Hyperbolic Sets Research Articles

Special homogeneous surfaces

Abstract We classify hyperbolic polynomials in two real variables that admit a transitive action on some component of their hyperbolic level sets. Such surfaces are called special homogeneous surfaces, and they are equipped with a natural Riemannian metric obtained by restricting the negative Hessian of their defining polynomial. Independent of the degree of the polynomials, there exist a finite number of special homogeneous surfaces. They are either flat, or have constant negative curvature.

Rigidity of Lyapunov exponents for derived from Anosov diffeomorphisms

Abstract For a class of volume-preserving partially hyperbolic diffeomorphisms (or non-uniformly Anosov) $f\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ homotopic to linear Anosov automorphism, we show that the sum of the positive (negative) Lyapunov exponents of f is bounded above (respectively below) by the sum of the positive (respectively negative) Lyapunov exponents of its linearization. We show this for some classes of derived from Anosov (DA) and non-uniformly hyperbolic systems with dominated splitting, in particular for examples described by Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math.115(1) (2000), 157–193]. The results in this paper address a flexibility program by Bochi, Katok and Rodriguez Hertz [Flexibility of Lyapunov exponents. Ergod. Th. & Dynam. Sys.42(2) (2022), 554–591].

Symbolic dynamics for pointwise hyperbolic systems on open regions

Abstract Under certain conditions, we construct a countable Markov partition for pointwise hyperbolic diffeomorphism $f:M\rightarrow M$ on an open invariant subset $O\subset M$ , which allows the Lyapunov exponents to be zero. From this partition, we define a symbolic extension that is finite-to-one and onto a subset of O that carries the same finite f-invariant measures as O. Our method relies upon shadowing theory of a recurrent-pointwise-pseudo-orbit that we introduce. As a canonical application, we estimate the number of closed orbits for f.

On non-trivial hyperbolic sets and their bifurcations in families of diffeomorphisms of a two-dimensional torus.

We propose a simple model-two-parameter family of diffeomorphisms of a two-dimensional torus. Combining analytical and numerical methods, we find regions in the parameter plane such that each diffeomorphism of the family is hyperbolic and describe the structure of the corresponding hyperbolic sets. We also study bifurcations on the boundaries of these regions, which lead to the change of hyperbolicity type (from Anosov diffeomorphisms to DA-diffeomorphisms).

From Anosov Closing Lemma to Global Data of Cohomological Nature

For diffeomorphisms with hyperbolic sets, the Anosov Closing Lemma ensures the existence of periodic orbits in the neighbourhood of orbits that return close enough to themselves. Moreover, it defines how is controlled the distance between corresponding points of an initial orbit and the constructed periodic orbits. In the essential, this article presents proof of the estimate of this distance. The Anosov Closing Lemma is crucial in the statement of Livschitz Theorem that, based only on the periodic data, provides a necessary and sufficient condition so that cohomological equations have sufficiently regular solutions, Hölder solutions. It is one of the main tools to obtain global data of cohomological nature based only on periodic data. As suggested by Katok and Hasselblat in \cite{KH}, it is demonstrated, in detail and in the cohomology context, the Livschitz Theorem for hyperbolic diffeomorphisms, where the mentioned distance control inequality is essential.

Automorphisms of ℂ2 with cycles of escaping Fatou components with hyperbolic limit sets

Automorphisms of ℂ² with cycles of escaping Fatou components with hyperbolic limit sets

Diffeomorphisms with infinitely many Smale horseshoes

ABSTRACT A class of planar diffeomorphims is formulated, with infinitely many coexisting Smale horseshoes, where the Lebesgue measure of the parameters with such strange dynamics is infinite. On each horseshoe, there exists a uniformly hyperbolic invariant set, on which the map is topologically conjugate to the two-sided full-shift on two symbols. Moreover, the topological entropy is infinite in certain parameter regions.

Dynamical coherence in isotopy classes of fibered lifted partially hyperbolic diffeomorphisms

Dynamical coherence in isotopy classes of fibered lifted partially hyperbolic diffeomorphisms

Physical measures for partially hyperbolic diffeomorphisms with mixed hyperbolicity * *Y Cao is partially supported by National Key R&D Program of China(2022YFA1005802) and NSFC 12371194. Z Mi is partially supported by National Key R&D Program of China(2022YFA1007800) and NSFC 12271260. J Yang was partially supported by CNPq, FAPERJ, PRONEX and NNSFC Grants 12271538, 12071202 and 12161141002.

We study the partially hyperbolic diffeomorphisms whose centre direction admits the u-definite property in the sense that all the central Lyapunov exponents of each ergodic Gibbs u-state are either all positive or all negative. We prove that for this kind of partially hyperbolic diffeomorphisms, there are finitely many physical measures, whose basins cover a full Lebesgue measure subset of the ambient space. We also provide examples of diffeomorphisms whose centres are u-definite, where the supports of different ergodic Gibbs u-states have nontrivial intersections.

Escaping Fatou components with disjoint hyperbolic limit sets

We construct automorphisms of C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {C}}}^2$$\\end{document} of constant Jacobian with a cycle of escaping Fatou components, on which there are exactly two limit functions, both of rank 1. On each such Fatou component, the limit sets for these limit functions are two disjoint hyperbolic subsets of the line at infinity. In the literature there are currently very few examples of automorphisms of C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {C}}}^2$$\\end{document} with rank one limit sets on the boundary of Fatou components. To our knowledge, this is the first example in which such limit sets are hyperbolic, and moreover different limit sets of rank 1 coexist.

Review of Research on the Qualitative Theory of Differential Equations at St. Petersburg University. I. Stable Periodic Points of Diffeomorphisms with Homoclinic Points and Systems with Weakly Hyperbolic Invariant Sets

Review of Research on the Qualitative Theory of Differential Equations at St. Petersburg University. I. Stable Periodic Points of Diffeomorphisms with Homoclinic Points and Systems with Weakly Hyperbolic Invariant Sets

Regularity of the Hausdorff dimension of some hyperbolic systems

We study the regularity of the Hausdorff dimension for certain hyperbolic basic sets, as a function of the dynamical system. The main results ensure that the Hausdorff dimension of certain multidimensional thin hyperbolic attractors and of certain nonconformal iterated function systems varies smoothly and analytically with respect to the diffeomorphism. Similar results hold for average conformal hyperbolic sets and average conformal repellers.

Stability estimates for the holonomy inverse problem

On a Riemannian manifold (M, g) with Anosov geodesic flow, the problem of recovering a connection from the knowledge of traces of its holonomies along primitive closed geodesics is known as the holonomy inverse problem. In this paper, we prove Hölder type stability estimates for this inverse problem: locally, near generic connections; globally, for line bundles, and for vector bundles satisfying a certain low-rank assumption over negatively curved base (M, g). The proofs are based on a combination of microlocal analysis along with a new non-Abelian approximate Livšic Theorem in hyperbolic dynamics.

On invariant holonomies between centers

Abstract We prove that for $C^{1+\theta }$ , $\theta $ -bunched, dynamically coherent partially hyperbolic diffeomorphisms, the stable and unstable holonomies between center leaves are $C^1$ , and the derivative depends continuously on the points and on the map. Also for $C^{1+\theta }$ , $\theta $ -bunched partially hyperbolic diffeomorphisms, the derivative cocycle restricted to the center bundle has invariant continuous holonomies which depend continuously on the map. This generalizes previous results by Pugh, Shub, and Wilkinson; Burns and Wilkinson; Brown; Obata; Avila, Santamaria, and Viana; and Marin.

An upper bound for the box dimension of the hyperbolic dynamics via unstable topological pressure

In this paper, we utilize the sub-additive unstable topological pressure to give an upper bound for the upper box dimension of the C1 hyperbolic set on local unstable manifold. As a by-product, we give a new expression of the topological pressure on symbolic spaces. This work is inspired by Barreira (1996), Feng and Simon (2023) and Zhang et al. (2022).

Numerical Evidence of Hyperbolic Dynamics and Coding of Solutions for Duffing-Type Equations with Periodic Coefficients

Numerical Evidence of Hyperbolic Dynamics and Coding of Solutions for Duffing-Type Equations with Periodic Coefficients

Transitive Centralizer and Fibered Partially Hyperbolic Systems

Abstract We prove several rigidity results about the centralizer of a smooth diffeomorphism, concentrating on two families of examples: diffeomorphisms with transitive centralizer, and perturbations of isometric extensions of Anosov diffeomorphisms of nilmanifolds. We classify all smooth diffeomorphisms with transitive centralizer: they are exactly the maps that preserve a principal fiber bundle structure, acting minimally on the fibers and trivially on the base. We also show that for any smooth, accessible isometric extension $f_{0}\colon M\to M$ of an Anosov diffeomorphism of a nilmanifold, subject to a spectral bunching condition, any $f\in \textrm{Diff}^{\infty }(M)$ sufficiently $C^{1}$-close to $f_{0}$ has centralizer a Lie group. If the dimension of this Lie group equals the dimension of the fiber, then $f$ is a principal fiber bundle morphism covering an Anosov diffeomorphism. Using the results of this paper, we classify the centralizer of any partially hyperbolic diffeomorphism of a $3$-dimensional, nontoral nilmanifold: either the centralizer is virtually trivial, or the diffeomorphism is an isometric extension of an Anosov diffeomorphism, and the centralizer is virtually ${{\mathbb{Z}}}\times{{\mathbb{T}}}$.

Effects of particle creation rate in two-fluid interacting cosmologies

ABSTRACT In this work, a two-fluid interacting model in a flat FLRW universe has been studied considering particle creation mechanism with a particular form of particle creation rate $\Gamma =\Gamma _0 H+\frac{\Gamma _1}{H}$ from different aspects. Statistical analysis with a combined data set of SNe Ia (Supernovae Type Ia) and Hubble data is performed to achieve the best-fitting values of the model parameters, and the model is compatible with current observational data. We also perform a dynamical analysis of this model to get an overall qualitative description of the cosmological evolution by converting the governing equations into a system of ordinary differential equations considering a proper transformation of variables. We find some non-isolated sets of critical points, among which some usually are normally hyperbolic sets of points that describe the present acceleration of the universe dominated by dark energy mimicking cosmological constant or phantom fluid. Scaling solutions are also obtained from this analysis, and they can alleviate the coincidence problem successfully. Statefinder diagnosis is also carried out for this model to compare it with the ΛCDM, and any other dark energy models byfinding various statefinder parameters. Finally, the thermodynamic analysis shows that the generalized second law of thermodynamics is valid in an irreversible thermodynamic context.

Computing parametrised large intersection sets of 1D invariant manifolds: a tool for blender detection

A dynamical system given by a diffeomorphism with a three-dimensional phase space may have a blender, which is a hyperbolic set Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} with, say, a one-dimensional stable invariant manifold that behaves like a surface. This means that the stable manifold of any fixed or periodic point in Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} weaves back and forth as a curve in phase space such that it is dense in some projection; we refer to this as the carpet property. We present a method for computing very long pieces of such a one-dimensional manifold so efficiently and accurately that a very large number of intersection points with a specified section can reliably be identified. We demonstrate this with the example of a family of Hénon-like maps H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document} on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}, which is the only known, explicit example of a diffeomorphism with proven existence of a blender. The code for this example is available as a Matlab script as supplemental material. In contrast to earlier work, our method allows us to determine a very large number of intersection points of the respective one-dimensional stable manifold with a chosen planar section and render each as individual curves when a parameter is changed. With suitable accuracy settings, we not only compute these parametrised curves for the fixed points of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document} over the relevant parameter interval, but we also compute the corresponding parametrised curves of the stable manifolds of a period-two orbit (with negative eigenvalues) and of a period-three orbit (with positive eigenvalues). In this way, we demonstrate that our algorithm can handle large expansion rates generated by (up to) the fourth iterate of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document}.

First-time Sensitive Homeomorphisms

AbstractWe introduce first-time sensitivity for a homeomorphism of a compact metric space, that is a condition on the first increasing times of open balls of the space. Continuum-wise expansive homeomorphisms, the shift map on the Hilbert cube, and also some partially hyperbolic diffeomorphisms satisfy this condition. We prove the existence of local unstable continua satisfying similar properties with the local unstable continua of continuum-wise expansive homeomorphisms, but assuming first-time sensitivity. As a consequence we prove that first-time sensitivity (with some additional technical assumptions) implies positive topological entropy.