Dynamics of elementary maps of dendrites

The notion of elementary map of a dendrite into itself is introduced. Arithmetical relations between the periods of periodic points are given; the structure ofω-limit sets, sets of periodic and nonwandering points is described; the topological entropy of elementary maps is shown to be equal to 0. Examples are given illustrating the differences in the entropic properties of continuous maps of dendrites with a countable set of branch points and continuous maps of their retracts which are finite trees. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 183–195, February, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01755.     

On the existence of maximal ω-limit sets for dendrite maps

For interval maps and also for graph maps, every ω-limit set is a subset of a maximal one. In this note we construct a continuous map on a dendrite with no maximal ω-limit set. Moreover, the set of branch points is nowhere dense, every ω-limit set of the map is nowhere dense, the set of periodic points and the set of recurrent points are equal and the set of ω-limit points is not closed (an example with the last property was constructed by the authors already in [Ko?an Z, Kornecká-Kurková V, Málek M. On the centre and the set of omega-limit points of continuous maps on dendrites. Topol Appl 2009;156:2923-2931]).     

On Ω-limit Sets of Discrete-time Dynamical Systems

The paper investigates z -limit sets for discrete-time dynamical systems of the form x n +1 = f n +1 ( x n ), n S 0, with each f n mapping an interval I of R into itself. For autonomous systems, i.e. f n = f for all n , and f continuous on I =[ a , b ], the case that all z -limit sets consist of one point only is characterized by several equivalent conditions, one being that f has no 2-periodic points. The non-autonomous case assumes that the functions f n converge uniformly to a continuous function f X that has no 2-periodic points. It is shown that the z -limit sets are closed intervals consisting of fixed points of f X only. Under certain conditions these closed intervals contain exactly one point each. This allows a treatment of certain discrete-time dynamical systems in R n .     

On lightly and countably compact spaces in ZF

Abstract

Given a topological space X = (X, T ), we show in the Zermelo-Fraenkel set theory ZF that:

1. Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.

2. Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.

3. The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T₄ and the countable axiom of choice holds true.

   We also show:

4. It is relatively consistent with ZF the existence of a non countably compact T₁ space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.

5. It is relatively consistent with ZF the existence of a countably compact T₄ space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.

     

On HCO spaces. An uncountable compactT 2 space,different from ℵ1+1, which is homeomorphic to each of its uncountable closed subspaces,which is homeomorphic to each of its uncountable closed subspaces

LetX be an Hausdorff space. We say thatX is a CO space, ifX is compact and every closed subspace ofX is homeomorphic to a clopen subspace ofX, andX is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems: Theorem (R. Bonnet):

1. Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β.
2. Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α.

Theorem (S. Shelah):Assume \(\diamondsuit _{\aleph _1 } \) . Then there is a HCO compact space X of Cantor-Bendixson rankω ₁} and of cardinality ?₁ such that:

1. X has only countably many isolated points,
2. Every closed subset of X is countable or co-countable,
3. Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and
4. X is retractive.

In particularX is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.     

Orders of accumulation of entropy on manifolds

For a continuous self-map T of a compact metrizable space with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the theory of entropy structure and symbolic extensions. Given any compact manifold M and any countable ordinal α, we construct a continuous, surjective self-map ofM having order of accumulation of entropy α. If the dimension of M is at least 2, then the map can be chosen to be a homeomorphism.     

Cohomology of permutative cellular automata

We examineU(d) valued cocycles for a ?²⁺ action generated by a mixing, permutative cellular automaton and show that the set of Hölder continuous cocycles (for a given Hölder order) which are cohomologous to constant cocycles is both open and closed in the appropriate topology. A continuous dimension function with values in {0, 1,…,d} is defined on cocycles; a cocycle is cohomologous to a constant precisely when the value isd. Whend=1 (the abelian case) the first (essential) cohomology group is countable. IfU(1)? circle is replaced by a finite subgroup, this cohomology group is finite.     

Reduced 2-to-1 maps and decompositions of graphs with no 2-to-1 cut sets

A graph has an increasing ear decomposition if it can be constructed from a simple closed curve by attaching arcs in stages with the endpoints of each arc attached to different points so that at least one new branch point is formed at each stage. A reduced 2-to-1 map is a 2-to-1 map that does not have a restriction that is 2-to-1. A 2-to-1 cut set of a graph G is a finite subset B such that G⧹B has at least 2|B| components. A graph has an increasing ear decomposition if and only if it does not have a 2-to-1 cut set, and a graph is the image of a reduced 2-to-1 map if and only if it does not have a 2-to-1 cut set.     

Regular and quasiregular isometric flows on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on smooth Riemannian manifolds. We describe the properties of the set of all points of finite (infinite) period for general isometric flows on Riemannian manifolds. It is shown that this flow is generated by an effective almost free isometric action of the group S ¹ if there are no points of infinite or zero period. In the last case, the set of periods is at most countable and generates naturally an invariant stratification with closed totally geodesic strata; the union of all regular orbits is an open connected dense subset of full measure.     

A correction to the definition of local center

It is known that in order to solve the minimax facility location problem on a graph with a finite set of demand points, only a finite set of possible location points, called ‘local centers’ must be considered.It has been shown that the continuous m-center problem on a graph can be solved by using a series of set covering problems in which each local center covers the demand points at a distance not greater than a corresponding number called ‘the range’ of the local center.However, all points which are at the same distance from more than two demand points, and from which there is no direction where all these distances are decreasing, must also be considered as local centers. This paper proves that, in some special cases, it is not sufficient to consider only the points where this occurs with respect to pairs of demand points. The definition of local center is corrected and the corresponding results and algorithms are revised.     

A Note on the Set of Periods of Transversal Homological Sphere Self-maps

Let M be a n -dimensional manifold with the same homology than the n -dimensional sphere. A C 1 map f : M M M is called transversal if for all m ] N the graph of f m intersects transversally the diagonal of M 2 M at each point ( x , x ) such that x is a fixed point of f m . We study the minimal set of periods of f by using the Lefschetz numbers for periodic points. In the particular case that n is even, we also study the set of periods for the transversal holomorphic self-maps of M .     

On the Distribution Behaviour of Sequences

This paper presents results concerning those sets of finite Borel measures μ on a locally compact Hausdorff space X with countable topological base which can be represented as the set of limit distributions of some sequence. They arc characterized by being nonanpty, closed, connected and containing only measures μ with μ(X) = 1 (if X is compact) or 0 ≤ μ(X) ≤ 1 (if X is not compact). Any set with this properties can be obtained as the set of limit distributions of a sequence even by rearranging an arbitrarily given sequence which is dense in the sense that the set of accumulation points is the whole space X. The typical case (in the sense of Baire categories) is that a sequence takes as limit distributions all possible measures of this kind. This gives new aspects for the recent theory of maldistribukd sequences.     

Centers and shore points of a dendroid

A bottleneck in a dendroid is a continuum that intersects every arc connecting two nonempty open sets. Every dendroid contains a point called a center which is contained in arbitrarily small bottlenecks. A subset A of a dendroid is a shore set if for every ε>0 there is a continuum in D?A with Hausdorff distance from D less than ε. If a shore set has only one point it is called a shore point. This paper explores the relationship between center points and shore points in a dendroid. We show that if a dendroid contains a strong center, then any finite union of the arc components of the set of shore points is a shore set.     

Universality of Embeddability Relations for Coloured Total Orders

Some examples of Σ₁ ¹-universal preorders are presented, in the form of various relations of embeddability between countable coloured total orders. As an application, strengthening a theorem of (Marcone, A. and Rosendal, C.: The Complexity of Continuous Embeddability between Dendrites, J. Symb. Log. 69 (2004), 663–673), the Σ₁ ¹-universality of continuous embeddability for dendrites whose branch points have order 3 is obtained.     

Exposed and Strongly Exposed Points of Finite Order in Some Banach Spaces

in this paper we study the exposed and strongly exposed points of finite order inl(D,X) as well as in L(ll (D,X), Y) and L(X, l(D,Y) ), where X and Y are real Banachspaces while D is a non empty index set. In particular, we characterize the exposed andstrongly exposed points in these spaces.     

Embeddings of countable closed sets and reverse mathematics

Summary If there is a homeomorphic embedding of one set into another, the sets are said to be topologically comparable. Friedman and Hirst have shown that the topological comparability of countable closed subsets of the reals is equivalent to the subsystem of second order arithmetic denoted byATR ₀. Here, this result is extended to countable closed locally compact subsets of arbitrary complete separable metric spaces. The extension uses an analogue of the one point compactification of .     

On totally periodic ω-limit sets in regular continua

Let f be a continuous self-mapping of a compact metric space X, an ω-limit set of f is said to be totally periodic if it is composed of periodic points. We prove that a totally periodic ω-limit set of one-to-one continuous self mapping of regular continuum is finite. In the other hand, we built a continuous self-mapping (not one-to-one) of a dendrite having a totally periodic ω-limit set with unbounded periods.     

Sul metodo delle approssimazioni successive: Convergenza globale e plus-convergenza globale

With reference to the notions of global convergence and plus-global convergence introduced in [4], we prove that, for a dendriteX with a finite number of end points and a continuous mapF ofX into itself, the successive approximations method relative to pair (X, F) converges globally if and only ifF has no periodic points; furthermore, we show that ifF has no periodic points and the set of fixed points ofF is totally disconnected then the successive approximations method relative to the pair (X, F) plus-converge globally. Some auxiliary propositions have been proved in more general topological areas. One of them refers to a compact metric space (Γ,d′), to a continuous map ? of Γ into itself and to a sequence \((x_n )_{n \in IN} \) of points of Γ such that the sequence \((d^1 (x_{n + 1} ,\varphi (x_n )))_{n \in IN} \) converges to zero, and concerns the set of the cluster points of the sequence \((x_n )_{n \in IN} \) . In the end the global convergence and plus-global convergence in any metric space are characterized.     

Geometric Models for Quasicrystals I. Delone Sets of Finite Type

This paper studies three classes of discrete sets X in ⁿ which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X-X . A finitely generated Delone set is one such that the abelian group [X-X] generated by X-X is finitely generated, so that [X-X] is a lattice or a quasilattice. For such sets the abelian group [X] is finitely generated, and by choosing a basis of [X] one obtains a homomorphism . A Delone set of finite type is a Delone set X such that X-X is a discrete closed set. A Meyer set is a Delone set X such that X-X is a Delone set. Delone sets of finite type form a natural class for modeling quasicrystalline structures, because the property of being a Delone set of finite type is determined by ``local rules.' That is, a Delone set X is of finite type if and only if it has a finite number of neighborhoods of radius 2R , up to translation, where R is the relative denseness constant of X . Delone sets of finite type are also characterized as those finitely generated Delone sets such that the map ϕ satisfies the Lipschitz-type condition ||ϕ (x) - ϕ (x')|| < C ||x - x'|| for x, x' ∈X , where the norms || . . . || are Euclidean norms on ^s and ⁿ , respectively. Meyer sets are characterized as the subclass of Delone sets of finite type for which there is a linear map and a constant C such that ||ϕ (x) - (x)|| for all x∈X . Suppose that X is a Delone set with an inflation symmetry, which is a real number η > 1 such that . If X is a finitely generated Delone set, then η must be an algebraic integer; if X is a Delone set of finite type, then in addition all algebraic conjugates | η ' | η; and if X is a Meyer set, then all algebraic conjugates | η ' | 1. Received May 9, 1997, and in revised form March 5, 1998.     

On pointwise approximation of arbitrary functions by countable families of continuous functions

The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyf _n :n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F?X and every real number ε>0 one can choosen∈ω such that ∥f(x)?f_n(x)∥<ε for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A?X, there exists a continuous mapf _A:X→R ^ω such that A=f _A ^?1 (A). Splitting spaces will be studied systematically.