Almost periodicity,chain recurrence and chaos

In this paper we consider a continuous mapf: X→X, whereX is a compact metric space. The existence of chaotic sets off is discussed. For the special caseX=[0,1], we prove thatf has a positive topological entropy iff it has an uncountable chaotic set in which each point is almost periodic, and iff it has an uncountable chaotic set in which each point is chain recurrent. As an application, a uniform proof for some known results will be given.     

Devaney’s chaos on uniform limit maps

Let (X, d) be a compact metric space and f_n : X → X a sequence of continuous maps such that (f_n) converges uniformly to a map f. The purpose of this paper is to study the Devaney’s chaos on the uniform limit f. On the one hand, we show that f is not necessarily transitive even if all f_n mixing, and the sensitive dependence on initial conditions may not been inherited to f even if the iterates of the sequence have some uniform convergence, which correct two wrong claims in [1]. On the other hand, we give some equivalence conditions for the uniform limit f to be transitive and to have sensitive dependence on initial conditions. Moreover, we present an example to show that a non-transitive sequence may converge uniformly to a transitive map.     

Oscillation for almost continuity

Let X be a topological space and (Y,d) be a metric space. If f: X → Y is a function then there is a function a _f : X → [0, ∞] such that f is almost continuous at x if and only if a _f (x) = 0. Some properties of this function are investigated. Supported by grant VEGA 2/6087/26 and APVT-51-006904.     

Order of a function on the Bruschlinsky group of a two-dimensional polyhedron

Homotopy classes of mappings of a compact polyhedron X to the circle T form an Abelian group B(X), which is called the Bruschlinsky group and is cananically isomorphic to H ¹ (X; ℤ), Let L be an Abelian group, and let f: B(X) → L be a function. One says that the order of f does not exceed r if for each mapping a: X → T the value f([a]) is ℤ-linearly expressed via the characteristic function I _r (a): (X × T) ^r → ℤ of (Γ _a ) ^r , where Γ _a ⊂ X × T is the graph of a. The (algebraic) degree of f is not greater than r if the finite differences of f of order r + 1 vanish. Conjecturally, the order of f is equal to the algebraic degree of f. The conjecture is proved in the case where dim X ≤ 2. Bibliography: 1 title.     

Weak local homeomorphisms and <Emphasis Type="Italic">B</Emphasis>-favorable spaces

Let X and Y be topological spaces such that an arbitrary mapping f: X → Y for which every preimage f ⁻¹ (G) of a set G open in Y is an F _σ-set in X can be represented in the form of the pointwise limit of continuous mappings f _n : X → Y. We study the problem of subspaces Z of the space Y for which the mappings f: X → Z possess the same property. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1189–1195, September, 2008.     

Homotopy invariants of mappings to the circle

Homotopy classes of mappings of a space X to the circle T form an Abelian group B(X) (the Bruschlinsky group). If a: X → T is a continuous mapping, then [a] denotes the homotopy class of a, and I _r (a): (X × T) _r → \mathbbZ \mathbb{Z} is the indicator function of the rth Cartesian power of the graph of a. Let C be an Abelian group and let f: B(X) → C be a mapping. By definition, f has order not greater than r if the correspondence I _r (a) → f([a]) extends to a (partly defined) homomorphism from the Abelian group of Z-valued functions on (X × T) ^r to C. It is proved that the order of f equals the algebraic degree of f. (A mapping between Abelian groups has degree at most r if all of its finite differences of order r +1 vanish.) Bibliography: 2 titles.     

On the almost monotone convergence of sequences of continuous functions

A sequence (f _n ) _n of functions f _n : X → ℝ almost decreases (increases) to a function f: X → ℝ if it pointwise converges to f and for each point x ∈ X there is a positive integer n(x) such that f _n+1(x) ≤ f _n (x) (f _n+1(x) ≥ f _n (x)) for n ≥ n(x). In this article I investigate this convergence in some families of continuous functions.     

Chaos caused by a strong-mixing measure-preserving transformation

The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m _i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer _i such that lim_i→∞ f ^r i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension. the National Natural Science Foundation of China.     

Convergence theorems for the Birkhoff integral

We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence (f _n ) of functions from a measure space to a Banach space. In one result the equi-integrability of f _n ’s is involved and we assume f _n → f almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of (f _n ) to f is assumed.     

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.     

Idealization of a decomposition theorem

In 1986, Tong [13] proved that a function f : (X,τ)→(Y,φ) is continuous if and only if it is α-continuous and A-continuous. We extend this decomposition of continuity in terms of ideals. First, we introduce the notions of regular-I-closed sets, A _I-sets and A _I -continuous functions in ideal topological spaces and investigate their properties. Then, we show that a function f : (X,τ,I)→(Y, φ) is continuous if and only if it is α-I-continuous and A _I-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

极限跟踪的不变性

It is showed that the LmSP (limit shadowing property) for flows is invariant under topological equivalence, and the suspension flow ϕ _f of a homeomorphism f under a continuous function ϑ: X → R>0 has LmSP₊ if and only if f has LmSP₊. As examples of application, the LmSP₊ of the ergodic flow on torus which has irrational number is discussed, and the characterization of a class of flows on torus with LmSP₊ is given. Supported by the National Natural Science Foundation of China (10371030) and the doctor’s foundation of Hebei Normal Univ. (L2003B05).     

Algebraically constructible functions and signs of polynomials

LetW be a real algebraic set. We show that the following families of integer-valued functions onW coincide: (i) the functions of the formω →λ(X _ω ), where X _ω are the fibres of a regular morphismf :X →W of real algebraic sets, (ii) the functions of the formω →χ(X _ω ), where X _ω are the fibres of a proper regular morphismf :X →W of real algebraic sets, (iii) the finite sums of signs of polynomials onW. Such functions are called algebraically constructible onW. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization. Research partially supported by an Australian Research Council Small Grant. Second author also partially supported by KBN 610/P3/94.     

Equivalent conditions for n-equivalences modulo a classc of groups

In this work we define the notion of a mapf:XY to be ann-equivalence modulo a classC of groups. Then we show an equivalent condition, which is more close to a homological condition, in order to a mapf:XY to be ann-equivalence modulo a classC of groups. Finally, at least for a complexK which is finite and is a suspension of a connected space, the notion above is also given in terms of the mapf _#: [K, X][K, Y].     

A Convenient Subcategory of Tych

A map f:XY between Hausdorff topological spaces is k-continuous if its restriction f| _K to every compact subspace K of X is continuous. X is called a k _R -space if every k-continuous function from X to a Tychonoff space is continuous. In this paper we investigate the category of Tychonoff k _R -spaces, and show that it is Cartesian closed (thus convenient in the sense of Wyler).     

The structure of LC-continuous functions

A function f is LC-continuous if the inverse image of any open set is a locally closed set; i.e., an intersection of an open set and a closed set. The aim of this paper is to prove the following theorem: Let f: X→Y be an LC-continuous function onto a separable metric space Y. Then X can be covered by countably many subsets T _n ⊂X such that each restriction f∣T _n is continuous at all points of T _n .     

A function space model approach to the rational evaluation subgroups

Let f : U → X be a map from a connected nilpotent space U to a connected rational space X. The evaluation subgroup G _*(U, X; f), which is a generalization of the Gottlieb group of X, is investigated. The key device for the study is an explicit Sullivan model for the connected component containing f of the function space of maps from U to X, which is derived from the general theory of such a model due to Brown and Szczarba (Trans Am Math Soc 349, 4931–4951, 1997). In particular, we show that non Gottlieb elements are detected by analyzing a Sullivan model for the map f and by looking at non-triviality of higher order Whitehead products in the homotopy group of X. The Gottlieb triviality of a fibration in the sense of Lupton and Smith (The evaluation subgroup of a fibre inclusion, 2006) is also discussed from the function space model point of view. Moreover, we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space. In consequence, the first Gottlieb group of the total space of each S ¹-bundle over the n-dimensional torus is determined explicitly in the non-rational case.      

Semi-stable models for rigid-analytic spaces

 Let R be a complete discrete valuation ring with field of fractions K and let X _K be a smooth, quasi-compact rigid-analytic space over S_p K. We show that there exists a finite separable field extension K' of K, a rigid-analytic space X' _K' over S_p K' having a strictly semi-stable formal model over the ring of integers of K', and an étale, surjective morphism f : X' _K' →X _K of rigid-analytic spaces over S_p K. This is different from the alteration result of A.J. de Jong [dJ] who does not obtain that f is étale. To achieve this property we have to work locally on X _K , i.e. our f is not proper and hence not an alteration. Received: 26 October 2001 / Revised version: 14 August 2002 Published online: 14 February 2003     

Quasi-polynomial functions over bounded distributive lattices

In [6] the authors introduced the notion of quasi-polynomial function as being a mapping f : X ⁿ → X defined and valued on a bounded chain X and which can be factorized as ${f(x_1,\ldots,x_n)=p(\varphi(x_1),\ldots,\varphi(x_n))}In [6] the authors introduced the notion of quasi-polynomial function as being a mapping f : X ⁿ → X defined and valued on a bounded chain X and which can be factorized as f(x₁,?,x_n)=p(j(x₁),?,j(x_n)){f(x_1,\ldots,x_n)=p(\varphi(x_1),\ldots,\varphi(x_n))} , where p is a polynomial function (i.e., a combination of variables and constants using the chain operations ù{\wedge} and ú{\vee}) and j{\varphi} is an order-preserving map. In the current paper we study this notion in the more general setting where the underlying domain and codomain sets are, possibly different, bounded distributive lattices, and where the inner function is not necessarily order-preserving. These functions appear naturally within the scope of decision making under uncertainty since, as shown in this paper, they subsume overall preference functionals associated with Sugeno integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-polynomial functions, we propose several generalizations of well-established properties in aggregation theory, as well as show that some of the characterizations given in [6] still hold in this general setting. Moreover, we investigate the so-called transformed polynomial functions (essentially, compositions of unary mappings with polynomial functions) and show that, under certain conditions, they reduce to quasi-polynomial functions.