ort braude college of engineering, 2013

Ort Braude College of Engineering, 2013

Final Project for the Applied MathematicsBachelor's Degree (B.Sc)

By Ariel Hoffman

Advisors: Dr. Fiana Yacobzon, Prof. Mark Elin

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Topics

What Are Semigroups of Holomorphic Mappings?

What Are Dynamical Systems?

New Results

Method of Proof

Definitions and key concepts

In-depth reviewInteresting questions

Summary of previous, known resultsExplanation of new results found in this project

A short summary of the methods used in the proof

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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.



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

5 Ort Braude College of Engineering, 2013



What Are Semigroups of Holomorphic Mappings?Differentiability

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z

)(zFt


What Are Semigroups of Holomorphic Mappings?Denjoy-Wolff Point



Semigroup Classifications

We will focus on the parabolic type.

1111

z

)(zFt


We will consider:

Asymptotic BehaviorWhat Are Semigroups of Holomorphic Mappings?

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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, 2005M. Elin, S. Reich, D. Shoikhet and F. Yacobzon, 2008M. Elin, D. Shoikhet and F. Yacobzon, 2008


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


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Parabolic-type semigroups




Previous results: rate of convergence

Elin and ShoikhetBoundery behavior and rigidity of semigroups of holomorphic mappings, 2011



Elin and ShoikhetBoundery behavior and rigidity of semigroups of holomorphic mappings, 2011



Elin, Shoikhet and YacobzonLinearization models for parabolic type semigroups, 2008


New results: a more general case


New result

New result


Method of proof

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Thank you for your attention!


I wish to thank:

and Prof. ElinDr. YacobzonFor all their help and guidance with this project

and mrs. ShmidovProf. KarpFor their patience and understanding

ort braude college of engineering, 2013

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