Distributionally scrambled set and minimal set

We investigate the relation between distributional chaos and minimal sets, and discuss how to obtain various distributionally scrambled sets by using least and simplest minimal sets. We show: i) an uncountable extremal distributionally scrambled set can appear in a system with just one simple minimal set: a periodic orbit with period 2; ii) an uncountable dense invariant distributionally scrambled set can occur in a system with just two minimal sets: a fixed point and an infinite minimal set; iii) infinitely many minimal sets are necessary to generate a uniform invariant distributionally scrambled set, and an uncountable dense extremal invariant distributionally scrambled set can be constructed by using just countably infinitely many periodic orbits.     

Devaney’s chaos implies distributional chaos in a sequence

Let X be a complete metric space without isolated points, and let f:X→X be a continuous map. In this paper we prove that if f is transitive and has a periodic point of period p, then f is distributionally chaotic in a sequence. Particularly, chaos in the sense of Devaney is stronger than distributional chaos in a sequence.     

On multi-transitivity with respect to a vector

A topological dynamical system(X,f)is said to be multi-transitive if for every n∈N the system(Xn,f×f2××fn)is transitive.We introduce the concept of multi-transitivity with respect to a vector and show that multi-transitivity can be characterized by the hitting time sets of open sets,answering a question proposed by Kwietniak and Oprocha(2012).We also show that multi-transitive systems are Li-Yorke chaotic.     

Connected level sets,minimizing sets,and uniqueness in optimization

Intimate relationships are investigated between connectedness properties of the lower level sets of a real functionf on a topological spaceX and the uniqueness of suitably defined minimizing sets forf. Two distinct theories are presented, the simpler one pertaining to the LE-level sets $$LE_\alpha (f) = \{ x \in X|f(x) \leqslant \alpha \} $$ and the other to the LT-level sets $$LT_\alpha (f) = \{ x \in X|f(x) \leqslant \alpha \} .$$ In each theory, a specific notion of minimizing set is defined in such a way that a functionf having connected level sets can have at most one minimizing set. That this uniqueness is not trivial, however, is shown by the converse result that, ifX is Hausdorff and the sets LE_α(f) are all compact, then, in each theory,f has a unique minimizing set only if it has connected level sets. The paper concludes by showing that functions with connected LT-level sets arise naturally in parametric linear programming.     

Syndetically proximal pairs

For continuous self-maps of compact metric spaces, we study the syndetically proximal relation, and in particular we identify certain sufficient conditions for the syndetically proximal cell of each point to be small. We show that any interval map f with positive topological entropy has a syndetically scrambled Cantor set, and an uncountable syndetically scrambled set invariant under some power of f. In the process of proving this, we improve a classical result about interval maps and establish that if f is an interval map with positive topological entropy and m?2, then there is n∈N such that the one-sided full shift on m symbols is topologically conjugate to a subsystem of fn² (the classical result gives only semi-conjugacy).     

On Gibson functions with connected graphs

A function f: X → Y between topological spaces is said to be a weakly Gibson function if $f(\bar G) \subseteq \overline {f(G)} $ for any open connected set G ? X. We call a function f: X → Y segmentary connected if X is topological vector space and f([a, b]) is connected for every segment [a, b] ? X. We show that if X is a hereditarily Baire space, Y is a metric space, f: X → Y is a Baire-one function and one of the following conditions holds:

1. X is a connected and locally connected space and f is a weakly Gibson function
2. X is an arcwise connected space and f is a Darboux function
3. X is a topological vector space and f is a segmentary connected function, then f has a connected graph.

     

Algorithms for finding connected separators between antipodal points

A set (or a collection of sets) contained in the Euclidean space Rm is symmetric if it is invariant under the antipodal map. Given a symmetric unicoherent polyhedron X (like an n-dimensional cube or a sphere) and an odd real function f defined on vertices of a certain symmetric triangulation of X, we algorithmically construct a connected symmetric separator of X by choosing a subcollection of the triangulation. Each element of the subcollection contains the vertices v and u such that f(v)f(u)?0.     

P-chaos implies distributional chaos and chaos in the sense of Devaney with positive topological entropy

Let f be a continuous map from a compact metric space X to itself. The map f is called to be P-chaotic if it has the pseudo-orbit-tracing property and the closure of the set of all periodic points for f is equal to X. We show that every P-chaotic map from a continuum to itself is chaotic in the sense of Devaney and exhibits distributional chaos of type 1 with positive topological entropy.     

On bounding,interior and covering functions

We prove three theorems yielding sufficient conditions for a continuous function f: X → Y to have no isolated bounding points (i.e., points at which it is not an open mapping), to be interior a z (i.e.,f(z) ?f(X)⁰), and to have its image cover a segment. In the first theorem X and Y are rather general topological spaces and some applications to optimal control are discussed. In the second and third theorems X an Y are finite-dimensional and we use the concepts of ordinary and unbounded derivate containers which are a form of set-valued derivatives. The proof of the second theorem also involves the topological degree of f.     

On the Form of Subobjects in Semi-Abelian and Regular Protomodular Categories

Let Gls denote the category of (possibly large) ordered sets with Galois connections as morphisms between ordered sets. The aim of the present paper is to characterize semi-abelian and regular protomodular categories among all regular categories ?, via the form of subobjects of ?, i.e. the functor ? → Gls which assigns to each object X in ? the ordered set Sub(X) of subobjects of X, and carries a morphism f : X → Y to the induced Galois connection Sub(X) → Sub(Y) (where the left adjoint maps a subobject m of X to the regular image of fm, and the right adjoint is given by pulling back a subobject of Y along f). Such functor amounts to a Grothendieck bifibration over ?. The conditions which we use to characterize semi-abelian and regular protomodular categories can be stated as self-dual conditions on the bifibration corresponding to the form of subobjects. This development is closely related to the work of Grandis on “categorical foundations of homological and homotopical algebra”. In his work, forms appear as the so-called “transfer functors” which associate to an object the lattice of “normal subobjects” of an object, where “normal” is defined relative to an ideal of null morphism admitting kernels and cokernels.     

Jensen's inequality for a convex vector-valued function on an infinite-dimensional space

Jensen's inequality f(EX) ≤ Ef(X) for the expectation of a convex function of a random variable is extended to a generalized class of convex functions f whose domain and range are subsets of (possibly) infinite-dimensional linear topological spaces. Convexity of f is defined with respect to closed cone partial orderings, or more general binary relations, on the range of f. Two different methods of proof are given, one based on geometric properties of convex sets and the other based on the Strong Law of Large Numbers. Various conditions under which Jensen's inequality becomes strict are studied. The relation between Jensen's inequality and Fatou's Lemma is examined.     

On the existence of continuous (approximate) roots of algebraic equations

The present paper considers the existence of continuous roots of algebraic equations with coefficients being continuous functions defined on compact Hausdorff spaces. For a compact Hausdorff space X, C(X) denotes the Banach algebra of all continuous complex-valued functions on X with the sup norm ∥⋅_∥∞. The algebra C(X) is said to be algebraically closed if each monic algebraic equation with C(X) coefficients has a root in C(X). First we study a topological characterization of a first-countable compact (connected) Hausdorff space X such that C(X) is algebraically closed. The result has been obtained by Countryman Jr, Hatori-Miura and Miura-Niijima and we provide a simple proof for metrizable spaces.Also we consider continuous approximate roots of the equation zn−f=0 with respect to z, where f∈C(X), and provide a topological characterization of compact Hausdorff space X with dimX?1 such that the above equation has an approximate root in C(X) for each f∈C(X), in terms of the first ?ech cohomology of X.     

Topological entropy of maps on regular curves

In [G.T. Seidler, The topological entropy of homeomorphisms on one-dimensional continua, Proc. Amer. Math. Soc. 108 (1990) 1025-1030], G.T. Seidler proved that the topological entropy of every homeomorphism on a regular curve is zero. Also, in [H. Kato, Topological entropy of monotone maps and confluent maps on regular curves, Topology Proc. 28 (2) (2004) 587-593] the topological entropy of confluent maps on regular curves was investigated. In particular, it was proved that the topological entropy of every monotone map on any regular curve is zero. In this paper, furthermore we investigate the topological entropy of more general maps on regular curves. We evaluate the topological entropy of maps f on regular curves X in terms of the growth of the number of components of f−n(y) (y∈X).     

On the Fell topology

Let 2 ^X denote the closed subsets of a Hausdorff topological space <X, {gt}>. The Fell topology τ_F on 2 ^X has as a subbase all sets of the form {A ∈ 2 ^X :A ∩V ≠ 0}, whereV is an open subset ofX, plus all sets of the form {A ∈ 2 ^X :A ?W}, whereW has compact complement. The purpose of this article is two-fold. First, we characterize first and second countability for τ_F in terms of topological properties for τ. Second, we show that convergence of nets of closed sets with respect to the Fell topology parallels Attouch-Wets convergence for nets of closed subsets in a metric space. This approach to set convergence is highly tractable and is well-suited for applications. In particular, we characterize Fell convergence of nets of lower semicontinuous functions as identified with their epigraphs in terms of the convergence of sublevel sets.     

Resolutions of topological linear spaces and continuity of linear maps

The main result of the paper is the following: If an F-space X is covered by a family of sets such that Eα⊂Eβ whenever α?β, and f is a linear map from X to a topological linear space Y which is continuous on each of the sets Eα, then f is continuous. This provides a very strong negative answer to a problem posed recently by J. Ka?kol and M. López Pellicer. A number of consequences of this result are given, some of which are quite curious. Also, inspired by a related question asked by J. Ka?kol, it is shown that if a linear map is continuous on each member of a sequence of compact sets, then it is also continuous on every compact convex set contained in the linear span of the sequence. The construction applied to prove this is then used to interpret a natural linear topology associated with the sequence as the inductive limit topology in the sense of Ph. Turpin, and thus derive its basic properties.     

Isomorphism and Embedding of Borel Systems on Full Sets

A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category, up to isomorphism, there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy <t, and if (X,T) is another free system obeying the same entropy condition then X embeds into Y off a null set. One gets a strictly t-universal system from mixing shifts of finite type of entropy ≥t by removing the periodic points and “restricting” to the part of the system of entropy <t. As a consequence, after removing their periodic points the systems in the following classes are completely classified by entropy up to Borel isomorphism off null sets: mixing shifts of finite type, mixing positive-recurrent countable state Markov chains, mixing sofic shifts, beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular any two equal-entropy systems from these classes are entropy conjugate in the sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.     

Implications of pseudo-orbit tracing property for continuous maps on compacta

We look at the dynamics of continuous self-maps of compact metric spaces possessing the pseudo-orbit tracing property (i.e., the shadowing property). Among other things we prove the following: (i) the set of minimal points is dense in the non-wandering set Ω(f), (ii) if f has either a non-minimal recurrent point or a sensitive minimal subsystem, then f has positive topological entropy, (iii) if X is infinite and f is transitive, then f is either an odometer or a syndetically sensitive non-minimal map with positive topological entropy, (iv) if f has zero topological entropy, then Ω(f) is totally disconnected and f restricted to Ω(f) is an equicontinuous homeomorphism.     

Topological-algebraic properties of function spaces with set-open topologies

For a Tychonoff space X, we denote by Cλ(X) the space of all real-valued continuous functions on X with set-open topology. In this paper, we study the topological-algebraic properties of Cλ(X). Our main results state that (1) Cλ(X) is a topological vector space (a topological group) iff λ is a family of C-compact sets and Cλ(X)=Cλ^′(X), where λ^′ consists of all C-compact subsets of every set of λ. In particular, if Cλ(X) is a topological group, then the set-open topology coincides with the topology of uniform convergence on a family λ; (2) a topological group Cλ(X) is ω-narrow iff λ is a family of metrizable compact subsets of X.