Shadowing and -limit sets of circular Julia sets

In this paper we consider quadratic polynomials on the complex plane${f}_{c} (z)= {z}^{2} + c$and their associated Julia sets,${J}_{c} $. Specifically, we consider the case that the kneading sequence is periodic and not an$n$-tupling. In this case${J}_{c} $contains subsets that are homeomorphic to the unit circle, usually infinitely many disjoint such subsets. We prove that${f}_{c} : {J}_{c} \rightarrow {J}_{c} $has shadowing, and we classify all$\omega $-limit sets for these maps by showing that a closed set$R\subseteq {J}_{c} $is internally chain transitive if, and only if, there is some$z\in {J}_{c} $with$\omega (z)= R$.

THE BUNDLE PLOT: EVOLUTION OF SYMBOLIC SPACE UNDER THE SYSTEM PARAMETER CHANGES

This paper provides a topological dynamics perspective on the full bifurcation unfolding in unimodal mappings. We present a bundle structure, visualized as a bundle plot, to show the evolution of symbolic space as we vary a system parameter. The bundle plot can be viewed as a limit process of an assignment plot, which are line assignments between points from two dynamical systems. Such line assignments are determined by a commuter, which is a coordinates transformation function that satisfies a commuting relationship but not necessarily a homeomorphism. The bundle structure is studied by understanding the implication of the system's qualitative changes. In addition, the case of the bundle plot with higher dimensional parameter variation is also considered. A main concern in the bundle plot is a special structure, called "joint", which determines a critical value of the parameter where the kneading sequence becomes periodic.

Bernoulli measure of complex admissible kneading sequences

Iterated quadratic polynomials give rise to a rich collection of different dynamical systems that are parametrized by a simple complex parameter $c$. The different dynamical features are encoded by the kneading sequence, which is an infinite sequence over $\{ \mathtt{0} , \mathtt{1} \} $. Not every such sequence actually occurs in complex dynamics. The set of admissible kneading sequences was described by Milnor and Thurston for real quadratic polynomials, and by the authors in the complex case. We prove that the set of admissible kneading sequences has positive Bernoulli measure within the set of sequences over $\{ \mathtt{0} , \mathtt{1} \} $.

COMPUTING THE TOPOLOGICAL ENTROPY OF UNIMODAL MAPS

We derive an algorithm to determine recursively the lap number (minimal number of monotone pieces) of the iterates of unimodal maps of an interval with free end-points. For this family of maps, the kneading sequence does not determine the lap numbers. The algorithm is obtained by the sign analysis of the itineraries of the critical point and of the boundary points of the interval map. We apply this algorithm to the estimation of the growth number and the topological entropy of maps with direct and reverse bifurcations.

K-theory for Cuntz-Krieger algebras arising from real quadratic maps

We compute theK-groups for the Cuntz-Krieger algebras𝒪A𝒦(fμ), whereA𝒦(fμ)is the Markov transition matrix arising from the kneading sequence𝒦(fμ)of the one-parameter family of real quadratic mapsfμ.

A RECYCLED CHARACTERIZATION OF KNEADING SEQUENCES

For any infinite sequence E on two symbols one can define two sequences of positive integers S(E) (the splitting times) and T(E) (the cosplitting times), which each describe the self-replicative structure of E. If E is the kneading sequence of a unimodal map, it is known that S(E) and T(E) carry a lot of information on the dynamics, and that they are disjoint. We show the reverse implication: A nonperiodic sequence E is the kneading sequence of some unimodal map if the sequences S(E) and T(E) are disjoint.

Stable and unstable manifold structures in the Hénon family

For parameter values $a$ where the quadratic map $f_a(x) = 1-ax^2$ has an attracting periodic point, and small values of $b$, the Hénon map $H_{a,b}(x, y) = (1 + y - ax^2, bx)$ has a periodic attractor and an attracting set that is homeomorphic with the inverse limit of $f_a|_{[1-a, 1]}$. This attracting set consists of the collection of unstable manifolds of a hyperbolic invariant set together with the attracting periodic orbit. In the case in which $f_a$ is also not renormalizable with smaller period and $b < 0$, we give a symbolic description of the collection of stable manifolds of this hyperbolic set and show that this collection is homeomorphic with the collection of unstable manifolds precisely when the attracting periodic orbit is accessible from the complement of the attracting set, a condition that can be characterized in terms of the kneading sequence of the quadratic map. As an application, we answer a question raised by Hubbard and Oberste-Vorth by proving that the basin boundaries corresponding to three distinct period five sinks in the Hénon family are non-homeomorphic.

Symbolic dynamics for angle-doubling on the circle III. Sturmian sequences and the quadratic map

By the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly connected with the angle-doubling maphon the circleT. In particular, a connected and locally connected Julia set can be considered as a topological factorT/ ≈ ofTwith respect to a specialh-invariant equivalence relation ≈ onT, which is called Julia equivalence by Keller. Following an idea of Thurston, Bandt and Keller have investigated a map α →αfromTonto the set of all Julia equivalences, which gives a natural abstract description of the Mandelbrot set. By the use of a symbol sequence called the kneading sequence of the point α, they gave a topological classification of the abstract Julia setsT/α. It turns out thatT/αcontains simple closed curves iff the point α has a periodic kneading sequence. The present article characterizes the set of points possessing a periodic kneading sequence and discusses this set in relation to Julia sets and to the Mandelbrot set.