Equicontinuity of maps on a dendrite with finite branch points

Let(T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote byω(x,f) and P(f) the ω-limit set of x under f and the set of periodic points of,respectively. Write Ω(x,f) = {y| there exist a sequence of points x_k E T and a sequence of positive integers n_1 n_2 … such that lim_(k→∞)x_k=x and lim_(k→∞)f~(n_k)(x_k) =y}. In this paper, we show that the following statements are equivalent:(1) f is equicontinuous.(2) ω(x, f) = Ω(x,f) for any x∈T.(3) ∩_(n=1)~∞f~n(T) = P(f),and ω(x,f)is a periodic orbit for every x ∈ T and map h : x→ω(x,f)(x ET)is continuous.(4) Ω(x,f) is a periodic orbit for any x∈T.     

高阶Hermite-Fejer型插值的渐近估计

1. Introduction Let f∈C[-1,1] and X_k=X_(kn)=COSθ_k=COS(2k-1)π/(2n)(k=1,…,n) be the zeros of the Chebyshev polynomial T_n(x)=cosnθ(x=cosθ). Let ω(t) be a given modulus of continuity and H_ω={f;ω(f,t)≤ω(t),for all.t≥0}. In this paper, c will always denote different constant independent of x, n and f and the sign"A～B" means that there exist two positive constants c_1相似文献   

Non-wandering sets of the powers of dendrite maps

Let(X, d) be a metric space and f be a continuous map from X to X. Denote by EP(f)and Ω(f) the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold:(1) If x ∈Ω(f)-Ω(f~n) for some n ≥ 2,then x ∈ EP(f).(2) Ω(f) is contained in the closure of EP(f). The aim of this note is to show that the above results do not hold for maps of dendrites D with Card(End(D)) = ?0(the cardinal number of the set of positive integers).     

整数集的加权表示的渐进基

Let k1, k2 be nonzero integers with(k1, k2) = 1 and k1k2≠-1. Let Rk1,k2(A, n)be the number of solutions of n = k1a1 + k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A  Z such that Rk1,k2(A, n) = 1 for all n ∈ Z. Let f : Z-→ N0∪ {∞} be a function such that f-1(0) is finite. In this paper, we generalize Xiong's result and prove that there exist uncountably many sets A  Z such that Rk1,k2(A, n) = f(n) for all n ∈ Z.     

Topological entropy of a graph map

Let G be a graph and f : G → G be a continuous map. Denote by h(f), P(f), AP(f), R(f)and ω(x, f) the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the ω-limit set of x under f, respectively. In this paper,we show that the following statements are equivalent:(1) h(f) 0.(2) There exists an x ∈ G such that ω(x, f) ∩ P(f) = ? and ω(x, f) is an infinite set.(3) There exists an x ∈ G such that ω(x, f)contains two minimal sets.(4) There exist x, y ∈ G such that ω(x, f)-ω(y, f) is an uncountable set and ω(y, f) ∩ω(x, f) = ?.(5) There exist an x ∈ G and a closed subset A ? ω(x, f) with f(A) ? A such that ω(x, f)-A is an uncountable set.(6) R(f)-AP(f) = ?.(7) f |P(f)is not pointwise equicontinuous.     

Non-wandering points and the depth for graph maps

Let G be a graph and f:G→G be continuous.Denote by R(f) andΩ(f) the set of recurrent points and the set of non-wandering points of f respectively.LetΩ_0(f) = G andΩ_n(f)=Ω(f|_(Ω_(n-1)(f))) for all n∈N.The minimal m∈NU {∞} such thatΩ_m(f)=Ω_(m 1)(f) is called the depth of f.In this paper,we show thatΩ_2 (f)=(?) and the depth of f is at most 2.Furthermore,we obtain some properties of non-wandering points of f.     

树映射的链回归点与拓扑熵

Let f be a tree map,P(f) the set of periodic points of f and CR(f) the set of chain recurrent points of f. In this paper,the notion of division for invariant closed subsets of a tree map is introduced. It is proved that: (1) fhas zero topological entropy if and only if for any x∈CR(f)-P(f) and each natural number s the orbit of x under f^5 has a division; (2) If f has zero topological entropy,then for any xECR(f)--P(f) the w-limit set of x is an infinite minimal set.     

Some Equivalent Representations of Nonsquare Constants and Its Applications

Let(X,‖·‖ ) be a normed space.Let S(X) ={ x∈X:‖x‖ =1 } and B(X) ={ x∈X:‖ x‖≤ 1 } be the unit sphere and unit ball of X,respectively.In 1 990 ,Gao andLau[1 ] introduced the following definition.　 Definition　 The Parameter CJ(X) of a normed space X,which will be called a non-square constant in the sense of James in this paper,is defined byCJ(X) =sup{‖ x + y‖∧‖ x -y‖ :x,y∈ S(X) } .The nonsquare constant CS(X) of a normed space X in the sense of Schaffer is definedbyC…     

INTERSECTION OF TRANSLATIONS OF CANTOR TRIADIC SET

Let C be the Cantor triadic set and let Cα=C+α={β+α: β∈C} for -1≤α≤ 1. Let Hp={α∈C: dim H(Cα∩C)=dim B(C α∩C)=(1-p)log 2/log 3}, 0＜ p＜ 1. The authors give the dimensions of C α∩C and Hp. In additionthe characteristic of Hp is described by means of some measure μ supported on C.     

Comonotone approximation with interpolation at the ends on an interval

Let a function f ∈ C[-1, 1], changes its monotonisity at the finite collection Y := {y1,… ,ys} of s points yi ∈ (-1, 1). For each n ≥ N(Y), we construct an algebraic polynomial Pn, of degree ≤ n, which is comonotone with f, that is changes its monotonisity at the same points yi as f, and |f(x)-Pn(x)|≤c(s)ω2(f,(√1-x2)/n), x∈[-1,1],where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s and ω2 (f, t) is the second modulus of smoothness of f.     

Non-wandering sets of the powers of maps of a tree

Let T be a tree and let Ω ( f ) be the set of non-wandering points of a continuous map f: T→ T. We prove that for a continuous map f: T→ T of a tree T: ( i) if x∈ Ω( f) has an infinite orbit, then x∈ Ω( fⁿ) for each n∈ ℕ; (ii) if the topological entropy of f is zero, then Ω( f) = Ω( fⁿ) for each n∈ ℕ. Furthermore, for each k∈ ℕ we characterize those natural numbers n with the property that Ω(f^k) = Ω(f^kn) for each continuous map f of T.     

Topological Structure of Non-wandering Set of a Graph Map

Let G be a graph （i.e., a finite one-dimensional polyhedron） and f ： G → G be a continuous map. In this paper, we show that every isolated recurrent point of f is an isolated non-wandering point; every accumulation point of the set of non-wandering points of f with infinite orbit is a two-order accumulation point of the set of recurrent points of f; the derived set of an ω-limit set of f is equal to the derived set of an the set of recurrent points of f; and the two-order derived set of non-wandering set of f is equal to the two-order derived set of the set of recurrent points of f.     

Conditions of C0 flows on closed surfaces having the pseudo-orbit tracing property

Let M be a closed surface, orientable or non-orientable, and letf be a C° flow onM of which all singular points are isolated. Thenf has the pseudo-orbit tracing property if and only if (i) for anyx ∈ M, both the ωlimit set ω(x) and the α-limit set α(x) ofx contain only one orbit; (ii) for any regular pointx off, if ω(x) is not quasi-attracting, then α(x) is quasi-exclusive; (iii) every saddle point off is strict, and at most 4-forked. Project supported by the National Natural Science Foundation of China.     

Sobolev functions whose inner trace at the boundary is zero

Let Ω⊂R ⁿ be an arbitrary open set. In this paper it is shown that if a Sobolev functionf∈W ^1,p (Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, thenf is weakly zero on ϖΩ in the sense thatf∈W ₀ ^1,p (Ω).     

Potential space estimates in local Hardy spaces for Green potentials in convex domains

Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)＜p≤1.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.     

Dynamics of neighborhoods of points under a continuous mapping of an interval

Let {I, f, Z ⁺} be a dynamical system induced by a continuous mapping f of a closed bounded interval I into itself. To describe the dynamics of neighborhoods of points unstable under the mapping f, we propose the concept of the εω-set ω ^{f, ε}(x) of a point x as the ω-limit set of the ε-neighborhood of the point x. We investigate the relationship between the εω-set and the domain of influence of a point. It is also shown that the domain of influence of an unstable point is always a cycle of intervals. The results obtained can be directly used in the theory of difference equations with continuous time and similar equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1534–1547, November, 2005.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities are finite for all if and only if ∂Ω and ∂Π do not contain isolated points. This work was supported by a grant of the Deutsche Forschungsgemeinschaft for F. G. Avkhadiev.