On the Topological Entropy of Continuous and Almost Continuous Functions ∗ Ryszard J

View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 158 (2011) 2022–2033 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol On the topological entropy of continuous and almost continuous functions ∗ Ryszard J. Pawlak , Anna Loranty, Anna Bakowska ˛ Faculty of Mathematics and Computer Science, Łód´z University, Banacha 22, 90-238 Łód´z, Poland article info abstract Dedicated to Professor Lawrence M. Brown We prove some results concerning the entropy of continuous and almost continuous on the occasion of his 70th birthday functions. We first introduce the notions of bundle entropy and (strong) entropy points and then we study properties of these notions in connection with the theory of multifunctions. MSC: Basedonthesefactswegivetheoremsaboutapproximationoffunctionsdefinedand 54C70 assuming their values on compact manifold by functions having strong entropy points. 54C35 © 37B40 2011 Elsevier B.V. All rights reserved. 54C60 54C65 Keywords: Almost continuity f -Bundle Multivalued function Entropy point Approximation Manifold 1. Introduction This paper was inspired by some problems which one can find in papers connected with topology and real analysis, es- pecially by results concerning the theory of dynamical systems (more precisely: entropy), fixed point theory, multifunctions and some approximation problems in function spaces. Studies of the entropies of various functions led to the observation that sometimes the entropy “is focused around one point of the domain” or speaking less precisely “at one point”. Although many mathematicians have noticed that (cf. e.g. [23]), there were no attempts to use multifunctions and its selections to describe this fact, as it is done in this paper. On the other hand, the development of fixed point theory has induced us to analyze a situation when the entropy of a function “is focused at a fixed point of the function” (a so-called strong entropy point). Recently problems connected with entropy and fixed points were examined with reference to discontinuous functions. Taking into consideration this fact we have extended our research to almost continuous functions (in the sense of Stallings). The choice of this class of functions was due to the fact that in a Hausdorff space with the fixed point property each almost continuous function has a fixed point (cf. Theorem 2.2). The first two sections are intended to introduce the notation and terminology. There are also presented useful properties and some additional explanations connected with them. * Corresponding author. E-mail addresses: [email protected] (R.J. Pawlak), [email protected] (A. Loranty), [email protected] (A. Bakowska). ˛ 0166-8641/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2011.06.049 R.J. Pawlak et al. / Topology and its Applications 158 (2011) 2022–2033 2023 In Section 3, using the notion of “ f -bundle”, we describe, in the terminology of multifunctions, some basic facts connected with entropy. The statements presented allow us to build the foundations of our considerations connected with “entropy point” and “strong entropy point”. Many papers analyze, from the topological point of view, the problem of the “size” of some sets of functions in the established function space. It is worth to mention here at least a few works connected with this problem in the context of fixed point theory: [2–4,12,21,25,26]. Since the property of having a strong entropy point seems to be a strong assumption, it is natural to try to characterize the “size” of the set of continuous or almost continuous functions having a strong entropy point with reference to the family of all continuous or all almost continuous functions. In Section 4 we describe the “size” of such set in the terminology of density, i.e. in the terminology connected with the possibility of approximation (“graph approximation” and “uniform approximation”) of an arbitrary continuous or almost continuous function by functions having a strong entropy point. In this part of the paper our purpose is to present a universal idea of reasoning, which (with slight changes) works for different kinds of approximations. We were able to create this idea (shown in the proof of Theorem 4.1) thanks to terminology built in the previous sections. The paper is completed by an open problem connected with the approximation by functions having finite entropy. 2. Preliminaries N R m Throughout the paper and denote the set of positive integers and real numbers, respectively. Moreover by ρE we will denote the Euclidean metric for Rm (m ∈ N). The cardinality of the set A we will denote by #A and the graph of afunction f by Γ(f ).FurthermoreifA, B are any subsets of a metric space (X, ρ) then by dist(A, B) we will denote the distance between A and B. Let (X, ρ) be a metric space, x0 ∈ X and r1 > 0. We will use the symbol B(x0, r1) (S(x0, r1))todenoteanopenball (a sphere) with center at x0 and radius r1.ThesetA(x0, r1, r2) ={y ∈ X: r1 < ρ(x0, y)<r2} is an open annulus with center at x0 and the radii r1 and r2 (r2 > r1 > 0). Moreover a closed ball and a closed annulus will be denoted by B(x0, r1) and A(x0, r1, r2), respectively. − Let f : X → X. Then we define f 0(x) = x and f i (x) = f ( f i 1(x)) for any positive integer i.Apointx ∈ X such that f (x) = x is called a fixed point of f .Thesetofallfixedpointsof f we will denote by Fix( f ).WewillsaythataspaceX has the fixed point property if Fix( f ) = ∅ for all continuous functions f : X → X. Topological entropy is an analogous notion to metric entropy, and is usually defined by means of the set of covers of a compact space (e.g. [1]). In a compact metric space a more intuitive definition is given by R. Bowen [6] and E. Dinaburg [9]. The Bowen–Dinaburg version of the definition is equivalent to the one connected with the families of covers. Moreover in [8] it has been shown that the Bowen–Dinaburg version can be applied for an arbitrary function (not necessarily continuous). Consequently, in our considerations it is convenient to use this kind of definition (we restrict the definition to “separated sets”, despite that one can also use the dual notion “span sets” (see [1,8])). Let ε > 0 and n be a positive integer. A set M ⊂ X is (n, ε)-separated if for each x, y ∈ M, x = y there is 0 i < n such that ρ( f i(x), f i (y)) > ε.Let maxsep[n, ε]=max #(M): M ⊂ X is (n, ε)-separated set . The topological entropy of the function f is the number 1 h( f ) = lim lim sup log maxsep[n, ε] . ε→0 n→∞ n Let f , g : X → X. The functions f and g are conjugate if there exists a homeomorphism ϕ : X → X such that ϕ ◦ f = g ◦ ϕ. The following result will be useful in the remainder of this paper. Theorem 2.1. ([8]) Let X be a compact metric space. If functions f : X → Xandg: X → X are conjugate then h( f ) = h(g). 2 Let f : X → X. We shall say that a set A ⊂ Xf-covers a set B ⊂ X (denoted by A → B)ifB ⊂ f (A).Moreover,wewill f use the symbol f|A to denote the restriction of f to A. Let X and Y be nonempty sets. We shall say that a function F : X Y is a multifunction if it is a function which associates each point x ∈ X with a nonempty set F (x) ⊂ Y . The graph of a multifunction F : X Y is defined by Γ(F ) = {(x, y) ∈ X × Y : y ∈ F (x)}. We say that a multifunction F is closed if Γ(F ) is closed as a subset of X × Y .Afunction f : X → Y is a selection of F if f (x) ∈ F (x) for any x ∈ X. From now on we will only consider a compact metric space (X, ρ). For simplicity of notation, we will write X instead of (X, ρ). Additionally, in Section 4 we will assume that X is an m-dimensional manifold with boundary. Our terminology and notations related to m-dimensional manifolds will coincide with these of [19]. We shall say that a topological space X is an m-dimensional topological manifold (manifold with boundary) if X is a second countable Hausdorff space and every point q ∈ X has a neighborhood that is homeomorphic to an open subset of Rm m m (the m-dimensional upper half space H ={(x1,...,xm) ∈ R : xm 0}). We will call any homeomorphism from an open subset U of X to an open subset of Hm a chart on U . 2024 R.J. Pawlak et al. / Topology and its Applications 158 (2011) 2022–2033 Fig. 1. Examples of discontinuous mappings having different entropies. If X is an m-dimensional topological manifold with boundary, a point which belongs to the inverse image of δHm = m {(x1,...,xm) ∈ R : xm = 0} under some chart is called a boundary point of X, and a point that is in the inverse image of the interior of Hm (denoted by Int Hm) is called an interior point. The boundary of X (the set of all its boundary points) we will denote by δ X.WewillusethesymbolIntm X to denote the set of all interior points of X. Since any open ball in Rm is homeomorphic to an open subset of Hm,anm-dimensional topological manifold is automatically an m-dimensional topological manifold with boundary (with empty boundary).

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