Piotr Oprocha

• Professor
• Professor at AGH University of Krakow

For every $0<\alpha\le\infty$ we construct a continuous pure mixing map (topologically mixing, but not exact) on the Gehman dendrite with topological entropy $\alpha$. It has been previously shown by \v{S}pitalsk\'y that there are exact maps on the Gehman dendrite with arbitrarily low positive topological entropy. Together, these results show that...

We prove that cw-hyperbolic homeomorphisms with jointly continuous stable/unstable holonomies satisfy the periodic shadowing property and, if they are topologically mixing, the periodic specification property. We discuss difficulties to adapt Bowen's techniques to obtain a measure of maximal entropy for cw-hyperbolic homeomorphisms, exhibit the uni...

In the present paper, we focus on dynamical systems on the Gehman dendrite G. It is well-known that the set of end points of this dendrite is homeomorphic to the standard Cantor ternary set C. For any given surjective dynamical system acting on C, we provide constructions of dynamical systems on G, which are (i) topologically mixing but not exact o...

In this work, we investigate the dynamics of homeomorphisms through the lens of the local shadowing theory. We study the influence of positively shadowable points and positively shadowable measures into the local entropy theory of homeomrphisms. Specifically, we use pointwise shadowing to approximate invariant measures by ergodic measures with bigg...

The aim of this paper is to study measure-theoretical rigidity and partial rigidity for classes of Cantor dynamical systems including Toeplitz systems and enumeration systems. We use Bratteli diagrams to control invariant measures that are produced in our constructions. This leads to systems with desired properties. Among other things, we show that...

The paper proposes a methodology for incorporating uncertainties of material behaviours in the microstructure evolution model for eutectoid steels. The stochastic model of phase transformations was developed. The model accounts for a random character of the nucleation of pearlite and bainite and the differential growth equations for these structura...

We show that if $G$ is a topological graph, and $f$ is continuous map, then the induced map $\tilde{f}$ acting on the hyperspace $C(G)$ of all connected subsets of $G$ by natural formula $\tilde{f}(C)=f(C)$ carries the same entropy as $f$. This is well known that it does not hold on the larger hyperspace of all compact subsets. Also negative exampl...

We prove that the Lelek fan admits a completely scrambled weakly mixing homeomorphism. This is then used to show that for every there is a continuum of topological dimension admitting a weakly mixing completely scrambled homeomorphism. This provides a final answer to a question from 2001.

The motivation for this research was the need for a reliable prediction of the distribution of microstructural parameters in steels during thermomechanical processing. The stochastic model describing the evolution of dislocation populations and grain size, which considers the random phenomena occurring during the hot forming of metallic alloys, was...

In the present note we focus on dynamics on the Gehman dendrite $\mathcal{G}$. It is well-known that the set of its endpoints is homeomorphic to a standard Cantor ternary set. For any given surjective Cantor system $\mathcal{C}$ we provide constructions of (i) a mixing but not exact and (ii) an exact map on $\mathcal{G}$, such that in both cases th...

The paper deals with dynamics of expanding Lorenz maps, which appear in a natural way as Poincarè maps in geometric models of well-known Lorenz attractor. Using both analytical and symbolic approaches, we study connections between periodic points, completely invariant sets and renormalizations. We show that some renormalizations may be connected wi...

The need for a reliable prediction of the distribution of microstructural parameters in metallic materials after processing was the motivation for this work. The model describing phase transformations, which considers the stochastic character of the nucleation of the new phase, was formulated. Numerical tests of the model, including sensitivity ana...

In this paper we provide a detailed topological and measure-theoretic study of Lebesgue measure-preserving continuous circle maps that are composed with independent rotations on each of the sides. In particular, we analyze the stability of the locally eventually onto and measure-theoretic mixing properties.

We consider continuous maps of the interval which preserve the Lebesgue measure. Except for the identity map or $1 - \id$ all such maps have topological entropy at least $\log2/2$ and generically they have infinite topological entropy. In this article we show that thegeneric map has zero measure-theoreticentropy. This implies that there are dramati...

A compact space Y is called homeo-product-minimal if given any minimal system (X, T), it admits a homeomorphism S:Y→Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S:Y...

The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group, we mean any probability measure \nu on the collection \widetilde{\mathcal O} of linear orders of type \mathbb Z on G , invariant under the natural action of G on such orders. Multi...

In this paper we construct a paramaterized family of annular homeomorphisms with Birkhoff-like rotational attractors that vary continuously with the parameter, are all homeomorphic to the pseudo-circle, display interesting boundary dynamics and furthermore preserve the induced Lebesgue measure from the circle. Namely, in the constructed family of a...

Modern construction materials, including steels, have to combine strength with good formability. In metallic materials, these features are obtained for heterogeneous multiphase microstructures. Design of such microstructures requires advanced numerical models. It has been shown in our earlier works that models based on stochastic internal variables...

A compact space $Y$ is called homeo-product-minimal if given any minimal system $(X,T)$, it admits a homeomorphism $S:Y\to Y$, such that the product system $(X\times Y,T\times S)$ is minimal. We show that a large class of cofrontiers is homeo-product-minimal. This class contains R. H. Bing's pseudo-circle, answering a question of Dirb\'{a}k, Snoha...

Advanced numerical models, which predict heterogeneity of microstructural features, are needed to design modern multiphase steels. Models based on stochastic internal variables meet this requirement. Our objective was to account for the random character of the recrystallization and to transfer this randomness to equations describing the evolution o...

We survey the current state-of-the-art about the dynamical behavior of continuous Lebesgue measure-preserving maps on one-dimensional manifolds.

Hom shifts form a class of multidimensional shifts of finite type (SFT) and consist of colorings of the grid Z2 where adjacent colours must be neighbors in a fixed finite undirected simple graph G. This class in- cludes several important statistical physics models such as the hard square model. The gluing gap measures how far any two square pattern...

We study the metric mean dimension of Φ-irregular set IΦ(f) in dynamical systems with the shadowing property. In particular we prove that for dynamical systems with shadowing containing a chain recurrent class Y , the values of topological entropy together with the values of lower and upper metric mean dimension of the set IΦ(f)∩ B(Y, ε) ∩ CR(f) ar...

In this paper we provide a detailed topological and measure-theoretic study of Lebesgue measure-preserving circle maps that are rotated with inner and outer rotations which are independent of each other. In particular, we analyze the stability of the locally eventually onto and measure-theoretic mixing properties.

Construction metallic materials combine strength with formability. These features are obtained for heterogeneous microstructures with hard constituents dispersed in a soft matrix. On the other hand, sharp gradients of properties between phases cause a local fracture. Advanced models are needed to design microstructures with smoother gradients of th...

Enhancing strength-ductility synergy of materials has been for decades an objective of research on structural metallic materials. It has been shown by many researchers that significant improvement of this synergy can be obtained by tailoring heterogeneous multiphase microstructures. Since large gradients of properties in these microstructures cause...

In this article we study dynamical behaviour of generic Lebesgue measure-preserving interval maps. We show that for each k ⩾ 1 the set of periodic points of period at least k is a Cantor set of Hausdorff dimension zero and of upper box dimension one. Moreover, we obtain analogous results also in the context of generic Lebesgue measure-preserving ci...

The main goal of this paper is to study topological and measure-theoretic properties of an intriguing family of strange planar attractors. Building toward these results, we first show that any generic Lebesgue measure-preserving map f generates the pseudo-arc as inverse limit with f as a single bonding map. These maps can be realized as attractors...

We propose a novel methodology for estimating the epidemiological parameters of a modified SIRD model (acronym of Susceptible, Infected, Recovered and Deceased individuals) and perform a short-term forecast of SARS-CoV-2 virus spread. We mainly focus on forecasting number of deceased. The procedure was tested on reported data for Poland. For some s...

The need for a reliable prediction of the distribution of microstructural parameters in metallic materials during processing was the motivation for this work. The model describing the evolution of dislocation populations, which considers the stochastic aspects of occurring phenomena, was formulated. The validation of the presented model requires th...

In this paper, we study dynamics of maps on quasi-graphs and characterise their invariant measures. In particular, we prove that every invariant measure of a quasi-graph map with zero topological entropy has discrete spectrum. Additionally, we obtain an analog of Llibre–Misiurewicz’s result relating positive topological entropy with existence of to...

In this paper we completely solve the problem of when a Cantor dynamical system \begin{document}$ (X, f) $\end{document} can be embedded in \begin{document}$ \mathbb{R} $\end{document} with vanishing derivative. For this purpose we construct a refining sequence of marked clopen partitions of \begin{document}$ X $\end{document} which is adapted to a...

This paper deals with the solution of delay differential equations describing evolution of dislocation density in metallic materials. Hardening, restoration, and recrystallization characterizing the evolution of dislocation populations provide the essential equation of the model. The last term transforms ordinary differential equation (ODE) into de...

We are interested in dendrites for which all invariant measures of zero-entropy mappings have discrete spectrum, and we prove that this holds when the closure of the endpoint set of the dendrites is countable. This solves an open question which has been around for awhile, and almost completes the characterization of dendrites with this property.

In this article we show that R.H. Bing’s pseudo-circle admits a minimal non-invertible map. This resolves a conjecture raised by Bruin, Kolyada and Snoha in the negative. The main tool is a variant of the Denjoy–Rees technique, further developed by Béguin–Crovisier–Le Roux, combined with detailed study of the structure of the pseudo-circle. This is...

In this paper we show that generic continuous Lebesgue measure-preserving circle maps have the s-limit shadowing property. In addition, we obtain that s-limit shadowing is a generic property also for continuous circle maps. In particular, this implies that classical shadowing, periodic shadowing and limit shadowing are generic in these two settings...

We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms of a class of annulus homeomorphisms we describe, Handel-Anosov-Katok (HAK) homeomorphisms, that generalize the homeomorphism first described by Handel. Given a HAK homeomorphism and a homeomorphism of the Cantor set, the pseudo-suspension yields a homeomorp...

The aim of this paper is to show that α-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on [0,1]. On the basis of provided examples, we also present how the performed study on the struc...

We show that graph map with zero topological entropy is Li-Yorke chaotic if and only if it has an NS-pair (a pair of non-separable points containing in a same solenoidal $\omega$-limit set), and a non-diagonal pair is an NS-pair if and only if it is an IN-pair if and only if it is an IT-pair. This completes characterization of zero topological sequ...

The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group we mean any probability measure $\nu$ on the collection $\mathcal O$ of linear orders of type $\mathbb Z$ on $G$, invariant under the natural action of $G$ on such orders. Every fre...

The main goal of this paper is to study topological and measure-theoretic properties of an intriguing family of strange planar attractors. Building towards these results, we first show that any generic Lebesgue measure preserving map $f$ generates the pseudo-arc as inverse limit with $f$ as a single bonding map. These maps can be realized as attrac...

In the paper we study what sets can be obtained as $\alpha$-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those $\alpha$-limit sets are $\omega$-limit sets and for all but finitely many points $x$, we can obtain every $\omega$-limits set as the $\alpha$-limit set of a backward trajectory starting in...

We prove the entropy conjecture of M. Barge from 1989: for every r ∈ [0, ∞] there exists a pseudo-arc homeomorphism h, whose topological entropy is r. Until now all pseudo-arc homeomorphisms with known entropy have had entropy 0 or ∞.

Microstructure evolution model based on the differential equation describing evolution of dislocations was proposed. Sensitivity analysis was performed and parameters with the strongest influence on the output of the model were revealed. Identification of the model coefficients was performed for various metallic materials using inverse analysis for...

In this paper, we completely solve the problem when a Cantor dynamical system $(X,f)$ can be embedded in $\mathbb{R}$ with vanishing derivative everywhere. For this purpose, we construct a refining sequence of marked clopen partitions of $X$ which is adapted to a dynamical system of this kind. It turns out that there is a huge class of such systems...

In this paper we show that generic continuous Lebesgue measure preserving circle maps have the s-limit shadowing property. In addition we obtain that s-limit shadowing is a generic property also for continuous circle maps. In particular, this implies that classical shadowing, periodic shadowing and limit shadowing are generic in these two settings...

We show that for the generic continuous maps of the interval and circle which preserve the Lebesgue measure it holds for each k>0 that the set of periodic points of period k is a Cantor set of Hausdorff dimension zero and of upper box dimension one. Furthermore, building on this result, we show that there is a dense collection of transitive Lebesgu...

We are interested in dendrites for which all invariant measures of zero-entropy mappings have discrete spectrum, and we prove that this holds when the closure of the endpoint set of the dendrite is countable. This solves an open question which was around for awhile, almost completing the characterization of dendrites with this property.

The paper deals with dynamics of expanding Lorenz maps, which appear in a natural way as Poincaré maps in geometric models of well-known Lorenz attractor. Using both analytical and symbolic approaches, we study connections between periodic points, completely invariant sets and renormalizations. We show that some renormalizations may be connected w...

We show that for the generic continuous maps of the interval and circle which preserve the Lebesgue measure it holds for each k $\ge$ 1 that the set of periodic points of period k is a Cantor set of Hausdorff dimension zero and of upper box dimension one. Furthermore, building on this result, we show that there is a dense collection of transitive L...

In the paper we study what sets can be obtained as \begin{document}$ \alpha $\end{document}-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those \begin{document}$ \alpha $\end{document}-limit sets are \begin{document}$ \omega $\end{document}-limit sets and for all but finitely many points \begin{docu...

This paper deals with the solution of delay differential equations describing evolution of dislocation density in metallic materials. Hardening, restoration, and recrystallization characterizing the evolution of dislocation populations provide the essential equation of the model. The last term transforms ordinary differential equation (ODE) into de...

We show that topological mixing, weak mixing, the strong property P, and total transitivity are equivalent for coded systems (shift spaces presented by labeling the edges of a countable irreducible graphs by symbols from a finite alphabet). We provide an example of a topologically mixing coded system which cannot be approximated by any increasing s...

In the paper we use a special geometric structure of selected one-dimensional continua to prove that some stronger versions of the shadowing property are generic (or at least dense) for continuous maps acting on these spaces. Specifically, we prove that (i) the periodic \(\mathscr {T}_{S}\)-bi-shadowing property, where \(\mathscr {T}_{S}\) means so...

We study the properties of $\Phi$-irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of $\Phi$-irregular set in terms of entropy on chain recurrent classes and prove that $\Phi$-irregular sets of full entropy are typical. We also consider $\Ph...

Continuous development of transport industry is associated with the search for new construction materials that combine high strength with good plastic properties. Intensive research during last few decades have shown that there is still a huge potential for improvement of properties of various metallic materials. New grades called Advanced High Str...

Recently Bernardes and Darji provided a very nice characterization of a residual set of maps of Cantor set in terms of covers of special type. Using their characterization, we provide a more direct description of this class. This way we are able to provide a further characterization of dynamical properties (e.g. shadowing properties, nullness) of m...

We answer the last question left open in [Kočan, 2012] which asks whether there is a relation between an infinite LY-scrambled set and ω-chaos for dendrite maps. We construct a continuous self-map of a dendrite without any infinite LY-scrambled set but containing an uncountable ω-scrambled set.

The aim of this paper is presentation of some optimization strategies applicable in the optimization of multi-stage and multi-thread chain structures (linear and tree-structured acyclic graphs) with multiple intermediate goal functions. The inspirations for this type of analysis are production chains often seen in industrial plants. Production in t...

We show that every (invertible, or noninvertible) minimal Cantor system embeds in $\mathbb{R}$ with vanishing derivative everywhere. We also study relations between local shrinking and periodic points.

We show that every (invertible, or noninvertible) minimal Cantor system embeds in R with vanishing derivative everywhere. We also study relations between local shrinking and periodic points.

Let Tf:[0,1]→[0,1] be an expanding Lorenz map, this means Tfx:=f(x)(mod 1) where f:[0,1]→[0,2] is a strictly increasing map satisfying inf⁡f′>1. Then Tf has two pieces of monotonicity. In this paper, sufficient conditions when Tf is topologically mixing are provided. For the special case f(x)=βx+α with β≥23 a full characterization of parameters (β,...

We study three notions of shadowing: classical shadowing, limit (or asymptotic) shadowing, and s-limit shadowing. We show that classical and s-limit shadowing coincide for tent maps and, more generally, for piecewise linear interval maps with constant slopes, and are further equivalent to the linking property introduced by Chen in 1991. We also con...

In this paper we propose a new sufficient condition for disjointness with all minimal systems. Using proposed approach we construct a transitive dynamical system (X; T) disjoint with every minimal system and such that the set of transfer times N(x,U) is not an IP∗-set for some nonempty open set U ⊂ X and every x ϵ X. This example shows that the new...

In this review paper we describe some consequences of the shadowing property for global and local aspects of dynamics. We will put additional emphasis on approximation of invariant measures by ergodic measures with additional properties of their supports (minimality, positive entropy, mixing).

In this article, we show that R.H. Bing's pseudo-circle admits a minimal non-invertible map. This resolves a problem raised by Bruin, Kolyada and Snoha in the negative. The main tool is the Denjoy-Rees technique, further developed by Béguin-Crovisier-Le Roux, combined with detailed study into the structure of the pseudo-circle.

In this article, we show that R.H. Bing's pseudo-circle admits a minimal non-invertible map. This resolves a problem raised by Bruin, Kolyada and Snoha in the negative. The main tool is the Denjoy-Rees technique, further developed by B\'eguin-Crovisier-Le Roux, combined with detailed study into the structure of the pseudo-circle.

We answer the last question left open in [Z.~Ko\v{c}an, \emph{Chaos on one-dimensional compact metric spaces}, Internat. J. Bifur. Chaos Appl. Sci. Engrg. \textbf{22}, article id: 1250259 (2012)] which asks whether there is a relation between an infinite LY-scrambled set and $\omega$-chaos for dendrite maps. We construct a continuous self-map of a...

In this paper, we study dynamics of maps on quasi-graphs characterizing their invariant measures. In particular, we prove that every invariant measure of quasi-graph map with zero topological entropy has discrete spectrum. We also obtain an analog of Llibre-Misiurewicz's result relating positive topological entropy with existence of topological hor...

We prove that when $f$ is a continuous selfmap acting on compact metric space $(X,d)$ which satisfies the shadowing property, then the set of irregular points (i.e. points with divergent Birkhoff averages) has full entropy. Using this fact we prove that in the class of $C^0$-generic maps on manifolds, we can only observe (in the sense of Lebesgue m...

The goal of the research is an attempt of optimization of the hydrometallurgy-based zinc production line, consisting of three stages: mixing of raw materials, oxidative roasting and leaching. The output product of one stage is an input to the next stage. Goal of mixing is preparation of zinc concentrates mix on the basis of zinc concentrates origin...

Niniejsza praca stanowi próbę przedstawienia istniejących definicji chaosu dla dyskretnych układów dynamicznych. Dyskusję zawężono do zagadnień związanych z dynamiką topologiczną. Przedstawiono i umotywowano definicje: wrażliwości na warunki początkowe, chaosu w sensie Li i Yorke’a, Auslandera i Yorke’a, Devaneya, chaosu dystrybucyjnego, entropii t...

We explore five variants of the Ramsey shadowing property. Each variant is a different way of formalizing the notion that, in a given dynamical system, every sequence of points looks like an orbit. The main goal is to relate these properties to more classical notions in topological dynamics.

We give a counterexample to Theorem 9 in [T.K. Subrahmonian Moothathu, Syndetically proximal pairs, J. Math. Anal. Appl. 379 (2011) 656--663]. We also provide sufficient conditions for the conclusion of Theorem 9 to hold.

We prove that the Sierpiński curve admits a homeomorphism with strong mixing properties. We also prove that the constructed example does not have Bowen's specification property.

Motivated by a recent result of Ciesielski and Jasiński we study periodic point free Cantor systems that are conjugate to systems with vanishing derivative everywhere, and more generally locally radially shrinking maps. Our study uncovers a whole spectrum of dynamical behaviors attainable for such systems, providing new counterexamples to the Conje...

We explore the dynamics of graph maps with zero topological entropy. It is shown that a continuous map $f$ on a topological graph $G$ has zero topological entropy if and only if it is locally mean equicontinuous, that is the dynamics on each orbit closure is mean equicontinuous. As an application, we show that Sarnak's M\"obius Disjointness Conject...

We explore the dynamics of graph maps with zero topological entropy. It is shown that a continuous map f on a topological graph G has zero topological entropy if and only if it is locally mean equicontinuous, that is the dynamics on each orbit closure is mean equicontinuous. As an application, we show that Sarnak's Möbius disjointness conjecture is...

It is proved that to every invariant measure of a compact dynamical system one can associate a certain asymptotic pseudo orbit such that any point asymptotically tracing in average that pseudo orbit is generic for the measure. It follows that the asymptotic average shadowing property implies that every invariant measure has a generic point. The pro...

We give a counterexample to Theorem 9 in [T.K. Subrahmonian Moothathu, Syndetically proximal pairs, J. Math. Anal. Appl. 379 (2011) 656--663]. We also provide sufficient conditions for the conclusion of Theorem 9 to hold.

Different optimization strategies applied to sequence of several stages of production chains were validated in this paper. Two benchmark problems described by ordinary differential equations (ODEs) were considered. A water tank and a passive CR-RC filter were used as the exemplary objects described by the first and the second order differential equ...

We propose a definition of average tracing of finite pseudo-orbits and show that in the case of this definition measure center has the same prop-erty as nonwandering set for the classical shadowing property. We also show that the average shadowing property trivializes in the case of mean equicon-tinuous systems, and that it implies distributional c...

The aim of this paper is a study on relations between ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-chaos and the structure of ω\documentclas...

Motivated by a recent result of Ciesielski and Jasinski we study periodic point free Cantor systems that are conjugate to systems with vanishing derivative everywhere, and more generally locally radially shrinking maps. Our study uncovers a whole spectrum of dynamical behaviors attainable for such systems, providing new counterexamples to the Conje...

In this paper we deal with uniformly rigid systems obtained by a method introduced by Katznelson and Weiss and show that such systems never contain DC2 pairs. On the other hand, we introduce a modification of this technique that leads to a uniformly rigid system with DC2 pairs. We also show that every dynamical system contains a pair of distinct po...

We study three notions of shadowing: classical shadowing, limit (or asymptotic) shadowing, and s-limit shadowing. We show that classical and s-limit shadowing coincide for tent maps and, more generally, for piecewise linear interval maps with constant slopes, and are further equivalent to the linking property introduced by Chen in 1991. We also con...

In this paper we study relations of various types of sensitivity between a t.d.s. (X, T ) and t.d.s. (M(X ), TM) induced by (X, T ) on the space of probability measures. Among other results, we prove that F -sensitivity of (M(X ), TM) implies the same of (X, T ) and the converse is also true when F is a filter. We show that (X, T ) is multi-sensiti...

We explore recurrence properties arising from dynamical approach to combinatorial problems like the van der Waerden Theorem. We describe relations between these properties, study their consequences for dynamics, and demonstrate connections to combinatorial problems. In particular, we present a measure-theoretical analog of a result of Glasner on mu...