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Title:  Iterates of functions defined in terms of digital representations of the integers 
Author(s):  Thiel, Johann A. 
Director of Research:  Hildebrand, A.J. 
Doctoral Committee Chair(s):  Berndt, Bruce C. 
Doctoral Committee Member(s):  Hildebrand, A.J.; Reznick, Bruce A.; Stolarsky, Kenneth B. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Conway's RATS
iterative process Lehmer palindromes problem 196 discrete dynamical systems quasiperiodic Erd ̋osKac base Lyndon words ReverseAddThenSort (RATS) 
Abstract:  For a fixed base, John H. Conway’s RATS sequences are generated by iterating the following procedure on an initial integer: Reverse the digits of the integer, Add the reversal to the original, Then Sort the resulting digits in increasing order. For example, 334+433=767, which gets sorted into 677. In base 10, Conway discovered the curious sequence: 12333334444, 55666667777, 123333334444, 556666667777, .... Although the sequence is not periodic, it does display some periodiclike behavior which we refer to as “quasiperiodic.” Conway conjectured that all RATS sequences in base 10 are either eventually periodic, or they eventually lead to the previously mentioned quasiperiodic sequence. In this thesis, we study RATS sequences in various bases. In particular, we prove an Erd ̋osKac type result for the periods of RATS sequences in base 3; we establish a connection between RATS sequences in general bases and Lyndon words; and we construct infinite families of bases for which there exist RATS sequences having certain prescribed periodicity properties, e.g., we show that there are infinitely many bases for which we can construct quasiperiodic RATS sequences all of a similar type. In the final chapter, we consider a similar iteration process, the reverseadd process. We present data and heuristic arguments on a problem of D.H. Lehmer asking whether every sequence obtained by this process contains a palindrome. 
Issue Date:  20110825 
URI:  http://hdl.handle.net/2142/26109 
Rights Information:  Copyright 2011 Johann A. Thiel 
Date Available in IDEALS:  20110825 
Date Deposited:  201108 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics