Accepted (23):
- Plenary
Dissipative Dynamics on the Disc — Sylvain Crovisier <sylvain.crovisier@universite-paris-saclay.fr>
The dynamics of continuous interval maps admit a remarkably rich topological description, including the structure of attractors, criteria for positive entropy, and the density of periodic points in the recurrent set. In recent years, several extensions of these results have been obtained for dissipative diffeomorphisms of the disc, including Hénon maps. In this talk, I will survey some of these developments and present a new closing lemma, proved in collaboration with Enrique Pujals.
View Submission
A Family of Wild Attractors for Unimodal Maps — Lori Alvin <lori.alvin@furman.edu>
It is well-known that a unimodal map $f$ has a unique metric attractor that is either an attracting periodic orbit, the union of $n$ open intervals that are cyclically permuted by $f$, or a Cantor set. A Cantor attractor that arises from a non-infinitely renormalizable map is called an _absorbing Cantor set_ or a _wild attractor_. We present a symbolic construction that can be used to generate the kneading sequences for a family of unimodal maps with embedded strange odometers and wild attractors. This is joint work with Jernej Činč.
View Submission
A Tits alternative for groups of surface homeomorphisms — Frédéric Le Roux <frederic.le-roux@imj-prg.fr>
I will present recent results using the natural action of Homeo(S), for a compact surface S, on the fine curve graph introduced by Bowden, Hansel and Webb, who also proved that this graph is Gromov hyperbolic.
View Submission
Area-preserving surface homeomorphisms without zero entropy — Fabio Armando Tal <fabiotal@ime.usp.br>
The dynamics of area-preserving flows on closed orientable surfaces is a well-understood topic, and there exists a canonical invariant decomposition of the phase space into a region (a collection of topological annuli) where the dynamics is integrable, and a finite number of pieces of positive genus where the dynamics is quasi-minimal (closely resembling the dynamics of an irrational flow on a torus). In this work, we show that a very similar canonical decomposition remains valid when dealing with conservative homeomorphisms with zero topological entropy. We present this decomposition while also exhibiting examples of different phenomena that may arise, as well as several properties of the “quasi-minimal” regions. Part of the work involves proving a Thurston–Nielsen–type reduction result, showing that maps homotopic to Dehn twists and with zero entropy actually possess invariant “Dehn-like” annuli. Time permitting, we will also discuss some applications to Reeb flows on three-dimensional manifolds.
View Submission
Birkhoff-like attractors — Martin Sambarino <martinsambarino@gmail.com>
Birkhoff attractors arise from the study of dissipative annulus maps that twist the vertical direction. When their rotation set is nontrivial, these attractors exhibit a complicated topological structure, namely that of an indecomposable continuum. In this talk, we introduce a class of Birkhoff-like attractors for dissipative annulus maps and study the continuity properties of these attractors and their rotation sets under perturbations of the dynamics.
View Submission
Classification of Henon maps with strange attractors via the topology of a stable manifold — Sonja Stimac <sonja@math.hr>
In an earlier work with Boronski, we classified (up to conjugacy) the Henon maps with strange attractors in terms of three invariants that we introduced for them: (a) kneading sequences, (b) pruned trees, and (c) folding patterns of the unstable manifold of the hyperbolic fixed point $X$ in the attractor. In my talk, I will introduce yet another way to determine conjugacy classes of these maps, this time purely from the topology of the stable manifold $W^s$ of $X$. We consider a region of dissipation $D$ for the Henon map and study the connected components of $D \cap W^s$. To each such component, we assign a separation type and prove that two Henon maps are conjugate if and only if their corresponding components share the same separation type. This is joint work with Jan Boronski.
View Submission
Continuation of attractors for discrete semidynamical systems and applications to generalized Hopf bifurcations — Héctor Barge <h.barge@upm.es>
In this talk we shall study continuations of attractors of embeddings in manifolds. We shall introduce the abstract basin of attraction of such attractors and we shall see that if two attractors are related by continuation, their abstract basins are homeomorphic. Moreover, if this continuation is achieved by a small perturbation, then, the homeomorphism can be chosen to be the identity close to the original attractor. We shall make use of this rigidity property in order to characterize the Cech cohomology of the attractors expelled after a generalized Hopf bifurcation of an attractor. These results have been obtained in collaboration with J.J. Sánchez-Gabites
View Submission
Entropy Maximizing Measures for Coded Shifts: Beyond Uniqueness — Tamara Kucherenko <tkucherenko@ccny.cuny.edu>
For transitive subshifts of finite type and sofic shifts, the measure of maximal entropy is unique. This fails for coded shifts, which form a natural generalization of these classes. While non-uniqueness is often viewed as pathological, it is still possible to obtain a detailed description of entropy maximizing measures in this setting. We discuss coded shifts that are not intrinsically ergodic and show that an ergodic measure of maximal entropy can be associated with a generator for which it is Bernoulli. This perspective provides a unified framework for understanding both uniqueness and non-uniqueness of entropy maximizing measures and yields explicit descriptions even in non-intrinsically ergodic settings.
View Submission
From Dust to Fences — Udayan Darji <ubdarj01@gmail.com>
Homeomorphisms of the Cantor set (“dust”) play a fundamental role in topology, dynamical systems, and descriptive set theory, where they are studied from different perspectives. Recently, various properties of so-called fence-like objects have attracted attention. These include the Lelek fan (from topology), the hairy Cantor set and Cantor bouquet (from dynamical systems), and the Fraïssé fence (from model theory). Several recent works investigate both the structure of these spaces and the dynamics of homeomorphisms defined on them. In this work, we develop a general technique that allows one to transfer—or lift—the dynamics of a given homeomorphism of the Cantor set to a homeomorphism of a fence of the types described above.
View Submission
Generalizing the Poincaré-Hopf index in the discrete case — Nelson Schuback <nelson.schuback@imj-prg.fr>
In this talk, we will present a generalization of the Poincaré-Hopf index between trajectories of a non-singular flow on the plane to the discrete case. The main ingredient of the proof is to show that the space of pairs of positively-acessible points of a planar foliation forms a Serre fibration.
View Submission
Generic distributional chaos — Lenka Rucká <lenka.rucka@math.slu.cz>
The main result of this talk states, that for a continuous interval map $f$, the set of all Li-Yorke chaotic pairs which are not distributionally chaotic (of any type) is always of the first category in $I \times I$. This result has several immediate applications. For example, the characterization of generic Li-Yorke chaos by Snoha in [1] is valid also for distributional chaos. Following Geschke et al. in [2] we can deduce, that the existence of an uncountable $DCi$ scrambled set implies the existence of a Cantor $DCi$ scrambled set for the interval map $f$, where $i=1,2,3$. [1] L. Snoha; Generic chaos, Comment. Math. Univ. Carol., Vol. 31 (1990), No. 4, 793-810. [2] S. Geschke, J. Grebík, B. D. Miller; Scrambled Cantor sets, Proceedings of AMS, Vol.149, 10 (2021).
View Submission
Mean dimension and finite-to-one maps — Yonatan Gutman <gutman@impan.pl>
We prove that any dynamical system $(X,T)$ that admits the marker property and has mean dimension strictly less than $d$ admits a continuous, finite-to-one equivariant map into $(([0,1]^d)^\mathbb{Z},\operatorname{shift})$. Moreover, in the above situation a generic continuous equivariant map from $X$ to $([0,1]^d)^\mathbb{Z}$ is finite-to-one. In particular when $\operatorname{mdim}(X,T) < \frac{1}{2}d$, we show that such a a generic continuous equivariant map is an embedding and this strengthens the optimal embedding theorem of Gutman, Qiao, and Tsukamoto (2019), for $\mathbb{Z}$-actions. Unlike earlier works, our proof relies on classical topological techniques originating in the work of Ostrand (1965), Kolmogorov (1957), and Arnold (1957). Based on a joint work with Michael Levin and Tom Meyerovitch.
View Submission
On continuum-wise hyperbolic dynamics on surfaces — Piotr Oprocha <piotr.oprocha@osu.cz>
Hyperbolicity is a central notion in the study of chaotic dynamical systems. Unfortunately, expansivity which is one if its main ingredients is very uncommon in typical dynamics. Because of this limitation, over the years some generalizations appeared in the literature, trying to preserve main features of hyperbolic, yet present in much more generality. In 1993 Kato introduced the notion of continuum-wise expansive homeomorphisms, and in 2024 it was used by Artigue, Carvalho, Cordeiro and Vieitez to define continuum-wise hyperbolicity. This definition combines cw-expansive with kind of local product structure, also expressed in terms of evolution of continua. In this talk we will survey selected results for surface dynamics and present new results obtained by the author jointly with several collaborators.
View Submission
On the growth of the number of periodic points of smooth maps — Luis Hernández-Corbato <luiherna@ucm.es>
A conjecture by Shub states that the asymptotic exponential growth rate of the number of periodic points of a C¹ map f : M → M is bounded from below by an algebraic topological quantity: the exponential growth rate of Lefschetz numbers L(f ⁿ). In particular, if M is a sphere the conjecture states that # Fix(f ⁿ) grows asymptotically at least as d ⁿ, where d denotes the degree of f. The conjecture is wide open in general, even in S². In the talk, we will review some results in very particular cases: maps on S³ leaving invariant a circle and maps preserving a singular foliation on a closed surface. This is joint work with H. Barge, A. Moreno, J. Sanchez-Gabites (Madrid).
View Submission
On visit numbers to semi-circles and automatic sequences — Henk Bruin <henk.bruin@univie.ac.at>
Some sequences related to circle rotations over simple quadratic irrationals turn up in the online encyclopaedia of integers sequences (OEIS). In this joint work with Robbert Fokkink some conjectures about A120243 are solved, using either automata theory or renormalization of circle rotations.
View Submission
Renormalization of regularly critical diffeomorphisms of the disk — Jonguk Yang <jongukyang@gmail.com>
A diffeomorphism of the disk is called _mildly dissipative_ if, for every invariant measure, the stable manifold of almost every point disconnects the domain. Crovisier, Pujals and Tresser proved that every mildly dissipative diffeomorphism with zero topological entropy that is not generalized Morse–Smale is infinitely renormalizable. In this talk, we survey a various regularity conditions under which such systems converge, under renormalization, to the universal renormalization attractor in the space of unimodal maps. We also discuss several ongoing projects and open directions related to this program. This talk is based on joint work with Sylvain Crovisier, Mikhail Lyubich, and Enrique Pujals.
View Submission
Rotational Axiom A homeomorphisms for higher genus surfaces — Pierre-Antoine Guihéneuf <pierre-antoine.guiheneuf@imj-prg.fr>
Consider a homeomorphism of closed surface of genus $g \ge 2$. I will explain that in the case its homological rotation set (a compact subset of $\mathbf R^{2g}$ capturing the rotational behaviour of the dynamics) is big enough, the whole rotational behaviour is contained in a compact set that resembles a finite union of homoclinic classes with some heteroclinic connections. This is related to $C^0$ rotational versions of properties like Markov partitions, rotational density of periodic orbits, stability under perturbations... as well as purely rotational features such as the description of the rotation set's shape or some bounded deviations properties. The whole thing is based on Le Calvez-Tal forcing theory but I will mainly focus on some examples.
View Submission
Tameness, nullness, and amorphic complexity of automatic systems — Maik Gröger <maik.groeger@im.uj.edu.pl>
In the study of low-complexity aperiodic behaviour, tame and null systems arise naturally, yet providing concrete and easily testable conditions to establish their existence in a canonical class of systems is often nontrivial. In the talk I will present a recent result completely characterising tameness and nullness for minimal automatic systems generated by primitive constant-length substitutions in terms of a single numerical invariant: amorphic complexity, a topological invariant tailor-made to study zero-entropy systems with discrete spectrum. We show that for infinite automatic systems, tameness and nullness are equivalent to its value being one.
View Submission
The Riemann-Hurwitz formula and indecomposable continua — Juliana Xavier <mariajules@gmail.com>
The Riemann-Hurwitz formula establishes a relation between the degree and the number of critical points of branched coverings $f:X\to Y$. This relation involves the Euler characteristic of the spaces $X$ and $Y$. A priori, it has no sense when the spaces are not locally connected.The formula holds for branched coverings between finite cell complexes and also between open connected subsets of the sphere. We use it to define the Euler characteristic of a continuum even if it is not locally connected. For example, $\chi (X)=0$ for a solenoid and $\chi(K)=1/2$, where $K$ is the Knaster continuum. We give applications to dynamics of sphere branched coverings and provide several examples illustrating the interest of the results obtained.
View Submission
- Plenary
Topological Criteria for Annular Chaos — Alejandro Passeggi <apasseggi@cmat.edu.uy>
Although paradigmatic models of chaotic dynamics in low-dimensional systems are well understood, proving that a given system exhibits chaotic behavior often remains a challenging task. Moreover, identifying the underlying mechanisms responsible for such dynamics is frequently beyond the scope of the classical literature on the subject. In recent years, several topological criteria have been established for systems whose Poincaré map is defined on the annulus. These criteria provide simple and robust conditions guaranteeing the existence of chaos in the form of a rotational horseshoe. Roughly speaking, it is enough to find two topological disks with different rotation behavior under one iteration and whose forward iterates visit each other. This approach yields rigorous proofs of chaotic dynamics while relying on elementary information about the system [1,2]. Furthermore, effective implementations of these criteria have led to several concrete applications [3,4]. In this talk, I will review these results and discuss recent progress toward a natural next step: obtaining explicit constructions of the rotational horseshoe once the above criteria (or related ones) have been verified. Such constructions not only yield a rigorous computation of the map's topological entropy, but also allow one to locate the rotational horseshoe and its associated essential instability region. [1] A. Passeggi and F. A. Tal, Conditions Implying Annular Chaos, accepted to Inventiones Mathematicae. [2] A. Passeggi and F. Pirán, Annular Chaos for Non-Wandering Homeomorphisms, arXiv. [3] M. J. Capiński, M. Gröger, A. Passeggi and F. A. Tal, Conditions Implying Annular Chaos: Qualitative Results and CAP, arXiv. [4] M. J. Capiński, S. Llavayol and A. Passeggi, Rotational Chaos in the Driven Pendulum (to appear).
View Submission
Topologically mildly dissipative homeomorphisms and Wang-Young Strange Attractors — Jan Boronski <jan.boronski@uj.edu.pl>
In this joint work with Sonja Štimac, we extend R.F. Williams' result on 1-dimensional hyperbolic attractors to the non-uniformly hyperbolic setting, by showing that each Wang-Young strange attractor in the plane is conjugate to the shift on the inverse limit of a baobab (Peano continuum that contains at most one Jordan curve), generalizing our earlier result on Hénon attractors. More generally, the result holds on the core of the maximal attractor of any mildly dissipative diffeomorphism (in the sense of Crovisier and Pujals). We also generalize these results to the C<sup>0</sup> setting, by introducing the class of topologically mildly dissipative surface homeomorphisms, providing a unified approach that covers many classes of dissipative dynamical systems scattered in the literature. Our purely topological conditions lead to a one-to-one correlation between the sets of ergodic measures of the one-dimensional and two-dimensional systems, as well as equality between the corresponding measure-theoretic entropies.
View Submission
Universal bounds on the entropy of toroidal attractors — Jaime Jorge Sánchez-Gabites <jajsanch@ucm.es>
A compact set $K \subseteq \mathbb{R}^3$ is called \emph{toroidal} if it has a neighbourhood basis of solid tori. This is a natural generalization of the well-known notion of a cellular set. To any toroidal set one can assign a finite set of prime numbers called its prime divisors. These reflect purely topological properties of $K$. Suppose $f$ is a diffeomorphism of $\mathbb{R}^3$ and $K$ is an attractor for $f$ which happens to be a toroidal set (solenoids are the canonical example of this). We prove the following: the entropy of $f$ on $K$ is bounded below by ${\rm log}(p_1 \cdot \ldots \cdot p_n)$, where the $p_i$ are the prime divisors of $K$. Since the latter depend only on $K$, this provides a universal lower bound on any $\mathcal{C}^{\infty}$ attracting dynamics on $K$. In the talk we will discuss the geometric techniques used to prove this and discuss its (plausible?) validity when $f$ is just a homeomorphism.
View Submission
Zero Entropy Locus of Lozi Maps Revisited — Kristijan Kilassa Kvaternik <kkkvaternik@fer.hr>
We consider orientation-reversing Lozi maps $L_{a,b}$ in a parameter region $\mathfrak{R}$ where there are no homoclinic points for the fixed point $X$ in the first quadrant, and the period-two cycle $\{P,P'\}$ is attracting. We first analyze the boundary of that parameter region: we show that all homoclinic points for $X$ on that boundary are tangential or there is a segment of homoclinic points, and we classify them. Moreover, we study the topological entropy $h_{top}$ of $L_{a,b}$ when $(a,b)\in\mathfrak{R}$. Consider the set $\ell$ of accumulation points of the unstable manifold of $X$. Misiurewicz and Štimac have recently proven that $h_{top}(L_{a,b})=0$ in a certain open subset of $\mathfrak{R}$; in that case, $\ell=\{P,P'\}$. We extend this result by showing that the $L_{a,b}$, restricted to the complement of $\ell$ in the plane, has zero entropy. Finally, we discuss that $\ell=\{P,P'\}$ does not hold in general for all parameters in $\mathfrak{R}$.
View Submission