Hyperbolic Sets Research Articles

On the structure of rotation sets in hyperbolic surfaces

AbstractSearching for a relation between the genus of a closed oriented surface and the possible geometries for homological rotation sets of its maps, we prove that this invariant for Smale diffeomorphisms is given by a union of at most convex sets, all of them containing zero. The classical theory of hyperbolic dynamics allows then to extend this bound to a ‐open and dense set of homeomorphisms, suggesting this to be a general fact. Examples showing the sharpness for this asymptotic order are provided.

The hyperbolic sets with the fitting shadowing property

We explain around C1-stable fitting shadowing property of a diffeomorphism f of C∞ (closed manifold M) denote by SFSP of invariant closed set B in M; we want to prove the main Theorem that f_(|B) satisfies a C^1- stable fitting shadowing property if and only if B is a hyperbolic set with the same index; if partially hyperbolic diffeomorphisms map contains q, r are types of saddle points with different indices, then f does not investigate the fitting shadowing property.

Sine hyperbolic fractional orthotriple linear Diophantine fuzzy aggregation operator and its application in decision making

<abstract><p>The idea of sine hyperbolic fractional orthotriple linear Diophantine fuzzy sets (sinh-FOLDFSs), which allows more uncertainty than fractional orthotriple fuzzy sets (FOFSs) is noteworthy. The regularity and symmetry of the origin are maintained by the widely recognized sine hyperbolic function, which satisfies the experts' expectations for the properties of the multi-time process. Compared to fractional orthotriple linear Diophantine fuzzy sets, sine hyperbolic fractional orthotriple linear Diophantine fuzzy sets (sinh-FOLDFSs) provide a significant idea for enabling more uncertainty. The objective of this research is to provide some reliable sine hyperbolic operational laws for FOLDFSs in order to sustain these properties and the significance of sinh-FOLDFSs. Both the accuracy and score functions for the sinh-FOLDFSs are defined. We define a group of averaging and geometric aggregation operators on the basis of algebraic t-norm and t-conorm operations. The basic characteristics of the defined operators are studied. Using the specified aggregation operators, a group decision-making method for solving real-life decision-making problem is proposed. To verify the validity of the proposed method, we compare our method with other existing methods.</p></abstract>

Comparative analysis of the secure communication schemes based on the generators of hyperbolic strange attractor and strange nonchaotic attractor

The purpose of this work is to analyse qualitative features of the information transmission process via several communication schemes based on the synchronization of transmitter and receiver, both being complex signal generators. For this purpose generators of the hyperbolic chaos and generators with the strange nonchaotic attractor are employed. Evaluation of advantages and disadvantages of such schemes is made comparing themselves with each other as well as with schemes based on the nonhyperbolic chaotic generators. Methods. The power spectra and the distributions of the largest finite-time Lyapunov exponent are used to confirm the complexity of the dynamics of the generators in use and to verify the wide-bandness, robustness and stochasticity of their signals. Confidentiality of the informational signal transmission is achieved using its nonlinear mixing to the dynamics of the transmitter. The special phase mixing is used since the model generators employed for the research demonstrate nontrivial dynamics for the angular variable — oscillations phase shift. The digital image is used as an information for transmission. Visual control during the transmission process allows to carry out the qualitative analysis of the success of the signal coding and its detecting by the receiver. Results. Successful transmission and decoding of information for all schemes under investigation are demonstrated for the case of identical transmitter and receiver. Parameter detuning of these generators leads to difficulties in separation of the informational signal from the chaotic/complex carrier due to loss of the full synchronization. For the nonhyperbolic chaos detuning of the parameter responsible for the amplitude of the signal leads to the bad quality of the detection while frequency detuning makes detection absolutely impossible. Schemes with the hyperbolic chaos and strange nonchaotic dynamics appear to demonstrate much better results. The information detection is much better in this case because of the robustness of the generalized synchronization. Conclusion. Robust chaotic and complex nonchaotic generators appear to have significant advantages for communication systems comparing to the chaotic generators of nonhyperbolic type.

On codimension one partially hyperbolic diffeomorphisms

On codimension one partially hyperbolic diffeomorphisms

Quenched limit theorems for random U(1) extensions of expanding maps

In this paper we provide quenched central limit theorems, large deviation principles and local central limit theorems for random U(1) extensions of expanding maps on the torus. The results are obtained as special cases of corresponding theorems that we establish for abstract random dynamical systems. We do so by extending a recent spectral approach developed for quenched limit theorems for expanding and hyperbolic maps to be applicable also to partially hyperbolic dynamics.

A paradifferential approach for hyperbolic dynamical systems and applications

We develop a paradifferential approach for studying non-smooth hyperbolic dynamics and related non-linear PDE from a microlocal point of view. As an application, we describe the microlocal regularity, i.e the $H^s$ wave-front set for all $s$, of the unstable bundle $E_u$ for an Anosov flow. We also recover rigidity results of Hurder-Katok and Hasselblatt in the Sobolev class rather than H\"older: there is $s_0>0$ such that if $E_u$ has $H^s$ regularity for $s>s_0$ then it is smooth (with $s_0=2$ for volume preserving $3$-dimensional Anosov flows). In the appendix by Guedes Bonthonneau, it is also shown that it can be applied to deal with non-smooth flows and potentials. This work could serve as a toolbox for other applications.

Katok’s Entropy Formula of Unstable Metric Entropy for Partially Hyperbolic Diffeomorphisms

Katok’s Entropy Formula of Unstable Metric Entropy for Partially Hyperbolic Diffeomorphisms

Case study for hospital-based Post-Acute Care-Cerebrovascular Disease using Sine Hyperbolic q-rung orthopair fuzzy Dombi aggregation operators

The Sine Hyperbolic q-rung orthopair fuzzy sets (sinh-q-ROFSs) are the important concept of accepting more uncertainty than the q-rung orthopair fuzzy sets (q-ROFSs). The well-known sine hyperbolic function preserves the origin’s periodicity and symmetric existence, and so fulfills the expert’s expectations for the multi-time process’ parameters. The aim of this study is to offer some robust sine hyperbolic Dombi operation laws (sinhDOLs) for q-ROFSs in order to preserve these qualities and the significance of sinh-q-ROFSs. Score and accuracy functions for the sinh-q-ROFSs’ are also defined. We define a number of new averaging/geometric aggregation operators (AOs) based on Dombi t-norm and conorm operations. The fundamental properties of the defined operators were discussed. Then, using defined AOs, we provide a group decision-making (DM) approach for solving DM problems. We demonstrate a numerical example to verify the defined method.

No hyperbolic sets in J∞ for infinitely renormalizable quadratic polynomials

No hyperbolic sets in J∞ for infinitely renormalizable quadratic polynomials

Hyperbolicity and abundance of elliptical islands in annular billiards

AbstractWe study the billiard dynamics in annular tables between two eccentric circles. As the center and the radius of the inner circle changes, a two-parameters map is defined by the first return of trajectories to the obstacle. We obtain an increasing family of hyperbolic sets, in the sense of the Hausdorff distance, as the radius goes to zero and the center of the obstacle approximates the outer boundary. The dynamics on each of these sets is conjugate to a shift with an increasing number of symbols. We also show that for many parameters, the system presents quadratic homoclinic tangencies whose bifurcation gives rise to elliptical islands (conservative Newhouse phenomenon). Thus, for many parameters, we obtain the coexistence of a ‘large’ hyperbolic set with many elliptical islands.

Horocycle averages on closed manifolds and transfer operators

We adapt to $C^r$ Anosov flows on compact manifolds a construction for $C^r$ discrete-time hyperbolic dynamics ($r>1$), obtaining anisotropic Banach or Hilbert spaces on which the resolvent of the generator of weighted transfer operators for the flow is quasi-compact. We apply this to study the ergodic integrals of the horocycle flows $h_\rho$ of $C^r$ codimension one mixing Anosov flows. In dimension three, for any suitably bunched $C^3$ contact Anosov flow with orientable strong-stable distribution, we establish power-law convergence of the ergodic average. We thereby implement the program of Giulietti-Liverani in the "real-life setting" of geodesic flows in variable negative curvature, where nontrivial resonances exist.

Center foliation rigidity for partially hyperbolic toral diffeomorphisms

We study perturbations of a partially hyperbolic toral automorphism L which is diagonalizable over \(\mathbb C\) and has a dense center foliation. For a small perturbation of L with a smooth center foliation we establish existence of a smooth leaf conjugacy to L. We also show that if a small perturbation of an ergodic irreducible L has smooth center foliation and is bi-Hölder conjugate to L, then the conjugacy is smooth. As a corollary, we show that for any symplectic perturbation of such an L any bi-Hölder conjugacy must be smooth. For a totally irreducible L with two-dimensional center, we establish a number of equivalent conditions on the perturbation that ensure smooth conjugacy to L.

Isoperimetric problem for the first curl eigenvalue

We consider an isoperimetric problem involving the smallest positive and largest negative curl eigenvalues on abstract ambient manifolds, with a focus on the standard model spaces. We in particular show that the corresponding eigenvalues on optimal domains, assuming optimal domains exist, must be simple in the Euclidean and hyperbolic setting. This generalises a recent result by Enciso and Peralta-Salas who showed the simplicity for axisymmetric optimal domains with connected boundary in Euclidean space. We then generalise another recent result by Enciso and Peralta-Salas, namely that the points of any rotationally symmetric optimal domain with connected boundary in Euclidean space which are closest to the symmetry axis must disconnect the boundary, to the hyperbolic setting, as well as strengthen it in the Euclidean case by removing the connected boundary assumption.

Exponential type orbitals with hyperbolic cosine function basis sets for isoelectronic series of the atoms Be to Ne

Abstract Exponential type orbital with hyperbolic cosine basis functions, proposed recently for Hartree–Fock–Roothaan calculations of neutral atoms, are studied in detail for the calculations of isoelectronic series of atoms from Be to Ne. Calculations are performed for the neutral and the first 20 cationic members of the isoelectronic series of each atom in its ground state. Three of the most popular exponential type orbitals (Slater type functions, B functions and ψ (α) functions with α = 2) are combined with modified hyperbolic cosine function cosh(βr + γ) to improve the basis function quality within the minimal basis sets framework. Performances of the basis functions are compared with each other by using the same number of variational parameters in them. The obtained results are also compared with numerical Hartree–Fock and extended Slater type basis set results. The presented accuracy of the minimal basis descriptions of atomic systems supports the usage of these unconventional basis functions in electronic structure and property calculations.

A local discontinuous Galerkin level set reinitialization with subcell stabilization on unstructured meshes

In this paper we consider a level set reinitialization technique based on a high-order, local discontinuous Galerkin method on unstructured triangular meshes. A finite volume based subcell stabilization is used to improve the nonlinear stability of the method. Instead of the standard hyperbolic level set reinitialization, the flow of time Eikonal equation is discretized to construct an approximate signed distance function. Using the Eikonal equation removes the regularization parameter in the standard approach which allows more predictable behavior and faster convergence speeds around the interface. This makes our approach very efficient especially for banded level set formulations. A set of numerical experiments including both smooth and non-smooth interfaces indicate that the method experimentally achieves design order accuracy.

Generalized hyperbolic sets on Banach spaces

Generalized hyperbolic sets on Banach spaces

Double hyperbolic cosine basis sets for LCAO calculations

Recently, we reported a new set of hyperbolic cosine type basis sets, which include the radial part of exponential type functions. In this study, it is show that the double hyperbolic cosine basis sets with non-integer Slater type orbitals give extremely accurate results, with the accuracy superior to that of the previous similar calculations. Total energy differences and comparison between double hyperbolic cosine and other similar basis sets suggested in literature are presented. Using double hyperbolic cosine orbitals, combined Hartree–Fock-Roothaan calculations have been carried out on the ground states of atoms and their ions within the minimal basis sets framework to compare the performance of the basis sets. The basis sets achieved the accuracy far beyond the double-zeta quality. Proposed basis sets can also play an effective role in ion-atom collisions problems and semi-empirical methods.

Dimension approximation for diffeomorphisms preserving hyperbolic SRB measures

For a C1+α diffeomorphism f preserving a hyperbolic ergodic SRB measure μ, Katok's remarkable results assert that μ can be approximated by a sequence of hyperbolic sets {Λn}n≥1. In this paper, we prove that the Hausdorff dimension for Λn on the unstable manifold tends to the dimension of the unstable manifold. Furthermore, if the stable direction is one dimension, then the Hausdorff dimension of μ can be approximated by the Hausdorff dimension of Λn.To establish these results, we utilize the u-Gibbs property of the conditional measure of the equilibrium measure of −ψs(⋅,fn) and the properties of the uniformly hyperbolic dynamical systems.

Ergodic optimization for some dynamical systems beyond uniform hyperbolicity

ABSTRACT In Ergodic optimization, one wants to find ergodic measures to maximize or minimize the integral of given continuous functions. This has been succefully studied for uniformly hyperbolic systems for generic continuous functions by Bousch and Brémon. In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full support. These interesting systems include singular hyperbolic attractors, surface diffeomorphisms and diffeomorphisms away from homoclinic tangencies.We try to give a uniform mechanism for these non-hyperbolic systems.