Jaroslav Smital

• Silesian University in Opava

For a topological dynamical system (X,f)(X,f) we consider the structure of the set F(f)F(f) of asymptotic distributions of the distances between pairs of trajectories. If f has the weak specification property then F(f)F(f) is closed and convex, and it can contain all nondecreasing functions [0,diamX]→[0,1]. We discuss relations to distributional ch...

We consider continuous solutions \({f : \mathbb{R}_{+} \rightarrow \mathbb{R}_{+} = (0, \infty)}\) of the functional equation \({f(xf(x)) = \varphi (f(x))}\) where \({\varphi}\) is a given continuous map \({\mathbb{R}_{+} \rightarrow \mathbb{R}_{+}}\). A solution f is singular if there are \({0 < a \leq b< \infty}\) such that \({f|_{(0,a)} > 1, f|_...

Let (X, f) be a topological dynamical system, where X is a compact metric space and f : X → X is a continuous map. Denote by the set of all invariant probability measures of f which are limit points of the sequence , where δx is the atomic probability measure on X with support {x}. We give a characterization of points x such that contains a meas...

We show that in the class T of the triangular maps (x,y)↦(f(x),gx(y)) of the square there is a map of type 2∞ with non-minimal recurrent points which is not DC3. We also show that every DC1 continuous map of a compact metric space has a trajectory which cannot be (weakly) approximated by trajectories of compact periodic sets. These two results make...

The aim of this paper is to give an account of some problems considered in the past years in the setting of Discrete Dynamical Systems and Iterative Functional Equations, some new research directions and also state some open problems.

Let f be a continuous map from a compact interval into itself. Continuing the work begun by Schweizer and Smítal [1994], we prove that the restriction of f to any basic set (i.e. any nonsolenoidal, infinite, maximal ω-limit set) satisfies a generalization of the specification property. We apply this generalization to establish several conjectures m...

We consider continuous solutions f:ℝ + →ℝ + =(0,∞) of the functional equation f(xf(x))=φ(f(x)) where φ is a given continuous map ℝ + →ℝ + . If φ is an increasing homeomorphism the solutions are completely described, if not there are only partial results. In this paper we bring some necessary conditions upon a possible range R f . In particular, if...

We provide a class of triangular maps of the square, (x,y)↦(f(x),gx(y)) of type 2∞, i.e., such that the periods of periodic points are the powers of 2, which has a minimal set supporting positive topological entropy. This improves the famous example by S. Kolyada from 1992 and contributes to the solution of an old problem by A.N. Sharkovsky.

We characterize functional equations of the form f(zf(z))=f(z)k+1,z Î \mathbb C{f(zf(z))=f(z)^{k+1},z\in\mathbb {C}}, with k Î \mathbb N{k\in\mathbb N}, like those generalized Dhombres equations f(zf(z))=j(f(z)){f(zf(z))=\varphi (f(z))}, z Î \mathbb C{z\in\mathbb C}, with given entire function j{\varphi}, which have a nonconstant polynomial s...

For a continuous map φ:X→X of a compact metric space, we study relations between distributional chaos and the existence of a point which is quasi-weakly almost periodic, but not weakly almost periodic. We provide an example showing that the existence of such a point does not imply the strongest version of distributional chaos, DC1. Using this we pr...

According to a well-known result, the collection of all ω-limit sets of a continuous map of the interval equipped with the Hausdorff metric is a compact metric space. In this paper, a similar result is proved for piecewise continuous maps with finitely many points of discontinuity, if the points of discontinuity are not periodic for any variant of...

We consider continuous solutions of the functional equation f(xf(x)) = (f(x)), where is a given continuous map + → + without any restrictions. It can be treated as a difference equation but the standard methods lead only to (usually very irregular) discontinuous solutions. Concerning continuous solutions, the case when is an increasing homeomorphis...

We study local analytic solutions f of the generalized Dhombres functional equation f(zf(z))=φ(f(z)), where φ is holomorphic at w0≠0, f is holomorphic in some open neighborhood of 0, depending on f, and f(0)=w0. After deriving necessary conditions on φ for the existence of nonconstant solutions f with f(0)=w0 we describe, assuming these conditions,...

In the class T of triangular maps of the square we consider the strongest notion of distributional chaos, DC1, originally introduced by Schweizer and Smítal [B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994) 737–854] for continuous maps of the interval. W...

We consider the set valued functions C, NW and ℒ taking f in 𝒞(I, I) to its centre C( f ), its set of nonwandering points NW( f ) and its collection of ω-limit sets ℒ( f ) = {ω (x, f ) : x ∈ I}, and consider how these sets are affected by pertubations of f. Our main results characterize those functions g in 𝒞(I, I) at which C, NW and ℒ are continuo...

The dynamics of triangular (or skew-product) maps of the square, given by became interesting since 1979 when Kloeden proved that Sharkovsky's theorem on coexistence of cycles of a continuous map of an interval is valid also for them. In this paper, we briefly recall the history of research. Then we provide a survey of recent results related to the...

The notion of distributional chaos was introduced by Schweizer, Smítal [Measures of chaos and a spectral decompostion of dynamical systems on the interval. Trans. Amer. Math. Soc. 344;1994:737–854] for continuous maps of the interval. For continuous maps of a compact metric space three mutually nonequivalent versions of distributional chaos, DC1–DC...

Key Words: iterative functional equation, equation of invariant curves, general solution Mathematical Reviews subject classification: 26A18, 39B12, 39B22

We study holomorphic solutions f of the generalized Dhombres equation f(zf(z))=φ(f(z)), z∈C, where φ is in the class E of entire functions. We show, that there is a nowhere dense set E0⊂E such that for every φ∈E∖E0, any solution f vanishes at 0 and hence, satisfies the conditions for local analytic solutions with fixed point 0 from our recent paper...

The notion of distributional chaos was introduced by Schweizer and Smítal [Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans Am Math Soc 1994;344:737–854] for continuous maps of the interval. For continuous maps of a compact metric space three mutually non-equivalent versions of distributional chaos, DC1–DC3...

Recently, Forti, Paganoni and Smítal constructed an example of a triangular map of the unite square, F(x,y)=(f(x),g(x,y)), possessing periodic orbits of all periods and such that no infinite ω-limit set of F contains a periodic point. In this note we show that the above quoted map F has a homoclinic orbit. As a consequence, we answer in the negativ...

We study local analytic solutions f of the generalized Dhombres equation f ðxfðxÞÞ ¼ 'ðf ðxÞÞ with f ð0 Þ¼ 0 in the complex domain. We give an existence result, describe the structure of the set of all local analytic solutions and solve the converse problem, i.e., we characterize those local analytic functions which are solutions of a generalized D...

Our main result is an example of a triangular map of the unite square, F(x,y)=(f(x),gx(y)), possessing periodic orbits of all periods and such that no infinite ω-limit set of F contains a periodic point. We also show that there is a triangular map F of type 2∞ monotone on the fibres such that any recurrent point of F is uniformly recurrent and F re...

The notion of distributional chaos was introduced by Schweizer and Smı́tal [Trans. Amer. Math. Soc. 344 (1994) 737] for continuous maps of the interval. However, it turns out that, for continuous maps of a compact metric space three mutually nonequivalent versions of distributional chaos, DC1–DC3, can be considered. In this paper we consider the we...

In this paper we exhibit a triangular map F of the square with the following properties: (i) F is of type 2∞ but has positive topological entropy; we recall that similar example was given by Kolyada in 1992, but our argument is much simpler. (ii) F is distributionally chaotic in the wider sense, but not distributionally chaotic in the sense introdu...

We consider the functional equation f(xf(x))=φ(f(x)) where φJ→J is a given increasing homeomorphism of an open interval J⊂(0,∞) and f(0,∞)→J is an unknown continuous function. In a series of papers by P. Kahlig and J. Smítal it was proved that the range of any non-constant solution is an interval whose end-points are fixed under φ and which contain...

Developping ideas of S. Li [Tran. Amer. Math. Soc. 301 (1993), 243-249] concerning the notion of ω-chaos we prove that any transitive continuous map f of the interval is conjugate to a map g of the interval which possesses an ω-scrambled set S of full Lebesgue measure. Thus, for any distinct x, y in S, ωg(x) ∩ ωg (y) is non-empty, and ωg(x)/ωg(y) i...

We consider the functional equation f(x f(x)) = phi(f(x)) where phi: J -> J is a given increasing homeomorphism of an open interval J (subset of the positive reals R+), and f: R+ -> J is an unknown continuous function. We give a characterization of the equations (or equivalently, of the functions phi) which have all continuous solutions monotone. I...

In 1992 Agronsky and Ceder proved that any finite collection of nondegenerate Peano continua in the unit square is an ω-limit set for a continuous map. We improve this result by showing that it is valid, with natural restrictions, for the triangular maps (x, y) → (f(x),g(x, y)) of the square. For example, we show that a non-trivial Peano continuum...

In this paper we show that there is a continuous map f : I ! I of the interval such that any !-limit set W of any continuous map g : I ! I can be transformed by a homeomorphism I ! I to an !-limit set ˜ W of f. Consequently, any nowhere-dense compact set and any finite union of compact intervals is a homeomorphic copy of an !-limit set of f.

A conjecture by Agronsky and Ceder [3], stating that a continuum is an orbit enclosing ω-limit set of a continuous map from the k-dimensional cube I k into itself if and only if it is arcwise connected, is disproved in both directions. Our main result is a general theorem allowing a construction of orbit enclosing ω-limit sets for triangular maps.

After surveying several earlier definitions of \lq\lq chaos\rq\rq , this paper is devoted to presenting the recently introduced notion of {\it distributional chaos} to a non-specialist audience. It is shown that the theory of distributional chaos avoids various shortcomings of the earlier theories and that it allows one not only to distinguish betw...

We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point xα∈I such tha...

The main goal of the paper is the construction of a triangular mapping F of the square with zero topological entropy, possessing a minimal set M such that F | M is a strongly chaotic homeomorphism, as well as other properties that are impossible for continuous maps on an interval. To do this we define a parametric class of triangular maps on Q × I...

In this paper we consider the functional equation f (x f (x)) = k f (x)2, where $ f : R^+ \rightarrow R^+ $ and k > 0 is a real parameter. We give a characterization of the class of its continuous solutions, and show that there are discontinuous solutions which are strongly irregular.

We give a characterization of the set of nonwandering points of a continuous map \(f\) of the interval with zero topological entropy, attracted to a single (infinite) minimal set \(Q\). We show that such a map \(f\) can have a unique infinite minimal set \(Q\) and an infinite set \(B\subset\Omega (f)\setminus\omega (f)\) (of nonwandering points tha...

Continuous solutions of the functional equation ƒ(xƒ(x)) = (ƒ(x))^2 for x ∈ [0,∞) have been characterized by Dhombres. They form a simple, two-parametric family and the result can be easily extended to solutions on the whole real line. However, the class of all solutions is much larger. We show that there is a solution ƒ whose graph has, among othe...

We show that continuous triangular maps of the square I ¹ , F : ( x, y ) → ( f ( x ), g ( x, y )), exhibit phenomena impossible in the one-dimensional case. In particular: (1) A triangular map F with zero topological entropy can have a minimal set containing an interval { a } × I , and can have recurrent points that are not uniformly recurrent; thi...

Let the upper and lower (distance) distribution functions, F*y and Fxy, are defined for any as the lim sup and lim inf as of the average number of times that the distance between the trajectories of x and y is less than t during the first n iterations. The spectrum of is the system 1(f) of lower distribution functions which is characterized by the...

Let $f: \lbrack 0, 1\rbrack \rightarrow \lbrack 0, 1\rbrack$ be continuous. For x, y ∈ [ 0, 1], the upper and lower (distance) distribution functions, F* xy and Fxy, are defined for any t ≥ 0 as the $\lim \sup$ and $\lim \inf$ as n → ∞ of the average number of times that the distance |fi(x) - fi(y)| between the trajectories of x and y is less than...

We prove that an infinite W (0, 1) is an ω-limit set for a continuous map ƒ of [0,1] with zero topological entropy iff W = Q P where Q is a Cantor set, and P is countable, disjoint from Q, dense in W if non-empty, and such that for any interval J contiguous to Q, card (J P) ≤ 1 if 0 or 1 is in J, and card (J P) ≤ 2 otherwise. Moreover, we prove a c...

We prove that a continuous map f of the interval is chaotic (in the sense of Li and Yorke) $\operatorname{iff}$ its sequence topological entropy hA(f) relative to a suitable increasing sequence A of times is positive. This result is interesting since the ordinary topological entropy h(f) of chaotic maps can be zero.

We prove, among others, the following relations between notions of chaos for continuous maps of the interval: (i) A map f is not chaotic in the sense of Li and Yorke $\operatorname{iff} f$ restricted to the set of its ω-limit points is stable in the sense of Ljapunov. (ii) The topological entropy of f is zero $\operatorname{iff} f$ restricted to th...

Recent results of the second author show that every continuous map of the interval to itself either has every trajectory approximable by cycles (sometimes this is possible even in the case when the trajectory is not asymptotically periodic) or is e-chaotic for some ∈ > 0. In certain cases, the first property is stable under small perturbations. Thi...

We find a class of C∞ maps of an interval with zero topological entropy and chaotic in the sense of Li and Yorke.

Consider the continuous mappings f from a compact real interval to itself. We show that when f has a positive topological entropy (or equivalently, when f has a cycle of order ≠ 2n, n = 0, 1, 2, …) then f has a more complex behaviour than chaoticity in the sense of Li and Yorke: something like strong or uniform chaoticity, distinguishable on a cert...

There is a continuous map of the unit interval which is chaotic in the sense of Li and Yorke and which has a scrambled set of Lebesgue measure arbitrarily close to 1.

Let I be a real compact interval, and let C be the space of continuous functions I → I with the uniform metric. For f ∈ C denote $\nu(f) = \sup_{x \in I}(\lim \sup_{n \rightarrow \infty}f^n(x) - \lim \inf_{n \rightarrow \infty}f^n(x))$, where fn is the nth iterate of f. Then for each positive d there is an open set C* dense in C such that the oscil...

Let I I be a real compact interval, and let C C be the space of continuous functions I → I I \to I with the uniform metric. For f ∈ C f \in C denote ν ( f ) = sup x ∈ I ( lim sup n → x f n ( x ) − lim inf n → x f n ( x ) ) \nu (f) = {\sup _{x \in I}}(\lim {\sup _{n \to x}}{f^n}(x) - \lim {\inf _{n \to x}}{f^n}(x)) , where f n {f^n} is the n n th it...

For a continuous function chaotic in the sense of Li and Yorke the continuum hypothesis implies the existence of a scrambled set which has among others full outer Lebesgue measure.

We find a class of weakly unimodal C ∞ maps of an interval with zero topological entropy such that no such map f is Lyapunov stable on the set Per(f) of its periodic points. This disproves a statement published in several books and papers, e.g., by V.