Dynamical behavior of endomorphisms on certain invariant sets

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Diana Putan ◽  
Diana Stan
Keyword(s):  
Hausdorff Dimension ◽  
Dynamical Behavior ◽  
Invariant Sets ◽  
Hyperbolic Polynomial ◽  
Stable Manifolds ◽  
Basic Set ◽  
Polynomial Endomorphisms ◽  
Complex Projective ◽  
Local Stable Manifolds ◽  
Basic Sets

AbstractWe study the Hausdorff dimension of the intersection between local stable manifolds and the respective basic sets of a class of hyperbolic polynomial endomorphisms on the complex projective space ℙ2. We consider the perturbation (z 2 +ɛz +bɛw 2, w 2) of (z 2, w 2) and we prove that, for b sufficiently small, it is injective on its basic set Λɛ close to Λ:= {0} × S 1. Moreover we give very precise upper and lower estimates for the Hausdorff dimension of the intersection between local stable manifolds and Λɛ, in the case of these maps.

On a class of stable conditional measures

2010 ◽  
Vol 31 (5) ◽  
pp. 1499-1515 ◽  
Author(s):  
EUGEN MIHAILESCU
Keyword(s):  
Invariant Measures ◽  
Reversible Systems ◽  
Absolutely Continuous ◽  
Stable Manifolds ◽  
Equilibrium Measures ◽  
Basic Set ◽  
Stable Dimension ◽  
Saddle Type ◽  
Local Stable Manifolds ◽  
Basic Sets

AbstractThe dynamics of endomorphisms (smooth non-invertible maps) presents many differences from that of diffeomorphisms or that of expanding maps; most methods from those cases do not work if the map has a basic set of saddle type with self-intersections. In this paper we study the conditional measures of a certain class of equilibrium measures, corresponding to a measurable partition subordinated to local stable manifolds. We show that these conditional measures are geometric probabilities on the local stable manifolds, thus answering in particular the questions related to the stable pointwise Hausdorff and box dimensions. These stable conditional measures are shown to be absolutely continuous if and only if the respective basic set is a non-invertible repeller. We find also invariant measures of maximal stable dimension, on folded basic sets. Examples are given, too, for such non-reversible systems.

Metric properties of some fractal sets and applications of inverse pressure

2009 ◽  
Vol 148 (3) ◽  
pp. 553-572 ◽  
Author(s):  
EUGEN MIHAILESCU
Keyword(s):  
Hausdorff Dimension ◽  
General Setting ◽  
Topological Pressure ◽  
Box Dimension ◽  
Stable Manifolds ◽  
Fractal Sets ◽  
Metric Properties ◽  
Basic Set ◽  
Unstable Set ◽  
Unstable Manifolds

AbstractWe consider iterations of smooth non-invertible maps on manifolds of real dimension 4, which are hyperbolic, conformal on stable manifolds and finite-to-one on basic sets. The dynamics of non-invertible maps can be very different than the one of diffeomorphisms, as was shown for example in [4,7,12,17,19], etc. In [13] we introduced a notion of inverse topological pressureP−which can be used for estimates of the stable dimension δs(x) (i.e the Hausdorff dimension of the intersection between the local stable manifoldWsr(x) and the basic set Λ,x∈ Λ). In [10] it is shown that the usual Bowen equation is not always true in the case of non-invertible maps. By using the notion of inverse pressureP−, we showed in [13] that δs(x) ≤ts(ϵ), wherets(ϵ) is the unique zero of the functiont→P−(tφs, ϵ), for φs(y):= log|Dfs(y)|,y∈ Λ and ϵ > 0 small. In this paper we prove that if Λ is not a repellor, thents(ϵ) < 2 for any ϵ > 0 small enough. In [11] we showed that a holomorphic s-hyperbolic map on2has a global unstable set with empty interior. Here we show in a more general setting than in [11], that the Hausdorff dimension of the global unstable setWu() is strictly less than 4 under some technical derivative condition. In the non-invertible case we may have (infinitely) many unstable manifolds going through a point in Λ, and the number of preimages belonging to Λ may vary. In [17], Qian and Zhang studied the case of attractors for non-invertible maps and gave a condition for a basic set to be an attractor in terms of the pressure of the unstable potential. In our case the situation is different, since the local unstable manifolds may intersect both inside and outside Λ and they do not form a foliation like the stable manifolds. We prove here that the upper box dimension ofWsr(x) ∩ Λ is less thants(ϵ) for any pointx∈ Λ. We give then an estimate of the Hausdorff dimension ofWu() by a different technique, using the Holder continuity of the unstable manifolds with respect to their prehistories.

Bi-invariant sets and measures have integer Hausdorff dimension

1999 ◽  
Vol 19 (2) ◽  
pp. 523-534 ◽  
Author(s):  
DAVID MEIRI ◽  
YUVAL PERES
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Ergodic Measure ◽  
Invariant Set ◽  
Invariant Sets ◽  
Dimensional Torus ◽  
Uniformly Distributed Sequence ◽  
Invariant Ergodic Measure

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus with corresponding eigenvalues relatively prime. We show that for any $A$-invariant ergodic measure $\mu$, there exists a projection onto a torus ${\mathbb T}^r$ of dimension $r\ge\dim\mu$, that maps $\mu$-almost every $B$-orbit to a uniformly distributed sequence in ${\mathbb T}^r$. As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer.

scholarly journals Qualitative Study of a 4D Chaos Financial System

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Fuchen Zhang ◽  
Gaoxiang Yang ◽  
Yong Zhang ◽  
Xiaofeng Liao ◽  
Guangyun Zhang
Keyword(s):  
Hausdorff Dimension ◽  
Chaos Synchronization ◽  
Chaos Control ◽  
Dynamical Behavior ◽  
Lyapunov Stability Theory ◽  
Attraction Domain ◽  
Ultimate Boundedness ◽  
Lyapunov Dimension ◽  
Simulation Results ◽  
Ultimate Bound

Some dynamics of a new 4D chaotic system describing the dynamical behavior of the finance are considered. Ultimate boundedness and global attraction domain are obtained according to Lyapunov stability theory. These results are useful in estimating the Lyapunov dimension of attractors, Hausdorff dimension of attractors, chaos control, and chaos synchronization. We will also present some simulation results. Furthermore, the volumes of the ultimate bound set and the global exponential attractive set are obtained.

scholarly journals Some compact invariant sets for hyperbolic linear automorphisms of torii

1988 ◽  
Vol 8 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Albert Fathi
Keyword(s):  
Hausdorff Dimension ◽  
Homology Group ◽  
Invariant Set ◽  
Finite Union ◽  
Invariant Sets ◽  
Pisot Number ◽  
Toral Automorphism ◽  
Lower Capacity

AbstractIf the action induced by a pseudo-Anosov map on the first homology group is hyperbolic, it is possible, by a theorem of Franks, to find a compact invariant set for the toral automorphism associated with this action. If the stable and unstable foliations of the Pseudo-Anosov map are orientable, we show that the invariant set is a finite union of topological 2-discs. Using some ideas of Urbański, it is possible to prove that the lower capacity of the associated compact invariant set is >2; in particular, the invariant set is fractal. When the dilatation coefficient is a Pisot number, we can compute the Hausdorff dimension of the compact invariant set.

Knaster-like continua and complex dynamics

1993 ◽  
Vol 13 (4) ◽  
pp. 627-634 ◽  
Author(s):  
Robert L. Devaney
Keyword(s):  
Complex Dynamics ◽  
Julia Set ◽  
Dynamical Behavior ◽  
Invariant Sets ◽  
Entire Plane ◽  
Limit Sets

AbstractIn this paper we discuss the topology and dynamics ofEλ(z) = λezwhen λ is real and λ > 1/e. It is known that the Julia set ofEλis the entire plane in this case. Our goal is to show that there are certain natural invariant subsets forEλwhich are topologically Knaster-like continua. Moreover, the dynamical behavior on these invariant sets is quite tame. We show that the only trivial kinds of α- and ω-limit sets are possible.

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