Research Overview

My research lies at the intersection of fractal geometry, quantization theory, and dynamical systems. I am interested in understanding the complexity of mathematical structures through geometric, probabilistic, and dynamical viewpoints. My current research focuses on dynamical systems, particularly on entropy theory, multifractal analysis, and the study of invariant measures. I investigate entropy-related invariants and their multifractal properties to understand the complexity of dynamical systems better.

Although these research areas have developed independently, my long-term goal is to develop quantization-based invariants for dynamical systems, investigate their mathematical properties, and explore their relationships with classical notions such as entropy, dimension, and other dynamical invariants. Through this research, I hope to establish new connections between quantization theory, fractal geometry, and dynamical systems, leading to a deeper understanding of the complexity of dynamical phenomena.

Initially, I worked on fractal interpolation and fractal dimension of different types of functions and also explored their fractional integrals and approximation properties. We introduced the concept of set-valued alpha fractal functions and extended the theory of fractal interpolation to the set-valued setting. Further, we introduced the concept of constrained and conditional quantization of probability measures, providing new perspectives in quantization theory.

Research Topics

Fractal Geometry

"Fractal geometry will make you see everything differently. You risk the loss of your childhood vision of clouds, forests, galaxies, leaves, feathers, flowers, rocks, mountains, and much else. Never again will your interpretations of these things be quite the same." -Michael Barnsley.

Fractal dimension and fractal interpolations are two of the most celebrated topics in fractal geometry. Classical interpolation methods typically produce smooth and differentiable functions, which are often inadequate for modeling irregular and self-similar phenomena.

Motivated by the pioneering work of Michael F. Barnsley, I worked on fractal interpolation functions (FIFs), which provide a flexible framework for constructing interpolants that may be nowhere differentiable and exhibit self-similarity. In this direction, I investigated the Hausdorff and box dimensions of graphs of vector-valued, multivariate, and set-valued fractal interpolation functions, along with their fractional integrals.

A key contribution of my work was the introduction of $\alpha$-fractal functions for set-valued maps [1] , where I established structural properties and derived dimensional results. I also obtained results on the fractal dimension of vector-valued [2] and multivariate maps [3], and studied invariant measures associated with fractal interpolation functions [4] , thereby extending the scope of fractal interpolation theory beyond classical real-valued settings.

Quantization Theory

Quantization theory studies the approximation of probability distributions by discrete measures with finite support and has a long history in information theory, signal processing, and approximation theory. From a mathematical viewpoint, quantization problems focus on the asymptotic behavior of optimal approximations of probability measures, including the structure of optimal point sets, quantization errors, and quantization dimensions.

With one of my collaborators, I introduced the notion of constrained quantization [5] [6], where the approximating discrete measure is supported on a prescribed set, thereby generalizing the classical unconstrained framework. We extended fundamental notions such as quantization error, quantization dimension, and quantization coefficient to this constrained setting and established foundational results. In particular, we analyzed constrained quantization for the Cantor distribution and showed that both the quantization dimension and coefficient depend on the underlying constraints. Interestingly, we demonstrated that, unlike the unconstrained case, the quantization coefficient can exist and coincide with the quantization dimension under constraints. Furthermore, we introduced the concept of conditional quantization [7] , which unifies and extends both constrained and unconstrained approaches, opening new directions for research in structured approximation of probability measures.

Dynamical System

The notion of dimension works as a fundamental invariant for quantifying the complexity of geometric structures in fractal geometry. However, in dynamical systems, complexity arises not only from the geometry of the underlying space but also from the evolution of points under repeated iteration. To capture this dynamical complexity, entropy has emerged as one of the central invariants, measuring the rate at which information is created and orbit structures become increasingly complex.

One of the fundamental challenges in modern dynamical systems is to develop mathematical invariants that capture the complexity of systems beyond the reach of classical entropy theory. While topological and measure-theoretic entropy have proved remarkably successful for many finite-dimensional systems, they often become insufficient for describing systems with unstable behaviour, rich multifractal structures, or infinite-dimensional phase spaces. These limitations motivate the search for new invariants and deeper connections between existing notions of complexity.

A central theme of my current research is understanding how complexity should be quantified in dynamical systems when classical entropy no longer provides a complete description. Many systems exhibit unstable behaviour, multifractal phenomena, or infinite-dimensional features that require new mathematical tools beyond traditional entropy theory. My work explores these questions through the study of unstable entropy, metric mean dimension, and the multifractal properties of invariant measures. By developing and relating these entropy-based invariants, I seek to reveal new aspects of the long-term behaviour and complexity of dynamical systems.