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Ort Braude College of Engineering, 2013

Asymptotic Behavior of Parabolic-Type Semigroups of Holomorphic Mappings. Final Project for the Applied Mathematics Bachelor's Degree ( B.Sc ). By Ariel Hoffman Advisors: Dr. Fiana Yacobzon , Prof. Mark Elin. Ort Braude College of Engineering, 2013. Topics. What Are Dynamical Systems?.

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Ort Braude College of Engineering, 2013

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  1. Asymptotic Behavior of Parabolic-Type Semigroups of Holomorphic Mappings Final Project for the Applied Mathematics Bachelor's Degree (B.Sc) By Ariel Hoffman Advisors: Dr. FianaYacobzon, Prof. Mark Elin Ort Braude College of Engineering, 2013

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

  3. 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, 2013

  4. 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. Ort Braude College of Engineering, 2013

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

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

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

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

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

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

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

  12. What Are Semigroups of Holomorphic Mappings? Limit tangent lines Hyperbolic case: Each semigroup trajectory has a limit tangent line at its Denjoy-Wolff point, which depends on the initial point of the trajectory. Parabolic case: If a trajectory has a limit tangent line at the Denjoy-Wolff point, then all the trajectories share the same tangent line. 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 τ τ 12 Ort Braude College of Engineering, 2013

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

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

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

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

  17. New results: a more general case Ort Braude College of Engineering, 2013

  18. New result New result Ort Braude College of Engineering, 2013

  19. New result New result Ort Braude College of Engineering, 2013

  20. Method of proof Ort Braude College of Engineering, 2013

  21. Method of proof Ort Braude College of Engineering, 2013

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

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