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The Role Of Potential Theory In Complex Dynamics

Electronic Theses of Indian Institute of Science

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Title The Role Of Potential Theory In Complex Dynamics
 
Creator Bandyopadhyay, Choiti
 
Subject Ordinary Differential Equations
Potential Theory
Dynamical Systems
Harmonic Functions
Dirichlet Problem
Hausdorff Measures
Complex Dynamics
Holomorphic Polynomials
Entropy
Subharmonic Functions
Hausdorff Dimension
Ergodic Theory
Mathematical Analysis
 
Description Potential theory is the name given to the broad field of analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, Green’s functions, potentials and capacity. In this text, our main goal will be to gain a deeper understanding towards complex dynamics, the study of dynamical systems defined by the iteration of analytic functions, using the tools and techniques of potential theory. We will restrict ourselves to holomorphic polynomials in C.
At first, we will discuss briefly about harmonic and subharmonic functions. In course, potential theory will repay its debt to complex analysis in the form of some beautiful applications regarding the Julia sets (defined in Chapter 8) of a certain family of polynomials, or a single one.
We will be able to provide an explicit formula for computing the capacity of a Julia set, which in some sense, gives us a finer measurement of the set. In turn, this provides us with a sharp estimate for the diameter of the Julia set. Further if we pick any point w from the Julia set, then the inverse images q−n(w) span the whole Julia set. In fact, the point-mass measures with support at the discrete set consisting of roots of the polynomial, (qn-w) will eventually converge to the equilibrium measure of the Julia set, in the weak*-sense. This provides us with a very effective insight into the analytic structure of the set.
Hausdorff dimension is one of the most effective notions of fractal dimension in use. With the help of potential theory and some ergodic theory, we can show that for a certain holomorphic family of polynomials varying over a simply connected domain D, one can gain nice control over how the Hausdorff dimensions of the respective Julia sets change with the parameter λ in D.
 
Contributor Verma, Kaushal
 
Date 2014-04-07T11:01:50Z
2014-04-07T11:01:50Z
2014-04-07
2012-05
 
Type Thesis
 
Identifier http://etd.iisc.ernet.in/handle/2005/2291
http://etd.ncsi.iisc.ernet.in/abstracts/2949/G25293-Abs.pdf
 
Language en_US
 
Relation G25293