Time Zone Notice. All talk times listed below are in US Central Time (America/Chicago). To view a talk in your local time zone, click the “view in your time zone” link under that talk.
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Talk 01 — Andrei Ghenciu (University of Wisconsin–Stout, USA) — [computing…]Local time: click here to view in your time zoneTitle: The extended Hausdorff dimension spectrum of a conformal iterated function system is maximal
Abstract
For any conformal iterated function system (CIFS) consisting of finitely or countably many maps, and any closed shift-invariant set of right-infinite sequences of such maps, one can associate a limit set, which we call a shift-generated conformal iterated construction. We define the extended Hausdorff dimension spectrum of a CIFS to be the set of Hausdorff dimensions of all such limit sets. We prove that for any CIFS with finitely or countably many maps, the extended Hausdorff dimension spectrum is maximal, i.e. all nonnegative dimensions less than or equal to the dimension of the limit set of the CIFS are realized. We also prove a version of this result even for so-called conformal graph directed Markov systems, obtained via nearest-neighbor restrictions on the CIFS. The main step of the proof is to show that for the family of so-called β-shifts, the Hausdorff dimension of the limit set associated to varies continuously as a function of β.
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Talk 02 — Anita Tomar (Pt. L. M. S. Campus, Shridev Suman Uttarakhand University, Rishikesh 249201, Uttarakhand, India) — [computing…]Local time: click here to view in your time zoneTitle: From Ancient India to Modern Fractals: Fibonacci Sequences, Symmetry, and Complex Dynamics
Abstract
The Fibonacci sequence is one of the most intriguing mathematical constructs, distinguished by its simple recursive definition in which each term is the sum of the two preceding ones. While often attributed to Leonardo of Pisa (Fibonacci), this sequence was known and systematically studied in India centuries earlier, notably by Acharya Pingala in the context of prosody. Over time, the Fibonacci sequence has emerged as a fundamental pattern underlying phenomena in nature, art, architecture, and financial systems, earning its reputation as a “number system of nature.”
This talk traces the historical evolution of the Fibonacci sequence from ancient Indian mathematics to its modern mathematical formulations and applications. The primary focus is on its contemporary role in fractal geometry, where we investigate Mandelbrot and Julia sets generated via Fibonacci–Mann iterations for various classes of functions. Using both standard and approximate escape criteria, we demonstrate that the resulting Mandelbrot and Julia sets exhibit n-fold rotational symmetry.
We further analyze the influence of the control parameter on the geometry of these fractals using numerical indicators such as Average Escape Time, Non-Escaping Area Index, and Average Number of Iterations. The results highlight subtle distinctions in fractal complexity and demonstrate the continuing relevance of Fibonacci-based models in modern mathematics and applications. -
Talk 03 — Tayba Arooj (Lahore College for Women University, Lahore, Pakistan) — [computing…]Local time: click here to view in your time zoneTitle: Scientific Data Visualization via Hybrid Model based on Fractal Spline Interpolation
Abstract
This research presents the development of a novel hybrid interpolation model and its application to the visualization of scientific data. The proposed model, termed GPRC-FIF, combines spline interpolation with fractal interpolation and is characterized by four shape parameters and one scaling factor. Data-dependent constraints are imposed on two parameters and the scaling factor to preserve the inherent positive shape of the data, while the remaining two parameters are kept free to provide enhanced shape flexibility to the user. The effectiveness of the proposed approach is illustrated through several numerical data sets, where the theoretical results are demonstrated graphically. In addition, a comparative analysis with two existing interpolation methods is carried out to highlight the advantages and improved performance of the proposed model.
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Talk 04 — Kirthana Rajasekar (KTH Royal Institute of Technology, Sweden) — [computing…]Local time: click here to view in your time zoneTitle: Negative Lyapunov exponent of circle maps forced by expanding circle endomorphisms
Abstract
Lyapunov exponents characterize the stability/instability of a system by quantifying the exponential rate at which the orbits of arbitrarily close points in a system, diverge or converge. Their continuity, bounds, and asymptotes of the exponent gives us valuable insights into the hyperbolicity, invariant measures and, attractors. For (well-behaved) linear cocycles, we have the classic result of Oseledets multiplicative ergodic theorem. Whereas, for general skew-product maps on a two-dimensional torus, such a characterisation of the asymptotic behavior of the system is not immediate. Thus, the existence of open classes of maps with non-zero Lyapunov exponents, continues to be of keen interest. In this talk, we present one such method and establish a class of skew-product maps with expanding base for which the Lyapunov exponents admit a uniform negative lower bound.
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Talk 05 — Carl Dettmann (University of Bristol, UK) — [computing…]Local time: click here to view in your time zoneTitle: Kronecker sequences with many distances
Abstract
The 3 gap theorem states that for any real alpha and natural N, the sequence n alpha (mod 1) for 1<=n<=N has at most three different sized gaps. Biringer and Schmidt (2008) generalise this to higher dimensions by replacing gaps by the distance from each point to its nearest neighbour. Haynes and Marklof (2021) give a 5 distance theorem in dimension d=2 using Euclidean distance (p=2). Haynes and Ramirez (2021) consider the maximum distance (p=infinity), and find 5 distances for d=2 and 9 distances for d=3. In both cases there is an exponential upper bound in higher dimensions. I will describe recent work giving lower bounds for d<=6 for p=2, all d for p=infinity and d=2 for all p, which show that the maximum number of distances varies with p for d>=11.
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Talk 06 — Tushar Das (University of Wisconsin - La Crosse, USA) — [computing…]Local time: click here to view in your time zoneTitle: Dimension theory of continued fraction Cantor sets and their fractal cousins from the Fatou--Sullivan dictionary
Abstract
The Fatou–Sullivan dictionary presents a web of intriguing analogies between conformal dynamical systems arising from iteration of holomorphic or meromorphic maps, discrete subgroups of isometries of negatively curved space, and conformal iterated function systems and graph-directed Markov systems acting on locally Euclidean spaces of finite or infinite dimension, cf. Table 1 in https://arxiv.org/abs/1504.01774 I will report on a small slice of dimension-theoretic research — see https://arxiv.org/a/das_t_4.html — regarding conformal fractal objects starting with the simplest examples, which are continued fraction Cantor sets, and point to puzzles concerning their conformal cousins across the Fatou–Sullivan dictionary, where much less is known.
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Talk 07 — William O'Regan (University of British Columbia, Canada) — [computing…]Local time: click here to view in your time zoneTitle: A short proof of Bourgain's discretised projection theorem
Abstract
Bourgain's discretised projection theorem from 2010, which is a fundamental tool in many recent breakthroughs in geometric measure theory, harmonic analysis, and homogeneous dynamics. In this talk, we give a short elementary argument that shows that a discretised subset of the reals satisfying a weak `two-ends' spacing condition is expanded by a polynomial to a set of positive Lebesgue measure. After, we explain how the projection theorem follows from this fact. Joint work with Pablo Shmerkin and Hong Wang.
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Talk 08 — Chenxi Wu (UW Madison, USA) — [computing…]Local time: click here to view in your time zoneTitle: Self similarity in the Thurston’s teapots
Abstract
Given a family of interval maps with the same combinatorial maps such that the critical points have periodic orbits, the logarithm of their topological entropy and their Galois conjugate can be used to form a geometrical object in R^3 called the “Thurston’s teapots”. I will show that Solomyak’s results on the Mandelbrot set of iterated function systems can be used to show asymptomatic self similarity around certain points of this object.
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Talk 09 — Chayce Hughes (University of British Columbia, Canada) — [computing…]Local time: click here to view in your time zoneTitle: Smoothness of Markov Partitions for Expanding Toral Endomorphisms: Part 1
Abstract
Let $A$ be an integer matrix for which all eigenvalues have modulus strictly bigger than 1. The induced expanding toral endomorphism admits a Markov partition, which yields a good symbolic description of the dynamics. For $A$ diagonalizable, we show that if all eigenvalues are roots of integers, then the induced endomorphism admits a Markov partition with smooth boundary. Moreover, the converse holds in dimension 2.
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Talk 10 — Huub de Jong (University of British Columbia, Canada) — [computing…]Local time: click here to view in your time zoneTitle: Smoothness of Markov Partitions for Expanding Toral Endomorphisms: Part 2
Abstract
Let $A$ be an integer matrix for which all eigenvalues have modulus strictly bigger than 1. The induced expanding toral endomorphism admits a Markov partition, which yields a good symbolic description of the dynamics. We show that a non-diagonalizable 2 by 2 matrix admits no Markov partition with everywhere smooth boundary. However, it possible that some component of the boundary is smooth. We exhibit similar mixed behaviour in higher dimension, and provide an estimate on the Hausdorff dimension of the boundary.
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Talk 11 — Suddhasattwa Das (Texas Tech University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Path-space of dynamical systems - a categorical approach
Abstract
Dynamical systems are related to each other by semi-conjugacies. These relations collectively give dynamical systems the structure of a category. As a result, several concepts such as orbit, invariance and ergodicity can be explained by universal properties. Moreover, discourses in Dynamical systems theory can be set in the diagrammatic language of Category theory. Every dynamical system and every stochastic process on a domain X leads to a sub-collection of all the possible paths on X. This sub-collection if called the path-space of the process. I will present new ideas in this field on how the path-space of a dynamical system may be re-interpreted as the largest common factor of the Transfer operator and the full-shift on the domain space. Such a characterization of the path-space reveals its precise role in data-generation and recursion. I will also discuss how a careful choice of the categorical context allows an indirect study of differential properties in a space which is inherently non-smooth.
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Talk 12 — Andreas Mountakis (Northwestern University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Recurrence for pretentious systems along generalized Pythagorean triples
Abstract
We establish multiple recurrence results for pretentious measure-preserving multiplicative actions along generalized Pythagorean triples, that is, solutions to the equation $ax^2 + by^2 =cz^2$. This confirms the ergodic-theoretic form of the generalized Pythagorean partition regularity conjecture in this critical case of structured measure-preserving actions. This is based on joint work with Nikos Frantzikinakis.
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Talk 13 — Boris Kalinin (The Pennsylvania State University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Global smooth rigidity for toral automorphisms
Abstract
We consider the question of regularity of a conjugacy between a toral automorphism $L$ and a $C^\infty$ diffeomorphism $f$. We briefly survey prior results and then discuss our new results. For a weakly irreducible hyperbolic automorphism $L$ we obtain that any $C^1$ conjugacy between $L$ and $f$ is $C^\infty$. For a weakly irreducible ergodic partially hyperbolic $L$ we show that any $C^{1+Holder}$ conjugacy is $C^\infty$. This talk is based on a joint work with Victoria Sadovskaya and Zhenqi Wang.
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Talk 14 — Victoria V Sadovskaya (The Pennsylvania State University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Rigidity of strong and weak foliations.
Abstract
We consider a perturbation of a hyperbolic toral automorphism and a dominated splitting of the stable foliation into weak and strong parts. We discuss rigidity results related to exceptional properties of these weak and strong foliations. This talk is based on a joint work with Boris Kalinin.
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Talk 15 — Patrick Shipman (University of Arizona, USA) — [computing…]Local time: click here to view in your time zoneTitle: Dynamical Systems Applied to Optimal Packing Problems
Abstract
Phyllotactic patterns—spiral arrangements of leaves, florets, or seeds—exhibit striking regularity while achieving near-optimal packing on curved and growing domains. Descriptions of these patterns reveal the appearance of continued fraction expansions, the cross ratio, and number-theoretic structures associated with optimal irrational angles, motivating a dynamical-systems approach to packing beyond static geometric constructions. We present a pattern-forming PDE framework that links dynamical evolution to geometric organization. Within this setting, we review results on Fermat spiral lattices, their associated Voronoi geometry, and related circle-packing properties, showing how these structures yield quantitative measures of packing efficiency. Guided by the PDE analysis, we further construct diamond-based packings on the disk that capture key features of optimality observed in biological systems.
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Talk 16 — Miaohua Jiang (Wake Forest University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Sinai-Ruelle-Bowen entropy as an analogue of Boltzmann entropy
Abstract
Motivated by the Gallavotti–Cohen Chaotic Hypothesis, we study whether the entropy of the Sinai–Ruelle–Bowen invariant measure for transitive Anosov systems can serve as an analogue of Boltzmann entropy in classical thermodynamics. In low-dimensional systems, we prove results that can be viewed as manifestations of the basic postulate and the second law of thermodynamics.
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Talk 17 — William Ott (University of Houston, USA) — [computing…]Local time: click here to view in your time zoneTitle: The impact of nonlinear projections on the dimension spectra of fractal measures in Banach spaces
Abstract
Consider a set $X$ or a measure $\mu$ in a Banach space $\mathfrak{B}$. In this talk, I will use prevalence, thickness exponents, and the theory of projection constants to investigate how well a typical nonlinear map $f : \mathfrak{B} \to \mathbb{R}^{m}$ preserves the structure of $X$ or $\mu$. Does a typical such $f$ embed $X$? If so, what is the Holder regularity of the inverse? Does a typical such $f$ preserve the Hausdorff dimension of $X$? For the embedding problem, I will present a new theorem that improves on recent work of Margaris and Robinson. I will present results on preservation of Hausdorff dimension and dimension spectra that build on work from the Hilbert-space setting.
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Talk 18 — Kenneth M. Golden (University of Utah, USA) — [computing…]Local time: click here to view in your time zoneTitle: Fractals in the sea ice system
Abstract
As a material sea ice displays complex composite structure over a vast range of length scales. In fact, fractals appear naturally throughout the sea ice system, from brine inclusions inside the ice that host microbial communities to labyrinthine melt ponds on its surface and the ice pack itself on the scale of the Arctic Ocean. We explore how the fractal dimensions of these structures depend on their dynamical evolution, leading us into percolation theory, statistical physics, and computational topology. We also consider related dynamical systems models of algae living in the brine inclusions to study bloom dynamics.