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Talk 01 — Runze Zhang (Stony Brook University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Parabolic implosion in the parameter space of cubic polynomials
Abstract
Parabolic implosion is a remarkable phenomenon in complex dynamics. It describes the enrichment of Julia sets when the parabolic point of a rational map is perturbed. It is also natural to study the parabolic implosion in parameter spaces. In particular, when one perturbs properly the family of cubic polynomials having a stable parabolic fixed point into nearby families, the enrichment of bifurcation loci occurs. We investigate the topology of such enrichment in the parameter space and relate it to the corresponding enrichment of Julia sets of quadratic polynomials, the latter of which has been studied systematically by P. Lavaurs in the 80s.
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Talk 02 — Fan Yang (Wake Forest University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Recent progress in singular flows
Abstract
In this talk, we will discuss recent progress in the theory of smooth star flows that contain singularities and consider their expansiveness, continuity of the topological pressure, and the existence and uniqueness of equilibrium states. We will prove an ergodic version of the Spectral Decomposition Conjecture: $C^1$ open and densely, every singular star flow has only finitely many ergodic measures of maximal entropy, and only finitely many ergodic equilibrium states for Holder continuous potentials satisfying a mild yet optimal condition. Joint with M.J. Pacifico and J. Yang.
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Talk 03 — Matthew Kvalheim (University of Maryland, Baltimore County, USA) — [computing…]Local time: click here to view in your time zoneTitle: On the universality of linear dynamics
Abstract
Which nonlinear dynamical systems can be embedded as invariant subsets of linear systems of equal or greater dimension? This question is a natural variant of one posed by Terence Tao regarding the universality of potential well dynamics, with the class of potential well systems replaced by linear systems. It is also a fundamentally important question for various numerical algorithms associated with "applied Koopman operator theory" that seek to model the time-evolution of observations of an unknown nonlinear system by a linear system. Linearizing or "Koopman" embeddings do not generally exist, posing a fundamental challenge for such algorithms. In this talk, I will present necessary and sufficient conditions for the existence of linearizing embeddings for a broad class of nonlinear systems. Along the way, I will present extensions to the classical Hartman-Grobman and Floquet normal form theorems, as well as strong counterexamples to the oft-repeated claim that nonlinear systems with multiple isolated equilibria do not admit continuous linearizing embeddings. Aspects of this talk are based on joint works with Philip Arathoon and with Eduardo D. Sontag.
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Talk 04 — Rodrigo Perez (IU Indianapolis, USA) — [computing…]Local time: click here to view in your time zoneTitle: Flexibility of Entropies in Piecewise Expanding Unimodal Maps
Abstract
We illustrate Katok's principle of flexibility in the family of piecewise expanding unimodal maps. These maps have unique absolutely continuous invariant probability measures, so one can consider both metric and topological entropies. Both values are positive, with topological entropy at most log 2, but no smaller than the metric entropy (by the variational principle). We will show that these are the only restrictions for the values of those entropies, so all allowable pairs of values are realizable. This is joint work with Ll. Alseda and M. Misiurewicz
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Talk 05 — Nicolai T A Haydn (University of Southern California, USA) — [computing…]Local time: click here to view in your time zoneTitle: Rwturn times distribution for dynamical systems
Abstract
We consider the limiting return times distribution to sets that shrink to a nullset that not necessarily has to be a simple point. In general such limiting distributions are compound Poisson where the straight Poisson distribution occurs when the limiting set is a non-periodic single point. The method presented applies to expanding maps (which can be extended to hyperbolic maps) that have at least polynomial decay of correlations. Similar results can also be obtained in the symbolic setting when the approximating sets are unions of cylinders.
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Talk 06 — Svetlana Katok (Penn State University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups
Abstract
Given a closed, orientable surface of constant negative curvature and genus g ≥ 2, we study a family of generalized Bowen–Series boundary maps and their two dynamical invariants: the topological entropy and the measure-theoretic entropy with respect to a smooth invariant measure. Each such map is defined for a particular fundamental polygon and a particular multi-parameter. We prove two strikingly different results: rigidity of topological entropy and flexibility of measure-theoretic entropy. The topological entropy is constant in this family and depends only on the genus of the surface. We give an explicit formula for the topological entropy and show that it stays constant both within our parameter space and within the Teichmüller space of the surface. We obtain an explicit formula for the measure-theoretic entropy for the smooth invariant probability measure on the circle that only depends on the genus of the surface and the perimeter of the (8g−4)-sided fundamental polygon, and prove that it varies in the Teichmüller space and takes all values between 0 and maximum that is achieved on the surface which admits a regular fundamental (8g-4)-gon, and stays constant on a subset of the parameter space. The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof - valid only on a subset of multi-parameters - uses the realization of the geodesic flow on the surface as a special flow over the natural extension of the boundary map, and the Reduction Theory for maps associated to Fuchsian groups. This is joint work with Adam Abrams and Ilie Ugarcovici.
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Talk 07 — Sovanlal Mondal (The Ohio State University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Ergodic averages along sequences of return times
Abstract
Motivated by Bourgain’s return times theorem, recently, Donoso, Maass, and Arya-Saavedra studied ergodic averages along sequences generated by return times to shrinking targets in rapidly mixing systems. In this talk, we will show that their results can be improved to multiple ergodic averages for commuting transformations, and a wider range of sequences. This talk is based on joint work with Sebastián Donoso, and Vicente Saavedra-Araya.
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Talk 08 — Connor Kennedy (ABSENT) (Brandeis University, USA) — [computing…]Local time: click here to view in your time zoneStatus: ABSENT — This talk was not delivered.Title: Koopman operators and measures of maximal entropy via symbolic dynamics
Abstract
The method of symbolic representations gives a powerful tool to describe the behavior of a dynamical system via a simple left shift action on an appropriate sequence space. Meanwhile the method of Extended Dynamic Mode Decomposition (EDMD) considers the action of the Koopman operator for a dynamical system via its behavior on some appropriately chosen finite subspace of the full function space, spanned by an appropriate dictionary of observables. In essence both methods trade a simpler representation of the system’s behavior in exchange for a more complicated space it acts upon. In this talk I describe a combination of the two methods via a particular choice of dictionary for EDMD, built upon an existing symbolic representation of a system. I will present how the corresponding approximation of the Koopman operator converges to the true Koopman operator, complete with bounds on the finite step errors. In particular I note the capacity of this method to estimate several dynamical properties of interest, including estimation of the measure of maximal entropy and the spectral gap for a system. Of note is that this can be performed without knowledge of the form for any invariant measures of the system, with simple uniform sampling methods being sufficient.
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Talk 09 — Karin Reinhold Larsson (University at Albany, USA) — [computing…]Local time: click here to view in your time zoneTitle: End point bounds for variation and jump inequalities for convolution operators
Abstract
We investigate end point bounds for variation and jump inequalities corresponding to the family of convolution operators $\mu f = \sum_k \mu(k) T^kf$ for measures that satisfy a bounded angular condition. This provides a finer understanding of convergence of the operators in $L^1$ and in $L^{\infty}$.
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Talk 10 — Philipp Kunde (Oregon State University, USA) — [computing…]Local time: click here to view in your time zoneTitle: (Anti-)classification results in Topological Dynamical Systems and Ergodic Theory
Abstract
A fundamental theme in dynamics is the classification of systems up to appropriate equivalence relations. For instance, the equivalence relation of topological conjugacy preserves the qualitative behavior of topological dynamical systems. Smale's celebrated program proposes to classify topological or smooth dynamical systems up to topological conjugacy. In Ergodic Theory the isomorphism problem dates back to von Neumann's foundational paper and asks to classify measure-preserving transformations up to measure isomorphism. These classification problems not only turn out to be hard but sometimes even to be impossible. In this talk, we give an overview of classification as well as anti-classification results.
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Talk 11 — Eduardo Dueñez (University of Texas at San Antonio, USA) — [computing…]Local time: click here to view in your time zoneTitle: Real-valued computations: a perspective via topology and model theory
Abstract
We study the complexity of “deep computations” (limits of iterations of classical computations such as feed-forward neural networks). Our framework is based on types spaces in real-valued logic (topologized as Cp-spaces) and semigroup actions. The theory of Rosenthal compacta and Shelah’s NIP (No-Independence Property) serve as “Rosetta stone” translating between the topological, model-theoretical and computational viewpoints. Our main result identifies complexity classes corresponding to Todorčević’s trichotomy; these classes distinguish levels of “tameness” based on topological properties: first-countability, hereditary separability, and metrizability. The classification speak to the inherent complexity and limits on the ability to effectively/feasibly capture features of deep/asymptotic computations.
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Talk 12 — Isaia Nisoli (UFRJ, Brazil) — [computing…]Local time: click here to view in your time zoneTitle: On a problem by Gauss
Abstract
In this talk I will discuss a problem by Gauss, whose first solution was given by Wirsig. I will present a computer assisted tecnique that allows to give a certified solution to the problem, with a mathematically proved error bound. This is one instance of an a posteriori eigenvalue enclosure method for transfer operators, that allows to prove some important quantities, as the mixing rates or spectral gap, by means of a posteriori computations on finite dimensional discretizations.
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Talk 13 — Pratyush Mishra (Wake Forest University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Dynamics of primitive elements under group actions
Abstract
There has been some substantial work on studying the structure of a group by analyzing the behavior of primitive elements, sometimes under strong assumptions. 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. As an application, we introduce the notion of Nielsen and Schreier girth and obtain a class of groups with finite Nielsen girth but having infinite girth.