Sequence Of Continuous Functions Research Articles

Topological pressure of a factor map for nonautonomous dynamical systems

In this paper, we give the definition of topological pressure on a noncompact subset for nonautonomous dynamical systems. Moreover, we discuss a relation for two topological pressures and establish an inequality formula for two topological pressures with a factor map of nonautonomous dynamical systems.

Three Types of Distributional Chaos for a Sequence of Uniformly Convergent Continuous Maps

Let h s s = 1 ∞ be a sequence of continuous maps on a compact metric space W which converges uniformly to a continuous map h on W . In this paper, some equivalence conditions or necessary conditions for the limit map h to be distributional chaotic are obtained (where distributional chaoticity includes distributional chaotic in a sequence, distributional chaos of type 1 (DC1), distributional chaos of type 2 (DC2), and distributional chaos of type 3 (DC3)).

LINEAR ODE’S WITH LAURENT POLYNOMIAL COEFFICIENTS

The forward and backward shift operators (and hence all Laurent polynomials in these operators) act on two-sided infinite sequences of continuous functions as well as the differentiation operator. One can define therefore linear ODEs with Laurent polynomial coefficients, where the unknowns are such sequences. It turns out that equations of this type can be treated easily, exactly as linear ODEs with constant coefficients. Our motivation for considering these ODEs, which seem to be quite natural on their own, has been the fact that the collection of all Bessel functions is characterized by a very simple first order equation of this kind.

Chaotic Behaviors of One-Dimensional Wave Equations with van der Pol Boundary Conditions Containing a Source Term

For one-dimensional wave equations with the van der Pol boundary conditions, there have been several different ways in the literature to characterize the complexity of their solutions. However, if the right-end van der Pol boundary condition contains a source term, then a considerable technical difficulty arises as to how to describe the complexity of the system. In this paper, we take advantage of a topologically dynamical method to characterize the dynamical behaviors of the systems, including sensitivity, transitivity and Li–Yorke chaos. For this end, we consider a system [Formula: see text] induced by a sequence of continuous maps and its functional envelope [Formula: see text], and show that, under some considerable condition, [Formula: see text] is transitive if and only if [Formula: see text] is weakly mixing of order [Formula: see text]; [Formula: see text] is Li–Yorke chaotic and sensitive if [Formula: see text] is strongly mixing. Those abstract results have their own significance and can be applied to such kind of equations.

Recurrence in non-autonomous dynamical systems

ABSTRACT We consider a sequence of continuous maps on a compact metric space X uniformly converging to a function f. This sequence forms a non-autonomous discrete dynamical system. In such case, the set of omega-limit points is invariant with respect to the limit function f. Here we give negative answer to questions whether the sets of recurrent points and non-wandering points are also invariant. We also discuss the relation of the set of recurrent points of and its limit function f.

Variational principle and zero temperature limits of asymptotically (sub)-additive projection pressure

Let $$\{S_i\}_{i=1}^l$$ be an iterated function system (IFS) on ℝd with an attractor K. Let (∑, σ) denote the one-sided full shift over the finite alphabet {1, 2,...,l}, and let π : ∑ → K be the coding map. Given an asymptotically (sub)-additive sequence of continuous functions ℱ = {fn}n⩾1; we define the asymptotically additive projection pressure Pπ(ℱ) and show the variational principle for Pπ(ℱ) under certain affne IFS. We also obtain variational principle for the asymptotically sub-additive projection pressure if the IFS satisfies asymptotically weak separation condition (AWSC). Furthermore, when the IFS satisfies AWSC, we investigate the zero temperature limits of the asymptotically sub-additive projection pressure Pπ(βℱ) with positive parameter β.

Sequences of maps and convergence properties of first-order difference equations with variable coefficients

We consider sequences of continuous functions on that converge to a continuous monotone limit function. We prove a convergence theorem for the iterative map This result is used to solve an open problem in the field concerning the convergence properties of first-order rational difference equations that are linear in numerator and denominator with variable coefficients. We also establish convergence properties for a certain class of systems of first-order difference equations.

Interpolation of bounded sequences by $\alpha $-dense curves

In 1905 Lebesgue showed that there is a sequence of continuous functions, put $f_{n}:[0,1]\longrightarrow [0,1]$, which interpolates any sequence in $[0,1]$, that is, given $(a_{n})_{n\geq 1}\subset [0,1]$ there is $t\in [0,1]$ such that $f_{n}(t)=a_{n}$ for each positive integer $n$. This result was improved (in the sense of Theorem ) in 1998 by Y. Benyamini. In this paper, we generalize the Benyamini's result in Theorem . The key tool for this goal are the so called $\alpha $-dense curves. We apply our results to approach the solution of a certain infinite-dimensional linear program with a countable number of constraints.

Baire classification of fragmented maps and approximation of separately continuous functions

We study a class of fragmented maps which contains all barely continuous maps, scatteredly continuous maps and maps which are pointwise discontinuous on each closed set. We prove that for any Hausdorff space X, a metric space Z and a fragmented map \(f:X\rightarrow Z\) there exists a pointwisely convergent to f sequence of continuous functions \(f_n:X\rightarrow Z\) with an additional condition (which is a weakening of the uniform convergence at each point) in the following cases: (a) X is stratifiable and Z is locally convex equiconnected; (b) X is stratifiable with \(\dim X<\infty \) and Z is equiconnected; (c) X is stratifiable with \(\dim X=0\). We show that for a hereditarily Baire space X, a compact space Y and a metric space Z every separately continuous map Open image in new window is a pointwise limit of a sequence of continuous maps which is uniformly convergent to f with respect to each variable at every point of Open image in new window in cases (a)–(c); (d) X is a perfect zero-dimensional compact space; and (e) X is the Sorgenfrey line.

On the continuity of global attractors

1. GlobalattractorsThe global attractor of a dynamical system is the unique compact invariant setthat attracts the trajectories starting in any bounded set at a uniform rate. Intro-duced by Billotti & LaSalle [3], they have been the subject of much research sincethe mid-1980s, and form the central topic of a number of monographs, includingBabin & Vishik [1], Hale [9], Ladyzhenskaya [13], Robinson [16], and Temam [18].The standard theory incorporates existence results [3], upper semicontinuity [10],and bounds on the attractordimension [7]. Global attractorsexist for many infinite-dimensional models [18], with familiar low-dimensional ODE models such as theLorenz equations providing a testing ground for the general theory [8].While upper semicontinuity with respect to perturbations is easy to prove, lowersemicontinuity (and hence full continuity) is more delicate, requiring structuralassumptions on the attractor or the assumption of a uniform attraction rate. How-ever, Babin & Pilyugin [2] proved that the global attractor of a parametrised setof semigroups is continuous at a residual set of parameters, by taking advantage ofthe known upper semicontinuity and then using the fact that upper semicontinuousfunctions are continuous on a residual set.Here we reprove results on equi-attraction and residual continuity in a moredirect way, which also serves to demonstrate more clearly why these results aretrue. Given equi-attraction the attractor is the uniform limit of a sequence ofcontinuous functions, and hence continuous (the converse requires a generalisedversion of Dini’s Theorem); more generally, it is the pointwise limit of a sequence ofcontinuous functions, i.e. a ‘Baire one’ function, and therefore the set of continuitypoints forms a residual set.

Функції зі зв'язним графіком та $B_1$-ретракти

Підмножина $E$ топологічного простору $X$ називається $B_1$-ретрактом цього простору, якщо існує відображення $r:X\to E$, яке є поточковою границею послідовності неперервних відображень $r_n:X\to E$, i таке, що $r(x)=x$ для всіх $x\in E$. Доводиться, що графік функції $f:\mathbb R\to Y$, де простір $Y$ - це об'єднання зростаючої послідовності континуумів, є $B_1$-ретрактом добутку $\mathbb R\times Y$ тоді і тільки тоді, коли функція $f$ неперервна.

A Limit Theorem at the Spectral Edge for Corners of Time-Dependent Wigner Matrices

For the eigenvalues of principal submatrices of stochastically evolving Wigner matrices, we construct and study the edge scaling limit: a random decreasing sequence of continuous functions of two variables, which at every point has the distribution of the Airy point process. The analysis is based on the methods developed by Soshnikov to study the extreme eigenvalues of a single Wigner matrix.

Bayesian Games with Baire Class 1 Payoff Functions

This paper considers Bayesian games where a player's payoff function is a Baire class 1 function. A Baire class 1 function (Baire (1905) is defined as a pointwise limit of a sequence of continuous functions and include semicontinuous functions, step functions, functions of bounded variations, and monotone functions. Bayesian games with Baire class 1 payoff's generalize the canonical model of Milgrom and Weber (1985) and encompass a large class of auction games and mechanism design problems. A Bayesian game with Baire class 1 payoff's has a Bayesian equilibrium via continuous approximations when the better reply security condition of Reny (1999) is satisfied. This continuous approximation approach can be regarded as a generalization of the path-following method to games with discontinuous payoff's. Using the continuous approximation method, we characterize the optimal auction mechanism with heterogeneous objects and multidimensional types with continuous distributions by unifying Myerson (1981), Mussa and Rosen (1978), and Rochet and Chone (1998). We illustrate the method by solving an example where the buyers private values are uniformly distributed on a square.

Cross Topology and Lebesgue Triples

The cross topology γ on the product of topological spaces X and Y is the collection of all sets G ⊆ X × Y such that the intersections of G with every vertical line and every horizontal line are open subsets of the vertical and horizontal lines, respectively. For the spaces X and Y from a class of spaces containing all spaces $ {{\mathbb{R}}^n} $ , it is shown that there exists a separately continuous function f : X × Y → (X × Y, γ) which is not a pointwise limit of a sequence of continuous functions. We also prove that each separately continuous function is a pointwise limit of a sequence of continuous functions if it is defined on the product of a strongly zero-dimensional metrizable space and a topological space and takes values in an arbitrary topological space.

On double sequences of continuous functions having continuous P-limits

On double sequences of continuous functions having continuous P-limits

Decomposition of Topologies Which Characterize the Upper and Lower Semicontinuous Limits of Functions

We present a decomposition of two topologies which characterize the upper and lower semicontinuity of the limit function to visualize their hidden and opposite roles with respect to the upper and lower semicontinuity and consequently the continuity of the limit. We show that (from the statistical point of view) there is an asymmetric role of the upper and lower decomposition of the pointwise convergence with respect to the upper and lower decomposition of the sticking convergence and the semicontinuity of the limit. This role is completely hidden if we use the whole pointwise convergence. Moreover, thanks to this mirror effect played by these decompositions, the statistical pointwise convergence of a sequence of continuous functions to a continuous function in one of the two symmetric topologies, which are the decomposition of the sticking topology, automatically ensures the convergence in the whole sticking topology.

On strongly countably continuous functions

Abstract A real-valued function f on R is strongly countably continuous provided that there is a sequence of continuous functions (fn)n∈N such that the graph of f is contained in the union of the graphs of fn. Some examples of interesting strongly countably continuous functions are given: one for which the inverse function is not strongly countably continuous, another which is an additive discontinuous function with a big image and a function which is approximately and I-approximately continuous, but it is not strongly countably continuous.

On [formula omitted]-limit sets of non-autonomous discrete systems on trees

Let T be a tree and f n be a sequence of continuous maps from T to T which converges uniformly to a continuous map f . In this note we show that if every periodic point of f is a fixed point, then for every x ∈ T , ω ( x , f n ) , ω -limit set of x under ( T , f n ) , is a closed connected subset of T and every point of ω ( x , f n ) is a fixed point of f .

Uniform convergence and chaotic behavior

We consider a compact metric space X , and a sequence of continuous functions f n : X → X that converges uniformly to a function f . We investigate the elements of the chaotic behavior possessed by f n that can be inherited by f .

The turnpike property of discrete-time control problems arising in economic dynamics

In this work we study the structure of approximate solutions of a nonautonomous discrete-time control system in a compact metric space $X$ which is determined by a sequence of continuous functions $v_i: X \times X \to R^1$, $i=0,\pm 1,\pm 2,$.... The main result in this paper deals with the turnpike property of optimal control problems. To have this property means that the approximate solutions of the problems are determined mainly by the the sequence $\{v_i\}_{i=-\infty}^{\infty}$, and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints.