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Non-recurrent dynamics in the exponential family

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dc.contributor.advisor Feldman, Stanley en_US
dc.contributor.author Benini, Anna en_US
dc.contributor.other Department of Mathematics en_US
dc.date.accessioned 2012-05-15T18:02:20Z
dc.date.accessioned 2015-04-24T14:45:12Z
dc.date.available 2012-05-15T18:02:20Z
dc.date.available 2015-04-24T14:45:12Z
dc.date.issued 2010-08-01
dc.identifier Benini_grad.sunysb_0771E_10189.pdf en_US
dc.identifier.uri http://hdl.handle.net/1951/55369 en_US
dc.identifier.uri http://hdl.handle.net/11401/70944 en_US
dc.description.abstract This dissertation deals with the dynamics of non-recurrent parameters in the exponential family $\{e^z+c\}$. One of the main open problems in one-dimensional complex dynamics is whether hyperbolic parameters are dense;this conjecture can be restated by saying that all fibers, i.e. classes of parameters with the same ray portrait, are single points unless they contain a hyperbolic parameter. The main goal of this dissertation was to prove some statements in this direction, usually referred to as rigidity statements.We prove that fibers are single points for post-singularly finite (Misiurewicz) parameters and for combinatorially non-recurrent parameters with bounded post-singular set. We also prove some slightly different rigidity statement for combinatorially non-recurrent parameters with unbounded postsingular set.We also add some understanding to the correspondence between combinatorics of polynomials and combinatorics of exponentials and we prove hyperbolicity of the postsingular set for non-recurrent parameters, generalizing a previous statement concerning only non-recurrent parameters with bounded post-singular set.We finally contribute to another open problem in transcendental dynamics, i.e. understanding whether repelling periodic orbits are landing points of dynamic rays, giving a positive answer to this question in the case on non-recurrent parameters with bounded post-singular set. The strategy used also gives a new, more elementary proof of the corresponding statement for polynomials, dating back to work of Douady. en_US
dc.description.sponsorship This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree. en_US
dc.format Monograph en_US
dc.format.medium Electronic Resource en_US
dc.language.iso en_US en_US
dc.publisher The Graduate School, Stony Brook University: Stony Brook, NY. en_US
dc.subject.lcsh Mathematics en_US
dc.subject.other combinatorics, complex exponential, density of hyperbolicity, holomorphic dynamics, rigidity, transcendental dynamics en_US
dc.title Non-recurrent dynamics in the exponential family en_US
dc.type Dissertation en_US
dc.mimetype Application/PDF en_US
dc.contributor.committeemember Marco Martens en_US
dc.contributor.committeemember John Milnor en_US
dc.contributor.committeemember Saeed Zakeri. en_US

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