scholarly journals How to identify a hyperbolic set as a blender

2020 ◽  
Vol 40 (12) ◽  
pp. 6815-6836
Author(s):  
Stefanie Hittmeyer ◽  
◽  
Bernd Krauskopf ◽  
Hinke M. Osinga ◽  
Katsutoshi Shinohara ◽  
...  
Keyword(s):  
Hyperbolic Set

Hyperbolic set covering problems with competing ground-set elements

2011 ◽  
Vol 134 (2) ◽  
pp. 323-348 ◽  
Author(s):  
Edoardo Amaldi ◽  
Sandro Bosio ◽  
Federico Malucelli
Keyword(s):  
Set Covering ◽  
Covering Problems ◽  
Set Covering Problems ◽  
Hyperbolic Set

scholarly journals Hartman’s theorem for hyperbolic sets

1977 ◽  
Vol 67 ◽  
pp. 41-52 ◽  
Author(s):  
Masahiro Kurata
Keyword(s):  
Fixed Point ◽  
Finite Type ◽  
Conjugacy Problem ◽  
Topological Conjugacy ◽  
Shift Of Finite Type ◽  
Linear Map ◽  
Hyperbolic Set ◽  
Slight Extension ◽  
Hyperbolic Fixed Point ◽  
Hyperbolic Sets

Hartman proved that a diffeomorphism is topologically conjugate to a linear map on a neighbourhood of a hyperbolic fixed point ([3]). In this paper we study the topological conjugacy problem of a diffeomorphism on a neighbourhood of a hyperbolic set, and prove that for any hyperbolic set there is an arbitrarily slight extension to which a sub-shift of finite type is semi-conjugate.

Normally Hyperbolic Sets in Discontinuous Dry Friction Oscillator

2015 ◽  
Vol 25 (06) ◽  
pp. 1550086
Author(s):  
Jeferson Cassiano ◽  
Maurício Firmino Silva Lima ◽  
André Fonseca
Keyword(s):  
Control Theory ◽  
Numerical Simulations ◽  
Dry Friction ◽  
Piecewise Smooth ◽  
Hyperbolic Set ◽  
Friction Oscillator ◽  
Hyperbolic Sets

In this paper, we study a four-parameters piecewise-smooth dry friction oscillator from Control theory. Using Filippov's convention, we prove the existence of a codimension-1 bifurcation which gives rise to a normally hyperbolic set composed by a family of attracting cylinders. This bifurcation exhibits interesting discontinuous oscillation phenomena. We also present consistent numerical simulations.

A property of vector fields without singularity in {\mathcal G}^1(M)

2001 ◽  
Vol 21 (1) ◽  
pp. 303-314 ◽  
Author(s):  
HIROYOSHI TOYOSHIBA
Keyword(s):  
Vector Fields ◽  
Hyperbolic Set ◽  
Stability Conjecture

We prove the following property: if X in \mathcal G^1(M) has no singularity and x \in \Sigma(X), then \overline{\operatorname{orbit}(x)} \cap \overline{\operatorname{per}(X)} \not = \emptyset. In addition, if we assume \overline{\operatorname{per}_i(X)} \cap \overline{\operatorname{per}_j(X)} = \emptyset for i \not = j, then \overline{\operatorname{per}(X)} = \bigcup_{i=0}^{n-1} \overline{\operatorname{per}_i(X)} is a hyperbolic set. Moreover, we shall give a proof of the \Omega-stability conjecture for flows.

Markov cell structures near a hyperbolic set

1993 ◽  
Vol 103 (491) ◽  
pp. 0-0 ◽  
Author(s):  
Tom Farrell ◽  
Lowell Jones
Keyword(s):  
Hyperbolic Set ◽  
Cell Structures

An attracting singular-hyperbolic set containing a non trivial hyperbolic repeller

2009 ◽  
Vol 30 (1) ◽  
pp. 12-16 ◽  
Author(s):  
D. Carrasco-Olivera ◽  
M. E. Chavez-Gordillo
Keyword(s):  
Hyperbolic Set

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

2021 ◽  
pp. 1-10
Author(s):  
ALINE CERQUEIRA ◽  
CARLOS G. MOREIRA ◽  
SERGIO ROMAÑA
Keyword(s):  
Hausdorff Dimension ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Riemannian Metric ◽  
Hausdorff Dimensions ◽  
Hyperbolic Set ◽  
Negatively Curved ◽  
Smooth Perturbation ◽  
Unit Tangent Bundle ◽  
Markov Spectrum

Abstract Let $g_0$ 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 $t\in \mathbb {R}$ .

INVARIANT SUBSETS OF VOLUME HYPERBOLIC SETS

2005 ◽  
Vol 07 (06) ◽  
pp. 839-848
Author(s):  
I. W. AGUILAR ◽  
E. H. APAZA ◽  
C. A. MORALES
Keyword(s):  
Three Dimensional ◽  
Closed Surface ◽  
Invariant Set ◽  
Lorenz Attractor ◽  
Invariant Subset ◽  
Dominated Splitting ◽  
Three Dimensional Flow ◽  
Hyperbolic Set ◽  
Hyperbolic Sets ◽  
Geometric Lorenz Attractor

A volume hyperbolic set is a compact invariant set with a dominated splitting whose external bundles uniformly contract and expand the volume respectively [1]. Examples of volume hyperbolic sets for diffeomorphisms or flows are the hyperbolic sets, the geometric Lorenz attractor [3] and the singular horseshoe [6]. We shall prove that no invariant subset of a volume hyperbolic set of a three-dimensional flow is homeomorphic to a closed surface.

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