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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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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 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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Ort Braude College of Engineering, 20133
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.
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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
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What Are Dynamical Systems?
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What Are Dynamical Systems?
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What Are Dynamical Systems?
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What Are Semigroups of Holomorphic Mappings?Differentiability
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z
)(zFt
Ort Braude College of Engineering, 2013
What Are Semigroups of Holomorphic Mappings?Denjoy-Wolff Point
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What Are Semigroups of Holomorphic Mappings?
Semigroup Classifications
We will focus on the parabolic type.
1111
z
)(zFt
Ort Braude College of Engineering, 2013
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
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?
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Parabolic-type semigroups
Ort Braude College of Engineering, 2013
What Are Semigroups of Holomorphic Mappings?
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Previous results: rate of convergence
Elin and ShoikhetBoundery behavior and rigidity of semigroups of holomorphic mappings, 2011
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Previous results: rate of convergence
Elin and ShoikhetBoundery behavior and rigidity of semigroups of holomorphic mappings, 2011
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Previous results: rate of convergence
Elin, Shoikhet and YacobzonLinearization models for parabolic type semigroups, 2008
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New results: a more general case
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New result
New result
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New result
New result
Ort Braude College of Engineering, 201320
Method of proof
Ort Braude College of Engineering, 201321
Method of proof
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Thank you for your attention!
Ort Braude College of Engineering, 2013
I wish to thank:
and Prof. ElinDr. YacobzonFor all their help and guidance with this project
and mrs. ShmidovProf. KarpFor their patience and understanding
