RECENT DEVELOPMENTS IN DYNAMICAL SYSTEMS: THREE PERSPECTIVES F. BALIBREA Universidad de Murcia, Dpto. de Matem´aticas, 30100-Murcia, Spain E-mail: [email protected] T. CARABALLO Universidad de Sevilla, Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Apdo. de Correos 1160, 41080-Sevilla (Spain) E-mail: [email protected] P.E. KLOEDEN Institut f¨ur Mathematik, Goethe-Universit¨at, D-60054 Frankfurt am Main, Germany E-mail: [email protected] J. VALERO Univ. Miguel Hern´andez de Elche, Centro de Investigaci´on Operativa, Avda. Universidad s/n, 03202 Elche (Alicante), Spain E-mail: [email protected] Received September 15, 2009; Revised October 30, 2009 Dedicated to the memory of Valery S. Melnik The aim of this paper is to give an account of some problems considered in past years in the setting of Dynamical Systems, some new research directions and also state some open problems Keywords: Topological entropy, Li-Yorke chaos, Lyapunov exponent, continua, non- autonomous systems, difference equations, ordinary and partial differential equations, set-valued dynamical system, global attractor, non-autonomous and random pullback attractors. Mathematics Subject Classifications (2000): 35B40, 35B41, 35K40, 35K55, 37B25, 37B30, 37B40, 37B45, 37B55, 37E05, 37E15, 39A20 58C06, 60H15. 1. Introduction nected with celestial mechanics. In particular, the theory of discrete dynamical systems, which mainly uses iteration theory, is one of the most relevant The theory of dynamical systems has a long prehis- topics in the subject. tory, but essentially started in its present form with the work of Poincar´eand Birkhoff on problems con- The purpose of this tutorial is to give a par- 1 2 Balibrea, Caraballo, Kloeden & Valero tial account of the progress obtained in discrete and centers and depth of centers [Ye, 1993, Kato, 1995, continuous systems is recent years, to present some Kato, 1998, Efremova & Makhrova, 2003] on open and new problems. It consists of four sections. trees, graphs and dendrites, and on the In Sections 2 and 3 we consider some results in the structure of ω-limit sets [Kocan et al., 2010, topological dynamics in non-autonomous discrete Balibrea & Garc´ıa Guirao, 2005], in particular, systems and in Section 4 we give a rather complete ω-limit sets on hereditarily locally connected review of multi-valued dynamical systems arising continua [Spitalsk´y, 2008], etc. from models involving partial differential equations. Within dynamical systems theory one of the Finally, in Section 5 we provide an overview of many most interesting topics is that of minimal systems, problems in non-autonomous and random dynami- i.e., systems that do not contain non-trivial sub- cal systems, in particular on new concepts of non- systems. A system (X,f) is minimal if there is no autonomous and random attractors. proper subset Y X which is non-empty, closed ⊆ and f-invariant, i.e. satisfies f(Y ) Y . It is im- ⊆ 2. Autonomous discrete dynamical systems mediate that (X,f) is minimal if and only if the forward orbit of all points in X are dense in X. We The development of the theory of topological dy- will say also that in this case f is also minimal. namics began in the earlier part of the last century. Here we will concentrate on the progress on mini- It focused in particular on problems related to au- mal systems in the case that the phase space X is tonomous discrete dynamical systems given by the one-dimensional. pair (X,f), where X is a topological space and f a continuous map of X into itself. The crucial prob- The topological characterization of minimal lem was the study of properties of all orbits of all sets of one-dimensional X has been carried out for points in the space state X. For x X, the orbit intervals, circles, trees, finite graphs and dendrites. n ∈ n of x by f is the sequence (f (x))n∞=0, where f = In the interval case, these are finite and Cantor n 1 0 f(f − ) for all n 1 and f = idX (identity on X). sets (see [Block & Coppel, 1992]), while for the cir- ≥ In most cases X is a compact metric space and, in cle case the circle itself can also be minimal. These particular, extensive results were obtained when X results can be generalized to graphs, where minimal = I = [a, b] because many phenomena from social, sets are characterized as finite sets, Cantor sets and natural and economical sciences can be formulated also unions of finitely many pairwise disjoint circles as systems evolving with time in a discrete way in [Balibrea et al., 2003, Mai, 2005]. such spaces. Moreover, when X = Rd, or a subset thereof, we generally speak of problems on differ- A dendrite is defined as a locally con- ence equations. nected continuum which contains no simple closed Although the number of new results has been curve. In this case, besides partial results in impressive, we will include here some of them from [Balibrea et al., 2003], a complete characterization one of the subjects more active in the field in last has been given recently in [Balibrea et al., 2009] as years, dynamical systems on continua. a consequence of a more general result based on the new notion of almost totally disconnected spaces. A space X is almost totally disconnected if the set of 2.1. Autonomous dynamical systems on its degenerate components is dense in X. The main continua results says that an almost totally disconnected One line of research that has been very active space admits a minimal map if and only if it is ei- in recent years is that of dynamical systems on ther a finite set or has no isolated point. By a brain continua, i.e., where X is a continuum (a compact we mean a cantoroid whose degenerate components and connected topological space) and f C(X, X). are dendrites and form a null family (for any ε> 0, ∈ For definitions and detailed account of results see only a finite number of its members have diameters [Nadler, 1995]. Problems such as periodic struc- greater than ε). A cantoroid is a compact metric ture using methods from combinatorial dynamics and almost totally disconnected space without iso- have been studied on circles, trees and finite lated points. With these ingredients we can now graphs (see [Alsed`a et al., 2000]), as well as on state the characterization in [Balibrea et al., 2009]. Three perspectives 3 Theorem 2.1. Let D be a dendrite and let M be crete system, which they denoted by h(f1, ), using ∞ a subset of D. Then M is a minimal set for some the technique of open covers of X as in the orig- dynamical system (D,f) if and only if M is either inal paper [Adler et al., 1965] on autonomous sys- a finite set or a brain. tems. In the case that X is metric or metrizable, they also used separated and spanning sets as in In addition, the following characterization was [Bowen, 1970]. It is easy to see that such defini- given in [Balibrea et al., 2009] for general almost tions give similar results when the system is in fact totally disconnected spaces. autonomous. What is really interesting, is that it is now possible to define the entropy h(f1, ,Y ), when ∞ Theorem 2.2. An almost totally disconnected Y is a subset of X which is not necessarily compact compact metric space admits a minimal dynamical or invariant under f(f1, ). Such an extension was ∞ system if and only if it is either a finite set or a necessary to deal with other problems considered in cantoroid. [Kolyada & Snoha, 1996]. An interesting and surprising consequence of Question 2.3. Is it possible to give a topological the paper [Kolyada & Snoha, 1996] was a proof of characterization of minimal sets in other families the commutativity of the entropy autonomous dy- of one-dimensional continua like arc-like, tree-like namical systems, i.e., the entropy of the composi- or circle-like ? (For definitions see [Nadler, 1995]). tion of two continuous self-maps on a compact space does not depend on the order in which they are 3. Non-autonomous discrete systems taken, i.e., h(f g) = h(g f). The first time that ◦ ◦ the commutativity of the entropy was mentioned We now consider the situation in which the map seems to have been in [Dinaburg, 1970]. describing the evolution of the dynamics is itself al- Inspired by [Kolyada & Snoha, 1996], the com- lowed to change with time. This admits the follow- mutativity or non-commutativity of other types ing formulation. Given a compact topological space of entropy such sequence entropy were proved in X and a sequence of continuous self-maps (f ) n n∞=1 [Balibrea et al., 1999a, Balibrea et al., 1999b]). = f1, from X into itself, the pair (X,f1, ) will be ∞ ∞ called a non-autonomous discrete system where the It is well known (see [Misiurewicz,1989]) that orbit of a point x X is described by the sequence for interval maps, the entropy is positive if and ∈ only if one of its iterates has a structure called x, f1(x), f2(f1(x)),...,fn(fn 1)...(f2(f1(x)))...) horseshoe. An interval map f has a horseshoe if − there are two disjoint intervals J and K such that We will use the notation f(J) f(K) J K. It is difficult to extend this n ∩ ⊃ ∪ f1 = fn fn 1 ... f2 f1 definition to non-autonomous interval systems be- ◦ − ◦ ◦ ◦ cause it is associated to a unique map and f1, is ∞ which was introduced in [Kolyada & Snoha, 1996].