Seminar iz topologije

Vodji seminarja: Tina Sovič in Jernej Činč.
Predavanja potekajo ob petkih ob 13:00 v predavalnici 01/20.

Bodoča predavanja

Minula predavanja

Invariant measures of minimal homeomorphisms on manifolds

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.

Maximal entropy Lebesegue measure, some results, examples and open questions

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.

Shadowing property, entropy, invariant measures and irregularity

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.

Polygonal billiards

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.

PTR-prostori

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.

Characterising Functions between Compact Connected Metric Spaces

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.

n-manifold and inverse limits of set-valued functions

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.

Dendroids and Selections

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.

Normal Numbers

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.

Solenoids and continua which are groups

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.

On recurrence and entropy in hyperspace of continua in dimension one

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.

Leo preslikave in množice funkcij z gosto periodičnostjo

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.

On pseudoarc and dynamics

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.

Dynamics and complicated continua

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.

A classification of strange attractors: the inverse limit approach

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.

Arcs, circles and inverse limits

Predavateljica: Sina Greenwood (Univerza v Aucklandu, Auckland, Nova Zelandija)

Povzetek: In this talk I will give a characterisation of inverse limits of continuous functions on intervals that are arcs. In addition, I will discuss inverse limits of set-valuad functions that are graphs, and in particular, arcs or circles.

Predavanje bo v prostoru 0/103 na Koroški cesti 160.

Štiri črte

Predavatelj: Iztok Banič
Predavanje bo na daljavo v MS teams – za vstop v predavalnico uporabite kodo w20dsjj

Dve črti

Predavatelj: Iztok Banič
Predavanje bo na daljavo v MS teams – za vstop v predavalnico uporabite kodo w20dsjj

Entropija (3. del)

Predavatelj: Iztok Banič
Predavanje bo na daljavo v MS teams – za vstop v predavalnico uporabite kodo w20dsjj

Entropija (2. del)

Predavatelj: Iztok Banič
Predavanje bo na daljavo v MS teams – za vstop v predavalnico uporabite kodo w20dsjj

Entropija (1. del)

Predavatelj: Iztok Banič
Predavanje bo na daljavo v MS Teams – za vstop v predavalnico uporabite kodo w20dsjj

 

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