Citation
Pavelich, Janet (2002) Commuting equivalence relations and scales on differentiable functions. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:01262012103823681
Abstract
This work consists of two independent chapters:
The first is a study of commuting countable Borel equivalence relations, where two equivalence relations R and 5 are said to commute if, as binary relations, they commute with respect to the composition operator , i.e., R ◦ S = S ◦ R. The primary problem considered is, to what extent does the complexity of E = R V S depend on the complexity of R and S , if R and S commute? This is considered both in the case where the underlying space supports no Einvariant probability measure, and the case where it supports at least one such measure. In the first case, the answer is 'not very much': any such aperiodic equivalence relation E can be written as R V S, where Rand 5 are smooth aperiodic. In the second case, we frame our study within the context of costs, a system of invariants for countable Borel equivalence relations with invariant probability measures, developed by G. Levitt [12] and D. Gaboriau [5]. One aspect of costs which is not well understood is the extent to which 'commutativity' within an equivalence relation (in a more general sense than the definition given above) trivializes its cost. We have shown that, under certain conditions, this is in fact the case. One of the consequences of these investigations is a new, elementary proof of the fact the group SL_2 (Z[^1_2]) is antitreeable.
The second chapter is motivated by the well known theorem of descriptive set theory that every П^1_1 subset of a Polish (separable, completely metrizable) space admits a П^1_1 scale. We construct a П^1_1 scale on the set of differentiable functions with domain [0,1], which is a П^1_1 subset of the Polish space C([0,1]) . This construction is based on the П^1_1 rank of differentiable functions given by Kechris and Woodin in [4], and, like this rank, is meant to reflect the intrinsic nature of DIFF, and so give a 'natural ' criterion for determining whether the uniform limit of differentiable functions is itself differentiable. We then attempt to further analyze this 'scale criterion' for a sequence of differentiable functions (ƒ_n) by comparing it to the criterion that the sequence (ƒ'_n) converges.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Restricted to Caltech community only 
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Thesis Committee: 

Defense Date:  5 December 2001 
Record Number:  CaltechTHESIS:01262012103823681 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:01262012103823681 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  6783 
Collection:  CaltechTHESIS 
Deposited By:  John Wade 
Deposited On:  26 Jan 2012 19:52 
Last Modified:  29 Jan 2014 23:40 
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