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Talk 01 — Karoly Simon (Budapest University of Technology and Economics, Hungary) — [computing…]Local time: click here to view in your time zoneTitle: Randomly perturbed iterated function systems
Abstract
We add a random additive error (independently in every step) when we apply an iterated function system. We investigate the dimension and Lebesgue measure of the attractor for these randomly perturbed IFSs based on resent papers of Dekking, Simon, Szekely, Szekeres and Barany, Rams.
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Talk 02 — Megala M (National Institute of Technology Andhra Pradesh, India) — [computing…]Local time: click here to view in your time zoneTitle: Self-Affine Measures: Spectral Classification and Beyond
Abstract
A fundamental question in fractal harmonic analysis is determining when a self-affine measure μ_(M,D) admits an orthonormal basis of exponential functions. This talk presents recent classification results characterizing spectral self-affine measures, including explicit spectrum constructions for specific matrix-digit set pairs. However, orthogonality requirements are often too restrictive for many important fractal measures. We explore modern alternatives that provide effective Fourier expansions beyond the classical spectral framework, presenting a broader perspective on harmonic analysis for self-affine fractals.
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Talk 03 — Kifayat Ullah Lone (Chandigarh University, India) — [computing…]Local time: click here to view in your time zoneTitle: Generation of attractor by weak Φ tupling iterated function system
Abstract
The objective of this talk is to introduce novel weak Φ m-tuple fractals through the application of the strong m-tuple fixed-point technique. Exemplary instances, along with computational analyses, will be presented to substantiate the derived outcomes.
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Talk 04 — Amlan Banaji (University of Jyväskylä, Finland) — [computing…]Local time: click here to view in your time zoneTitle: Fourier decay for nonlinear self-conformal measures
Abstract
Given a fractal measure on the real line, it is natural to ask whether it satisfies the Rajchman property (which means that its Fourier transform decays to 0), and if so, to estimate the rate of decay. As a general heuristic, if a conformal iterated function system (IFS) is "sufficiently nonlinear" in some appropriate sense, then one often expects the resulting self-conformal measures to have good Fourier decay. For instance, if every contraction in a conformal IFS is real-analytic and at least one of them is non-affine, then every non-atomic self-conformal measure for the IFS has polynomial Fourier decay. However, when the regularity of the contractions is merely $C^{\infty}$, the situation can be very different: we provide nonlinearity conditions on such IFSs which ensure that the resulting self-conformal measures are Rajchman, but show that the rate of Fourier decay can be extremely slow. This talk is based on joint work with Simon Baker.
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Talk 05 — Senthil Raani KS (IISER Berhampur, India) — [computing…]Local time: click here to view in your time zoneTitle: Quantitative existence of chains in sparse subsets of $\mathbb{R}^d$
Abstract
Building on a quantitative Steinhaus-type theorem for distances proved in [https://doi.org/10.1016/j.aim.2025.110752], we show that if a set $E\subset[0,1]^d$ has Hausdorff content sufficiently close to $1$ and Hausdorff dimension sufficiently close to $d$, then $E$ contains nondegenerate $k$-chains whose edge lengths may be chosen from a fixed nontrivial interval. This proves quantitative existence results for finite chains inside sparse subsets of $\mathbb{R}^d$.
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Talk 06 — Stéphane Seuret (Université Paris-Est Créteil, France) — [computing…]Local time: click here to view in your time zoneTitle: About the level sets of functions
Abstract
We investigate the possible size of the level sets of a (not necessarily) continuous function. We indicate some previous results, new ones, and some links with multifractal analysis.
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Talk 07 — Amit Priyadarshi (Indian Institute of Technology Delhi, India) — [computing…]Local time: click here to view in your time zoneTitle: Using Operator Theoretic methods in estimating the Hausdorff Dimension
Abstract
Estimating the Hausdorff dimension of the attractor of an iterated function system is in general a difficult task. For self-similar fractals under appropriate separation condition the Hausdorff dimension is the same as the similarity dimension by the famous work by Hutchinson. Later people have generalized his results to several other settings. In this talk we will discuss how Perron-Frobenius operators and spectral radius can be used to estimate the Hausdorff dimension of attractors of certain iterated function systems. We will apply it to some interesting examples.
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Talk 08 — Bikramjit Acharjee (Indian Institute of Technology Guwahati, India) — [computing…]Local time: click here to view in your time zoneTitle: Structural Results for One-parameter Iterated Function System
Abstract
One-parameter iterated function systems provide a fundamental framework for studying the behaviour of the attractors at the critical parameter, where contractivity ceases. At this critical parameter, two transition attractors arise: the lower and the upper transition attractor. In general, these transition attractors need not coincide, however, the former is always contained in the latter, reflecting the transition phenomena at the loss of contractivity. In this work, we show that taking convex geometry into account, this discrepancy is resolved. We introduce the notion of the convexified Hutchinson operator and show that every attractor has a canonical convexified counterpart. Within this framework, the lower and the upper transition attractor become indistinguishable at the level of their metrically convex hull. Moreover, we establish that the convexified upper transition attractor is unique, whenever upper transition attractor exists. Our main result states that under mild and natural conditions, the metrically convex hull of the lower and upper transition attractors coincides. This establishes a convex-geometric unification of transition attractors and reflects a fundamental structural stability at the contractivity threshold. These findings provide a fundamental insight into the transition phenomena for the one-parameter iterated function systems and highlight the central role of convexification in restoring uniqueness and coherence.
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Talk 09 — Sunil Mathews (National Institute of Technology, Calicut, India) — [computing…]Local time: click here to view in your time zoneTitle: Product Iterated Function Systems and their Separation Properties
Abstract
Fractals are objects with mathematical patterns, consisting of irregular curves, that repeat and does not terminate on finite scale magnification. Most of the fractals identified are found to follow a relatively simple recursive mathematical formula in comparison to the complexity of its structure. Fractal structures are often considered as the manifestation of chaos arising in dynamical systems. The similarity in patterns followed by fractals to the minute scale is termed in mathematical literature as self-similarity. There are several methods available to generate these objects mathematically, of which a widely used method is the method of iterated function systems(IFSs). A quantum of theory has been developed in mathematical analysis concerning the complex geometrical patterns of fractals. It is recognized as an emerging branch of mathematical sciences and is called the fractal theory. A dynamical system consists of a state-space with different states, a time-space through which the states can change following a rule defined by a function called the evolution operator. An iterated function system can be identified with a dynamical system with the underlying state space as a complete metric space, and the evolution functions as contractions. There have been many generalizations to the idea of the iterated function system. We explore more on the recent generalizations to the theory of iterated function systems. We discuss the space of iterated function systems consisting of a fixed number of contractions. It also gives an ordering of an IFS with respect to another IFS in this space. Further, it defines several types of sequences of IFSs, such as decreasing, eventually decreasing, Cauchy, convergent, etc. Later, this basic structure is used to develop certain results connecting the attractors of IFSs in a sequence of IFSs. The idea of sequences of IFSs then motivates to establish the notion of a dynamical system of iterated function systems, and the work further discusses discrete and continuous dynamical systems of IFSs. Later, different dynamical properties of the IFS right shift map are explained. Then the work looks into (n,m)- iterated function systems existing in the literature by replacing contraction mappings with the more general notion of contraction called convex contraction.