The paper is based on the Curriculum Vitae of Győrgy Targoński (March 27. 1928 - January 10. 1998) written by the second author, and is completed by a survey of his domains of research with special attention to selected papers. Professor Targonski's scientific legacy consists of over hundred research papers, books, lecture notes and reports, contributions to meetings (problems, remarks, abstracts of lectures) and of proceedings of the ECIT. These are all documented in the three lists attached to the article.

In this paper we show that triangular maps of the unit square can have properties that are impossible in the one-dimensional case. In particular, we find a map with infinite spectrum; a distributionally chaotic map whose principal measure of chaos is not generated by a pair of points and which has the empty spectrum; a distributionally chaotic map that is not chaotic in the sense of Li and Yorke.

In this paper we study the periodic structure of the antitriangular maps on I² = [0,l]×[0,l] and obtain an ordering on the periods of these maps of a Sarkovskii type.

In this paper we obtain some formulas which allow us to compute topological and metric entropy and topological pressure for a new class of maps. It is also shown that similar formulas do not hold for metric and topological sequence entropy and a new commutativity problem is posed.

In this paper we study the finite-dimensionality of the global attractor of a discrete dynamical system generated by a reaction-diffusion equation with non-differentiable nonlinear term and periodic right-hand side. The existence of an exponential attractor is also proved. Explicit estimates of the fractal dimension are given.

Based on a previous theoretical result of the same authors the presen paper deals with discrete perturbed two-dimensional maps having a semi-hyperbolic fixed point. We give applicable sufficient conditions assuring a particular kind of bifurcation of homoclinic orbits when the perturbative parameter μ varies in a small neighborhood of zero: no homoclinic orbits when μ is on one side of zero, one homoclinic orbit when μ = 0, and infinite homoclinics when μ is on the other side of zero.

In this paper we consider two-dimensional maps, defined in the whole plane, with the property of mapping a whole curve δ into a single point Q. We relate such property to the fact that at least one inverse map exists with a denominator which can vanish, and assumes the form 0/0 in Q. This allows us to apply the properties of focal points and prefocal curves in order to explain the dynamic phenomena observed. By an example we show that the iteration of such maps may generate discrete dynamical systems with peculiar attracting sets, characterized by the presence of "knots", where infinitely many phase curves shrink into a single point.

Maps with a denominator which vanishes in a subset of the phase space may generate unbounded trajectories which are not divergent, i.e. trajectories involving arbitrarily large values of the dynamic variables but which are not attracted to infinity. In this paper we propose some simple one-dimensional and two-dimensional recurrences which generate unbounded chaotic sequences, and through these examples we try to explain the basic mechanisms and bifurcations leading to the creation of unbounded sets of attraction.

The aim of this paper is to give a necessary and sufficient condition for conjugacy of some iteration groups ???? = {F^t : S↦S, t∈ℝ} and ???? = {G^t : S↦S, t∈ℝ} defined on the unit circle S. Our basic assumption is that the yare non-singular, that is at least one element of ???? and ???? has no periodic point. Moreover, under some further restrictions, we determine all orientation-preserving homeomorphisms Γ : S↦S such that
Γ◦F^t = G^t◦Γ, t∈ℝ.

In this paper we consider a two-dimensional map with a denominator which can vanish, obtained by applying Bairstow's method, an iterative algorithms to find the real roots of a polynomial based on Newton's method. The complex structure of the basins of attraction of the fixed points is related to the existence of singularities specific to maps with a vanishing denominator, such as sets of non definition, focal points and prefocal curves.

An elementary proof of the existence of iteration groups consisting of additive bijections of the real line onto itself and containing nonlinear mappings is presented. An embeddability problem of that kind is also considered. These results are used to get a description of some semigroup automorphisms that are embeddable into an iteration group of automorphisms of the same semigroup.

Let Ψ: ℝ×T ↦ T be a continuous dynamical system on the two-dimensional torus T. The aim of this paper is to prove some characterizations about the existence of a Siegel curve, i.e. a simple closed curve which cuts every half-trajectory of the dynamical system (T,Ψ). This result completes and precises some results obtained in our article [7].

We introduce the notion of iterative equivalence of two classes of mappings on metric spaces and we demonstrate its utility in metric fixed-point theory. In particular, we show that the fixed-point theorem for Matkowski's contractions can be derived from the corresponding theorem for Browder's contractions, though the first class of mappings is essentially wider than the second one.

By some geometrical considerations we formulate a functional equation which is related to the ancient isoperimetric problem of Dido. The continuous solution of this Dido functional equation depends on an arbitrary function. However, we show that in a class of functions of suitable asymptotic behavior at infinity, the Dido functional equation has a one-parameter family of "principal" solutions. Some applications are given.

Relations between the following properties of triangular maps F: I2→I2 are studied in this paper: (P1) the period of any cycle is a power of two; (P2) every cycle is simple; (P3) the topological entropy of F restricted to the set of periodic points is 0; (P4) the topological entropy of F is 0; (P5) every ω-limit set contains a unique minimal set; (P6) F has no homoclinic trajectory; (P7) every ω-limit set either is a cycle or contains no cycle; (P8) no infinite ω-limit set contains a cycle. It is known that for continuous maps of the interval these properties are mutually equivalent. In the case of triangular maps of the square we prove that the properties (P1), (P2) and (P3) are mutually equivalent, and that (P7) is equivalent to (P8). Moreover, we show which of the remaining implications are true and which not. The problem is completely solved, with the following exceptions: we conjecture that (P7)⇒(P6) but we do not provide the argument; validity of (P7)⇒(P1) remains open. Our paper gives a partial solution of the problem stated in 1989 by A.N. Sharkovsky.

This communication will discuss the dynamics of iterated cubic maps from the real line to itself, and will describe the renormalization of the parameter space for such maps using methods of symbolic dynamics.

We show that theory of distributional chaos for continuous functions on the unit interval as developed recently by Schweizer and Smítal remains essentially true for continuous mappings of the circle, with natural exceptions.

It is shown that, under some general conditions, the sequence of iterates of every mean-type mapping on a finite dimensional cube converges to a unique invariant mean-type mapping. Some properties of the invariant means and their applications are presented.

On donne la revue des résultats au sujet des équations de pré-Schröder introduites par G. Targonski. On formule aussi des problèmes ouverts.

Consider a continuous and strictly increasing function f: [0,1]→[0,2], and define T_fx = f(x)(mod 1). Then T_f is a monotonie mod one transformation with two monotonic pieces, if and only if f(0) < 1 < f(1). It is proved that T_f is topologically transitive, if f is piecewise differentiable and inf_x∈[0,1]f'(x) ≥ √2.

It is known there are boundary value problems investigation of which is directly reduced to study of iteration of functions. This allows to carry on detailed and deep analysis of original problems. We consider such a class of simple (in form) problems, solutions of which demonstrate, nevertheless, very complicated behavior, make possible the simulation of self-birthing structures, including self-similar ones, and self-stochasticity phenomenon.

The aim of this paper is to establish the existence of a "box-within-a-box" bifurcation structure for monotone families of Lorenz maps and to study its combinatorics.

Let K be a convex cone in a real normed space X. A one-parameter family {F_t : t ≥ 0} of set-valued functions F_t :K→n(K), where n(K) := {D : D ⊂ K, D ≠ ⌀}, is called cosine iff F_t+s + F_t-s = 2F_t◦F_s, whenever 0 ≤ s ≤ t and F₀ is the identity map. A cosine family {F_t : t > 0} is regular iff lim_t→0+F_t(x) = {x} for every x.
The growth and the continuity of regular cosine families are investigated.