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Talk 01 — Zhiming Li (Northwest University, China) — [computing…]Local time: click here to view in your time zoneTitle: Ergodic optimization for random dynamical systems
Abstract
The theory of ergodic optimization for random dynamical systems has been developed. For every (continuous) random dynamical system $\varphi$ on a compact metric space $(\mathscr{X},d)$, associated with a Lebesgue space $(\mathscr{Q},\mathscr{F},\mathbb{P})$, a concept of maximum ergodic average is introduced, along with several equivalent characterizations. It is demonstrated that, in a separable Banach space of continuous random functions, a maximizing measure is generically unique. Furthermore, each ergodic measure can be uniquely maximized by a continuous random function. Additionally, it is shown that the maximizing measure is generically fully supported in $\mathscr{Q}\times \mathscr{X}$, provided that $\varphi$ satisfies that all random periodic measure is dense in the set of all invariant measures with respect to the narrow topology. Lastly, the zero-temperature limits of equilibrium states and their connections with maximizing measures for random dynamical systems are also explored.
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Talk 02 — Srijanani Anurag Prasad (Indian Institute of Technology Tirupati, India) — [computing…]Local time: click here to view in your time zoneTitle: Edelstein Hidden Variable Fractal Interpolation: Construction, Smoothness, and Dimension
Abstract
In this talk, we present a method for constructing hidden-variable fractal interpolation functions using Edelstein contractions within an iterated function system generated by a finite set of data points. The formulation employs variable functions as vertical scaling factors, leading to a generalized vector-valued fractal interpolation function. We further investigate the smoothness properties of the constructed function and derive an explicit upper bound for the box-counting dimension of its graph, highlighting the analytical and geometric features of the proposed framework.
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Talk 03 — Radu MICULESCU (Transilvania University of Brasov, Romania) — [computing…]Local time: click here to view in your time zoneTitle: On the range of fractal interpolation functions
Abstract
In this talk we show how to generate coverings (consisting of finite families of rhombi) of the graphs of fractal interpolation functions. In this way, we give an answer to the important problem concerning the spatial extent of such functions’ image, which appears in geometric aided design. As a by-product we obtain estimations for the range of these functions derived solely from interpolation data and scaling parameters. We should stress upon the fact that the existing papers dealing with this subject focus on restrictions imposed on parameters of the iterated function systems generating the fractal function based on a fixed range of it. Some concrete examples and graphical representations are provided.
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Talk 04 — Rim Achour (University of Monastir, Tunisia) — [computing…]Local time: click here to view in your time zoneTitle: The General Fractal Measures
Abstract
This presentation aims to clarify the importance of fractal geometry and to discuss several fundamental problems associated with classical fractal geometry. Subsequently, a new definition of fractal measures is introduced in order to overcome these limitations.
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Talk 05 — Priyanka T M C (Guizhou University, China) — [computing…]Local time: click here to view in your time zoneTitle: Multifractal Analysis of Chaotic Dynamics in Hopfield Neural Networks
Abstract
This talk focuses on the multifractal characteristics of chaos in Hopfield neural networks. Neural networks are basically dynamical systems, and when their long-term behavior exhibits sensitive dependence on initial conditions along with fractal properties, the resulting attractors are referred to as strange or chaotic attractors. Multifractal measure provides a more advanced and detailed description than single fractal dimensions. Multifractal analysis of time responses associated with chaotic attractors quantitatively captures irregularity and heterogeneity in network dynamics.
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Talk 06 — Harsha Gopalakrishnan (Indian Institute of Technology Tirupati, India) — [computing…]Local time: click here to view in your time zoneTitle: Box dimension of the graph of harmonic functions and the comparability of domains of energy on the Vicsek fractal
Abstract
This talk focuses on harmonic functions and various domains of energy on the Vicsek fractal. We investigate the oscillatory behavior of harmonic functions on the cells of the Vicsek fractal and analyze the box-counting dimension of their graphs corresponding to different energy domains. It is shown that, although the energy domains on the Vicsek fractal are mutually comparable, the energies of a given function arising from different energy domains are not necessarily related. In addition, we establish results on the box-counting dimension of the graph of functions of finite energy on the Vicsek fractal.
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Talk 07 — Md Nasim Akhtar (Presidency University Kolkata, India) — [computing…]Local time: click here to view in your time zoneTitle: Fractal interpolation on projective space
Abstract
In this talk, we study the contraction properties of an iterated function system (IFS) on the real projective plane that generates a fractal interpolation function for a prescribed set of data points. To this end, we introduce a suitable linear structure together with a metric on the real projective plane that avoids a hyperplane. We then prove that the resulting metric space is complete.
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Talk 08 — Alamgir Hossain (Teaching Fellow, Ashoka University, Haryana, India) — [computing…]Local time: click here to view in your time zoneTitle: Fractal Approximation on the Real Projective Plane
Abstract
The study of fractal theory on Euclidean spaces has emerged as an intriguing research area in recent times. The concept of fractal interpolation yields a method to approximate functions that are both self-affine or non-self-affine, and consequently allows substantial flexibility and diversity of the fractal modeling problem. In this talk, we introduce non-affine fractal functions on the non-Euclidean real projective plane. To this end, we consider a real projective plane endowed with a suitable linear structure and investigate classical approximation results in this setting. Subsequently, by constructing an appropriate iterated function system (IFS) on the real projective plane, we develop a class of non-affine fractal functions. We then establish fractal analogues of classical approximation theorems on the projective plane. Finally, we show that the attractor of an IFS defined on the dual space of the real projective plane can be identified with the graph of a fractal function.
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Talk 09 — Sangita Jha (NIT Rourkela, India) — [computing…]Local time: click here to view in your time zoneTitle: General recurrent fractal functions and their box-dimension
Abstract
Fractal interpolation function (FIF) is a recent technique that generalizes the classical interpolation methods. Recurrent fractal interpolation functions (RFIFs) generalize classical fractal interpolation functions and provide a more efficient framework for reconstructing data that exhibit piecewise self-affine structures. The aim of this talk is to discuss the construction of the RFIF associated with a weak iterated function system (IFS), and to estimate the lower and upper bounds of their box dimension.
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Talk 10 — Michael Hinz (Bielefeld University, Germany) — [computing…]Local time: click here to view in your time zoneTitle: On first order operators on metric graphs and fractals
Abstract
We will talk about analysis on fractal spaces. One remarkable feature of this theory is that, while "second order differential operators" have now been understood for quite some time, an understanding of "first order differential operators" and first order equations on fractals is still at the very beginning. We will give a short introduction and then explain recent results, joint with Waldemar Schefer (Bielefeld). These results generalize earlier results of Hino, Strichartz and others and led to novel integration by parts formulas and to first well-posedness results for equations of continuity type on fractals.
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Talk 11 — Vasileios Drakopoulos (Department of Computer Science and Biomedical Informatics, University of Thessaly, Greece) — [computing…]Local time: click here to view in your time zoneTitle: A fixed-point theorem for 𝜌−𝐹∗−contraction in complete 𝑏-metric spaces
Abstract
We investigate the existence and uniqueness of fixed points for 𝜌−𝐹∗-contractions in complete 𝑏-metric spaces. Firstly, we introduce the notion of a 𝜌−𝐹∗-contraction in a complete 𝑏-metric space and prove that every such mapping admits a unique fixed point. Next, we show that the Hutchinson–Barnsley operator associated with an iterated function system built from these contractions possesses a unique attractor (i.e., a unique fixed point).
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Talk 12 — Tanmoy Som (Indian Institute of Technology (BHU) Varanasi, India) — [computing…]
Cancelled: This talk has been cancelled.
Local time: click here to view in your time zoneTitle: Generalized multivariate Fractal Interpolation Function and α-Fractal FunctionAbstract
In the present work, a new approach is introduced for constructing multivariate fractal interpolation functions and α-fractal functions associated with multivariate functions. Unlike the existing methods that rely on the Banach contraction principle, here the construction is based on Matkowski and Rakotch contractions. While numerous methods for constructing multivariate fractal interpolation functions have been explored in the literature, the approach given in this paper is distinct in the sense that it generalizes all previously known techniques and provides a broader framework for such constructions. We propose a technique to develop nonlinear iterated function systems using the generalized contractions and establish that the attractors of such systems are the graphs of continuous multivariate functions interpolating theoretical data points. Furthermore, for the Rakotch contractions, the existence of an invariant Borel probability measure supported on the graph of the associated multivariate fractal interpolation function is explored.
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Talk 13 — Takayuki Watanabe (Chubu University, Japan) — [computing…]Local time: click here to view in your time zoneTitle: On the topology of the limit set of non-autonomous IFS
Abstract
Fractals are ubiquitous in nature, and since Mandelbrot's seminal insights into their structure, there has been growing interest in them. While the topological properties of the limit sets of IFSs have been studied— notably in the pioneering work of Hata — many aspects remain poorly understood, especially in the non-autonomous setting. In this talk, we present a homological framework which captures the structure of the limit sets. We apply our abstract theory to the concrete analysis of the so-called fractal square, and provide an answer to a variant of Mandelbrot’s percolation problem. This talk is based on our preprint arXiv:2510.23255, coauthored with Yuto Nakajima (Doshisha University, Japan).
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Talk 14 — Abhilash Sahu (Panchayat College, Bargarh, Odisha, India) — [computing…]
Cancelled: This talk has been cancelled.
Local time: click here to view in your time zoneTitle: Box-counting dimension of graphs of fractal functions on the Sierpinski gasketAbstract
The objective of this talk is to study the box-counting dimension of graphs of fractal interpolation functions on the Sierpiński gasket. Firstly, we give construction of a fractal interpolation function on the Sierpiński gasket and then with the help of fractal interpolation functions we show the existence of fractal functions in the space dom(E) consisting of all finite energy functionals on the Sierpiński gasket. Also, we obtain upper and lower bounds for the box-counting dimension of graphs of functions that belong to dom(E). Later, we provide bounds for the box-counting dimension of graphs of some functions belonging to the family of continuous functions which arise as fractal interpolation functions.
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Talk 15 — Ali Raza (Abdus Salam School of Mathematical Sciences, GC University Lahore, Pakistan) — [computing…]Local time: click here to view in your time zoneTitle: Interpolative contractive results in metric spaces
Abstract
In this talk I will present the results related to the interpolative Kannan type contraction in m-metric spaces. I will discuss different cases depending upon the sum of the interpolative exponents, including the case when sum is equal to one less than one and greater than one. Further, I will also provide some examples for better understanding of the results. Moreover, I will discuss the analysis for two scenarios, first when the underlying mappings are single valued and secondly when the mappings are multi-valued.
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Talk 16 — Akash Banerjee (Presidency University Kolkata, India) — [computing…]Local time: click here to view in your time zoneTitle: Advancing Local $\alpha$-Fractal Interpolation: From Compact Intervals to Standard Function Spaces
Abstract
Fractal interpolation functions (FIFs) have emerged as powerful tools for approximating non-smooth and irregular phenomena, offering advantages over traditional piecewise differentiable interpolation methods. While classical FIFs, introduced by Barnsley, utilize global iterated function systems (IFS), recent advancements by Massopust introduced "local" fractal functions defined via local IFS, allowing for greater flexibility in modelling local irregularities. The work by A. Banerjee, Md. N. Akhtar and M. A. Navascués bridges these two concepts by constructing Local $\alpha$-Fractal Functions, a generalization that integrates the analytical properties of Navascués’s $\alpha$-fractal functions with the localized structure of Massopust’s framework. The work on local $\alpha$-fractal interpolation functions has successfully established their existence and uniqueness on compact intervals within the space of bounded functions $B(A, \mathbb{R})$. While robust, the current theoretical infrastructure leaves several significant avenues unexplored. This abstract outlines critical open problems that must be addressed to elevate this framework to a broadly applicable analytical tool. The previous construction relies on the supremum norm. A primary open problem is the rigorous extension of the definition and proof of contractivity for the Read-Bajactarević (RB) operator $T$ to standard mathematical analysis spaces, including Lebesgue spaces ($L^p$) and Sobolev spaces ($W^{k,p}$). This extension is crucial for applying local $\alpha$-fractal operators to variational problems and the weak solutions of differential equations. The existing theorems rely on the compactness of the interval $A$. The formulation of local $\alpha$-fractal functions on non-compact domains (e.g., $[0, \infty)$ or $\mathbb{R}$) remains an open problem. Future work must determine the necessary decay conditions for the scaling functions $\alpha_i$ and the base function $b$ to ensure the contraction property of the local IFS on the real line.
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Talk 17 — Bogdan Anghelina (Transilvania University of Brasov, Romania) — [computing…]Local time: click here to view in your time zoneTitle: Covers of fractal interpolation surfaces with finite families of octahedrons
Abstract
In a previous work, On the localization of Hutchinson-Barnsley fractals, Chaos Solitons Fractals, 173 (2023), 113-674, we presented a method for finding a finite family of closed balls whose union contains the attractor of a given iterated function system. Our aim, for the particular framework of fractal interpolation surfaces, is to provide an improved version of it. This approach is more efficient, from the computational point of view, as it is based on finding the maximum of certain sets, in contrast to the previous method which uses a sorting algorithm.