffeomorphism on the 3-torus which is homotopic to Anosov must be accessible.

On existence of Morse energy function for topological flows

It is a well known fact that any smooth manifold admits a Morse function, whereas the problem of existence of a Morse function for a topological manifold stated by Marston Morse in 1959 ([6]) is still open. In the present paper we prove that a topological manifold admits a continuous Morse function if it admits a topological flow with a finite hyperbolic chain recurrent set. We construct this function as a Lyapunov function whose set of the critical points coincides with the chain recurrent set of the flow.

Partially volume expanding diffeomorphisms

We call a partially hyperbolic diffeomorphism \emph{partially volume expanding} if the Jacobian restricted to any hyperplane that contains the unstable bundle $E^u$ is larger than $1$. This is a $C^1$ open property. We show that any $C^{1+}$ partially volume expanding diffeomorphisms admits finitely many physical measures, the union of whose basins has full volume.

A landing theorem for entire functions with bounded post-singular sets

The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function f with bounded postsingular set. If f has finite order of growth, then it is known that the escaping set I(f) contains certain curves called periodic hairs; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function f of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected subsets of I(f), called dreadlocks. We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.

Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence

Let $M$ be a closed 3-manifold which admits an Anosov flow. In this paper we develop a technique for constructing partially hyperbolic representatives in many mapping classes of $M$. We apply this technique both in the setting of geodesic flows on closed hyperbolic surfaces and for Anosov flows which admit transverse tori. We emphasize the similarity of both constructions through the concept of $h$-transversality, a tool which allows us to compose different mapping classes while retaining partial hyperbolicity. In the case of the geodesic flow of a closed hyperbolic surface $S$ we build stably ergodic, partially hyperbolic diffeomorphisms whose mapping classes form a subgroup of the mapping class group $\mathcal{M}(T^1S)$ which is isomorphic to $\mathcal{M}(S)$. At the same time we show that the totality of mapping classes which can be realized by partially hyperbolic diffeomorphisms does not form a subgroup of $\mathcal{M}(T^1S)$. Finally, some of the examples on $T^1S$ are absolutely partially hyperbolic, stably ergodic and robustly non-dynamically coherent, disproving a conjecture by F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures.