# NCMW - Ergodic Theory and Fractals (2024)

## Venue: IIT, Tirupati

## Dates: 9 Dec 2024 to 21 Dec 2024

**Convener(s)**

Name: |
Dr Srijanani Anurag Prasad | Dr Shilpak Banerjee |

Mailing Address: |
Assistant Professor Department of Mathematics and Statistics, Indian Institute of Technology Tirupati, Yerpedu – Venkatagiri Road, Yerpedu Post, Tirupati District, Andhra Pradesh. Postcode - 517619. |
Assistant Professor Department of Mathematics and Statistics, Indian Institute of Technology Tirupati, Yerpedu – Venkatagiri Road, Yerpedu Post, Tirupati District, Andhra Pradesh. Postcode - 517619. |

Email: |
srijanani at iittp.ac.in | shilpak at iittp.ac.in |

Fractals and Ergodic Theory are two independent but interconnected disciplines of mathematics, each contributing to our knowledge of complex systems and patterns in nature.

Fractals, which exhibit self-similarity at multiple scales, provide a geometric foundation for describing irregular and complicated structures. Fractals were widely known in mathematics thanks to the groundbreaking work of Mandelbrot in the 1970s, which also revealed their prevalence in natural phenomena such as coastlines, clouds, and ferns. The recursive nature of fractals allows for the representation of complexity through simple repeating methods, encouraging discoveries into the intrinsic order of seemingly chaotic systems.

Ergodic Theory, on the other hand, is concerned about the statistical behaviour of dynamic systems over time. It delves into the concept of ergodicity, which occurs when a system's time average equals its ensemble average. This theory offers a basis for comprehending the long-term behaviour of dynamic systems and has applications in information theory, probability, and physics.

In the study of modern theory of dynamical systems, of which Ergodic Theory can be considered as a branch, we often seek out patterns that appear with the evolution of time and sometimes such patterns can even be considered `static’. Certain examples of interest from the world of complex dynamical systems would be Julia sets and Mandelbrot sets. Also, in the study of hyperbolic dynamical systems, structures such as Smale’s horseshoes appear as invariant sets and are often used as crucial tools to further our understanding of such systems. One thing common with these patterns is that they are self-repeating, or fractals! Hence a deeper knowledge of fractals is essential in the understanding of dynamical systems. In this workshop we seek to bring both the communities together and hope it will serve as a launching platform for many future collaborations.