Research

Broad Research Area: Dynamical Systems, Fractals, Polynomials

Specialization: Holomorphic dynamics

Iteration of entire and meromorphic functions in the complex plane is known as Holomorphic Dynamics. The main challenge in dynamical investigation of transcendental functions is the presence of an essential singularity at infinity which is not the case for rational functions. The primary issue is to decide the subset of the complex plane where the sequence of iterates of a given function behaves stably. This set is called Fatou set of the function and its complement in the extended complex plane is called the Julia set. Determining and studying the nature of these sets is the main theme in Holomorphic Dynamics. Currently, we investigate the iteration of transcendental functions and rational functions on the Riemann sphere.

Publications

  1. Polynomials through pictures

  2. (With Subhasis Ghora), On periods of Herman rings and relevant poles

  3. (With Ankit Pradhan and Sasmita Barik), Independence fractals of graphs, Bulletin of Malayasian Mathematical Society, 2020

  4. (with Tarun Kumar Chakra and Gorachand Chakraborty ), On Herman rings with small periods and omitted values, Acta Mathematica Scientia (2018)

  5. (with Tarun Kumar Chakra) Iteration of translated tangents, Bulletin of the Malaysian Mathematical Sciences Society (2018)

  6. (with Tarun Kumar Chakra and Kedarnath Senapati) Julia sets of certain exponential like meromorphic functions, Proceedings of Indian Academy of Sciences (Mathematical Sciences), (2018)

  7. (with Tarun Kumar Chakra and Gorachand Chakraborty) Baker Omitted value, Complex Variables and Elliptic Equations, An International Journal, 61 (2016), 10, 1353-1361

  8. On Fatou components and omitted values, in Geometry, Groups and Dynamics, Proceedings of ICTS Program: Geometry, Groups and Dynamics held December 3--16, 2012, CEMS, Kumaun University, Almora, India, Edited by C. S Aravinda et al., Contemporary Mathematics, Volume 639, American Mathematical Society, Providence, RI, 2015, ISBN 978-0-8218-9882-6

  9. Herman rings of meromorphic maps with an omitted value, Proceedings of the American Math. Society, 2015

  10. (with M. G. P. Prasad) Julia sets of Joukowski-exponential maps, Complex Analysis and Operator Theary, Nov 2013

  11. (with Jian Hua Zheng) Omitted values and dynamics of meromorphic functions, Journal of London Math. Society, (2), 83, 2011, page 121-136

  12. (with M. G. P. Prasad) Iteration of certain meromorphic functions with unbounded singular values, Ergodic Theory & Dynamical Systems, 30,3, 2010, 877-891

  13. (with M. G. P. Prasad) Dynamics of certain class of critically bounded entire transcendental functions, J. Math. Anal. Appl. 329, no.2, 2007, 1446-1459

  14. (with M. G. P. Prasad) Dynamics of $\lambda \tanh(e^z)$, Discrete and Continuous Dynamical Systems- Series A, 19, September 2007,1, 121-138

  15. (with M. G. P. Prasad and Ashis Kumar Roy) Exploding Julia sets in the dynamics of $f_{\lambda}(z)= \lambda J_{1}(iz)/iz$, Proceedings of the Third National Conference on Non-linear Systems and Dynamics (NCNSD-2006), 171- 174, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, February 6-8, 2006, Allied Publishers PVT LTD

  16. (with M. G. P. Prasad) Siegel Discs in Complex Dynamics, Proceedings of the Second National Conference on Non-linear Systems and Dynamics (NCNSD-2005), 202-206, Aligarh Muslim University, Aligarh, February 24-26, 2005

  17. (with M. G. P. Prasad) Bifurcation and chaotic burst in the dynamics of $\lambda e^{az+be^z}$, Journal of Combinatorics, Information and System Sciences, 28-29, 2003-2004, 211-227

Research Supervision

  1. Iteration of meromorphic functions with an omitted value, PhD thesis work by Tarun Chakra at IIT Bhubaneswar

  2. Fractal transformation using Iterated function system, M.Sc. project by Chavan Ratan Madan, IIT Bhubaneswar during 2017-18.

  3. Quasiconformal mappings, M.Sc. project by Garima Gupta, IIT Bhubaneswar during 2016-17.

  4. A study of roots of Independence polynomials, M.Sc. project by Vijay Kumar Meena, IIT Bhubaneswar during 2016-17.

  5. The Independence fractal of a graph, M.Sc. project by Kartik Pandey, IIT Bhubaneswar during 2016-17.

  6. Finding all roots of a polynomial by Newton's Method, M.Sc. project by Soumen Pal, IIT Bhubaneswar during 2015-16.

  7. A study on palindromic polynomials, M.Sc. project by Priti Prasanna Mondal, IIT Bhubaneswar during 2015-16.

  8. Polynomial Root Finding, MSc project by Anudeep Nain, IIT Bhubaneswar during 2014-15

  9. Approximating Voronoi Diagram via Basic Family, MSc project by Satyabrata Jana, IIT Bhubaneswar during 2014-15

  10. Baker Wandering Doamins, MSc project by Yogender Singh, IIT Bhubaneswar during 2014-15

  11. On Herman rings of Transcendental meromorphic functions, a summer project by Pranav Kumar Upadrasta, B Tech., IIT Kharagpur during 2014

  12. When two fractals are same? Summer project by Prashanta Kumar Ojha, National Institute of Technology Rourkela during 2012

Geometry & Dynamics Lab

Sponsored Projects

  1. Title: Omitted Values in Complex Dynamics, Duration: 2014-2017, Supported by Dept. of Science & Technology (Govt. of India)

  2. Title: Julia sets of transcendental meromorphic functions, Duration: 2010--2012, Supported by IIT Bhubaneswar, Status: Completed

  3. Title: Topological Aspects of Transcendental Dynamics, Duration: 2009--2010, Supported by: China Postdoctoral Science Foundation (Grant no. 20090450420), Status: Complete

PhD and Post Doc details

PhD Thesis: Dynamics of certain transcendental entire and meromorphic functions Supervisor: Prof. M. Guru Prem Prasad

Postdoc Thesis: Omitted values and conformal measures in complex dynamics

Supervisor: Prof. Zheng Jian-Hua