Non-autonomous Julia sets with escaping critical points

We consider non-autonomous iteration which is a generalization of standard polynomial iteration where we deal with Julia sets arising from composition sequences for arbitrarily chosen polynomials with uniformly bounded degrees and coefficients. In this paper, we look at examples where all the critical points escape to infinity. In the classical case, any example of this type must be hyperbolic and there can be only one Fatou component, namely the basin at infinity. This result remains true in the non-autonomous case if we also require that the dynamics on the Julia set be hyperbolic or semi-hyperbolic. However, in general it fails and we exhibit three counterexamples of sequences of quadratic polynomials all of whose critical points escape but which have bounded Fatou components.     

Generic points for dynamical systems with average shadowing

We study the Besicovitch pseudometric \(D_B\) for compact dynamical systems. The set of generic points of ergodic measures turns out to be closed with respect to \(D_B\). It is proved that the weak specification property implies the average asymptotic shadowing property and the latter property does not imply the former one nor the almost specification property. Furthermore an example of a proximal system with the average shadowing property is constructed. It is proved that to every invariant measure \(\mu \) of a compact dynamical system one can associate a certain asymptotic pseudo orbit such that any point asymptotically tracing in average that pseudo orbit is generic for \(\mu \). A simple consequence of the theory presented is that every invariant measure has a generic point in a system with the asymptotic average shadowing property.     

Central sets and substitutive dynamical systems

In this paper we establish a new connection between central sets and the strong coincidence conjecture   for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of N$\mathbb{N}$ possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone–?ech compactification βN$\beta \mathbb{N}$. This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture  . The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone–?ech compactification of N$\mathbb{N}$.     

Rest points of generalized dynamical systems

In this paper we consider generalized dynamical systems whose integral vortex (that is, the set of all trajectories of the system starting at a given point) is an acyclic set in the corresponding space of curves. For such systems we apply the theory of fixed points for multi-valued maps in order to prove the existence of rest points. In this way we obtain new existence theorems for rest points of generalized dynamical systems. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 28–36, January, 1999.     

Linear systems on curves with no Weierstrass points

We study order-sequences of linear systems on smooth curves and establish the formula:b _j +b _N−j ≤b _N for allj, where {b ₀<b ₁<...<b _N } is the order-sequence of a linear system on a curve. As an application of the formula, we describe all linear systems on curves which have no Weierstrass points.     

Localizing sets for invariant compact sets of continuous dynamical systems with a perturbation

A functional method of localization of invariant compact sets, which was earlier developed for autonomous continuous and discrete systems, is generalized to continuous dynamical systems with perturbations. We describe properties of the corresponding localizing sets. By using that method, we construct localizing sets for positively invariant compact sets of the Lorenz system with a perturbation.     

Localizing sets for invariant compact sets of discrete dynamical systems with perturbation and control

We consider the localization problem for the invariant compact sets of a discrete dynamical system with perturbation and control, that is, the problem of constructing domains in the system state space that contain all invariant compact sets of the system. The problem is solved on the basis of a functional method used earlier in localization problems for time-invariant continuous and discrete systems and also for control systems. The properties of the corresponding localizing sets are described.     

Critical points with lack of compactness and singular dynamical systems

Sunto Si prova l'esistenza di punti critici di funzionali che non verificano la condizione (PS).I teoremi astratti vengono applicati per trovare soluzioni periodiche di sistemi dinamici con potenziali sia limitati sia con singolarità.

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This work has been done during a visit at the Math. Dept. of the University of Chicago. The Authors whish to thank for the kind hospitality.

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Supported by Ministero Pubblica Istruzione, 40%, Gruppo Nazionale «Calcolo delle Variazioni».     

Misiurewicz points on the Mandelbrot-like set concerning renormalization transformation

We study Misiurewicz points on the parameter space about a family of rational maps Tλ concerning renormalization transformation in statistical mechanic. We determine the intersection points of the Julia set J(Tλ) and the positive real axis R+and discuss the continuity of the Hausdorff dimension HD(J(f)) about real parameter λ.     

Limit sets of trajectories of regions in dynamical systems

Leningrad State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 3, pp. 82–83, July–September, 1989.     

F-limit points in dynamical systems defined on the interval

Given a free ultrafilter p on ? we say that x ∈ [0, 1] is the p-limit point of a sequence (x _n ) _n∈? ? [0, 1] (in symbols, x = p -lim _n∈? x _n ) if for every neighbourhood V of x, {n ∈ ?: x _n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f ^p : [0, 1] → [0, 1] is defined by f ^p (x) = p -lim _n∈? f ⁿ (x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f ^p . For a filter F we also define the ω ^F -limit set of f at x. We consider a question about continuity of the multivalued map x → ω _f ^F (x). We point out some connections between the Baire class of f ^p and tame dynamical systems, and give some open problems.     

Thickness of Julia sets of Feigenbaum polynomials with high order critical points

We consider unimodal polynomials with Feigenbaum topological type and critical points whose orders tend to infinity. It is shown that the hyperbolic dimensions of their Julia set go to 2; furthermore, that the Hausdorff dimensions of the basins of attraction of their Feigenbaum attractors also tend to 2. The proof is based on constructing a limiting dynamics with a flat critical point. To cite this article: G. Levin, G. ?wi?tek, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Ultrafilters and non-Cantor minimal sets in linearly ordered dynamical systems

It is well known that infinite minimal sets for continuous functions on the interval are Cantor sets; that is, compact zero dimensional metrizable sets without isolated points. On the other hand, it was proved in Alcaraz and Sanchis (Bifurcat Chaos 13:1665–1671, 2003) that infinite minimal sets for continuous functions on connected linearly ordered spaces enjoy the same properties as Cantor sets except that they can fail to be metrizable. However, no examples of such subsets have been known. In this note we construct, in ZFC, non-metrizable infinite pairwise non-homeomorphic minimal sets on compact connected linearly ordered spaces.      

Approximation by smooth functions with no critical points on separable Banach spaces

We characterize the class of separable Banach spaces X such that for every continuous function and for every continuous function there exists a C¹ smooth function for which |f(x)−g(x)|?ε(x) and g^′(x)≠0 for all x∈X (that is, g has no critical points), as those infinite-dimensional Banach spaces X with separable dual X^∗. We also state sufficient conditions on a separable Banach space so that the function g can be taken to be of class Cp, for p=1,2,…,+∞. In particular, we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces ?p(N) and Lp(Rn). Some important consequences of the above results are (1) the existence of a non-linear Hahn-Banach theorem and the smooth approximation of closed sets, on the classes of spaces considered above; and (2) versions of all these results for a wide class of infinite-dimensional Banach manifolds.     

Measure of minimal sets for certain discrete dynamical systems

Let X be a separable metric space, μ a complete Borel measure on X that is finite on balls, and f a closed discrete dynamical system on X that preserves μ and has the diameters of all orbits bounded. We prove that almost every point in X (in the sense of measure μ) has its orbit contained in its ω-limit set.