scholarly journals STABLE SETS OF CERTAIN NON-UNIFORMLY HYPERBOLIC HORSESHOES HAVE THE EXPECTED DIMENSION

2019 ◽  
pp. 1-25
Author(s):  
Carlos Matheus ◽  
Jacob Palis ◽  
Jean-Christophe Yoccoz
Keyword(s):  
Hausdorff Dimension ◽  
Stable Sets ◽  
Surface Diffeomorphisms ◽  
Stable And Unstable Sets

We show that the stable and unstable sets of non-uniformly hyperbolic horseshoes arising in some heteroclinic bifurcations of surface diffeomorphisms have the value conjectured in a previous work by the second and third authors of the present paper. Our results apply to first heteroclinic bifurcations associated with horseshoes with Hausdorff dimension ${<}22/21$ of conservative surface diffeomorphisms.

Interactions of the Julia Set with Critical and (Un)Stable Sets in an Angle-Doubling Map on ℂ\{0}

2015 ◽  
Vol 25 (04) ◽  
pp. 1530013 ◽  
Author(s):  
Stefanie Hittmeyer ◽  
Bernd Krauskopf ◽  
Hinke M. Osinga
Keyword(s):  
Critical Point ◽  
Julia Set ◽  
Saddle Points ◽  
Period Doubling ◽  
Stable Sets ◽  
Critical Set ◽  
Critical Circle ◽  
Quadratic Family ◽  
Doubling Map ◽  
Stable And Unstable Sets

We study a nonanalytic perturbation of the complex quadratic family z ↦ z2 + c in the form of a two-dimensional noninvertible map that has been introduced by Bamón et al. [2006]. The map acts on the plane by opening up the critical point to a disk and wrapping the plane twice around it; points inside the disk have no preimages. The bounding critical circle and its images, together with the critical point and its preimages, form the so-called critical set. For parameters away from the complex quadratic family we define a generalized notion of the Julia set as the basin boundary of infinity. We are interested in how the Julia set changes when saddle points along with their stable and unstable sets appear as the perturbation is switched on. Advanced numerical techniques enable us to study the interactions of the Julia set with the critical set and the (un)stable sets of saddle points. We find the appearance and disappearance of chaotic attractors and dramatic changes in the topology of the Julia set; these bifurcations lead to three complicated types of Julia sets that are given by the closure of stable sets of saddle points of the map, namely, a Cantor bouquet and what we call a Cantor tangle and a Cantor cheese. We are able to illustrate how bifurcations of the nonanalytic map connect to those of the complex quadratic family by computing two-parameter bifurcation diagrams that reveal a self-similar bifurcation structure near the period-doubling route to chaos in the complex quadratic family.

scholarly journals The entropy of C2 surface diffeomorphisms in terms of Hausdorff dimension and a Lyapunov exponent

1985 ◽  
Vol 5 (2) ◽  
pp. 273-283 ◽  
Author(s):  
Leonardo Mendoza
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Ergodic Measure ◽  
Positive Lyapunov Exponent ◽  
Stable Manifolds ◽  
Surface Diffeomorphisms ◽  
The Family ◽  
Surface Diffeomorphism

AbstractIn this paper we prove that if the entropy of an ergodic measure preserved by a C2 surface diffeomorphism is positive then it is equal to the product of the Hausdorff dimension of the quotient measure defined by the family of stable manifolds and the positive Lyapunov exponent.

Dimensions of stable sets and scrambled sets in positive finite entropy systems

2011 ◽  
Vol 32 (2) ◽  
pp. 599-628 ◽  
Author(s):  
CHUN FANG ◽  
WEN HUANG ◽  
YINGFEI YI ◽  
PENGFEI ZHANG
Keyword(s):  
Dynamical System ◽  
Riemannian Manifold ◽  
Continuum Hypothesis ◽  
Large Set ◽  
Metric Entropy ◽  
Positive Entropy ◽  
Stable Sets ◽  
Stable And Unstable Sets ◽  
The Continuum ◽  
Hyperbolic Points

AbstractWe study the dimensions of stable sets and scrambled sets of a dynamical system with positive finite entropy. We show that there is a measure-theoretically ‘large’ set containing points whose sets of ‘hyperbolic points’ (i.e. points lying in the intersections of the closures of the stable and unstable sets) admit positive Bowen dimension entropies; under the continuum hypothesis, this set also contains a scrambled set with positive Bowen dimension entropies. For several kinds of specific invertible dynamical systems, the lower bounds of the Hausdorff dimension of these sets are estimated. In particular, for a diffeomorphism on a smooth Riemannian manifold with positive entropy, such a lower bound is given in terms of the metric entropy and Lyapunov exponent.

scholarly journals Self-similar hyperbolicity

2017 ◽  
Vol 38 (7) ◽  
pp. 2422-2446
Author(s):  
ALFONSO ARTIGUE
Keyword(s):  
Hausdorff Dimension ◽  
Hyperbolic Metric ◽  
Canonical Coordinates ◽  
Compact Spaces ◽  
Ergodic Measures ◽  
Self Similar ◽  
Stable And Unstable Sets

In this paper we consider expansive homeomorphisms of compact spaces with a hyperbolic metric presenting a self-similar behavior on stable and unstable sets. Several applications are given related to Hausdorff dimension, entropy, intrinsically ergodic measures and the transitivity of expansive homeomorphisms with canonical coordinates.

scholarly journals Minimal F2-flows

1985 ◽  
Vol 39 (2) ◽  
pp. 187-193
Author(s):  
Daniel Berend
Keyword(s):  
Hausdorff Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Affine Transformations ◽  
Dimensional Torus ◽  
Minimal Sets ◽  
Necessary And Sufficient

AbstractLet σ be an ergodic endomorphism of the r–dimensional torus and Π a semigroup generated by two affine transformations lying above σ. We show that the flow defined by Π admits minimal sets of positive Hausdorff dimension and we give necessary and sufficient conditions for this flow to be minimal.

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