Digital lines, Sturmian words, and continued fractions

In this thesis we present and solve selected problems arising from digital geometry and combinatorics on words. We consider digital straight lines and, equivalently, upper mechanical words with positive irrational slopes a<1 and intercept 0. We formulate a continued fraction (CF) based descri...

Full description

Bibliographic Details
Main Author: Uscka-Wehlou, Hanna
Format: Doctoral Thesis
Language:English
Published: Uppsala universitet, Matematiska institutionen 2009
Subjects:
Online Access:http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-107274
http://nbn-resolving.de/urn:isbn:978-91-506-2090-0
id ndltd-UPSALLA1-oai-DiVA.org-uu-107274
record_format oai_dc
spelling ndltd-UPSALLA1-oai-DiVA.org-uu-1072742013-01-08T13:05:25ZDigital lines, Sturmian words, and continued fractionsengUscka-Wehlou, HannaUppsala universitet, Matematiska institutionenUppsala : Matematiska institutionen2009digital geometrydigital linehierarchy of runscombinatorics on wordsSturmian wordupper mechanical wordcharacteristic wordirrational slopecontinued fractionGauss mapfixed pointDiscrete mathematicsDiskret matematikIn this thesis we present and solve selected problems arising from digital geometry and combinatorics on words. We consider digital straight lines and, equivalently, upper mechanical words with positive irrational slopes a<1 and intercept 0. We formulate a continued fraction (CF) based description of their run-hierarchical structure. Paper I gives a theoretical basis for the CF-description of digital lines. We define for each irrational positive slope less than 1 a sequence of digitization parameters which fully specifies the run-hierarchical construction. In Paper II we use the digitization parameters in order to get a description of runs using only integers. We show that the CF-elements of the slopes contain the complete information about the run-hierarchical structure of the line. The index jump function introduced by the author indicates for each positive integer k the index of the CF-element which determines the shape of the digitization runs on level k. In Paper III we present the results for upper mechanical words and compare our CF-based formula with two well-known methods, one of which was formulated by Johann III Bernoulli and proven by Markov, while the second one is known as the standard sequences method. Due to the special treatment of some CF-elements equal to 1 (essential 1's in Paper IV), our method is currently the only one which reflects the run-hierarchical structure of upper mechanical words by analogy to digital lines. In Paper IV we define two equivalence relations on the set of all digital lines with positive irrational slopes a<1. One of them groups into classes all the lines with the same run length on all digitization levels, the second one groups the lines according to the run construction in terms of long and short runs on all levels. We analyse the equivalence classes with respect to minimal and maximal elements. In Paper V we take another look at the equivalence relation defined by run construction, this time independently of the context, which makes the results more general. In Paper VI we define a run-construction encoding operator, by analogy with the well-known run-length encoding operator. We formulate and present a proof of a fixed-point theorem for Sturmian words. We show that in each equivalence class under the relation based on run length on all digitization levels (as defined in Paper IV), there exists exactly one fixed point of the run-construction encoding operator. Doctoral thesis, comprehensive summaryinfo:eu-repo/semantics/doctoralThesistexthttp://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-107274urn:isbn:978-91-506-2090-0Uppsala Dissertations in Mathematics, 1401-2049 ; 65application/pdfinfo:eu-repo/semantics/openAccess
collection NDLTD
language English
format Doctoral Thesis
sources NDLTD
topic digital geometry
digital line
hierarchy of runs
combinatorics on words
Sturmian word
upper mechanical word
characteristic word
irrational slope
continued fraction
Gauss map
fixed point
Discrete mathematics
Diskret matematik
spellingShingle digital geometry
digital line
hierarchy of runs
combinatorics on words
Sturmian word
upper mechanical word
characteristic word
irrational slope
continued fraction
Gauss map
fixed point
Discrete mathematics
Diskret matematik
Uscka-Wehlou, Hanna
Digital lines, Sturmian words, and continued fractions
description In this thesis we present and solve selected problems arising from digital geometry and combinatorics on words. We consider digital straight lines and, equivalently, upper mechanical words with positive irrational slopes a<1 and intercept 0. We formulate a continued fraction (CF) based description of their run-hierarchical structure. Paper I gives a theoretical basis for the CF-description of digital lines. We define for each irrational positive slope less than 1 a sequence of digitization parameters which fully specifies the run-hierarchical construction. In Paper II we use the digitization parameters in order to get a description of runs using only integers. We show that the CF-elements of the slopes contain the complete information about the run-hierarchical structure of the line. The index jump function introduced by the author indicates for each positive integer k the index of the CF-element which determines the shape of the digitization runs on level k. In Paper III we present the results for upper mechanical words and compare our CF-based formula with two well-known methods, one of which was formulated by Johann III Bernoulli and proven by Markov, while the second one is known as the standard sequences method. Due to the special treatment of some CF-elements equal to 1 (essential 1's in Paper IV), our method is currently the only one which reflects the run-hierarchical structure of upper mechanical words by analogy to digital lines. In Paper IV we define two equivalence relations on the set of all digital lines with positive irrational slopes a<1. One of them groups into classes all the lines with the same run length on all digitization levels, the second one groups the lines according to the run construction in terms of long and short runs on all levels. We analyse the equivalence classes with respect to minimal and maximal elements. In Paper V we take another look at the equivalence relation defined by run construction, this time independently of the context, which makes the results more general. In Paper VI we define a run-construction encoding operator, by analogy with the well-known run-length encoding operator. We formulate and present a proof of a fixed-point theorem for Sturmian words. We show that in each equivalence class under the relation based on run length on all digitization levels (as defined in Paper IV), there exists exactly one fixed point of the run-construction encoding operator.
author Uscka-Wehlou, Hanna
author_facet Uscka-Wehlou, Hanna
author_sort Uscka-Wehlou, Hanna
title Digital lines, Sturmian words, and continued fractions
title_short Digital lines, Sturmian words, and continued fractions
title_full Digital lines, Sturmian words, and continued fractions
title_fullStr Digital lines, Sturmian words, and continued fractions
title_full_unstemmed Digital lines, Sturmian words, and continued fractions
title_sort digital lines, sturmian words, and continued fractions
publisher Uppsala universitet, Matematiska institutionen
publishDate 2009
url http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-107274
http://nbn-resolving.de/urn:isbn:978-91-506-2090-0
work_keys_str_mv AT usckawehlouhanna digitallinessturmianwordsandcontinuedfractions
_version_ 1716508539235598336