Recent results on iteration theory: iteration groups and semigroups in the real case

Marek Cezary Zdun Pawel Solarz In this survey paper we present some recent results in iteration theory. Mainly, we focus on the problems concerning real iteration groups (flows) and semigroups (semiflows) such as existence, regularity and embeddability. We also discuss some issues associated to the problem of embeddability, i.e. iterative roots and approximate iterability. The topics of this paper are: (1) measurable iteration semigroups; (2) embedding of diffeomorphisms in regular iteration semigroups in the space Rn; (3) iteration groups of fixed point free homeomorphisms on the plane; (4) embedding of interval homeomorphisms with two fixed points in a regular iteration group; (5) commuting functions and embeddability; (6) iterative roots; (7) the structure of iteration groups on an interval; (8) iteration groups of homeomorphisms of the circle; (9) approximately iterable functions; (10) set-valued iteration semigroups; (11) iterations of mean-type mappings; (12) Hayers-Ulam stability of the translation equation. Most of the results presented here were obtained by means of functional equations. We indicate the relations between iteration theory and functional equations. 1. Introduction Iteration groups and semigroups are the main objects of study in iteration theory. They are also called flows and semiflows in the theory of dynamical systems. In this survey paper we present some selected achievements concerning iteration groups and semigroups that have been made during the last years. Let us recall the general definition. Definition 1. Let G be an additive subgroup or subsemigroup of R or C and X = be a given set. A family of mappings {f t : X X, t G} is said to be an iteration group or iteration semigroup on X over G respectively if f t(f s(x)) = f t+s(x), s, t G, x X. Usually we assume that G = R or G = R+ and then we call {f t, t G} an iteration group (flow) or semigroup (semiflow), respectively. Extending the domain of the iterative index t on an arbitrary algebraic structure G and introducing the notation F (x, t) := f t(x) we write (1) as the translation equation F (F (x, t), s) = F (x, t + s), s, t G, x X. The translation equation on algebraic structures was studied by Moszner and his collaborators. More information on this topic may be found in the survey article [102]. General enough but still a very useful condition is that X is a Banach space. Under suitable differentiability conditions the translation equation can be transformed into differential equations. Introducing we obtain three equations g(x) := = g = g F, = g F called Jabotinsky equations. The exact relations between the translation equation and Jabotinsky equations have been described by Aczel and Gronau [2]. For detailed results we refer the reader to the articles [3537]. One origin of the notion of iteration groups is a natural extension of the domain of indices of discrete iterations f n of a given mapping f : X X to the real space. Another is a description of deterministic processes originated in the theory of ordinary differential equations. If we interpret an iterative index as time, then iteration semigroups and groups are the models of deterministic processes. A deterministic process, roughly speaking, describes the change of the position in time of the points in a given space X in such a way that after the given time they are uniquely determined. Usually one assumes additionally that the positions of the points are uniquely determined also in the past. In other words, the state of each object in the future and in the past is uniquely determined by the state of the object in the present moment independently of its itinerary. The points from the space X are called objects and their positions after time t are called the states at the moment t. Denote by f s(x) the state of x after time s. Then f t(f s(x)) means the state of f s(x) after time t. Hence it is the state of x after time s + t. On the other hand, f t+s(x) is also the state of x after time t + s thus, by the uniqueness, we get the equality f t(f s(x)) = f t+s(x). This gives equation (1) for G = R or G = R+. In iteration theory we generally assume that X is a metric space and the mappings f t are of a suitable regularity. One can say that iteration theory is a part of the translation equation theory but with suitable regularities. Iteration groups are strictly connected to dynamical systems. Their significance lies in the fact that they describe deterministic processes. For more information on iteration theory see [6, 64, 152154, 180, 66]. This note is a continuation of the above papers. In the present survey we concentrate on selected topics connected to real iteration groups. Complex iteration groups have their own specificity which follows from the properties of holomorphic functions and the rings of formal power series. This is a very large domain and the subject is beyond the scope of this paper. A lot of information and references of this topic can be found in the survey articles [1, 120, 121]. We focus here on the selected problems of iteration theory strictly related to functional equations. We point out the role of functional equations as the basic research tools in the theory of iteration groups. We consider the following topics. 1. Measurable iteration semigroups. 2. Embedding of diffeomorphisms in regular iteration semigroups in the Rn space. 3. Iteration groups of fixed point free homeomorphisms on the plane. 4. Embedding of interval homeomorphisms with two fixed points in a regular iteration group. 5. Commuting functions and embeddability. 6. Iterative roots. 7. The structure of iteration groups of homeomorphisms on an interval. 8. Iteration groups of homeomorphisms of the circle. 9. Approximately iterable functions. 10. Set-valued iteration semigroups. 11. Iterations of mean-type mappings. 12. HayersUlam stability of the translation equation. 2. Measurable iteration semigroups First we discuss the problem of continuity of measurable iteration semigroups. Let X be a metric space. We begin with the following. Definition 2. An iteration semigroup {f t : X X, t > 0} is said to be continuous if all functions f t are continuous and for every x X the mapping t f t(x) is continuous. Definition 3. An iteration semigroup {f t : X X, t > 0} is said to be measurable if all functions f t are continuous and for every x X the mapping t f t(x) is Lebesgue measurable. The question is, when a measurable iteration semigroup is continuous. For an arbitrary metric space the problem is still unsolved, but the answer is positive for particular wide classes of metric spaces. In 1979 in [159] the fact that a measurable iteration semigroup is continuous was proved for closed bounded intervals. The first such result was extended for compact metric spaces (see [163]). Next Baron and Jarczyk in [8] generalized this result for locally compact metric spaces. The same authors jointly with Chojnacki in pap (...truncated)