Ort Braude College of Engineering, 2013 Final Project for the Applied Mathematics Bachelor's Degree (B.Sc) By Ariel Hoffman Advisors: Dr. Fiana Yacobzon, Prof. Mark Elin

2 Ort Braude College of Engineering, 2013 Topics What Are Semigroups of Holomorphic Mappings? What Are Dynamical Systems? New Results Method of Proof Definitions and key concepts In-depth review Interesting questions Summary of previous, known results Explanation of new results found in this project A short summary of the methods used in the proof 2

Ort Braude College of Engineering, What Are Dynamical Systems? Outline Dynamical systems are evolving processes. They are useful constructs, able to describe many different natural and imaginary systems, as well as predict their future states or discern their origins.

Ort Braude College of Engineering, What Are Dynamical Systems? Impact Dynamical systems arise in many different fields of study, and the theories governing their behavior have been applied successfully to numerous natural phenomena. Robotics, engineering, fluid dynamics, chaos theory, neuroscience and economics.

A few key concepts 5Ort Braude College of Engineering, 2013 What Are Dynamical Systems?

6 Ort Braude College of Engineering, 2013 What Are Dynamical Systems?

7 Ort Braude College of Engineering, 2013 What Are Dynamical Systems?

8 Ort Braude College of Engineering, 2013 What Are Semigroups of Holomorphic Mappings? Differentiability

9 Ort Braude College of Engineering, 2013 What Are Semigroups of Holomorphic Mappings? Denjoy-Wolff Point

10Ort Braude College of Engineering, 2013 What Are Semigroups of Holomorphic Mappings? Semigroup Classifications We will focus on the parabolic type.

11 Ort Braude College of Engineering, 2013 We will consider: Asymptotic Behavior What Are Semigroups of Holomorphic Mappings?

12 Limit tangent lines τ Hyperbolic case: Each semigroup trajectory has a limit tangent line at its Denjoy-Wolff point, which depends on the initial point of the trajectory. This was shown in the works of: M. D. Contreras and S. Díaz-Madrigal, 2005 M. Elin, S. Reich, D. Shoikhet and F. Yacobzon, 2008 M. Elin, D. Shoikhet and F. Yacobzon, 2008 Ort Braude College of Engineering, 2013 Parabolic case: If a trajectory has a limit tangent line at the Denjoy-Wolff point, then all the trajectories share the same tangent line. τ What Are Semigroups of Holomorphic Mappings?

13 Parabolic-type semigroups Ort Braude College of Engineering, 2013 What Are Semigroups of Holomorphic Mappings?

14Ort Braude College of Engineering, 2013 Previous results: rate of convergence Elin and Shoikhet Boundery behavior and rigidity of semigroups of holomorphic mappings, 2011

15Ort Braude College of Engineering, 2013 Previous results: rate of convergence Elin and Shoikhet Boundery behavior and rigidity of semigroups of holomorphic mappings, 2011

16Ort Braude College of Engineering, 2013 Previous results: rate of convergence Elin, Shoikhet and Yacobzon Linearization models for parabolic type semigroups, 2008

Ort Braude College of Engineering, New results: a more general case

18Ort Braude College of Engineering, 2013 New result

19Ort Braude College of Engineering, 2013 New result

Ort Braude College of Engineering, Method of proof

Ort Braude College of Engineering, Method of proof

22 Thank you for your attention! Ort Braude College of Engineering, 2013 I wish to thank: and Prof. ElinDr. Yacobzon For all their help and guidance with this project and mrs. Shmidov Prof. Karp For their patience and understanding