Piotr Oprocha homepage

2024

Renormalization in Lorenz maps – completely invariant sets and periodic orbits

Advances in Mathematics

456 (2024), article id: 109890

Journal PaperRepresentative publications Ł. Cholewa, P. Oprocha

Renormalization in Lorenz maps – completely invariant sets and periodic orbits

Ł. Cholewa, P. Oprocha

Journal PaperRepresentative publications

About The Publication

The paper deals with dynamics of expanding Lorenz maps, which appear in a natural way as Poincar\`e maps in geometric models of well-known Lorenz attractor. We study connections between periodic points, completely invariant sets and renormalizations. We show that in general, renormalization cannot be fully characterized by a completely invariant set, however there are various situations when such characterization is possible. This way we provide a better insight into the structure of renormalizations in Lorenz maps, correcting some gaps existing in the literature and completing to some extent the description of possible dynamics in this important field of study.

2024

Internal Maps With Dense Periodicity

Arxiv

Unpublished manuscript

Preprint J. Bobok, J. Cinc, P. Oprocha, S. Troubetzkoy

Internal Maps With Dense Periodicity

J. Bobok, J. Cinc, P. Oprocha, S. Troubetzkoy

Preprint

About The Publication

We consider the class of interval maps with dense set of periodic points CP and its closure Cl(CP) equipped with the metric of uniform convergence. Besides studying basic topological properties and density results in the spaces CP and Cl(CP) we prove that Cl(CP) is dynamically characterized as the set of interval maps for which every point is chain-recurrent. Furthermore, we prove that a strong topological expansion property called topological exactness (or leo property) is attained on the open dense set of maps in CP and on a residual set in Cl(CP). Moreover, we show that every second category set in CP and Cl(CP) is rich in a sense that it contains uncountably many conjugacy classes. An analogous conclusion also holds in the setting of interval maps preserving any fixed non-atomic probability measure with full support. Finally, we give a detailed description of the structure of periodic points of generic maps in CP and Cl(CP) and show that generic maps in CP and Cl(CP) satisfy the shadowing property.

2024

A note on homeo-product-minimality

Qualitative Theory of Dynamical Systems

23 (2024), article id: 140.

Journal Paper J. P. Boroński, Magdalena Foryś-Krawiec, Piotr Oprocha

A note on homeo-product-minimality

J. P. Boroński, Magdalena Foryś-Krawiec, Piotr Oprocha

Journal Paper

About The Publication

A compact space Y is called homeo-product-minimal if given any minimal system (X,T), it admits a homeomorphism S:Y→Y, such that the product system (X x Y, S x T) 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 Dirbák, Snoha and Spitalsky from [Minimal direct products, Trans. Amer. Math. Soc. 375 (2022)]

2024

Multiorders in amenable group actions

Groups, Geometry, and Dynamics

18 (2024), 25–65.

Journal Paper Tomasz Downarowicz, Piotr Oprocha, Mateusz Więcek, Guohua Zhang

Multiorders in amenable group actions

Tomasz Downarowicz, Piotr Oprocha, Mateusz Więcek, Guohua Zhang

Journal Paper

About The Publication

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 ν on the collection O of linear orders of type Z on G, invariant under the natural action of G on such orders. Every free measure-preserving G-action (X,μ,G) has a multiorder (O,ν,G) as a factor and has the same orbits as the Z-action (X,μ,S), where S is the successor map determined by the multiorder factor. The sub-sigma-algebra ΣO associated with the multiorder factor is invariant under S, which makes the corresponding Z-action (O,ν,S) a factor of (X,μ,S). We prove that the entropy of any G-process generated by a finite partition of X, conditional with respect to ΣO, is preserved by the orbit equivalence with (X,μ,S). Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to the one known for Z-actions. This fact is applied to prove a variant of a result by Rudolph and Weiss. The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a sub-sigma-algebra Σ, as soon as the “orbit change” is Σ-measurable. In our variant, we replace the measurability assumption by a simpler one: Σ should be invariant under both actions and the actions on the resulting factor should be free. In conclusion we prove that the Pinsker sigma-algebra of any G-process can be identified (with probability 1) using the following algorithm: (1) fix an arbitrary multiorder on G, (2) select any order from the support of that multiorder, (3) in the process, find the “remote past” along the selected order

2024

Are generic dynamical properties stable under composition with rotations?

Proceedings of the American Mathematical Society

152 (2024), 3011-3026.

Journal Paper J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy

Are generic dynamical properties stable under composition with rotations?

J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy

Journal Paper

About The Publication

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.

2024

Metric mean dimension of irregular sets for maps with shadowing

Journal of Difference Equations and Applications

30(5) (2024), 603–618.

Journal Paper Magdalena Foryś-Krawiec, Piotr Oprocha

Metric mean dimension of irregular sets for maps with shadowing

Magdalena Foryś-Krawiec, Piotr Oprocha

Journal Paper

About The Publication

We study the metric mean dimension of g-irregular set I(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 I(f)∩B(Y,g)∩CR(f) are bounded from below by the respective values for class

2024

Tranched graphs: consequences for topology and dynamics

Unpublished manuscript

Preprint M. Kowalewski, P. Oprocha

Tranched graphs: consequences for topology and dynamics

M. Kowalewski, P. Oprocha

Preprint

About The Publication

We compare quasi graphs and generalized sin(1/x)-type continua – two classes of continua that generalize topological graphs and contain the Warsaw circle as a non-trivial common element. We show that neither class is the subset of the other, provide some characterizations and present illustrative examples. We unify both approaches by considering the class of tranched graphs, compare it to concepts known from the literature and describe how the topological structure of its elements restricts possible dynamics.

2024

Short-range and long-range order: a transition in block-gluing behavior in Hom shifts

Journal d'Analyse Mathématique

Accepted for publication

Journal PaperPreprint S. Gangloff, B. Hellouin de Menibus, P. Oprocha

Short-range and long-range order: a transition in block-gluing behavior in Hom shifts

S. Gangloff, B. Hellouin de Menibus, P. Oprocha

Journal PaperPreprint

About The Publication

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 patterns of size
n can be glued, which can be seen as a measure of the range of order,
and affects the possibility to compute the entropy (or free energy per
site) of a shift. This motivates a study of the possible behaviors of the
gluing gap. The class of Hom shifts has the interest that mixing type
properties can be formulated in terms of algebraic graph theory, which
has received a lot of attention recently. Improving some former work of
N. Chandgotia and B. Marcus, we prove that the gluing gap either de-
pends linearly on n or is dominated by log(n). We also find a Hom shift
with gap Θ(log(n)), infirming a conjecture formulated by R. Pavlov and
M. Schraudner. The physical interest of these results is to better under-
stand the transition from short-range to long-range order (respectively
sublogarithmic and linear gluing gap), which is reflected in whether some
parameter, the square cover, is finite or infinite.

2024

Parameterized family of annular homeomorphisms with pseudo-circle attractors

Journal of Differential Equations

407 (2024), 102-132.

Journal Paper Jernej Činč, Piotr Oprocha

Parameterized family of annular homeomorphisms with pseudo-circle attractors

Jernej Činč, Piotr Oprocha

Journal Paper

About The Publication

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 attractors the outer prime ends rotation number vary continuously with the parameter through the interval [0,1/2]. This, in particular, answers a question from [J. London Math. Soc. (2) {\bf 102} (2020), 557–579]. To show main results of the paper we first prove a result of an independent interest, that Lebesgue-measure preserving circle maps generically satisfy the crookedness condition which implies that generically the inverse limits of Lebesgue measure-preserving circle maps are hereditarily indecomposable. For degree one circle maps, this implies that the generic inverse limit in this context is the R.H. Bing’s pseudo-circle

2024

Accounting for random character of recrystallisation and uncertainty of process parameters in the modelling of phase transformations in steels

Canadian Metallurgical Quarterly

63 (2024), 460–467.

Journal Paper Danuta Szeliga, Natalia Czyżewska, Jan Kusiak, Paweł Morkisz, Piotr Oprocha, Maciej Pietrzyk, Paweł Przybyłowicz

Accounting for random character of recrystallisation and uncertainty of process parameters in the modelling of phase transformations in steels

Danuta Szeliga, Natalia Czyżewska, Jan Kusiak, Paweł Morkisz, Piotr Oprocha, Maciej Pietrzyk, Paweł Przybyłowicz

Journal Paper

About The Publication

The problem of multiphase microstructure heterogeneity was considered in this paper. The stochastic model of multi-step hot deformation was applied to calculate histograms of the dislocation density and the grain size at the beginning of phase transformations. These histograms were used as input data for simulation of phase transformations using conventional deterministic modelling. Heterogeneity of the phase composition was calculated for various constant cooling rates, as well as for the industrial laminar cooling process. In the latter case, uncertainty of the boundary conditions was additionally accounted for. The results showing an influence of the heterogeneity of the dislocation density and the grain size on the microstructure heterogeneity in the final product can be considered reliable, because they are based on the material models, which were identified and verified experimentally in earlier publications. The effect of the uncertainty of the boundary conditions is presented in a qualitative manner only, and may be the subject of future research.

2023

On limit sets and equicontinuity in hyperspace of continua in dimension one

Unpublished manuscript

Preprint Domagoj Jelić and Piotr Oprocha

On limit sets and equicontinuity in hyperspace of continua in dimension one

Domagoj Jelić and Piotr Oprocha

Preprint

2023

Continuous Lebesgue measure-preserving maps on one-dimensional manifolds: a survey

Topology and its Applications

Accepted for publication

Journal PaperPreprint Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy

Continuous Lebesgue measure-preserving maps on one-dimensional manifolds: a survey

Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy

Journal PaperPreprint

About The Publication

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

2023

Lelek fan admits completely scrambled weakly mixing homeomorphism

Unpublished manuscript

Preprint P. Oprocha

Lelek fan admits completely scrambled weakly mixing homeomorphism

P. Oprocha

Preprint

About The Publication

We prove that Lelek fan admits completely scrambled weakly mixing homeomorphism. This is then used to show

that for every n there is a continuum of topological dimension $n$ admitting weakly mixing completely scrambled homeomorphism.

This provides a final answer to a question from 2001.

2023

Various questions around finitely positively expansive dynamical systems

ArXiv

Unpublished manuscript

Preprint Silvère Gangloff, Pierre Guillon, Piotr Oprocha

Various questions around finitely positively expansive dynamical systems

Silvère Gangloff, Pierre Guillon, Piotr Oprocha

Preprint

About The Publication

It is well-known that when a positively expansive dynamical system is invertible then its underlying space is finite. C.Morales has introduced a decade ago a natural way to generalize positive expansiveness, by introducing other properties that he called positive n-expansiveness, for all n≥1, positive 1-expansiveness being identical to positive expansiveness. Contrary to positive expansiveness, positive n-expansiveness for n>1 does not enforce that the space is finite when the system is invertible. In the present paper we call finitely positively expansive dynamical systems as the ones which are positively n-expansive for some integer n, and prove several results on this class of systems. In particular, the well-known result quoted above is true if we add the constraint of shadowing property, while it is not if this property is replaced with minimality. Furthermore, finitely positively expansive systems cannot occur on certain topological spaces such as the interval, when the system is assumed to be invertible finite positive expansiveness implies zero topological entropy. Overall we show that the class of finitely positively expansive dynamical systems is quite rich and leave several questions open for further research.

2021

Beyond 0 and ∞: A solution to the Barge Entropy Conjecture

Arxiv

Unpublished manuscript

Preprint J. Boroński, J. Činč, P. Oprocha

Beyond 0 and ∞: A solution to the Barge Entropy Conjecture

J. Boroński, J. Činč, P. Oprocha

Preprint

About The Publication

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 ∞.

2023

S-limit shadowing is generic for continuous Lebesgue measure preserving circle maps

Ergodic Theory and Dynamical Systems

43(1) (2023), 78–98.

Journal Paper Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy

S-limit shadowing is generic for continuous Lebesgue measure preserving circle maps

Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy

Journal Paper

About The Publication

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 as well.

2023

On recurrence and entropy in hyperspace of continua in dimension one

Fundamenta Mathematicae

263 (2023), 23–50.

Journal Paper Domagoj Jelić and Piotr Oprocha

On recurrence and entropy in hyperspace of continua in dimension one

Domagoj Jelić and Piotr Oprocha

Journal Paper

About The Publication

We show that if G is a topological graph, and f is contin-
uous map, then the induced map f acting on the hyperspace C(G) of
all connected subsets of G by natural formula 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 examples were given
for the hyperspace C(X) on some continua X, including dendrites.
Our work extends previous positive results obtained first for much
simpler case of compact interval by completely different tools.

2023

New exotic minimal sets from pseudo-suspensions of Cantor systems

Journal of Dynamics and Differential Equations

35(2) (2023), 1175–1201.

Journal Paper J. Boroński, A. Clark, P. Oprocha

New exotic minimal sets from pseudo-suspensions of Cantor systems

J. Boroński, A. Clark, P. Oprocha

Journal Paper

About The Publication

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 homeomorphism of a new space that admits a homeomorphism that combines features of both of the original homeomorphisms. This allows us to answer a well known open question by providing examples of hereditarily indecomposable continua that admit homeomorphisms of intermediate complexity. Additionally, we show that such examples occur as minimal sets of volume preserving smooth diffeomorphisms of 4-dimensional manifolds. We also use our techniques to exhibit the first examples of minimal, uniformly rigid and weakly mixing homeomorphisms in dimension 1, and these can also be realized as invariant sets of smooth diffeomorphisms of a 4-manifold. Until now the only known examples of spaces that admit minimal, uniformly rigid and weakly mixing homeomorphisms were modifications of those given by Glasner and Maon in dimension at least 2.

2023

Dendrites and measures with discrete spectrum

Ergodic Theory and Dynamical Systems

43(2) (2023), 545–555.

Journal Paper M. Foryś-Krawiec, J. Hantáková, J. Kupka, P. Oprocha, S. Roth

Dendrites and measures with discrete spectrum

M. Foryś-Krawiec, J. Hantáková, J. Kupka, P. Oprocha, S. Roth

Journal Paper

About The Publication

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.

2023

A Comparative Study of Deterministic and Stochastic Models of Microstructure Evolution during Multi-Step Hot Deformation of Steels

Materials

16 (2023), article id: 3316.

Journal Paper Piotr Oprocha, Natalia Czyżewska, Konrad Klimczak, Jan Kusiak, Paweł Morkisz, Maciej Pietrzyk, Paweł Potorski, Danuta Szeliga

A Comparative Study of Deterministic and Stochastic Models of Microstructure Evolution during Multi-Step Hot Deformation of Steels

Piotr Oprocha, Natalia Czyżewska, Konrad Klimczak, Jan Kusiak, Paweł Morkisz, Maciej Pietrzyk, Paweł Potorski, Danuta Szeliga

Journal Paper

About The Publication

Modern construction materials, including steels, have to combine strength with good forma- bility. 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 meet this requirement. The focus of the present paper is on deterministic and stochastic approaches to modelling hot deformation of multiphase steels. The main aim was to survey recent advances in describing the evolution of disloca- tions and grain size accounting for the stochastic character of the recrystallization. To present a path leading to this objective, we reviewed several papers dedicated to the application of internal variables and statistical approaches to modelling recrystallization. Following this, the idea of the model with dislocation density and grain size being the stochastic internal variables is described. Experiments composed of hot compression of cylindrical samples are also included for better presentation of the utility of this approach. Firstly, an empirical data describing the loads as a function of time during compression and data needed to create histograms of the austenite grain size after the tests were collected. Using the measured data, identification and validation of the models were performed. To present possible applications of the model, it was used to produce a simulation imitating industrial hot-forming processes. Finally, calculations of the dislocation density and the grain size distribution were utilized as inputs in simulations of phase transformations during cooling. Distributions of the ferrite volume fraction and the ferrite grain size after cooling recapitulate the paper. This should give readers good overview on the application of collected equations in practice.

2023

Through process stochastic model of hot strip rolling

Materials Research Proceedings

28 (2023), 1631 - 1640.

Journal Paper Danuta Szeliga, Natalia Czyżewska, Jan Kusiak, Piotr Oprocha, Maciej Pietrzyk, Paweł Przybyłowicz

Through process stochastic model of hot strip rolling

Danuta Szeliga, Natalia Czyżewska, Jan Kusiak, Piotr Oprocha, Maciej Pietrzyk, Paweł Przybyłowicz

Journal Paper

About The Publication

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 of the dislocations and the grain size during hot deformation. The idea of the internal variable model with the dislocation density and the grain size being stochastic variables is described briefly in the paper. Histograms of the grain size measured in the experimental compression tests were used to identify the coefficients in the model. Inverse analysis with the objective function based on the distance between histograms was applied. The model was used to simulation of the various technological routes in the industrial process of the hot strip rolling.

2022

Parametrized family of pseudo-arc attractors: physical measures and prime end rotations

PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY

125(2) (2022), 318-357.

Journal PaperRepresentative publications Jernej Činč, Piotr Oprocha

Parametrized family of pseudo-arc attractors: physical measures and prime end rotations

Jernej Činč, Piotr Oprocha

Journal PaperRepresentative publications

About The Publication

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 attractors of disc homeomorphisms in such a way that the attractors vary continuously (in Hausdorff distance on the disc) with the change of bonding map as a parameter. Furthermore, for generic Lebesgue measure preserving maps f the background Oxtoby-Ulam measures induced by Lebesgue measure for f on the interval are physical on the disc and in addition there is a dense set of maps f defining a unique physical measure. Moreover, the family of physical measures on the attractors varies continuously in the weak* topology; i.e. the parametrized family is statistically stable. We also find an arc in the generic Lebesgue measure preserving set of maps and construct a family of disk homeomorphisms parametrized by this arc which induces a continuously varying family of pseudo-arc attractors with prime ends rotation numbers varying continuously in [0,1/2]. It follows that there are infinitely many dynamically non-equivalent embeddings of the pseudo-arc in this family of attractors

2022

On mathematical aspects of evolution of dislocation density in metallic materials

IEEE Access

10 (2022), 86793 - 86812.

Journal Paper N. Czyżewska, J. Kusiak, P. Morkisz, P. Oprocha, M. Pietrzyk, P. Przybyłowicz, Ł. Rauch, D. Szeliga

On mathematical aspects of evolution of dislocation density in metallic materials

N. Czyżewska, J. Kusiak, P. Morkisz, P. Oprocha, M. Pietrzyk, P. Przybyłowicz, Ł. Rauch, D. Szeliga

Journal Paper

About The Publication

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 trans- forms ordinary differential equation (ODE) into delay differential equation (DDE) with strong (in general, Hölder) nonlinearity. We prove upper error bounds for the explicit Euler method, under the assumption that the right- hand side function is Hölder continuous and monotone which allows us to com- pare accuracy of other numerical methods in our model (e.g. Runge-Kutta), in particular when explicit formulas for solutions are not known. Finally, we test the above results in simulations of real industrial process.

2022

Periodic points and shadowing property for generic Lebesgue measure preserving interval maps

Nonlinearity

35(5) (2022), article id: 2534

Journal Paper Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy

Periodic points and shadowing property for generic Lebesgue measure preserving interval maps

Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy

Journal Paper

About The Publication

We show that for the generic continuous maps of the interval and circle which preserve the Lebesgue measure it holds for each k ≥ 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 Lebesgue measure preserving interval map whose periodic points have full Lebesgue measure and whose periodic points of period k have positive measure for each k ≥ 1. Finally, we show that the generic continuous maps of the interval which preserve the Lebesgue measure satisfy the shadowing and periodic shadowing property.

2022

Stochastic model describing evolution of microstructural parameters during hot rolling of steel plates and strips

ARCHIVES OF CIVIL AND MECHANICAL ENGINEERING

22(3) (2022), Article id: 139.

Journal Paper D. Szeliga, N. Czyzewska, K. Klimczak, J. Kusiak, R. Kuziak, P. Morkisz, P. Oprocha, M. Pietrzyk, L Poloczek, P. Przybylowicz

Stochastic model describing evolution of microstructural parameters during hot rolling of steel plates and strips

D. Szeliga, N. Czyzewska, K. Klimczak, J. Kusiak, R. Kuziak, P. Morkisz, P. Oprocha, M. Pietrzyk, L Poloczek, P. Przybylowicz

Journal Paper

About The Publication

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 a decrease of the local fracture resistance, the objective of research is to obtain smoother gradients of properties by control of the manufacturing process. Advanced material models are needed to design such microstructures with smooth gradients. These models should supply information about distributions of various microstructural features, instead of their average values. 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 into equations describing the evolution of dislocations and grain size during hot deformation and during interpass times. The idea of this stochastic model is described in the paper. Experiments composed of uniaxial compression tests were performed to supply data for the identification and verification of the model in the hot deformation and static recrystallization parts. Histograms of the grain size were measured after hot deformation and at different times after the end of deformation. Identification and validation of the model were performed. The validated model, which predicts evolution of heterogeneous multiphase microstructure, is the main output of our work. The model was implemented in the finite element program for hot rolling of plates and sheets and simulations of these processes were performed. The model’s capability to compare and evaluate various rolling strategies are demonstrated in the paper.

2022

Formulation, identification and validation of a stochastic internal variables model describing the evolution of metallic materials microstructure during hot forming

INTERNATIONAL JOURNAL OF MATERIAL FORMING

15(4) (2022), Article id: 53.

Journal Paper D. Szeliga, N. Czyzewska, K. Klimczak, J. Kusiak, R. Kuziak, P. Morkisz, P. Oprocha, V. Pidvysots'kyy, M. Pietrzyk, P. Przybylowicz.

Formulation, identification and validation of a stochastic internal variables model describing the evolution of metallic materials microstructure during hot forming

D. Szeliga, N. Czyzewska, K. Klimczak, J. Kusiak, R. Kuziak, P. Morkisz, P. Oprocha, V. Pidvysots'kyy, M. Pietrzyk, P. Przybylowicz.

Journal Paper

About The Publication

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 their features. 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 of dislocations and grain size. The idea of the internal variable model with dislocation density and grain size being stochastic variables is described in the paper. Experiments composed of uniaxial compression tests were performed to supply data for the identification and verification of the model. The loads as a function of time during compression and histograms of the grain size after deformation were measured in each test. Identification and validation of the model were performed. Finally, the developed model was applied to simulate selected industrial hot forming processes.

2022

Thermal-mechanical finite element simulation of flat bar rolling coupled with a stochastic model of microstructure evolution

Computer Methods in Materials Science

22(1) (2022), 127-138

Journal Paper D. Szeliga, N. Czyżewska, J. Kusiak, R. Kuziak, P. Morkisz, P. Oprocha, M. Pietrzyk, M. Piwowarczyk, Ł Poloczek, P. Przybyłowicz, Ł Rauch, N. Wolańska

Thermal-mechanical finite element simulation of flat bar rolling coupled with a stochastic model of microstructure evolution

D. Szeliga, N. Czyżewska, J. Kusiak, R. Kuziak, P. Morkisz, P. Oprocha, M. Pietrzyk, M. Piwowarczyk, Ł Poloczek, P. Przybyłowicz, Ł Rauch, N. Wolańska

Journal Paper

About The Publication

It is generally recognized that the kinetics of phase transformations during the cooling of steel products depends to a large extent on the state of the austenite after rolling. Austenite deformation (when recrystallization is not complete) and grain size have a strong influence on the nucleation and growth of low-temperature phases. Thus, the general objective of the present work was the formulation of a numerical model which simulates thermal, mechanical and microstructural phenomena during multipass hot rolling of flat bars. The simulation of flat bar rolling accounting for the evolution of a heterogeneous microstructure was the objective of the work. A conventional finite-element program was used to calculate the distribution of strains, stresses, and temperatures in the flat bar during rolling and during interpass times. The FE program was coupled with the stochastic model describing austenite microstructure evolution. In this model, the random character of the recrystallization was accounted for. Simulations supplied information about the distributions of the dislocation density and the grain size at various locations through the thickness of the bars.

2022

Quasi-graphs, zero entropy and measures with discrete spectrum

Nonlinearity

35(3) (2022), article id: 1360.

Journal Paper J. Li, P. Oprocha, G. Zhang

Quasi-graphs, zero entropy and measures with discrete spectrum

J. Li, P. Oprocha, G. Zhang

Journal Paper

About The Publication

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. Additionally, we obtain an analog of Llibre-Misiurewicz’s result relating positive topological entropy with existence of topological horseshoes. We also study dynamics on dendrites and show that if a continuous map on a dendrite, whose set of all endpoints is closed and has only finitely many accumulation points, has zero topological entropy, then every invariant measure supported on an orbit closure has discrete spectrum.

2022

On the structure of alpha-limit sets of backward trajectories for graph maps

Discrete and Continuous Dynamical Systems

42(3) (2022), 1435-1463.

Journal Paper M. Foryś-Krawiec, J. Hantáková, P. Oprocha

On the structure of alpha-limit sets of backward trajectories for graph maps

M. Foryś-Krawiec, J. Hantáková, P. Oprocha

Journal Paper

About The Publication

In the paper we study what sets can be obtained as α-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those α-limit sets are ω-limit sets and for all but finitely many points x, we can obtain every ω-limits set as the α-limit set of a backward trajectory starting in x. For zero entropy maps, every α-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.

2021

Graph maps with zero topological entropy and sequence entropy pairs

Proceedings of the American Mathematical Society

149(11) (2021), 4757–4770

Journal Paper J. Li, X. Liang, P. Oprocha

Graph maps with zero topological entropy and sequence entropy pairs

J. Li, X. Liang, P. Oprocha

Journal Paper

About The Publication

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 ω-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 sequence entropy for graph maps.

2021

On alpha-limit sets in Lorenz maps

Entropy

23(9) (2021), article id: 1153

Journal Paper Łukasz Cholewa, Piotr Oprocha

On alpha-limit sets in Lorenz maps

Łukasz Cholewa, Piotr Oprocha

Journal Paper

About The Publication

The aim of this paper is to show that alpha-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 closely performed study on the structure of alpha-limit sets is connected with calculation of the topological entropy.

2021

Sensitivity analysis, identification and validation of the dislocation density-based model for metallic materials

Metallurgical Research & Technology

118 (2021). article id: 317.

Journal Paper D. Szeliga, N. Czyżewska, K. Klimczak, J. Kusiak, P. Morkisz, P. Oprocha, M. Pietrzyk, P. Przybyłowicz

Sensitivity analysis, identification and validation of the dislocation density-based model for metallic materials

D. Szeliga, N. Czyżewska, K. Klimczak, J. Kusiak, P. Morkisz, P. Oprocha, M. Pietrzyk, P. Przybyłowicz

Journal Paper

About The Publication

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 experimental data. The model was implemented in the finite element code and simulations of various hot forming processes were performed.

2021

A Cantor dynamical system is slow if and only if all its finite orbits are attracting

Discrete and Continuous Dynamical Systems

42(6) (2022), 3039-3064.

Journal Paper S. Gangloff, P. Oprocha

A Cantor dynamical system is slow if and only if all its finite orbits are attracting

S. Gangloff, P. Oprocha

Journal Paper

About The Publication

In this paper we completely solve the problem when a Cantor dynamical system (X,f) can be embedded in 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 turn out that there is a huge class of such systems.

2021

Inverse problem in stochastic approach to modelling of microstructural parameters in metallic materials during processing

Mathematical Problems in Engineering

Volume 2022, Article ID 9690742

Journal Paper K. Klimczak, P. Oprocha, J. Kusiak, D. Szeliga, P. Morkisz, P. Przybyłowicz, N. Czyżewska, M. Pietrzyk

Inverse problem in stochastic approach to modelling of microstructural parameters in metallic materials during processing

K. Klimczak, P. Oprocha, J. Kusiak, D. Szeliga, P. Morkisz, P. Przybyłowicz, N. Czyżewska, M. Pietrzyk

Journal Paper

2021

Minimal non-invertible maps on the pseudo-circle

Journal of Dynamics and Differential Equations

33 (2021), 1897–1916.

Journal Paper J. Boroński, J. Kennedy, X. Liu, P. Oprocha

Minimal non-invertible maps on the pseudo-circle

J. Boroński, J. Kennedy, X. Liu, P. Oprocha

Journal Paper

About The Publication

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 the first example of a planar 1-dimensional space that admits both minimal homeomorphisms and minimal noninvertible maps.

2021

On entropy of Φ-irregular and Φ-level sets in maps with the shadowing property

Discrete and Continuous Dynamical Systems - A

41(3) (2021), 1271-1296.

Journal Paper M. Foryś-Krawiec, J. Kupka, P. Oprocha, X. Tian

On entropy of Φ-irregular and Φ-level sets in maps with the shadowing property

M. Foryś-Krawiec, J. Kupka, P. Oprocha, X. Tian

Journal Paper

About The Publication

We study the properties of

Φ

-irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of

Φ

-irregular set in terms of entropy on chain recurrent classes and prove that

Φ

-irregular sets of full entropy are typical. We also consider

Φ

-level sets (sets of points whose Birkhoff average is in a specified interval), relating entropy they carry with the entropy of some ergodic measures. Finally, we study the problem of large deviations considering the level sets with respect to reference measures.

2020

Shadowing is generic on various one-dimensional continua with a special geometric structure

Journal of Geometric Analysis

30 (2020), 1836–1864.

Journal Paper P. Kościelniak, M. Mazur, P. Oprocha, Ł. Kubica

Shadowing is generic on various one-dimensional continua with a special geometric structure

P. Kościelniak, M. Mazur, P. Oprocha, Ł. Kubica

Journal Paper

About The Publication

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

T_{S}

-bi-shadowing property, where

T_{S}

means some class of continuous methods, is generic as well as the s-limit shadowing property is dense in the space of all continuous maps (and all continuous surjective maps) of any topological graph; (ii) the

T_{S}

-bi-shadowing property is generic as well as the s-limit shadowing property is dense in the space of all continuous maps of any dendrite; (iii) the

T_{S}

-bi-shadowing property is generic in the space of all continuous maps of chainable continuum that can by approximated by arcs from the inside. The results of the paper extend ones obtained over the last few decades by various authors (see, e.g., Kościelniak in J Math Anal Appl 310:188–196, 2005; Kościelniak and Mazur in J Differ Equ Appl 16:667–674, 2010; Kościelniak et al. in Discret Contin Dyn Syst 34:3591–3609, 2014; Mazur and Oprocha in J Math Anal Appl 408:465–475, 2013; Mizera in Generic Properties of One-Dimensional Dynamical Systems, Ergodic Theory and Related Topics, III, Springer, Berlin, 1992; Odani in Proc Am Math Soc 110:281–284, 1990; Pilyugin and Plamenevskaya in Topol Appl 97:253–266, 1999; and Yano in J Fac Sci Univ Tokyo Sect IA Math 34:51–55, 1987) for both homeomorphisms and continuous maps of compact manifolds, including (in particular) an interval and a circle, which are the simplest examples of one-dimensional continua. Moreover, from a technical point of view our considerations are a continuation of those carried out in the earlier work by Mazur and Oprocha in J. Math. Anal. Appl. 408:465–475, 2013.

2020

Multi-sensitivity, multi-transitivity and Δ-transitivity

Contemporary Mathematics

In: Dynamics: Topology and Numbers
Contemporary Mathematics vol. 744
2020, pp. 231-244.

Book Chapter T. Yu, P. Oprocha, G. Zhang

Multi-sensitivity, multi-transitivity and Δ-transitivity

T. Yu, P. Oprocha, G. Zhang

Book Chapter

About The Publication

In thispaper firstly we construct for each m∈N\{1} a weakly mixing system which is (1,…,m-1)-sensitive but not (1,…,m−1,m)-sensitive, and a minimal system which is (1,2)-sensitive but not (1,2,3)-sensitive. It is known that ∆-(1, 2) transitivity implies weak mixing. We will show that, though (a, b)-transitivity implies total transitivity for all a,b∈N, for a general vector a∈Nr, r ∈ N, ∆-a transitivity implies weak mixing if and only if the vector a satisfies certain conditions. Furthermore, we prove that this difference will disappear for measurable systems, that is, for measure-theoretical setting multi-ergodicity is equivalent to weak mixing.

2019

All minimal Cantor systems are slow

Bulletin of the London Mathematical Society

51(6) (2019), 937-1128.

Journal Paper J. Boroński, J. Kupka, P. Oprocha

All minimal Cantor systems are slow

J. Boroński, J. Kupka, P. Oprocha

Journal Paper

About The Publication

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.

2019

Mixing properties in expanding Lorenz maps

Advances in Mathematics

343 (2019), 712-755.

Journal PaperRepresentative publications P. Oprocha, P. Potorski, P. Raith

Mixing properties in expanding Lorenz maps

P. Oprocha, P. Potorski, P. Raith

Journal PaperRepresentative publications

About The Publication

Let $T:[0,1]\to [0,1]$ be an expanding Lorenz map, this means $T x=f(x) \mod 1$ where $f: [0,1]\to [0,2]$ is a strictly increasing map satisfying $\inf f’>1$. Then $T$ has two pieces of monotonicity. In this paper, sufficient conditions when $\T$ is topologically mixing are provided. For the special case $f(x)=\beta x +\alpha$ with $\beta\geq \sqrt[3]{2}$ a full characterization of parameters $(\beta,\alpha)$ leading to mixing is given. Furthermore relations between renormalizability and $T$ being locally eventually onto are considered, and some gaps in classical results on the dynamics of Lorenz maps are corrected.

2019

Mixing properties in coded systems

Springer Proceedings in Mathematics & Statistics

In: New Trends in One-Dimensional Dynamics
Springer Proceedings in Mathematics & Statistics vol 285.
2019, pp. 183-200

Conference proceedings J. Epperlein, D. Kwietniak, P. Oprocha

Mixing properties in coded systems

J. Epperlein, D. Kwietniak, P. Oprocha

Conference proceedings

About The Publication

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 sequence of topologically mixing shifts of finite type, has only periodic points of even period and each set of its generators consists of blocks of even length. We prove that such an example cannot be a synchronized system.

2019

On the dynamics of generic maps on the Cantor set

Topology and its Applications

263 (2019), 330-342.

Journal Paper J. Kupka, P. Oprocha

On the dynamics of generic maps on the Cantor set

J. Kupka, P. Oprocha

Journal Paper

About The Publication

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 maps in the class and further study of what features (e.g. prescribed minimal sets or the values of topological entropy) we can get by small perturbations of a given homeomorphism.

2019

ω-chaos without infinite LY-scrambled set on Gehman dendrite

International Journal of Bifurcation and Chaos

29(5) (2019), Article id: 05,195007.

Journal Paper T. Drwięga, P. Oprocha

ω-chaos without infinite LY-scrambled set on Gehman dendrite

T. Drwięga, P. Oprocha

Journal Paper

About The Publication

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.

2019

Multi-criteria optimization strategies for tree-structured production chains

nternational Journal of Material Forming

12 (2019), 185-196.

Journal Paper P. Jarosz, J. Kusiak, S. Małecki, P. Morkisz, P. Oprocha, W. Piertucha and Ł. Sztangret

Multi-criteria optimization strategies for tree-structured production chains

P. Jarosz, J. Kusiak, S. Małecki, P. Morkisz, P. Oprocha, W. Piertucha and Ł. Sztangret

Journal Paper

About The Publication

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 these plants consists of sequences of multiple units connected linearly or in tree-structured graphs in which, at various production stages, intermediate quality criteria, related to the semi-products, may have to be considered. Optimization strategies as well as the results of their application for different hypothetical problems and the real metallurgical problem of lead production are presented.

2019

Shadowing, asymptotic shadowing and s-limit shadowing

Fundamenta Mathematicae

244 (2019), 287-312.

Journal Paper C. Good, P. Oprocha, M. Puljiz

Shadowing, asymptotic shadowing and s-limit shadowing

C. Good, P. Oprocha, M. Puljiz

Journal Paper

About The Publication

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 construct a system which exhibits shadowing but not limit shadowing, and we study how shadowing properties transfer to maximal transitive subsystems and inverse limits (sometimes called natural extensions). Where practicable, we show that our results are best possible by means of examples.

2019

Double minimality, entropy and disjointness with all minimal systems

Discrete and Continuous Dynamical Systems - A

39 (2019), 263-275.

Journal Paper P. Oprocha

Double minimality, entropy and disjointness with all minimal systems

P. Oprocha

Journal Paper

About The Publication

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 \subset X

and every

x \in X

. This example shows that the new condition sharpens sufficient conditions for disjointness below previous bounds. In particular our approach does not rely on distality of points or sets.

2019

Mixing completely scrambled system exists

Ergodic Theory and Dynamical Systems

39(1) (2019), 62-73.

Journal PaperRepresentative publications J. Boroński, J. Kupka, P. Oprocha

Mixing completely scrambled system exists

J. Boroński, J. Kupka, P. Oprocha

Journal PaperRepresentative publications

About The Publication

We prove that there exists a topologically mixing homeomorphism which is completely scrambled. We also prove that, for any integer n>0, there is a continuum of topological dimension n supporting a transitive completely scrambled homeomorphism and an n-dimensional compactum supporting a weakly mixing completely scrambled homeomorphism. This solves a 15-year-old open problem.

2019

Prediction of Distribution of Microstructural Parameters in Metallic Materials Described by Differential Equations with Recrystallization Term

International Journal for Multiscale Computational Engineering

17 (2019), 361-371.

Journal Paper P. Morkisz, P. Oprocha, P. Przybyłowicz, N. Czyżewska, J. Kusiak, D. Szeliga, Ł. Rauch, M. Pietrzyk

Prediction of Distribution of Microstructural Parameters in Metallic Materials Described by Differential Equations with Recrystallization Term

P. Morkisz, P. Oprocha, P. Przybyłowicz, N. Czyżewska, J. Kusiak, D. Szeliga, Ł. Rauch, M. Pietrzyk

Journal Paper

About The Publication

Continuous development of the transport industry is associated with the search for new construction materials that combine high strength with good plastic properties. Intensive research during the last few decades has shown that there is still a huge potential for improvement of properties of various metallic materials. New grades called advanced high strength steels (AHSS) with multiphase structures have been developed and widely used mainly in the automotive industry. These microstructures are characterized by large gradients of properties, which cause poor local formability. It is expected that materials with a more heterogeneous microstructure will have superior formability. More detailed models of the microstructure evolution are needed to answer this question. A hypothesis was made that application of the models based on internal variables allows for predictions of gradients of final product properties. The objectives of the present paper were two-fold. The first was to investigate the possibility of the analytical and numerical solutions of the evolution equation for the internal variable and evaluation of these solutions. We propose two numerical methods which give us an accurate approximation of the solution with relatively low computational cost at the same time. Implementation of the developed solutions in the finite element (FE) code and performing multiscale simulation of the evolution of internal variables during thermomechanical processing was the second objective of this paper. This solution supplied information about the distribution of the dislocation density in the volume of the material. Case studies for selected metal forming processes recapitulate the paper.

2018

A compact minimal space Y such that its square YxY is not minimal

Advances in Mathematics

335 (2018), 261-275.

Journal PaperRepresentative publications J. Boroński, A. Clark, P. Oprocha

A compact minimal space Y such that its square YxY is not minimal

J. Boroński, A. Clark, P. Oprocha

Journal PaperRepresentative publications

About The Publication

The following well known open problem is answered in the negative: Given two compact spaces X and Y that admit minimal homeomorphisms, must the Cartesian product admit a minimal homeomorphism as well? Moreover, it is shown that such spaces can be realized as minimal sets of torus homeomorphisms homotopic to the identity. A key element of our construction is an inverse limit approach inspired by combination of a technique of Aarts & Oversteegen and the construction of Slovak spaces by Downarowicz & Snoha & Tywoniuk. This approach allows us also to prove the following result. Let be a continuous, aperiodic minimal flow on the compact, finite-dimensional metric space M. Then there is a generic choice of parameters , such that the homeomorphism admits a noninvertible minimal map as an almost 1-1 extension

2018

On the irregular points for systems with the shadowing property

Ergodic Theory and Dynamical Systems

38(6) (2018), 2108-2131.

Journal Paper Y. Dong, P. Oprocha, X. Tian

On the irregular points for systems with the shadowing property

Y. Dong, P. Oprocha, X. Tian

Journal Paper

About The Publication

We prove that when f is a continuous self-map acting on a compact metric space (X, d) that 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 measure) points with convergent Birkhoff averages. In particular, the time average of atomic measures along orbits of such points converges to some Sinai-Ruelle-Bowen-like measure in the weak* topology. Moreover, such points carry zero entropy. In contrast, irregular points are non-observable but carry infinite entropy.

2018

Properties of invariant measures in dynamical systems with the shadowing property

Ergodic Theory and Dynamical Systems

38(6) (2018), 2257-2294.

Journal Paper J. Li, P. Oprocha

Properties of invariant measures in dynamical systems with the shadowing property

J. Li, P. Oprocha

Journal Paper

About The Publication

For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost one-to-one extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every and the collection of ergodic measures (supported on almost one-to-one extensions of odometers) with entropy between and is dense in the space of invariant measures with entropy at least . Moreover, if in addition the entropy function is upper semi-continuous, then, for every , ergodic measures with entropy are generic in the space of invariant measures with entropy at least c.

2018

Ultrafilters and Ramsey-type shadowing phenomena in topological dynamics

Israel Journal of Mathematics

227 (2018), 423–453.

Journal Paper W. Brian, P. Oprocha

Ultrafilters and Ramsey-type shadowing phenomena in topological dynamics

W. Brian, P. Oprocha

Journal Paper

About The Publication

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.

2018

Edrei’s Conjecture Revisited

Annales Henri Poincaré

19 (2018), 267–281.

Journal Paper J. P. Boronski, J. Kupka, P. Oprocha

Edrei’s Conjecture Revisited

J. P. Boronski, J. Kupka, P. Oprocha

Journal Paper

About The Publication

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 Conjecture of Edrei from 1952, first disproved by Williams in 1954.

2018

Corrigendum to “Syndetically proximal pairs” [J. Math. Anal. Appl. 379 (2011) 656–663]

Journal of Mathematical Analysis and Applications

459(1) (2018), 614-617.

Journal Paper J. Li, .K. Subrahmonian Moothathu, P. Oprocha

Corrigendum to “Syndetically proximal pairs” [J. Math. Anal. Appl. 379 (2011) 656–663]

J. Li, .K. Subrahmonian Moothathu, P. Oprocha

Journal Paper

About The Publication

We give a counterexample to Theorem 9 in “Syndetically proximal pairs” [J. Math. Anal. Appl. 379 (2011) 656–663]. We also provide sufficient conditions for the conclusion of Theorem 9 to hold.

2024

On dynamics of the Sierpinski carpet

Comptes Rendus Mathematique

356(3) (2018), 340-344.

Journal Paper J.P. Boronski, P. Oprocha

On dynamics of the Sierpinski carpet

J.P. Boronski, P. Oprocha

Journal Paper

About The Publication

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

2018

Failures prediction based on performance monitoring of a gas turbine: a binary classification approach

Schedae Informaticae

26 (2018), article id: 10832.

Journal Paper B. Mulewicz, M. Marzec, P. Morkisz, P. Oprocha

Failures prediction based on performance monitoring of a gas turbine: a binary classification approach

B. Mulewicz, M. Marzec, P. Morkisz, P. Oprocha

Journal Paper

About The Publication

This paper is dedicated to employ novel technique of deep learning for machines failures prediction. General idea of how to transform sensor data into suitable data set for prediction is presented. Then, neural network architecture that is very successful in solving such problems is derived. Finally, we present a case study for real industrial data of a gas turbine, including results of the experiments.

2018

An attempt of optimization of zinc production line

Archives of Civil and Mechanical Engineering

18(4) (2018), 1116-1122.

Journal Paper P. Jarosz, J. Kusiak, S. Małecki, P. Morkisz, P. Oprocha, W. Piertucha and Ł. Sztangret

An attempt of optimization of zinc production line

P. Jarosz, J. Kusiak, S. Małecki, P. Morkisz, P. Oprocha, W. Piertucha and Ł. Sztangret

Journal Paper

About The Publication

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 originated from different mines. The output semi-product of the next stage, the oxidative roasting process, is calcine, which is the input of the leaching. The result of the leaching is zinc sulfate solution and the goal of leaching is to carry out the maximum amount of zinc to solution. The preliminary step of any optimization is modeling of the analyzed processes. Modeling of considered three stages of zinc production line, based on the real industrial data of one of zinc production plants, was performed using different techniques. The elaborated models were the basis of the optimization for given objective functions of each of the production stages. The optimization methodology of multi-stage processes developed by the authors was applied. Obtained results of modeling and optimization are presented

2018

Shadowing, Entropy and Minimal Sets

Springer Proceedings in Mathematics & Statistics

In: Dynamical Systems in Theoretical Perspective
Springer Proceedings in Mathematics & Statistics, vol. 248
pp. 249-259

Conference proceedings P. Oprocha

Shadowing, Entropy and Minimal Sets

P. Oprocha

Conference proceedings

About The Publication

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 er- godic measures with additional properties of their supports (minimality, positive entropy, mixing).

2017

On dynamics of graph maps with zero topological entropy

Nonlinearity

30 (2017), article id: 4260.

Journal Paper J. Li, P. Oprocha, Y. Yang, T. Zeng

On dynamics of graph maps with zero topological entropy

J. Li, P. Oprocha, Y. Yang, T. Zeng

Journal Paper

About The Publication

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 true for graph maps with zero topological entropy. We also extend several results known in interval dynamics to graph maps. We show that a graph map has zero topological entropy if and only if there is no 3-scrambled tuple if and only if the proximal relation is an equivalence relation; a graph map has no scrambled pairs if and only if it is null if and only if it is tame.

2017

Validation of optimization strategies using the linear structured production chains

AIP Conference Proceedings

AIP Conference Proceedings, vol. 1836
Article id: 020067

Conference proceedings J. Kusiak, P. Morkisz, P. Oprocha, W. Pietrucha, Ł. Sztangret

Validation of optimization strategies using the linear structured production chains

J. Kusiak, P. Morkisz, P. Oprocha, W. Pietrucha, Ł. Sztangret

Conference proceedings

About The Publication

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 equations, respectively. Considered in the work optimization problems serve as the validators of strategies elaborated by the Authors. However, the main goal of research is selection of the best strategy for optimization of two real metallurgical processes which will be investigated in an on-going projects. The first problem will be the oxidizing roasting process of zinc sulphide concentrate where the sulphur from the input concentrate should be eliminated and the minimal concentration of sulphide sulphur in the roasted products has to be achieved. Second problem will be the lead refining process consisting of three stages: roasting to the oxide, oxide reduction to metal and the oxidizing refining. Strategies, which appear the most effective in considered benchmark problems will be candidates for optimization of the mentioned above industrial processes.

2016

Multi-criteria optimization strategies for production chains

MATEC Web of Confonferences

MATEC Web of Conferences, vol. 80
Article id: 10008

Conference proceedings J. Kusiak, P. Morkisz, P. Oprocha, W. Pietrucha, Ł. Sztangret

Multi-criteria optimization strategies for production chains

J. Kusiak, P. Morkisz, P. Oprocha, W. Pietrucha, Ł. Sztangret

Conference proceedings

About The Publication

The aim of this paper is presentation of some optimization strategies applicable in the optimization of multi-stage and multi-threads chain structures (linear or 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 these plants consists of sequences of multiple units connected linearly or in tree which, at various production stages, intermediate quality criteria, related to the semi-products, may occur.

2014

On a Cipher Based on Pseudo-random Walks on Graphs

Communications in Computer and Information Science,

In: Cryptography and Security Systems
Communications in Computer and Information Science, vol. 448
pp. 59-73.

Conference proceedings W. Foryś, Ł. Jęda, P. Oprocha

On a Cipher Based on Pseudo-random Walks on Graphs

W. Foryś, Ł. Jęda, P. Oprocha

Conference proceedings

About The Publication

In the paper a cipher which uses transition graph in the encryption and decryption processes is presented. It is created based on some ideas of dynamic systems (symbolic dynamics). The introduced cipher is tested according to the suggestions of FIPS 140-2 standard and the results are presented in the paper.

2014

Topological aspects of dynamics of pairs, tuples and sets

Atlantis Press

Recent Progress in General Topology III
K.P. Hart, J. van Mill, P. Simon (Eds.)
Atlantis Press, 2014, pp. 649-692.

Book Chapter P. Oprocha, G.H. Zhang

Topological aspects of dynamics of pairs, tuples and sets

P. Oprocha, G.H. Zhang

Book Chapter

About The Publication

The book presents surveys describing recent developments in most of the primary subfields of General Topology, and its applications to Algebra and Analysis during the last decade, following the previous editions (North Holland, 1992 and 2002). The book was prepared in connection with the Prague Topological Symposium, held in 2011. During the last 10 years the focus in General Topology changed and therefore the selection of topics differs from that chosen in 2002. The following areas experienced significant developments: Fractals, Coarse Geometry/Topology, Dimension Theory, Set Theoretic Topology and Dynamical Systems.

2013

On weak product recurrence and synchronization of return times

Advances in Mathematics

244 (2013), 395-412.

Journal PaperRepresentative publications Piotr Oprocha, Guohua Zhang

On weak product recurrence and synchronization of return times

Piotr Oprocha, Guohua Zhang

Journal PaperRepresentative publications

About The Publication

The paper is devoted to a study of product recurrence. First, we prove that notions of and are exactly the same as product recurrence, completing that way results of [P. Dong, S. Shao, X. Ye, Product recurrent properties, disjointness and weak disjointness, Israel J. Math. 188 (1) (2012) 463–507], and consequently, extending the characterization of return times of distal points which originated from the works of Furstenberg. We also study the structure of the set of return times of weakly mixing sets. As a consequence, we obtain new sufficient conditions for and also find a short proof that weakly mixing systems are disjoint with all minimal distal systems (in particular, our proof does not involve Furstenberg’s structure theorem of minimal distal systems).

2009

Optimization: Selected methods and their applications

Polish Scientific Publishers PWN

Warsaw, 2009

Book J. Kusiak, A. Danielewska-Tułecka, P. Oprocha

Optimization: Selected methods and their applications

J. Kusiak, A. Danielewska-Tułecka, P. Oprocha

Book

ABOUT

About me

Academic databases

RESEARCH

Entropy, shadowing and attractors

Limit sets of discrete dynamical systems

LISEDIDYS

RESEARCH TEAM

PUBLICATIONS

All publications

Renormalization in Lorenz maps – completely invariant sets and periodic orbits

Advances in Mathematics

Renormalization in Lorenz maps – completely invariant sets and periodic orbits

About The Publication

Internal Maps With Dense Periodicity

Arxiv

Internal Maps With Dense Periodicity

About The Publication

A note on homeo-product-minimality

Qualitative Theory of Dynamical Systems

A note on homeo-product-minimality

About The Publication

Multiorders in amenable group actions

Groups, Geometry, and Dynamics

Multiorders in amenable group actions

About The Publication

Are generic dynamical properties stable under composition with rotations?

Proceedings of the American Mathematical Society

Are generic dynamical properties stable under composition with rotations?

About The Publication

Metric mean dimension of irregular sets for maps with shadowing

Journal of Difference Equations and Applications

Metric mean dimension of irregular sets for maps with shadowing

About The Publication

Tranched graphs: consequences for topology and dynamics

Tranched graphs: consequences for topology and dynamics

About The Publication

Short-range and long-range order: a transition in block-gluing behavior in Hom shifts

Journal d'Analyse Mathématique

Short-range and long-range order: a transition in block-gluing behavior in Hom shifts

About The Publication

Parameterized family of annular homeomorphisms with pseudo-circle attractors

Journal of Differential Equations

Parameterized family of annular homeomorphisms with pseudo-circle attractors

About The Publication

Accounting for random character of recrystallisation and uncertainty of process parameters in the modelling of phase transformations in steels

Canadian Metallurgical Quarterly

Accounting for random character of recrystallisation and uncertainty of process parameters in the modelling of phase transformations in steels

About The Publication

On limit sets and equicontinuity in hyperspace of continua in dimension one

On limit sets and equicontinuity in hyperspace of continua in dimension one

Continuous Lebesgue measure-preserving maps on one-dimensional manifolds: a survey

Topology and its Applications

Continuous Lebesgue measure-preserving maps on one-dimensional manifolds: a survey

About The Publication

Lelek fan admits completely scrambled weakly mixing homeomorphism

Lelek fan admits completely scrambled weakly mixing homeomorphism

About The Publication

Various questions around finitely positively expansive dynamical systems

ArXiv

Various questions around finitely positively expansive dynamical systems

About The Publication

Beyond 0 and ∞: A solution to the Barge Entropy Conjecture

Arxiv

Beyond 0 and ∞: A solution to the Barge Entropy Conjecture

About The Publication

S-limit shadowing is generic for continuous Lebesgue measure preserving circle maps

Ergodic Theory and Dynamical Systems

S-limit shadowing is generic for continuous Lebesgue measure preserving circle maps

About The Publication

On recurrence and entropy in hyperspace of continua in dimension one

Fundamenta Mathematicae

On recurrence and entropy in hyperspace of continua in dimension one

About The Publication

New exotic minimal sets from pseudo-suspensions of Cantor systems

Journal of Dynamics and Differential Equations

New exotic minimal sets from pseudo-suspensions of Cantor systems

About The Publication

Dendrites and measures with discrete spectrum

Ergodic Theory and Dynamical Systems