Predavateljica: Klára Karasová (Charles University)

Povzetek: Among all notions of chaos, there are three widely accepted: Devaney chaos, Li-Yorke chaos and (positive) topological entropy. It is known that exact Devaney chaos – an exact map with dense set of periodic points – implies all three.

Various results establish the existence of maps with properties related to chaos (e.g., transitivity) for specific spaces such as the interval, the Cantor set or the Lelek fan, as well as for broader classes, including manifolds and dendrites. Moreover, chaotic behavior often emerges as a generic phenomenon in the sense of Baire category.

Inspired by some of the previous results, jointly with B. Vejnar we prove that every Peano continuum (i.e. a locally connected continuum) admits exact Devaney chaos. Additionally, we generalize some prior results by showing that if a Peano continuum X satisfies the condition that selfmaps locally constant on some dense open subset form a dense subset of all selfmaps, then:
• exactly Devaney chaotic maps form a dense subset of chain transitive selfmaps of X,
• mixing is generic among chain transitive selfmaps of X,
• shadowing is generic among all selfmaps of X.

Given that all Peano continua admit exact Devaney chaos, a natural question arises: do they also admit a mixing but non-exact Devaney chaotic map? We answer this affirmatively jointly with M. Kowalewski and P. Oprocha by analyzing the role of local cut points in Peano continua. We also derive consequences for the topological entropy of the constructed map. We say that a map f : X → X is strongly mixing, if for every nonempty open U ⊂ X and every x ∈ X there exists n_0 such that for every n ≥ n_0 it holds that x ∈ f^n(U). It is easy to see that strong mixing is stronger than mixing but weaker than exactness. Further, it is easy to prove that every strong mixing selfmap of a tree is already exact. A natural question whether strong mixing and exact selfmaps coincide for some other class of spaces has been of interest since then.

Recently, Illanes and Rito proved that every dendrite that is not a tree admits a strongly mixing but non-exact selfmap. Since the same holds for the circle, the case of finite graphs became of interest. We generalize the result of Illanes and Rito, resolve the question for finite graphs, and establish several related results.

This is a joint work with M. Kowalewski, P. Oprocha, and B. Vejnar.

Predavanje bo 17. 10. 2025 ob 14:00 v predavalnici 01/20.

Predavateljica: Veronika Ryzova (University of Opava)

Povzetek: n topological dynamics, the study of recurrence and limit sets is primarily oriented toward forward iteration. In this talk, we consider the backward perspective and focus on the notion of special alpha-limit points, which capture accumulation behaviour along backward orbit branches. We present an example showing that the set of special alpha-limit points is not comparable (by inclusion) with either the Birkhoff centre or the union of all omega-limit sets. This observation leads to an updated hierarchy of sets for general compact metric spaces.

Predavanje bo 21. 11. 2025 ob 13:00 v predavalnici 01/20.

Predavatelj: Hector Barge (Universidad Politécnica de Madrid)

Povzetek: Let f_λ : M → M with λ ∈ [0, 1] be a parametrized family of homeomorphisms of a manifold M. We say that an attractor K ⊆ M of f_0 undergoes a Hopf bifurcation at λ = 0 provided that K is a repeller for f_λ for every λ > 0. Whenever an attractor undergoes a Hopf bifurcation, there appears a family of attractors K_λ that converges to K upper semicontinously as λ → 0. In this talk we shall see that in many interesting situations we can characterize the Borsuk homotopy type of these attractors. These results have been obtained in collaboration with J. M. R. Sanjurjo and J. J. Sánchez-Gabites.

Predavanje bo 17. 10. 2025 ob 13:00 v predavalnici 01/20.

Predavateljica: Michaela Záškolná (University of Opava)

Povzetek: This talk will be based mostly on the paper “Stronger forms of sensitivity for dynamical systems” by T. K. Subrahmonian Moothathu that was published in 2007. We will study properties of cofinite sensitivity and Li-Yorke chaos on examples and look at the possible connection between these two notions.

Predavanje bo 10. 10. 2025 ob 13:00 v predavalnici 01/20.

Predavatelj: Rene Gril Rogina

Povzetek: We define canals of plane continua and give motivation for the main research problem. Then, we present our work on this problem using the notion of dead ends and quotients of plane continua.

Predavanje bo 20. 6. 2025 ob 13:00 v predavalnici 01/20.

Predavatelj: Jernej Činč

Povzetek: V sklopu teh predavanj bomo naredili pregled različnih pristopov k študiji ravninskih vložitev enodimenzionalnih kontinuumov. Najprej bomo uvedli inverzne limite in dokazali Anderson-Choquetov vložitveni izrek. Nadeljevali bomo s študijo dostopnih točk ravninskih kontinuumov. Nato bomo predstavili Carathéodoryevo teorijo prakoncov in povezane rezultate. Zaključili bomo s pregledom nedavnih rezultatov, kjer so se uporabljale predstavljene tehnike.

Predavanje bo 6. 6. 2025 ob 13:00 v predavalnici 01/20.

Predavatelj: Jernej Činč

Povzetek: V sklopu teh predavanj bomo naredili pregled različnih pristopov k študiji ravninskih vložitev enodimenzionalnih kontinuumov. Najprej bomo uvedli inverzne limite in dokazali Anderson-Choquetov vložitveni izrek. Nadeljevali bomo s študijo dostopnih točk ravninskih kontinuumov. Nato bomo predstavili Carathéodoryevo teorijo prakoncov in povezane rezultate. Zaključili bomo s pregledom nedavnih rezultatov, kjer so se uporabljale predstavljene tehnike.

Predavanje bo 30. 5. 2025 ob 13:00 v predavalnici 01/20.

Predavatelj: Jernej Činč

Povzetek: V sklopu teh predavanj bomo naredili pregled različnih pristopov k študiji ravninskih vložitev enodimenzionalnih kontinuumov. Najprej bomo uvedli inverzne limite in dokazali Anderson-Choquetov vložitveni izrek. Nadeljevali bomo s študijo dostopnih točk ravninskih kontinuumov. Nato bomo predstavili Carathéodoryevo teorijo prakoncov in povezane rezultate. Zaključili bomo s pregledom nedavnih rezultatov kjer so se uporabljale predstavljene tehnike.

Predavanje bo 23. 5. 2025 ob 13:00 v predavalnici 01/20.

Predavatelj: Jernej Činč

Povzetek: V sklopu teh predavanj bomo naredili pregled različnih pristopov k študiji ravninskih vložitev enodimenzionalnih kontinuumov. Najprej bomo uvedli inverzne limite in dokazali Anderson-Choquetov vložitveni izrek. Nadeljevali bomo s študijo dostopnih točk ravninskih kontinuumov. Nato bomo predstavili Carathéodoryevo teorijo prakoncov in povezane rezultate. Zaključili bomo s pregledom nedavnih rezultatov kjer so se uporabljale predstavljene tehnike.

Predavanje bo 22. 5. 2025 ob 14:30 v predavalnici 01/20.

Predavatelj: Peter Goričan

Povzetek: Na seminarju bomo spoznali Ripsove komplekse, definirali kritične simplekse in kritične vrednosti v 0-dimenzionalni vztrajni homologiji. Nato pa podobne rezultate pokazali za 1-dimenzionalno vztrajno homologijo, dokazali, kdaj je lahko H_1(Rips(X, r)) končno generiran, in podali nekaj primerov.

Predavanje bo 9. 5. 2025 ob 13:30 v predavalnici 0/80.

Predavatelj: Peter Goričan

Povzetek: Na seminarju bomo spoznali Ripsove komplekse, definirali kritične simplekse in kritične vrednosti v 0-dimenzionalni vztrajni homologiji. Nato pa podobne rezultate pokazali za 1-dimenzionalno vztrajno homologijo, dokazali, kdaj je lahko H_1(Rips(X, r)) končno generiran, in podali nekaj primerov.

Predavanje bo 25. 4. 2025 ob 13:00 v predavalnici 0/80.

Predavatelj: Iztok Banič

Povzetek: A fan is an arcwise-connected continuum, which is hereditarily unicoherent and has exactly one ramification point. Many of the known examples of fans were constructed as 1-dimensional continua that are unions of arcs which intersect in exactly one point. Borsuk proved in 1954 that each fan is a 1-dimensional continuum which is the union of arcs intersecting in exactly one point. But it is not yet known if this property is equivalent to being a fan. We show that under two additional assumptions, every such union of arcs is a fan.

Predavanje bo 18. 4. 2025 ob 13:30 v predavalnici 0/80.

Predavateljica: Stonja Štimac (Univerza v Zagrebu)

Predavanje bo 24. 1. 2025 ob 13:00 v predavalnici 0/80.

Predavateljica: Stonja Štimac (Univerza v Zagrebu)

Predavanje bo 23. 1. 2025 ob 14:00 v predavalnici 0/46.2.

Predavateljica: Stonja Štimac (Univerza v Zagrebu)

Predavanje bo 17. 1. 2025 ob 13:00 v predavalnici 0/80.

Predavateljica: Stonja Štimac (Univerza v Zagrebu)

Predavanje bo 16. 1. 2025 ob 11:30 v predavalnici 0/46.2.

Predavateljica: Lori Alvin (Furman University, ZDA)

Povzetek: We will discuss unimodal maps and explore how we may use symbolic sequences to study unimodal maps.

Predavanje bo 13. 1. 2025 ob 13:00 v predavalnici 0/103.

Predavateljica: Lori Alvin (Furman University, ZDA)

Povzetek: We will introduce some facts and definitions about symbolic dynamics.

Predavanje bo v predavalnici 0/80.

Predavatelj: Iztok Banič

Povzetek: We discuss work in progress about Lelek-like fans and dynamics on them. This is joint work with Judy Kennedy, Goran Erceg, and Ivan Jelić. 

Predavanje bo v predavalnici 0/80.

Predavateljica: Lori Alvin (Furman University, ZDA)

Povzetek: We investigate the dynamical property of shadowing in shift spaces generated by upper semicontinuous set-valued functions over a compact countable domain. In particular, we will discuss how shadowing in this setting differs from shadowing in shift spaces where the underlying metric on the alphabet is the discrete metric. We then discuss properties that the generating set-valued function can possess in order to guarantee shadowing.

Predavanje bo v predavalnici 01/20.

Predavatelj: Rene Gril Rogina

Predavateljica: Lori Alvin (Furman University, ZDA)

Povzetek:There are  many types of recurrence in the family of unimodal maps. We discuss the concept of persistent recurrence with the goal of better understanding how persistent recurrence relates to various topological properties of the map. We utilize kneading maps and Hofbauer towers, tools developed by F. Hofbauer and G. Keller in the 1970s and 1980s, to explore persistent recurrence.

Predavateljica: Tina Sovič

Predavateljica: Sejal Babel (Jeggielonian University Krakow, Poland)

Povzetek: It is well known that a topological dynamical system’s set of invariant measures is a nonempty metrizable Choquet simplex. In 1991, Downarowicz showed that all such simplices arise as the set of invariant measures of a class of minimal subshifts. Hence, we can ask: Which nonempty Choquet simplices can be realized as sets of invariant measures for minimal homeomorphisms on manifolds? It is known that for one-dimensional manifolds, the geometry of manifolds restricts the available dynamics. In particular, minimal homeomorphisms of the circle are uniquely ergodic. During the talk, we will discuss a class of minimal homeomorphisms of the Cantor set whose dynamics can be quite faithfully realized as minimal homeomorphisms on manifolds. Consequently, we will prove that every Choquet simplex is realized as a set of invariant measures of a minimal homeomorphism on every manifold of dimension two and higher carrying a minimal uniquely ergodic homeomorphism. The talk is based on results obtained in cooperation with Jernej Činč, Till Hauser, Dominik Kwietniak and Piotr Oprocha.

Predavanje bo ob 14:00 v predavalnici 0/46.2.

Predavatelj: Rene Gril Rogina

Predavatelj: Rene Gril Rogina

Predavatelj: Jernej Činč

Predavatelj: Rene Gril Rogina

Predavatelj: Josef Bobok

Povzetek: We are interested in interval unimodal maps preserving the Lebesgue measure and equipped with the supremum norm. In our investigation we focus on the case when the Lebesgue measure has maximal entropy. We describe the basic properties of the subspace of such maps and also provide challenging examples and state still unsolved questions.

Predavatelj: Piotr Oprocha

Povzetek: In 1970s Bowen related hyperbolic dynamics with specification property and used this to show that there exists a unique measure of maximal entropy. Almost the same time Sigmund used specification property as a tool in characterization of simplex of invariant measures. Since then, these results were inspiration for numerous mathematicians in various studies of dynamics. Several relations between entropy, structure of the space of invariant measures and historical behavior of trajectories were discovered. In this talk we will present selected questions and results inspired by the above framework of research, with shadowing property as main tool in our considerations.

Opomba: predavanje bo izjemoma v ponedeljek 8. 4. 2024 ob 15:00 v predavalnici 0/46.1.

Predavatelj: Serge Troubetzkoy

Povzetek: Polygonal billiards come in two classes; rational or irrational. There are strong tools (Teichmuller dynamics) to study billiards in rational polygons. On the other hand the only tools which researchers have used to study irrational polygons are elementary geometry, combinatorics, … The author will present a survey of the state of the art on polygonal  billiards, with an emphasis on billiards in irrational polygons.

Opomba: predavanje bo izjemoma v ponedeljek 8. 4. 2024 ob 14:00 v predavalnici 0/46.1.

Predavateljica: Teja Kac

Povzetek: Za poljuben metričen prostor X se vprašamo: ali obstaja pozitivno realno število a, da je za vsaki različni točki x in y iz X razdalja med njima večja ali enaka a, natanko tedaj, ko je vsaka funkcija f iz X v realna števila z evklidsko metriko zvezna. Definiramo pojme oddaljen prostor, vse-zvezen prostor in PTR-prostor ter dokažemo nekatere lastnosti le-teh prostorov.

Predavatelj: Iztok Banič

Predavatelj: Iztok Banič

Predavatelj: Iztok Banič

Predavatelj: Rene Gril Rogina

Predavatelj: Bradley Windelborn  (University of Auckland)

Povzetek: We consider the following question: When can an arbitrary function between sets be turned into a continuous function between compact connected metric spaces by adding metrics to the sets? We discuss our conjectured solution to this. We will focus on the bijective self-map case and through looking at examples build a classifying conjecture/theorem. Following that, we briefly discuss the general case.

Predavanje bo ob 15:00 v predavalnici 0/103.

Predavateljica: Sina Greenwood (University of Auckland)

Povzetek: I will discuss inverse sequences of set-valued functions whose inverse limits are manifolds, with and without boundary. It turns out that the only such manifolds (without boundary), are tori.

Predavanje bo ob 14:00 v predavalnici 0/103.

Predavateljica: Veronica Martinez de la Vega (Universidad Nacional Autónoma de México)

Povzetek: Let X be a nonempty, compact, connected metric space (a contiunuum) and C(X) the hyperspace of subcontinua of X, i.e. C(X) = {A in X : A is a continuum in X} and F1(X) = {{a} in C(X) : a in X}. It is easy to show that F1(X) is homeomorphic to X. Define a selection, s, as a continuous retraction s : C(X) —> F1(X) such that s(A) is contained in A, or s can be defined as a map s : C(X) —> X such that s(A) in A. We show in this talk for which continua selections can be defined and some interesting results around this type of maps.

Predavanje bo ob 13:00 v predavalnici 0/103.

Predavateljica: Teja Kac

Pradavanje bo izjemoma v predavalnici 01/33.

Predavatelj: William Mance (Adam Mickiewicz University in Poznan)

Abstract: Informally, a real number is normal in base $b$ if in its $b$-ary expansion all digits and blocks of digits occur as often as one would expect them to uniformly at random. Borel introduced normal numbers in 1909 and proved that Lebesgue-almost every real number is normal in all bases $b \geq 2$. Even though this shows that, in some sense, normal numbers are “typical,” no example of a number normal in all bases was given until 1939 by Turing. In the last 100 years, the study of normal numbers has spread over a wide breadth of seemingly unrelated disciplines. Normality is closely related to number theory, ergodic theory, theoretical computer science, probability theory, fractal geometry, descriptive set theory, and others areas of math. We will explore the basic properties of normal numbers and surprising connections they have, depending on the interest of the audience.

Predavatelj: Udayan B. Darji (University of Louisville)

Abstract: In this talk we discuss generic behaviour of groups which are continua. In fact, we show that such an object is the universal solenoid. This is analogous to Bing’s classics result that a generic continuum is the pseudoarc. This is joint work with my Hungarian colleagues M. Elekes, T. Kátay, A. Kocsis, M. Palfy.

Predavateljica: Teja Kac

Predavatelj: Domagoj Jelić

Abstract: We show that if $G$ is a topological graph, and $f$ is a 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 $f$. It is well known that this 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 a much simpler case of compact interval by completely different tools.

This is joint work with Piotr Oprocha.

Predavateljica: Teja Kac

Predavatelj: Jernej Činč

Izvleček: Intervalna funkcija f na [0,1] se imenuje lokalno eventualno surjektivna (leo), če za vsako odprto množico U iz [0,1] velja, da obstaja neko naravno število n, tako da n-ta iteracija f od U pokrije celoten interval [0,1]. V diskretnih topoloških dinamičnih sistemih je leo lastnost ena izmed najstriktnejših lastnosti, ki se jo lahko zahteva in med drugim implicira tranzitivnost dinamičnega sistema.

Na tem seminarju bom govoril o pogostosti pojavljanja leo preslikav v množici funkcij, ki imajo gosto množico periodičnih točk.

Predavatelj: Iztok Banič

Predavatelj: Piotr Oprocha (AGH University of Science and Technology, Krakow, Poland)

Abstract: The pseudoarc is an intriguing continuum that was discovered by Knaster 100 years ago. While its structure is very complicated, Bing proved that from topological perspective, pseudoarc is the typical continuum to encounter in the plane.
In this talk I will present selected results connecting pseudoarc as topological object with selected properties of dynamical systems.

Predavatelj: Udayan B. Darji (University of Louisville)

Abstract: Suppose we have a continuum X and f a continuous surjection of X. In this talk we will discuss some results in which dynamics of f imply complicated structure of X as well as the inverse limit determined by (X,f). All notions will be defined. The talk will be accessible to graduate students and nonspecialists.

Predavatelj: Jan Boroński (AGH University of Science and Technology)

I will present my joint work with Sonja Štimac on classification of strange attractors. Among the many results, our work relies on the notion of mild dissipation [CP], which allows for an inverse limit description, in terms of densely branching trees [BS].

For a large family of attractors, that include those of Lozi and Henon, two strange attractors are equivalent if and only if their kneading sequences coincide, if and only if their folding patterns agree.

References:
[BC] M. Benedicks, L.A.E. Carleson, The dynamics of the Henon map, Annals of Mathematics 133 (1991), 73–169.
[BS] J. Boroński, S. Štimac, Densely branching trees as models for Henon-like and Lozi-like attractors, preprint 2021, arXiv:2104.14780
[CP] S. Crovisier, E. Pujals, Strongly dissipative surface diffeomorphisms, Commentarii Mathematici Helvetici 93 (2018), 377–400.
[MS] M. Misiurewicz, S. Štimac, Symbolic dynamics for Lozi maps, Nonlinearity 29 (2016), 3031–3046.
[WY] Wang, Q.; Young, L.-S. Strange attractors with one direction of instability. Comm. Math. Phys. 218 (2001), 1–97.