There is a family of birational self-mappings of the plane arising from the theory of cluster algebra mutations that was studied previously by Machacek-Ovenhouse from the perspective of real dynamics. We study this family of mappings from the perspective of complex dynamics and, in particular, show that is most cases there is no conserved quantity. No background on cluster algebras is expected from the audience. This is the joint-work with Andrei Grigorev, Andres Quintero and Roland Roeder.
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There has been some substantial work on studying the structure of a group by analyzing the behavior of primitive elements, sometimes under strong assumptions by work of Platonov, Potapchik, Shpilrain and many others. We formulate and study a conjecture of Platonov and Potapchik for general group actions via analyzing the dynamics of primitive elements for a given action. Such studies led us to further afield to produce results that combines computational, dynamical, geometric, and purely algebraic viewpoints.
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Connected Julia sets of a polynomial generally correspond to laminations, sets of chords of the unit disc that reflect the dynamics of the Julia set. If the circle is measured in revolutions and the polynomials studied are of degree $d$, then the dynamics on the lamination are given by the covering map $\sigma_d(t) := td \pmod 1$ where chords are mapped by their end points. Every lamination has at least one laminational invariant set, which is loosely an invariant complementary component of the lamination. That set has a significant impact on the shape of the Julia set. James Malaugh showed how to topologically transform a laminational invariant set in one degree into one in another degree, but he provided no way to execute these operations concretely. In this talk, we show how to calculate those operations.
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Computability in dynamical systems is a relatively young field that has attracted significant attention in recent years. One of its central questions is whether dynamically relevant objects can be algorithmically represented by a Turing machine. While this question has been extensively studied in symbolic dynamics, where computability results are known for various thermodynamical quantities such as entropy, pressure, equilibrium states and zero-temperature measures, a corresponding general theory for broader classes of topological and smooth dynamical systems is lacking. In this talk, we present an approach to bridging this gap by introducing the concept of computable Markov partitions. This framework allows us to establish far-reaching computability results for several classes of topological and smooth dynamical systems. The results presented in this talk are joint work with Michael Burr and Tamara Kucherenko.
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Critical orbit relation curves of rational maps acting on the Riemann sphere are dynamically natural complex one-dimensional slices of moduli space. The aim of the talk is to review some known and new results (work in progress with Caroline Davis and Alex Kapiamba) about degenerations along these curves.
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An entire map is said to be post-singularly finite if the forward orbit of its set of singular values is finite. Such maps play a crucial role in understanding natural families of entire maps. Motivated by previous work of Devaney, Goldberg and Hubbard, we ask the following question: _Given a post-singularly finite entire function f, can f be realized as the limit of a sequence of post-singularly finite polynomials?_ In joint work with Nikolai Prochorov and Bernhard Reinke, using techniques from Teichmüller theory, we show how we may answer this question in the affirmative.
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A new advancement is presented in a broad ongoing effort to develop Dynamical systems theory in the language of Category theory. A new idea will be presented to describe a general dynamical systems as an enriched functor, and change of variables as enriched natural transformations. This framework is essential to establish the equivalence of three descriptions of dynamics -- a semigroup action on the domain; a parameterized family of endomorphisms; and a transformation of time-space into the collection of endomorphisms. A collection of categorical axioms are presented that provides a complete categorical language to develop dynamical systems theory. True to the philosophy of dynamical systems, none of these assumptions are rooted in specific contexts such as topology and measure spaces. The equivalence of the three descriptions is further used to construct other related notions,such as transfer operators, orbits and sub-shifts. All of these objects are defined by their structural role and universal properties, instead of their usual pointwise definitions. Source : https://arxiv.org/pdf/2509.05900
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TBA
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We consider all continuous maps of the interval preserving the Lebesgue measure $\lambda$ equipped with the uniform topology. Except for the identity map or $1 - id$ all such maps have topological entropy at least $\log2/2$ and generically they have infinite topological entropy. In this talk we discuss two generic properties: (i) invertibility $\lambda$-a.e. implied by the zero measure-theoretic entropy with respect to $\lambda$, and (ii) complicated structure of level sets. We also recall that there are Besicovitch maps (having no finite or infinite unilateral derivative at any point) preserving $\lambda$ and show that each such map has positive measure-theoretic entropy with respect to $\lambda$.
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We investigate the dynamics of maps of the real line whose behavior on an invariant interval is close to a rational rotation on the circle. We focus on a specific two-parameter family, describing the dynamics arising from models in game theory, mathematical biology and machine learning. If one parameter is a rational number, k/n, with k, n coprime, and the second one is large enough, we prove that there is a periodic orbit of period n. It behaves like an orbit of the circle rotation by an angle 2 π k/n and attracts trajectories of Lebesgue almost all starting points. We also discover numerically other interesting phenomena.
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In this talk we will compare three versions of attractors with large (in the sense of Lebesgue measure) basins of attraction: Milnor, statistical, and physical (in the sense of Ilyashenko). The emphasis will be put on typical continuous (i.e. in topology of uniform convergence) dynamical systems on the unit interval. We will go beyond what is known so far about characteristics of these attractors. We will also explain why in the typical family the attractors depends continuously on the map with respect to the Hausdorff metric. The talk is based on joint work with Magdalena Forys-Krawiec, Jana Hantakowa and Michal Kowalewski
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We study the Holomorphic and Random Dynamics of some rank 2 free groups generated by two Hénon type maps. For these simply constructed examples we prove that the Fatou set is non-empty and that the stationary measures are supported on a compact set. With some further care this allows us to construct examples having no stationary measures. These examples illustrate the types of phenomena that may arise when studying holomorphic group actions on non-compact manifolds.
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The Mandelbrot set encodes the dynamical dependence of quadratic polynomials on a parameter. The MLC Conjecture (asserting that the Mandelbrot set is locally connected) is a rigidity property that yields a satisfactory topological description of the Mandelbrot set. In this talk, we describe the historical motivation for the conjecture, explain how it became a central topic in Renormalization Theory (analyzing first-return maps to small neighborhoods of special points), and outline some of the ideas behind the most recent advances toward MLC and related questions.
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Classic theory of Parabolic Implosion deals with perturbation around parabolic point using Lavaurs theory. Here I will present a new approach using perturbation and rescaling limit of Blaschke product. As a consequence I will show a necessary and sufficient condition for continuity of Julia set around $z^2 + z$, allowing only movement within main hyperbolic component.
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Given two continuous self-maps f and g on the interval which have all periodic orbits in common (that is, O(x)={x,f(x),...,f^(p-1)(x)} is a p-periodic orbit of f if and only if it is a p-periodic orbit of g but a priori, f may permute the elements of O(x) in a different fashion than g does), it is natural to ask whether f=g on the closure of the periodic points (which is known to coincide with the closure of the recurrent points!). We show this is the case wherever orbits with prime periods are dense. Specifically, we show that mixing interval maps are uniquely determined by (the location of) their periodic orbits.
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For a complex single variable polynomial $f$ of degree $d$, let $K(f)$ be its filled Julia set, i.e., the union of all bounded orbits. Assume that $K(f)$ has an invariant component $K^{\*}$ on which $f$ acts as a degree $d^{\*}<d$ map. This is a simplest instance of _holomorphic polynomial-like renormalization_ (Douady-Hubbard): the dynamics of a higher degree (degree $d$) polynomial $f$ near $K^*$ can be understood in terms of a suitable lower degree (degree $d^{\*}$) polynomial to which the restriction $f{\|}_{K^{\*}}$ is conjugate. One can associate a certain Cantor-like subset $G’$ of the circle with $K^{\*}$; the latter is defined in a combinatorial way. We will describe a role the Hausdorff dimension of $G’$ and the respective Hausdorff measure play in geometry of $K^{\*}$.
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We discuss the linking structure of the attractor-repeller pairs in simple Smale flows on the 3-sphere in which the chaotic saddle set is modeled by four- band templates with twisted bands. We obtain new theorems which illustrate that the dynamics of simple Smale flows are sensitive to half-twists in the bands of the embedded template. Haynes and Sullivan showed that the attractor- repeller pair a∪r in a simple Smale flow with chaotic saddle set modeled by embedded template U^+ is either a Hopf link or a trefoil and meridian. By placing a single half-twist in a selected band of U^+, we obtain new templates that model chaotic saddle sets of Smale flows. For simple Smale flows on S^3 with chaotic saddle sets modeled by those templates, we find that such simple Smale flows are realizable and that a∪r must be a Hopf link, a figure-8 knot and meridian, or a trefoil and meridian. This is joint work with Michael Sullivan.
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A metric space is called minimal if it admits a minimal (not necessarily invertible) map. The question of which metric spaces are minimal remains largely open and may be intractable in full generality. Numerous examples of specific minimal spaces are known -- those admitting a minimal homeomorphism, a minimal noninvertible map, or both. However, only a few general results identify minimal spaces within broad and significant classes, establish obstructions to minimality, or provide methods for constructing new minimal spaces from known ones. In this lecture, we will discuss a selection of classical and recent results that we find particularly important or interesting, highlighting those we especially like or find appealing.
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We study the problem of relating cycles on a *triod* *Y* to *circle rotations*. We prove that a *triod-twist* cycle *P*, the simplest cycle on a *triod* with a given *rotation number* ρ, is *conjugate* to *circle rotation*, by angle ρ, restricted to one of its cycles *Q*. Further, the conjugacy Ψ : *P* → *Q* is *piece-wise monotone* and its modality exceeds the modality *m* of *P* by *at-most* 3. This explicit bound *m*+3 serves as a *combinatorial distortion* principle, where the additive constant "+3" represents the "topological cost" imposed by the *valence* of the *branching point* *a*.
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