A note on shadowing with chain transitivity

Let f : X → X be a continuous map of a compact metric space X. The map f induces in a natural way a map f_M on the space M(X) of probability measures on X, and a transformation f_K on the space K(X) of closed subsets of X. In this paper, we show that if (X, f) is a chain transitive system with shadowing property, then exactly one of the following two statements holds:

(a)

fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1.

(b)

fⁿ and (f_K)ⁿ are equicontinuous for all n ? 1.

In particular, we show that for a continuous map f : X → X of a compact metric space X with infinite elements, if f is a chain transitive map with the shadowing property, then fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1. Also, we show that if f_M (resp. f_K) is chain transitive and syndetically sensitive, and f_M (resp. f_K) has the shadowing property, then f is sensitive.In addition, we introduce the notion of ergodical sensitivity and present a sufficient condition for a chain transitive system (X, f) (resp. (M(X), f_M)) to be ergodically sensitive. As an application, we show that for a L-hyperbolic homeomorphism f of a compact metric space X, if f has the AASP, then fⁿ is syndetically sensitive and multi-sensitive for all n ? 1.     

Contractive maps on normed linear spaces and their applications to nonlinear matrix equations

In this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach’s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X = I_n + A^∗f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). This will give conditions on A and f under which the equation has a unique solution in a certain set. We will consider two examples of f in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A.     

Uniform convergence and sequence of maps on a compact metric space with some chaotic properties

Recently, C. Tain and G. Chen introduced a new concept of sequence of time invariant function. In this paper we try to investigate the chaotic behavior of the uniform limit function f : X → X of a sequence of continuous topologically transitive (in strongly successive way) functions fn : X → X, where X is a compact interval. Surprisingly, we find that the uniform limit function is chaotic in the sense of Devaney. Lastly, we give an example to show that the denseness property of Devaney’s definition is lost on the limit function.     

On strong ergodicity and chaoticity of systems with the asymptotic average shadowing property

Let X be a compact metric space and f: X → X be a continuous map. In this paper, we investigate the relationships between the asymptotic average shadowing property (Abbrev. AASP) and other notions known from topological dynamics. We prove that if f has the AASP and the minimal points of f are dense in X, then for any n ? 1, f × f × ? × f(n times) is totally strongly ergodic. As a corollary, it is shown that if f is surjective and equicontinuous, then f does not have the AASP. Moreover we prove that if f is point distal, then f does not have the AASP. For f: [0, 1] → [0, 1] being surjective continuous, it is obtained that if f has two periodic points and the AASP, then f is Li-Yorke chaotic.     

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.     

Maps preserving the nilpotency of products of operators

Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:

(a)

There is a bijective bounded linear or conjugate-linear operator S:X→X such that ? has the form A?S[f(A)A]S^-1.

(b)

The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S⁻¹.

If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra M_n of n × n complex matrices, then there exist a map f:M_n→C?{0}, a field automorphism ξ:C→C, and an invertible S ∈ M_n such that ? has one of the following forms:

     

On topological entropy of transitive triangular maps

In the paper of Alsedà, Kolyada, Llibre and Snoha [L. Alsedà, S.F. Kolyada, J. Llibre, L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc. 351 (1999) 1551-1573] there was—among others—proved that a nonminimal continuous transitive map f of a compact metric space (X,ρ) can be extended to a triangular map F on X×I (i.e., f is the base for F) in such a way that F is transitive and has the same entropy as f. The presented paper shows that under certain conditions the extension of minimal maps is guaranteed, too: Let (X,f) be a solenoidal dynamical system. Then there exist a transitive triangular map F such that h(F)=h(f).     

Central limit theorem for integrated square error of kernel estimators of spherical density

LetX ₁,…,X _n be iid observations of a random variableX with probability density functionf(x) on the q-dimensional unit sphere Ω_q in R^q+1,q ⩾ 1. Let be a kernel estimator off(x). In this paper we establish a central limit theorem for integrated square error off _n under some mild conditions.     

Topological groups and convex sets homeomorphic to non-separable Hilbert spaces

Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover of X there is a sequence of maps (f _n : X → X) _nεgw such that each f _n is -near to the identity map of X and the family {f _n (X)} _n∈ω is locally finite in X. Also we show that a metrizable space X of density dens(X) < is a Hilbert manifold if X has gw-LFAP and each closed subset A ⊂ X of density dens(A) < dens(X) is a Z _∞-set in X.      

Resolving Extensions of Finitely Presented Systems

In this paper we extend certain central results of zero dimensional systems to higher dimensions. The first main result shows that if (Y,f) is a finitely presented system, then there exists a Smale space (X,F) and a u-resolving factor map π ₊:X→Y. If the finitely presented system is transitive, then we show there is a canonical minimal u-resolving Smale space extension. Additionally, we show that any finite-to-one factor map between transitive finitely presented systems lifts through u-resolving maps to an s-resolving map.     

Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory

Let (X_n) be a positive recurrent Harris chain on a general state space, with invariant probability measure π. We give necessary and sufficient conditions for the geometric convergence of λPⁿf towards its limit π(f), and show that when such convergence happens it is, in fact, uniform over f and in L¹(π)-norm. As a corollary we obtain that, when (Xn) is geometrically ergodic, ∝ π(dx)6Pⁿ(x,·)-π6 converges to zero geometrically fast. We also characterize the geometric ergodicity of (X_n) in terms of hitting time distributions. We show that here the so-called small sets act like individual points of a countable state space chain. We give a test function criterion for geometric ergodicity and apply it to random walks on the positive half line. We apply these results to non-singular renewal processes on [0,∞) providing a probabilistic approach to the exponencial convergence of renewal measures.     

Bernstein operators for extended Chebyshev systems

Let U_n ⊂ Cⁿ[a, b] be an extended Chebyshev space of dimension n + 1. Suppose that f₀ ∈ U_n is strictly positive and f₁ ∈ U_n has the property that f₁/f₀ is strictly increasing. We search for conditions ensuring the existence of points t₀, …, t_n ∈ [a, b] and positive coefficients α₀, …, α_n such that for all f ∈ C[a, b], the operator B_n:C[a, b] → U_n defined by satisfies B_nf₀ = f₀ and B_nf₁ = f₁. Here it is assumed that p_n,k, k = 0, …, n, is a Bernstein basis, defined by the property that each p_n,k has a zero of order k at a and a zero of order n − k at b.     

Some properties and applications of weakly equicompact sets

Let X and Y be Banach spaces. A set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x _n) in X, there exists a subsequence (x _k(n)) so that (Tx_k(n)) is uniformly weakly convergent for T ∈ M. In this paper, the notion of weakly equicompact set is used to obtain characterizations of spaces X such that X ↩̸ ℓ₁, of spaces X such that B _X* is weak* sequentially compact and also to obtain several results concerning to the weak operator and the strong operator topologies. As another application of weak equicompactness, we conclude a characterization of relatively compact sets in when this space is endowed with the topology of uniform convergence on the class of all weakly null sequences. Finally, we show that similar arguments can be applied to the study of uniformly completely continuous sets. Received: 5 July 2006     

Stability of the geometric Ekeland variational principle: Convex case

In geometric terms, the Ekeland variational principle says that a lower-bounded proper lower-semicontinuous functionf defined on a Banach spaceX has a point (x ₀,f(x ₀)) in its graph that is maximal in the epigraph off with respect to the cone order determined by the convex coneK _λ = {(x, α) ∈X × ?:λ ∥x∥ ≤ ? α}, where λ is a fixed positive scalar. In this case, we write (x ₀,f(x ₀))∈λ-extf. Here, we investigate the following question: if (x ₀,f(x ₀))∈λ-extf, wheref is a convex function, and if 〈f _n 〉 is a sequence of convex functions convergent tof in some sense, can (x ₀,f(x ₀)) be recovered as a limit of a sequence of points taken from λ-extf _n ? The convergence notions that we consider are the bounded Hausdorff convergence, Mosco convergence, and slice convergence, a new convergence notion that agrees with the Mosco convergence in the reflexive setting, but which, unlike the Mosco convergence, behaves well without reflexivity.     

On the Stability and Accuracy of Least Squares Approximations

We consider the problem of reconstructing an unknown function f on a domain X from samples of f at n randomly chosen points with respect to a given measure ρ _X . Given a sequence of linear spaces (V _m ) _m>0 with dim(V _m )=m≤n, we study the least squares approximations from the spaces V _m . It is well known that such approximations can be inaccurate when m is too close to n, even when the samples are noiseless. Our main result provides a criterion on m that describes the needed amount of regularization to ensure that the least squares method is stable and that its accuracy, measured in L ²(X,ρ _X ), is comparable to the best approximation error of f by elements from V _m . We illustrate this criterion for various approximation schemes, such as trigonometric polynomials, with ρ _X being the uniform measure, and algebraic polynomials, with ρ _X being either the uniform or Chebyshev measure. For such examples we also prove similar stability results using deterministic samples that are equispaced with respect to these measures.     

Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

This paper studies the convergence of the sequence defined by x0∈C,xn 1=αnu (1-αn)Txn,n=0,1,2,…, where 0 ≤αn ≤ 1, limn→∞αn = 0, ∑∞n=0 αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.     

Estimation de la transition de probabilité d'une chaîne de Markov doëblin-récurrente. étude du cas du processus autorégressif général d'ordre 1

We consider an homogenous Markov chain {X_n}. We estimate its transition probability density with kernel estimators. We apply these methods to the estimation of the unknown function f of the process defined by X₁ and X_n+1 = f(X_n) + ε_n, where {ε_n} is a noise (sequence of independent identically distributed random variables) of unknown law. The mean quadratic integrated rates of convergence are identical to those of classical density estimations. These risks are used here because we want some global informations about our estimates. We also study the average of those risks when the variance changes; it is shown that they reach a minimal value for some optimal variance. We study uniform convergence of our estimators. We finally estimate the variance of the noise and its density.     

A New Class of Harmonic Measure Distribution Functions

Let Ω be a planar domain containing 0. Let h _Ω (r) be the harmonic measure at 0 in Ω of the part of the boundary of Ω within distance r of 0. The resulting function h _Ω is called the harmonic measure distribution function of Ω. In this paper we address the inverse problem by establishing several sets of sufficient conditions on a function f for f to arise as a harmonic measure distribution function. In particular, earlier work of Snipes and Ward shows that for each function f that increases from zero to one, there is a sequence of multiply connected domains X _n such that \(h_{X_{n}}\) converges to f pointwise almost everywhere. We show that if f satisfies our sufficient conditions, then f=h _Ω , where Ω is a subsequential limit of bounded simply connected domains that approximate the domains X _n . Further, the limit domain is unique in a class of suitably symmetric domains. Thus f=h _Ω for a unique symmetric bounded simply connected domain Ω.