C0generic properties of stable and unstable sets of diffeomorphisms*

Nonlinearity ◽  
2001 ◽  
Vol 15 (1) ◽  
pp. 89-114
Author(s):  
Jana Rodriguez Hertz
Keyword(s):  
Stable And Unstable Sets

KNOT POINTS IN TWO-DIMENSIONAL MAPS AND RELATED PROPERTIES

2009 ◽  
Vol 19 (02) ◽  
pp. 545-555 ◽  
Author(s):  
F. TRAMONTANA ◽  
L. GARDINI ◽  
D. FOURNIER-PRUNARET ◽  
P. CHARGE
Keyword(s):  
Focal Point ◽  
Singular Line ◽  
Two Dimensional ◽  
Focal Points ◽  
Singular Set ◽  
One Dimensional ◽  
Attracting Set ◽  
Straight Line ◽  
Special Character ◽  
Stable And Unstable Sets

We consider the class of two-dimensional maps of the plane for which there exists a whole one-dimensional singular set (for example, a straight line) that is mapped into one point, called a "knot point" of the map. The special character of this kind of point has been already observed in maps of this class with at least one of the inverses having a vanishing denominator. In that framework, a knot is the so-called focal point of the inverse map (it is the same point). In this paper, we show that knots may also exist in other families of maps, not related to an inverse having values going to infinity. Some particular properties related to focal points persist, such as the existence of a "point to slope" correspondence between the points of the singular line and the slopes in the knot, lobes issuing from the knot point and loops in infinitely many points of an attracting set or in invariant stable and unstable sets.

Homoclinic and Heteroclinic Situations Specific to Two-Dimensional Noninvertible Maps

1997 ◽  
Vol 07 (01) ◽  
pp. 39-70 ◽  
Author(s):  
Gilles Millerioux ◽  
Christian Mira
Keyword(s):  
Qualitative Change ◽  
Closed Curve ◽  
Stable Set ◽  
Complex Structures ◽  
Two Dimensional ◽  
Parameter Region ◽  
Critical Curves ◽  
Rank One ◽  
Stable And Unstable Sets ◽  
Noninvertible Maps

These situations are put in evidence from two examples of (Z0 - Z2) maps. It is recalled that such maps (the simplest type of non-invertible ones) are related to the separation of the plane into a region without preimage, and a region each point of which has two rank-one preimages. With respect to diffeomorphisms, non-invertibility introduces more complex structures of the stable and unstable sets defining the homoclinic and heteroclinic situations, and the corresponding bifurcations. Critical curves permit the analysis of this question. More particularly, a basic global contact bifurcation (contact of the map critical curve with a non-connected saddle stable set Ws) plays a fundamental role for explaining the qualitative change of Ws, which occurs between two boundary homoclinic bifurcations limiting a parameter region related to the disappearing of an attracting invariant closed curve.

Expansive homeomorphisms and hyperbolic diffeomorphisms on 3-manifolds

1996 ◽  
Vol 16 (3) ◽  
pp. 591-622 ◽  
Author(s):  
José L. Vieitez
Keyword(s):  
Basic Result ◽  
Three Dimensional ◽  
Classification Problem ◽  
Topological Manifold ◽  
Anosov Diffeomorphism ◽  
Hyperbolic Diffeomorphisms ◽  
Topological Manifolds ◽  
Stable And Unstable Sets

AbstractThis paper is a contribution to the classification problem of expansive homeomorphisms. Let M be a compact connected oriented three dimensional topological manifold without boundary and f: M → M an expansive homeomorphism.We show that if the topologically hyperbolic period points of f are dense in M then M = , and f is conjugate to an Anosov diffeomorphism. This follows from our basic result: for such a homeomorphism, all stable and unstable sets are (tamely embedded) topological manifolds.

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 Real-expansive flows and topological dimension

1981 ◽  
Vol 1 (2) ◽  
pp. 179-195 ◽  
Author(s):  
H. B. Keynes ◽  
M. Sears
Keyword(s):  
Topological Dimension ◽  
One Dimensional ◽  
Axiom A ◽  
Finite Dimensional ◽  
Expansive Homeomorphism ◽  
Slight Generalization ◽  
Expansive Flow ◽  
Stable And Unstable Sets ◽  
Expansive Flows

AbstractWe examine generalizations of R. Mañé's results on the topological dimension of spaces supporting an expansive homeomorphism to the case of real-expansive flows. We show that a space supporting a real-expansive flow must be finite dimensional, and a minimal real-expansive flow not exhibiting a type of spiral behaviour must be one-dimensional. This latter class includes all known examples and a slight generalization of Axiom A flows. These results are obtained by introducing a new concept of stable and unstable sets for real flows, and examining real-expansive flows in terms of these sets.

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