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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 Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Conway's RATS
iterative process
Lehmer
palindromes
problem 196
discrete dynamical systems
quasiperiodic
Erd ̋os-Kac
base
Lyndon words
Reverse-Add-Then-Sort (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 periodic-like 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 ̋os-Kac 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 reverse-add 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:2011-08-25
URI:http://hdl.handle.net/2142/26109
Rights Information:Copyright 2011 Johann A. Thiel
Date Available in IDEALS:2011-08-25
Date Deposited:2011-08


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