Dynamical systems are mathematical models of real-life evolutionary processes. They appear in various applications, from weather forecasts to the study of planetary motions to the evolution of species population numbers. The aim of the theory of dynamical systems is to understand the mechanisms behind certain fundamental phenomena that can be shared by various seemingly quite different dynamical systems. Such phenomena may include questions of stability (Is our solar system stable?) or the onset of chaos in completely deterministic systems (Why can’t we always give an accurate weather forecast?). One approach is to consider simplified versions of the real-life systems in the hope of “extracting” certain phenomena and then examining them without unnecessary obstructions. Nevertheless, it turns out that even very simple-looking dynamical systems can behave in mysterious ways that are yet to be understood.

Publications in Dynamical Systems 

1. (with N. Goncharuk) Renormalization and scaling of bubbles, (submitted) 2023, 30 pages   arXiv:2312.11308

2. (with X. Buff and V. Huguin) Entire maps with rational preperiodic points and multipliers, (submitted) 2023, 19 pages   arXiv:2304.13674

3. (with A. Dudko and W. Tucker) Lower bounds on the Hausdorff dimension of some Julia sets, Nonlinearity 36 (2023), no. 5, 2867-2893   arXiv:2204.07880

4. (with M. Yampolsky) Rigidity of analytic and smooth bi-cubic multicritical circle maps with bounded type rotation numbers, (submitted) 2021, 41 pages   arXiv:2112.05845

5. (with T. Firsova) Accumulation set of critical points of the multipliers in the quadratic family, Ergodic Theory Dynam. Systems 43 (2023), no. 5, 1548-1569   arXiv:2005.00665

6. (with D. Gaidashev) Complex a priori bounds for Lorenz maps, Nonlinearity 34 (2021), no. 3, 1263-1287   arXiv:2002.05807

7. (with T. Firsova) Equidistribution of critical points of the multipliers in the quadratic family, Adv. Math. 380, (2021), Paper No. 107591, 29 pages   arXiv:1903.00062

8. (with A. Belova) Critical points of the multiplier map for the quadratic family, Experimental Mathematics 31, (2022), no. 1, 337-345, DOI: https://doi.org/10.1080/10586458.2019.1627682,         arXiv:1902.10444

9. (with M. Yampolsky) Renormalization of unimodal maps with non-integer exponents, Arnold Math. Journal 4 (2018), no. 2, 179-191.   arXiv:1702.01214

10. (with M. Yampolsky) Rigidity, universality, and hyperbolicity of renormalization for critical circle maps with non-integer exponents,  Ergodic Theory Dynam. Systems 40 (2020), no. 5, 1282-1334.   arXiv:1505.00686 

11. Algebraic independence of multipliers of periodic orbits in the space of rational maps of the Riemann sphere, Mosc. Math. J. 2015, (15), no. 1, 73-87. arXiv:1401.4713 

12. Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable, Ergodic Theory Dynam. Systems 36 (2016), no. 4, 1156-1166.   arXiv:1305.0867 

13. On multi-dimensional Fatou bifurcation, Bull. Sci. Math. 2014, (138), no. 3, 356-375. 

14. Parameterizing degree n polynomials by multipliers of periodic orbits, C. R. Math. Acad. Sci. Soc. R. Can. 2013, (35), no. 4, 148-154.   arXiv:1310.6830 

15. Normal forms of families of maps in the Poincaré domain, Proc. Steklov Inst. Math. 2006, no. 3 (254), 94-102.

Publications in Discrete Geometry 

1. The central set and its application to the Kneser-Poulsen conjecture, Discrete Comput. Geom. 59 (2018), no. 4, 784-801   arXiv:1511.08134 

2. Kneser-Poulsen conjecture for a small number of intersections, Contrib. Discrete Math. 2014, (9), no. 1, 1-10.  arXiv:1006.0529 

3. Strict Kneser-Poulsen conjecture for large radii, Geom. Dedicata 2013, (162), 95-107.   arXiv:1006.0531 

Igors Gorbovickis on Google Sites

Igor Gorbovickis on ResearchGate