• No results found

Decision problems in groups of homeomorphisms of Cantor space

N/A
N/A
Protected

Academic year: 2020

Share "Decision problems in groups of homeomorphisms of Cantor space"

Copied!
172
0
0

Loading.... (view fulltext now)

Full text

(1)

DECISION PROBLEMS IN GROUPS OF HOMEOMORPHISMS

OF CANTOR SPACE

Feyisayo Aderonle Olukoya

A Thesis Submitted for the Degree of PhD

at the

University of St Andrews

2018

Full metadata for this item is available in

St Andrews Research Repository

at:

http://research-repository.st-andrews.ac.uk/

Please use this identifier to cite or link to this item:

http://hdl.handle.net/10023/15885

(2)

Decision problems in groups of homeomorphisms of

Cantor space

Feyisayo Aderonle Olukoya

University of

St Andrews

This thesis is submitted in partial fulfilment for the degree of

Doctor of Philosophy (PhD)

at the University of St Andrews

(3)

Abstract

The Thompson groups F,T andV are important groups in geometric group theory: T andV

being the first discovered examples of finitely presented infinite simple groups. There are many generalisations of these groups including, fornandrnatural numbers and 1< r < n, the groups

Fn,Tn,randGn,r(T =∼ T2,1andV=∼ G2,1). Automorphisms ofFandT were characterised in the seminal paper of Brin ([16]) and, later on, Brin and Guzman ([17]) investigate automorphisms of

Tn,n−1andFnforn >2. However, their techniques give no information about automorphisms of

Gn,r.

The second chapter of this thesis is dedicated to characterising the automorphisms of

Gn,r. Presenting results of the author’s article [10], we show that automorphisms ofGn,r are homeomorphisms of Cantor space induced by transducers (finite state machines) which satisfy a strong synchronizing condition.

In the rest of Chapter 2 and early sections of Chapter 3 we investigate the group Out(Gn,r) of outer automorphisms ofGn,r. Presenting results of the forthcoming article [6] of the author’s, we show that there is a subgroupHnof Out(Gn,r), independent ofr, which is isomorphic to the group of automorphisms of the one-sided shift dynamical system. Most of Chapter 3 is devoted to the order problem inHnand is based on [44]. We give necessary and sufficient conditions for an element ofHnto have finite order, although these do not yield a decision procedure.

Given an automorphismφof a group G, two elements f,g ∈ G are said to beφ-twisted conjugate to one another if for someh∈G,g=h−1f(h)φ. This defines an equivalence relation on

(4)

Declaration

Candidate’s declaration

I, Feyisayo Aderonle Olukoya, do hereby certify that this thesis, submitted for the degree of PhD, which is approximately 61,319 words in length, has been written by me, and that it is the record of work carried out by me, or principally by myself in collaboration with others as acknowledged, and that it has not been submitted in any previous application for any degree.

I was admitted as a research student at the University of St Andrews in September 2014.

I, Feyisayo Aderonle Olukoya, received assistance in the writing of this thesis in respect of language, grammar, spelling and syntax, which was provided by Dr Martyn Quick.

I received funding from an organisation or institution and have acknowledged the funder(s) in the full text of my thesis.

Date Signature of candidate

Supervisor’s declaration

I hereby certify that the candidate has fulfilled the conditions of the Resolution and Regulations appropriate for the degree of PhD in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree.

Date Signature of supervisor

Permission for publication

In submitting this thesis to the University of St Andrews we understand that we are giving permission for it to be made available for use in accordance with the regulations of the University Library for the time being in force, subject to any copyright vested in the work not being affected thereby. We also understand, unless exempt by an award of an embargo as requested below, that the title and the abstract will be published, and that a copy of the work may be made and supplied to any bona fide library or research worker, that this thesis will be electronically accessible for personal or research use and that the library has the right to migrate this thesis into new electronic forms as required to ensure continued access to the thesis.

I, Feyisayo Aderonle Olukoya, confirm that my thesis does not contain any third-party material that requires copyright clearance.

The following is an agreed request by candidate and supervisor regarding the publication of this thesis:

Printed copy

(5)

Electronic copy

No embargo on electronic copy.

Date Signature of candidate

(6)

Underpinning Research Data or Digital Outputs

Candidate’s declaration

I, Feyisayo Aderonle Olukoya, hereby certify that no requirements to deposit original research data or digital outputs apply to this thesis and that, where appropriate, secondary data used have been referenced in the full text of my thesis.

(7)

Acknowledgements

Over the course of the last three to four years, I have been challenged more than anything else by the extent of my ‘blindness’ and the process of learning to overcome it. I am grateful to those who helped me along the way.

First of all I would have to thank my supervisor Dr Collin Bleak: his patient advice on mathematics and non-mathematics related things have been invaluable as I often groped around in the dark.

Next I would like to thank my other collaborators Prof. Peter Cameron, Dr. Casey Donoven, Dr. Yonah Maissel and Prof. Andres Navas for leading me to places I might never have gone alone.

The thing about ‘blindness’ is that often one looses perspective on things, and so I would like to thank my friends of the ‘Beast of St Andrews’ and latterly the ‘Beats of no Nathan’, Ebe Bamboye, Anna Billson, and Nathan Price, who helped me many times to see the bigger picture. I am also grateful to Steve Brooks for dragging me to football on a Friday and to the other members who participated and made it a fun time.

I am greatly indebted to my parents for their constant support and encouragement and also to my brothers and sister who always believe in me.

I would like to thank my church family for their prayers and for helping me maintain the right spiritual perspective.

Finally I am grateful to God not least of all for creating a world where mathematics is possible.

Funding

(8)

“I try all things, I achieve what I can.”— Herman Melville, Moby-Dick.

(9)

Contents

Preface 8

1 The Rational GroupRnand Related groupsRn,r 12

1.1 Words . . . 12

1.2 Graphs and trees . . . 14

1.3 Automata and Transducers . . . 17

1.3.1 Automata . . . 17

1.3.2 Transducers . . . 18

1.4 Cantor Space from different points of view . . . 20

1.5 Transducers and continuous functions on Cantor space . . . 23

1.5.1 Transducers and continuous functions onCn . . . 23

1.5.2 Transducers and continuous functions onCn,r . . . 25

1.6 Minimising Transducers . . . 28

1.7 The groupsRnandRn,r . . . 31

1.7.1 Multiplying Transducers . . . 32

1.7.2 Inverting transducers . . . 37

2 Subgroups ofRnandRn,r 46 2.1 The synchronizing property . . . 46

2.2 Detecting membership ofBen,r[Ben] inRn,r[Rn] . . . 51

2.2.1 Other checks for detecting whether a transducer is bi-synchronizing . . . . 56

2.3 Higman-Thompson groupsGn,rand their automorphism groups Aut(Gn,r). . . . 58

2.3.1 Higman-Thompson groupsGn,ras prefix exchange maps onCn,r. . . 59

2.3.2 Higman-Thompson groupsGn,rand their action onTn,r . . . 61

2.3.3 The Higman-Thompson groupsGn,ras subgroups ofBn,r . . . 62

2.4 Automorphisms of the Higman-Thompson groupsGn,r. . . 64

2.5 The group Out(Gn,r)and some of its subgroups . . . 69

3 The monoidPenand the groupHn 76 3.1 The monoidPenand the group of automorphisms of the one-sided shift dynamical system . . . 77

3.2 The Curtis, Hedlund, Lyndon theorem . . . 78

3.3 From endomorphisms of the shift dynamical system to elements ofPenvia foldings of De Bruijn graphs . . . 80

3.4 The monoidePn . . . 85

3.5 The order problem, finiteness problem and groups and semigroups generated by transducers inPen . . . 96

3.5.1 Level one synchronizing transducers . . . 98

3.5.2 The equivalence of the finiteness and order problems for groups generated by synchronizing, synchronous transducers to the order problem inHn . . 100

3.5.3 The order problem inHn . . . 103

3.6 Combining elements ofPenand some embedding results . . . 117

3.6.1 Two element case . . . 117

(10)

3.7 The difference between the synchronizing and bi-synchronizing level . . . 123

3.8 Level one synchronizing transducers and Picantin’s conjecture . . . 124

3.9 The automaton group generated by a synchronizing transducer has exponential growth . . . 128

3.9.1 Further conditions for having infinite order: avoiding loops . . . 131

3.9.2 The growth rate of Cayley machines . . . 134

3.10 Growth rates of the core of elements ofPen. . . 136

4 φ-Twisted Conjugacy andRinGn,rfor certainφ∈Aut(Gn,r) 144 4.1 Introduction . . . 144

4.2 Gn,rhas infinitely manyφ-twisted conjugacy classed forφ∈BHn,r . . . 145

4.2.1 Twisted Conjugacy inGn,r . . . 145

4.2.2 The elementsgτ. . . 145

4.2.3 Revealing Pairs . . . 146

4.2.4 Gn,rhas infinitely manyφ-twisted conjugacy classes forφ∈BHn,r . . . . 150

4.3 Thompson’s groupV=G2,1 . . . 152

4.3.1 Revealing pairs for elements ofVR. . . 153

4.3.2 New Rollings . . . 153

4.3.3 Effects Of New Rollings . . . 156

4.3.4 Conjugacy inVR . . . 158

(11)

Preface

This thesis investigates certain subgroups of therational groupRnof [30] and its generalisations

Rn,rfor natural numbersnandrsatisfying 16r < n. It is a fact, which we do not make use of in this work, thatRn,randRnare isomorphic to the groupR2. Hence, in this preface we shall refer to both by the name ‘the rational group’ and phrase our discussion in terms of the groupRn,r. In subsequent paragraphs we shall use the notationDn,r, for some symbolD, to represent certain subgroups ofRn,r, however, to have uniformity in our notation and to make correspondences clear, whenr=1 we setDn,r = Dn; for instanceRn,1=Rn.

Since the rational groupRn,r was first introduced in [30] it has generated a lot of research activity. It has been shown to have insoluble order problem ([4]), and demonstrated to be simple ([5]) and not finitely-generated ([5],[30]). Moreover, all of its finitely generated subgroups have soluble word problem ([30] ). The groupRn,ralso contains many important classes of groups. For instance it contains the Thompson groupsF,T andVand the Higman-Thompson groupsGn,r generalisingV([30]); the Brin-Thompson groupnV([4]); the groups of automorphisms of the shift dynamical system ([30]); and groups generated by automata.

The groupsT andVwere the first discovered examples of finitely presented infinite simple groups, whilst Thompson’s groupF, also finitely presented, has a simple derived subgroup. Groups generated by automata are a vast source of groups with interesting properties. For instance the Grigorchuk group, which is an infinite torsion group of intermediate growth ([29], [31]), belongs to the class of groups generated by automata. The automorphisms of the shift dynamical system are an important class of groups in symbolic dynamics. These are defined as those homeomorphisms of the Cantor space of one-sided or two-sided infinite sequences over a finite alphabet, which commute with the shift map. The group Aut({0, 1, . . . ,n−1}ω,σn) of shift commuting homeomorphisms of the one-sided infinite sequence space is called the group of automorphisms of the one-sided shift (dynamical system). In a similar manner, the group Aut({0, 1, . . . ,n−1}Z,σn)of shift commuting homeomorphisms of the two-sided infinite sequence space is called thegroup of automorphisms of the two-sided shift dynamical system.

This thesis will mainly be concerned with the subgroupGn,r, its automorphism group, and a subgroup of its outer automorphisms group which is isomorphic to the group of automorphisms of the one-sided shift dynamical system. We also consider the monoid consisting of continuous functions from the Cantor space of one or two-sided infinite sequences to itself which commute with the shift map. These are the so calledendomorphisms of the one or two-sided shift dynamical system

We begin in Chapter 1 by defining, for 1 6 r < n, the groups Rn,r. These are groups of homeomorphisms of Cantor spaceCn,r, the disjoint union of r copies of Cn, which may be represented by finite state machines called transducers. Hence, in Chapter 1, we also introduce automata and transducers and present various algorithms for multiplying, inverting and minimising transducers. We distinguishsynchronous transducers, which always write a single output letter on consuming an input letter, as a subclass ofasynchronous transducerswhich are allowed to write strings (including the empty string) on reading a single input. Additionally, we show how to construct from an element ofRn,r, a transducer representing this homeomorphism. This is a technical chapter carrying out the fundamental constructions from [30] in full detail in the context of the groupsRn,r. These constructions are particularly relevant to later work in Chapter 2.

(12)

groupV. More specifically, in this chapter we characterise the automorphisms of the groupGn,r, presenting the results of the author’s article [10]. This completes a line of research first begun by Brin in his seminal paper [16] characterising the automorphisms of the Thompson groupsFandT. Later on, Brin together with Guzman, in the article [17], also investigate automorphisms of groups generalising Thompson’s groupsFandT. However their techniques do not apply to the groups

Gn,r.

The article [10], making use of the transitivity of the action ofGn,ronCn,rand a deep result of Rubin ([47]), shows that automorphisms ofGn,rare elements ofRn,rfor which the transducers representing these homeomorphisms have a particular property. This property is called the synchronizing property. The set of all rational homeomorphisms ofCn,rwhose transducers have the synchronizing property forms a submonoidBen,r. The largest inverse closed subset ofBen,ris the groupBn,r. The results of [10] demonstrate that Aut(Gn,r)is isomorphic toBn,r.

We do not present all the results of the article [10] in Chapter 2, we instead show that the normalizer ofGn,rin the rational group is the groupBn,r. By appealing to Rubin’s Theorem and finiteness results on local actions of rational homeomorphisms in Yonah Maissel’s thesis ([39]), which also appear in [10]), we conclude that the map which takes an elementτ∈ Bn,rto the automorphism ofGn,r induced by conjugation byτis an isomorphism. Additionally, in this chapter we explore various consequences and characterisations of the synchronizing property. We close the chapter by examining the outer automorphisms of the groupGn,r.

It is immediate from the characterisation of Aut(Gn,r)as those homeomorphisms inRn,rthat can be represented by transducers with the synchronizing property, that Out(Gn,r)is best thought of as a group consisting of non-initial transducers with the synchronizing property. (Observe that since groups are inverse closed, then inverses of elements of Out(Gn,r)also have the synchronizing property.) Perhaps surprisingly, it turns out that forr =n−1, Out(Gn,r)contains a subgroup which is isomorphic to the quotient of the group of automorphisms of the two-sided shift dynamical system, by the group generated by the shift map. We observe that, by Ryan’s Theorem ([48]), the group generated by the shift map is the centre of the group of automorphisms of the two-sided shift dynamical system.

The author’s forthcoming paper [6] explores this connection between Out(Gn,r)and the group of automorphisms of the two-sided shift dynamical system further. One of the results contained in this article, is that shift commuting homeomorphisms of the two-sided infinite sequence space can be represented by elements of Out(Gn,r), together with natural combinatorial data arising from the structure of the non-initial transducers. A consequence of the results of [6], is that there is a subgroupHnindependent ofr, that isHn6∩16r6n−1Out(Gn,r), which is isomorphic to the group of automorphisms of the one-sided shift dynamical system. This subgroup is the focus of Chapter 3.

In Chapter 3, we begin our investigation ofHnby first demonstrating the isomorphism between

Hnand the group of automorphisms of the one-sided shift dynamical system. As a somewhat natural starting point, we show that a submonoid of endomorphisms of the shift dynamical system (those requiring no ‘future information’), is isomorphic to a monoid of non-initial transducersPn which contains the groupHn. We then deduce, as a corollary of this fact, that the groupHnis isomorphic to the automorphisms of the one-sided shift dynamical system.

The rest of the chapter, is based on the paper [44] which explores the order problem inHn. Now, elements ofHnare precisely those elements of Out(Gn,r)which can be represented bysynchronous transducerswith the synchronizing property. We observe that, as a synchronous transducer on reading a single input letter, will always write a single output letter, inverses of elements ofHn are also synchronous; they are synchronizing since Out(Gn,r)is a group. It is a standard result in the literature that each state of an invertible synchronous transducer induces an element of

Rn. Thus, invertible synchronous transducers naturally generate subgroups ofRnby taking the group generated by the homeomorphisms ofCninduced by the states of such transducers. Groups generated in this way are calledautomata groups, and, as we mentioned in an earlier paragraph, this class of groups contain many groups with interesting properties.

(13)

the transducer is finite. The order problem asks if there is an algorithm, which, given an invertible synchronous transducer and an element of the group as a product of the generators, decides in finite time if the given element has finite order.

Bleak and Belk in [4] investigate the finiteness and order problems in their full generality. For groups generated by finitely many elements of the rational group, they show that the finiteness and order problem are undecidable in general. However, the decidability of these problems for groups generated by invertible, synchronous transducers, remained open.

Recently, the order problem for groups generated by invertible, synchronous, transducers was shown to be insoluble in general. This was done independently by Gillibert ([27]), and Bartholdi and Mitrofanov ([3]). The finiteness problem remains open for synchronous transducers. As it turns out, the finiteness problem for transducers inHnis equivalent to the order problem for automata groups generated by elements ofHn, and to the order problem inHn. The equivalence of the finiteness problem for transducers inHn to the order problem inHnwas shown in the paper [23]. However, we provide an independent proof all three equivalences in Chapter 3.

We also provide in Chapter 3 some new and sufficient conditions for when an element ofHn has finite or infinite order. These conditions focus on studying properties of thedual transducer for elements ofHn. This is a technique which has been used effectively in investigating the order problem, as in the paper [1]. We prove some new results about the dual transducer of finite order elements ofHnwhich go some way towards resolving a conjecture of Picantin in [40]. These properties of the dual automata for elements ofHnlead us to study a new combinatorial object we call thegraph of bad pairs. For each element ofHn, we associate infinitely many such graphs. Properties of this graph, for instance if it contains a circuit, help in determining whether an element ofHnhas infinite or finite order. In fact we conjecture that whenever an element ofHnhas infinite order, then eventually one of its graph of bad pairs contains a circuit. It is a consequence of our structure results for the dual transducer of a finite order element ofHnthat for a given finite element ofHn, eventually all of its graphs of bad pairs are precisely the empty graph.

To close Chapter 3, we investigate the growth rate of automata groups generated by transducers inHn. As indicated above, this is a standard question to ask about groups generated by invertible, synchronous transducers. We show that all such groups, whenever they are infinite, have exponential growth rate. This result, as it happens, is already implied by work of Chou in [21] and results of Silva and Steinberg in [51] showing, amongst other things, that groups generated by transducers inHn are elementary amenable groups. (We should perhaps also remark that Chou’s proof showing that finitely generated elementary amenable groups either have polynomial or exponential growth rate contains a gap. The gap, however, is fixed in a paper by Rosset ([46]) from which one also deduces that, in the case where a group generated by an element ofHnhas polynomial growth, it is finite. The author is grateful to Bartholdi for drawing his attention to the work of Chou and for pointing out both the error in Chou’s proof and the fix by Rosset.)

Chapter 3 also contains results which do not fit under the umbrella of order problem and growth, but which naturally arise in our consideration of these problems: for instance we present certain embedding results for the groupsHn, and some conjugacy invariants inHn.

In Chapter 4, we again consider the group Gn,r. Having now an understanding of the automorphisms of the groupGn,r, we consider thetwisted conjugacy probleminGn,r. The twisted conjugacy problem is a generalisation of the conjugacy problem and asks for a finitely presented groupG, if there is an algorithm which, given elementsf,g∈Gand an automorphismφofG, decides in finite time if there is an elementh ∈Gsuch thatf = h−1g(h)φ. If such anhexists thenfandgare said to beφ-twisted conjugate. Moreover ifφis the identity automorphism ofG, thenfandgare conjugate inG. For a givenφin Aut(G), the relation onGdefined by,f∼φgif and only iffandgareφ-twisted conjugate to one another, is an equivalence relation. A natural question to ask then, is if there are finitely many or infinitely manyφ-twisted conjugacy classes for

φan automorphism of a finitely presented group. We note that a group which has infinitely many twisted conjugacy classes for all automorphisms is said to have theRproperty.

The twisted conjugacy problem and corresponding Rquestion have been investigated for Thompson’s groupFandT, whose automorphisms were classified by Brin in the article [16]. The paper [7] shows thatFhas the R∞property, whilst the paper [19] shows that the twisted conjugacy problem inFis soluble, and demonstrates that bothFandT have the R∞-property.

(14)

property for the Higman-Thompson groupsGn,r. We demonstrate that for automorphismsφ

ofGn,r such that the image ofφin the quotient Out(Gn,r)is inHn, there are infinitely many

φ-twisted conjugacy classes. The remainder of the chapter focuses on the groupG2,1, and, for automorphismsφwhose image in Out(Gn,r)is inH2, we solve theφ-twisted conjugacy problem.

The majority of the computations with transducers appearing in this work were vastly helped by the GAP software ([25]) together with the GAP packages “AutomGrp” ([41]) and “aaa”. The former package deals only with synchronous transducers and was helpful in the computations appearing in Chapter 3; the latter package handles asynchronous transducers and is currently still being developed by Collin Bleak (the author’s supervisor), Fernando Flores-Brito, Plamena Minerva, the author, and Angela Richardson. Already implemented in this package are many of the algorithms in the paper [30], moreover, this package allows for graphical visualization of transducers which was a great help to the author as a starting point for many of the figures of transducers appearing in this document.

(15)

Chapter 1

The Rational Group

R

n

and Related

groups

R

n

,

r

This chapter shall be concerned primarily with defining the groupsRnandRn,r. Along the way we shall introduce certain algorithms that will play a key role in future discussion. We shall keep the exposition as self-contained and detailed as possible, illustrating key ideas with examples so the reader can become as familiar as possible with these groups and some of the ways we will be working with them.

We begin by introducing the notion of words, then graphs and trees, then automata and finally transducers. Afterwards, we define the groupsRnandRn,rwith which this thesis will be primarily concerned. In the process we establish some notation that we will be using throughout this work. Subsequent notation and terminology shall be introduced as we require them. The contents of the next few sections should therefore be thought of as containing the ‘essentials’. Our exposition throughout this chapter is based on those given in the author’s article [10] and the paper [30].

For a givenjan element ofZorN, we often need to refer to the subsets ofZorNconsisting of those elements which are greater than or equal toj. We establish the following general notation to refer to these subsets.

Notation 1.0.1. LetIbe one ofZ,NorR, and leti ∈ I, then we denote byIi the subset of I consisting of those elements greater than or equal toi.

1.1

Words

Definition 1.1.1. A finite set of symbolsX:={x1,x2, . . . ,xn}will be called analphabet. An element

x∈Xwill be called aletter. LetXbe an alphabet, then thesize ofX, denoted|X|is the number of symbols inX.

Definition 1.1.2. LetX:={x1,x2, . . . ,xn}be an alphabet, aword or string(over the alphabetX) is a finite or infinite sequencew1w2w3. . . of elements ofX. Thelength of a wordΓoverX, is the length of the sequence representing the word, and we denote it by|Γ|. Theempty word, denoted by the symbol, is the unique word of length zero. LetΓ be a word overX, then we callΓ aninfinite word if|Γ|=∞otherwise we callΓ afinite word.

Notation 1.1.3. LetXbe a finite alphabet. SetX∗to be the set of all finite words overX, and set

X+:=X∗\{}. Fork∈N, setXk:={Γ ∈X∗n| |Γ|=k}.

Remark 1.1.4. Forn∈N1letnbe the ordered set{0, . . . ,n−1}. LetN∈{n|n∈N1}t{N}and

X:={x1, . . . ,xn}be an alphabet. A functionf:N→Xdefines a non-empty wordf(0)f(1). . . over

X. In particular, every non-empty word overXcan be identified with a functionf :N →Xfor someN∈{n|n∈N1}t{N}with cardinality equal to the length of the word.

Definition 1.1.5. LetXbe a finite alphabet and letΓ ∈X∗be a word. Suppose thatΓ is the sequence

(16)

Definition 1.1.6. Given two wordsΓ1,Γ2∈X∗theconcatenation of Γ1withΓ2is the wordΓ1Γ2. Remark 1.1.7. Concatenation with the empty word returns the original word. ThusX∗forms a monoid under concatenation.

Notation 1.1.8. LetXbe a finite alphabet and letΓ ∈X∗be a word. Forj∈ N,j>1, we shall denote byΓjthe wordΓ1Γ2. . .ΓjwhereΓi=Γ for all 16i6j. SetΓ0:=.

Definition 1.1.9. LetXbe a finite alphabet. By Definition 1.1.4 we may identify a mapf:N→X

with the infinite wordf(0)f(1)f(2). . .. Analogously, a mapg: Z →Xdefines abi-infinite word consisting of the sequence . . .g(−2)g(−1)g(0)g(1)g(2). . .. We shall identify the mapgwith the bi-infinite sequence . . .g(−2)g(−1)g(0)g(1)g(2). . ..

Remark 1.1.10. We shall sometimes omit the prefixes infinite or bi-infinite when it is clear from the context that the word in question is infinite or bi-infinite.

Notation 1.1.11. LetXbe a finite alphabet. SetXωto be the set of all infinite words overXand set

XZto be the set of all bi-infinite words overX. Given a bi-infinite wordg∈XZandi∈Zwe shall sometimes denote bygithe letterg(i)and we shall call this letter theithletter ofg.

Definition 1.1.12. Given an alphabet X and two words u,v over X such that u ∈ X∗ and

v ∈ X∗tXω, we say thatuis a prefixof v ifv = uv1 for some word v1 ∈ X∗tXω; we say thatv1is asuffixofv.

Remark 1.1.13. The empty word is a prefix and suffix of every word.

Definition 1.1.14. Define a relation ‘6’ onX∗tXωbyν6η, forν∈X∗andη∈X∗tXω, if and only ifνis a prefix ofη. Ifν 66ηandη 66ν, then we say thatνandηareincomparableand we denote this byν⊥η.

Remark 1.1.15. It is easy to see that the relation ‘6’ onX∗tXωof Definition 1.1.14 is a partial order.

Notation 1.1.16. Letν,µ∈X∗and supposeν6µ. Letτ∈X∗be such thatµ= ντ, then we set

µ−ν:=τ. LetU⊂X∗tXω. Ifν∈X∗is such thatνis a prefix of every element ofUthen we shall indicate this by writingν6U. Furthermore, forU⊂X∗tXωandν∈X∗such thatν6U, set

U−ν:={δ∈X∗tXω|νδ∈U}.

Definition 1.1.17. LetXbe an alphabet, and letδ∈XωtX∗. Leti ∈Nbe such that 16i6|δ|, and letδi∈Xibe the unique word of lengthisuch thatδi6δ, then we callδithelengthiprefix of

δ.

Definition 1.1.18. Given two alphabetsXandY, and subsetsV⊂X∗, andW⊂Y∗tYωwe shall denote byVWthe set{vw|v∈V,w∈W}.

We now introduce the notion of antichains.

Definition 1.1.19. A finite, ordered, subsetu ⊂X∗is called anantichainif for any distinct pair

ν,η∈uwe haveν⊥η. An antichainuis calledcompleteif for anyτ∈X∗there is someν∈usuch thatτ6νorν6τ.

Remark 1.1.20. Observe that the definition of an antichain only depends on the partial order on the setX∗. Thus, for an arbitrary setYwith a partial order≺onYwe may define antichains onY

as in Definition 1.1.19.

Definition 1.1.21. LetX = {x1,x2, . . . ,xn}be a finite alphabet and letu = {u1,u2, . . . ,ul}⊂ X+ be an antichain of lengthl. For 1 6 i 6 l, a single expansion ofu(over ui) is the ordered set

(17)

Remark 1.1.22. Letu be a (complete) antichain over some finite alphabetX. If u0 is a single expansion ofuover some elementv∈u, thenu0is still a (complete) antichain. Thus any expansion ofuis a (complete) antichain. Moreover, ifwis an expansion ofuthen|u|≡|w| mod |X|−1; ifw is ak-fold expansion ofuthen|w|=|u|+k(|X|−1).

Now that we have the necessary preliminaries on words and related notions, in the next section we establish the essential definitions and facts concerning graphs and trees that we will require in this work.

1.2

Graphs and trees

This section shall be concerned with introducing some of the main geometric objects we will be working with.

Definition 1.2.1. Adirected graphis a tupleG= (V,E,ι,τ)whereVis a set of symbols and is called thevertex setofG,Eis aset of edgesofGandι,τ:E→Vare maps. For an edgee∈E,(e)ιis called thestart of the edgeand(e)τis called theend or terminus of the edge. Elements ofVare calledvertices or nodesand elements ofEare callededges. Thesize ofG, denoted|G|, is the size of the vertex set of

G. If|G|is finite then we say thatGisfinite, otherwise we say thatGisinfinite.

Definition 1.2.2. LetG= (V,E,ι,τ)be a directed graph and letV0⊂V. ThesubgraphofGinduced byV0is the graphG0= (V0,E0,ι0,τ0)whereE0⊂Econsists precisely of those edgesesuch that

(e)ι,(e)τ∈V0; the mapsι0andτ0are the restrictions ofιandτtoE0.

Definition 1.2.3. LetG = (V,E,ι,τ)be a directed graph. A path (inG)is a sequence of edges (alternatively a finite or infinite word over the set of edges)p:=e1,e2,e3. . . such that, for any 16i6|p|,(ei+1)ι= (ei)τ. If the sequence is finite then we say that the path isfiniteotherwise we say that the path isinfinite. Thelength of the path, denoted|p|, is the length of the sequence. We call

(e1)ιtheinitial vertex ofpand, if|p|<∞, we call(e|p|)τtheterminal vertex ofp.

Definition 1.2.4. LetG = (V,E,ι,τ)be a directed graph. Two verticesuandvofGare called connectedif there is a pathpinGwith initial vertexuand final vertexv. The directed graphGis calledconnected, if for any pairu,vof distinct vertices ofG, there is a pathpwith initial and final verticesµandν, respectively, such that{µ,ν}={u,v}. If for any pair of verticesu,vofGthere is a path with initial vertexuand final vertexvthen we say thatGisstrongly connected.

Definition 1.2.5. An(undirected) graphis a directedG= (V,E,ι,τ)such that for every edgee∈E, there is an edgee−1∈E, called theinverse ofe, such that(e−1)ι= (e)τand(e−1)τ= (e)ι.

Remark 1.2.6. For a graphG= (V,E,ι,τ)we shall identify every edgee∈Ewith its inversee−1

and denote both bye. Thus a graph can be described by a tripleG = (V,E,ξ), whereV is the vertex set ofG,Eis the set of edges, andξ: E→ V(2)whereV(2) :={{u,v} | u,v ∈ V}. For an edgee ∈E,(e)ξ = {u,v}is called theends ofeand we sayeis an edge betweenuandv. For an undirected graphG= (V,E,ξ), a path (inG) is a sequencep:=e1,e2,e3, . . . of edges with ends

{v1,v2},{v2,v3},{v3,v4}. . .. Notice that an undirected graph is connected if and only if it is strongly connected.

Definition 1.2.7. LetG= (V,E,ξ)be a graph, and letv1,v2∈V. Then we say thatv1isincidentto

v2if there is an edgee∈Ewith(e)ξ={v1,v2}. A vertexv∈Vis said tobelong toan edgee∈Eif

v∈(e)ξ. If a vertexv1is incident to a vertexv2then we callv2aneighbourofv1.

Definition 1.2.8. LetG= (V,E,ξ)be a graph. An edgeeis called aloopif(e)ξis a singleton. Definition 1.2.9. LetG= (V,E,ξ)be a graph. A pathp=e1,e2. . . defines a sequencev1,v2. . . of vertices of length|p|+1 (|p|for an infinite path), where, for 16i6p,(ei)ξ={vi,vi+1}. We call this sequence thevertex sequence ofp. For a pathpofGand a vertexv∈V, we say thatpvisitsvor

(18)

Definition 1.2.10. LetG= (V,E,ξ)be a graph. Then two verticesv1andv2are said to beconnected if there is a path inGwith start vertexv1and final vertexv2. A graphG= (V,E,ξ)is said to be connectedif any pair of distinct vertices are connected.

Definition 1.2.11. Let G1 = (V1,E1,ξ1) and G2 = (V2,E2,ξ2) be graphs. A pair of maps

γE:E1→E2,γV :V1→V2is called agraph homomorphismif whenevere∈E1with(e)ξ1={u,v}, then((e)γE)ξ2={(u)γV,(v)γV}. If the mapsγEandγV are also injective/ surjective/ bijective then we say thatγis agraph monomorphism/ epimorphism/ isomorphism.

Definition 1.2.12. LetG= (V,E,ι,τ)be a directed or undirected graph and letSbe an alphabet. A mapL :E→Sis called anedge labelling ofG. The pair(G,L)is called alabelled graph.

Definition 1.2.13. A graphG= (V,E,ξ)is calledsimpleif it has no loops and for any pairv1,v2of vertices ofV, there is at most one edgee∈Efor which(e)ξ={v1,v2}.

Remark 1.2.14. For a simple graphG= (V,E,ξ), an edgee∈Eis characterised precisely by(e)ξ. In particular, a simple graph can be represented by a pairG= (V,E)whereE⊂V(2) ={{u,v} | u,v∈V}. From henceforth, we denote all simple graphs by pairsG= (V,E)of vertices and edges, whereE⊂V(2). For a simple graphG= (V,E)we represent a pathpas a sequencev1,v2,v3, . . . of vertices where for all 16i6|p|,{vi,vi+1}∈E.

Definition 1.2.15. A simple graphGis called atreeif it is connected and contains no basic circuits. A treeGis said to berootedif there is a distinguished vertex; we call this distinguished vertex the root (of the tree). LetGbe a rooted tree, then we call a vertex ofGwith only one neighbour aleafof the treeG. A vertex which is neither a leaf nor the root will be called aninternal vertex.

Remark 1.2.16. Observe that in a treeT there is a unique geodesic connecting any two vertices. Definition 1.2.17. A rootedn-ary treeTn, is a rooted tree with an infinite set of vertices such that every vertex apart from the root has preciselyn+1 neighbours, and the root hasndistinct neighbours.

Remark 1.2.18. There is only one rootedn-ary tree up to isomorphism, thus, we fix a representative treeTn. We phrase all subsequent discussion with regards to this tree, and refer to it asthe rooted

n-ary tree.

Definition 1.2.19. LetTnbe the rootedn-ary tree. Fori∈N1, letvbe a vertex such that there is a geodesic from the rootofTntovof lengthi, then we say thatvis at levelior the level ofvisi. We set the level of the root to be 0. For a vertexvofTn, we denote byl(v)the level of the vertexv. Remark 1.2.20. LetTnbe the rootedn-ary tree, as there is a unique geodesic connecting any two vertices it follows that every vertex ofTnhas a unique level. Moreover, fori∈N1, every vertex at leveliis incident to precisely one vertex at leveli−1. Thus, since every vertex apart from the root is incident ton+1 distinct vertices, the number of vertices at leveliofTn, fori∈N, is precisely

ni.

Definition 1.2.21. LetTnbe the rootedn-ary tree with root. Letvbe a vertex ofTnnot equal to the root, then we call the unique neighbour ofvat levell(v) −1, theparent ofv. Letvbe any vertex ofTn, thechildren ofvare thendistinct neighbours ofvat levell(v) +1. Achildofvis therefore a neighbour ofvat levell(v) +1.

Definition 1.2.22. LetTnbe the rootedn-ary tree and letvbe an internal vertex ofTn. Thesubtree

Tn(v)ofTnrooted atv, is the rooted subtree ofTn, with rootv, induced byvand all vertices ofTn which are connected tovby a path which does not visit the parent ofv.

Remark 1.2.23. LetTnbe the rootedn-ary tree and letvbe a vertex ofTn, thenTn(v)is a rooted

(19)

Remark 1.2.25. LetTnbe the rootedn-ary tree, and let{v1,v2, . . .vk}be an antichain. Observe that the subtreesTn(v1), . . . ,Tn(vk)have no vertices in common.

Definition 1.2.26. LetTn= (V,E)be the rootedn-ary tree with root, and letv:={v1,v2, . . .vk} be an antichain. The subtreeTnv is the rooted subtree ofTn, with root, induced by the all the vertices ofTnexcept those which are internal vertices of someTn(vi)for some 16i6k. We may also refer toTv

nasthe complement of the subtreesTn(v1), . . . ,Tn(vk).

Remark 1.2.27. LetTnbe the rootedn-ary tree, and letvbe an antichain ofTn. All elements of vare vertices ofTnv, moreover, ifvis not the antichain consisting only of the root, anyv∈ vis a leaf. All other vertices of the treeTvn, apart from the root which hasnneighbours, haven+1 neighbours. Thus the leaves ofTnv are precisely the vertices inv. Ifvis the antichain consisting only of the root, thenvis the rooted tree with precisely one vertex. Ifvis a complete antichain, thenTnv is a finite rooted tree.

.. . .. . .. . .. . .. . .. . .. . .. .

(a) The binary treeT2.

(b) A finite rooted tree.

Figure 1.1: The binary tree and a finite subtree

Remark 1.2.28. All rooted trees will be drawn with the root at the top, and with the neighbours of a vertexvat levell(v) +1, immediately below the vertexv. Thus, as in the examples in Figure 1.1, we mainly leave the roots of such trees unlabelled.

Definition 1.2.29. LetTnbe the rootedn-ary tree, then afinite (rooted) subtreeofTnis a (rooted) subtreeTnv for some complete antichainvofTn.

Remark 1.2.30. Note that the definition of finite subtrees that we have above is stronger than definitions that occur elsewhere in the literature, since in our definition, all internal vertices of a subtree havenchildren.

Definition 1.2.31. A simple graphG= (V,E)is called aforestif it is a disjoint union of trees. Definition 1.2.32. Anr-rootedn-ary forestTn,r, is a forest which is the disjoint union ofrrooted

n-ary trees.

Remark 1.2.33. There is a uniquer-rooted,n-ary forest up to isomorphism. We shall thus fix a representativeTn,rfor ther-rootedn-ary forest. We refer to this representative forest as ther-rooted

n-ary forestin subsequent discussions.

Definition 1.2.34. LetTn,rbe ther-rootedn-ary forest. Afiniter-rooted subforestofTn,ris the disjoint union ofrfinite subtrees with roots corresponding to the roots of ther n-ary trees making up the forest.

Remark 1.2.35. We partially order the nodes ofTn,ras follows. Leti, 16i6rbe the roots of the rootedn-ary trees making up the forestTn,r. For eachn-ary tree in the forest, we order the nodes of the tree as in Definition 1.2.24. Fori6=j,i,j∈{1, . . . ,r}, set any node connected to the rootito be incomparable to any node connected to the rootj. This gives a partial ordering ofTn,r. We may thus refer to antichains and complete antichains ofTn,r. By definition a complete antichain uofTn,r can be written as a disjoint unionu1tu2t. . .turwhereui, 16i6r, is a complete antichain for then-ary tree with rooti. ThusTui

(20)

i. HenceTui

n t. . .tTnuris finite subforest ofTn,r. Observe that, as in Remark 1.2.27, the leaves oft16i6rTuni are precisely the vertices ofunot equal to a root. Moreover, by Definition 1.2.34, it follows that all finite subforests ofTn,rare obtained in this way. Thus, for a complete antichainu ofTn,r, we writeTun,rfor the finite subforestt16i6rTnui, where theui(16i6r) are disjoint and eachui(16i6r) is a complete antichain for then-ary tree with rooti.

1

.. . .. . .. . .. . .. . .. . .. . .. . .. .

2

.. . .. . .. . .. . .. . .. . .. . .. . .. .

(a) The forestT3,2

1 2

(b) A finite rooted subforest ofT3,2

Figure 1.2: Ther-rootedn-ary forest and a subforest

In the next section we introduce special types of labelled graphs calledtransducersandautomata. These may also be viewed as machines with a set of states which process inputs according to certain rules.

1.3

Automata and Transducers

In this section we describe some of the key machinery that we use to understand the various groups of homeomorphisms that this work considers. The ‘machinery’ referred to are called automata and transducers. We begin by first introducing automata, then we define transducers. We introduce in some sense the basics of these objects, further concepts will be revealed as needed in relevant sections.

1.3.1

Automata

As a transducer is a special type of automaton we shall begin by first defining automata. Definition 1.3.1. Anautomatonis a tripleA:=hX,QA,πAiwhere:

(1) Xis a finite alphabet,

(2) QAis a finite or infinite set of states of the automaton, (3) πA:X×QA→QAis thetransition function.

Remark 1.3.2. Note that finite automata as defined in Definition 1.3.1 are elsewhere in the literature called deterministic finite automata, see for instance [35].

Inductively we may extend the domain ofπAtoX∗according to the following rules:

πA(,q) = qfor allq∈QA, (1.1)

forΓ ∈X∗andx∈Xwe haveπA(Γ x,q) =πA(x,π(Γ,q))for allq∈QA. (1.2) Hence an automaton can be thought of as a machine with a finite set of states which reads letters from an input tape and changes states according to some rules. Notice that by (1.1) and (1.2) above, all states of an automata process words from left to right.

(21)

Usually we represent a finite automatonAby a finite, labelled directed graph as in Figure 1.3. The vertices of the graph correspond to the states of the automaton. There is an edgefroma state

q1toa stateq2 labelled with anx, whenever, in the automataAwe have πA(x,q1) = q2. We shall identify the automataAwith the labelled graph representing it. Thus when we refer to an automaton,A, we shall mean both the labelled graph representing it, and the tuplehX,QA,πAi. Definition 1.3.4. LetA=hX,QA,πAiandB=hX,QB,πbibe automata over the same alphabetX. We say thatAandBareisomorphicif there is a bijectionφ:QA→QBsatisfying, for anyi∈Xn,

πA(i,q) =pif and only ifπB(i,(q)φ) = (p)φ.

Given an automatonA=hX,QA,πAiwe may fix a stateqofAfrom which to begin reading inputs, in this case we say thatAisinitialisedat the stateqand we denote this byAq. In the graph of the automaton, we indicate thatAis initialised atqby double circling the stateq.

Below we have an example of an initial automaton:

q0

q1 q2

0

1 2

1

2

0

2

[image:21.595.215.383.259.394.2]

0 1

Figure 1.3: An example of an initial automaton.

1.3.2

Transducers

Definition 1.3.5. Atransduceris a quintupleT =hXI,XO,QT,πT,λTisuch that: (1) XIandXOare finite alphabets called theinputandoutputalphabets respectively. (2) QT is a set consisting of states of the transducer.

(3) πT :XI×QT →QT is thetransition function. (4) λT :XI×QT →X∗Ois there-writefunction.

A transducer can be thought of as an automaton which, as well as reading inputs from an input tape, may also write strings from the output alphabet onto the output tape.

As in (1.1) and (1.2) we may extend the domain ofπT toX∗I×QT. We may also analogously extend the domain ofλT toX∗I×QT. We do this as follows. First setλT(,q) =andπT(,q) =q for all statesq∈QT. Now forΓ ∈X∗Iandx∈XIwe have:

λT(Γ x,q) =λT(Γ,q)λT(x,πT(Γ,q)). (1.3)

Notice that we process inputs from left to right.

Now letδ∈XωI , and, fori∈N, letδi∈XiIbe the lengthiprefix ofδ. Observe that, for a given stateq∈QT, there is a unique elementρofXωI tX∗Isatisfying the following conditions:

(i) λT(δi,q)6ρfor alli∈N,

(22)

Notation 1.3.6. LetT =hXI,XO,QT,πT,λTibe a transducer. Forδ∈XωI we denote byλT(δ,q) the unique wordρ∈XωI tX∗Isatisfying conditions (i) and (ii).

Given a transducerT =hXI,XO,QT,πT,λTiand an elementγ∈X∗Iwe shall use the language readγthrough a stateq1(to a stateq2) orreadγfrom a stateq1(to a stateq2)to indicate the transition

πT(γ,q1) =q2. We may also append the phrasewith outputξ, for someξ ∈X∗O, if in addition

λT(γ,q1) =ξ.

Notation 1.3.7. Given a transducer T = hXI,XO,QT,πT,λTi, denote byA(T)the automaton

hXI,QT,πTi. We callA(T)theunderlying automatonofT. Given a wordγ∈X∗Iandq∈QT, it will sometimes be convenient to use the notation(γ)Tqfor the wordλT(γ,q). On rare occasions we also extend the notation(γ)Tqto wordsγ∈XωI .

If we fix a stateq∈QT from which we begin processing inputs, then we say thatTis initialised at stateqand we denote this byTq. We callTqaninitial transducer. We say the transducerT is finiteif the underlying automatonA(T)is finite, otherwise we say thatT isinfinite. Thesize of a transducerT is the size of the underlying automatonA(T).

Definition 1.3.8. LetTq0be an initial transducer and letqbe any state ofT not equal toq0. We call

qanon-initial state (ofTq0).

Notation 1.3.9. IfTq0 is an initial transducer, then we shall writeT for the transducerTq0 with no

states initialised. We shall sometimes call the transducerT theunderlying transducer ofTq0. This is

not to be confused with the underlying automatonA(T)ofT.

Notation 1.3.10. For a transducerTwith input and output alphabet both equal to an alphabetX, we writeT =hX,QT,πT,λTi. We call such a transducer atransducer over the (alphabet)X.

Usually we represent a finite transducerT =hXI,XO,QTTTiby a finite labelled graph. The vertices of the graph will correspond to the states ofT. For every state,q∈Q, and for every

x∈XI, there is a directed edge fromqtoπ(x,q)labelled by ‘x|λ(x,q)’. If we fix an initial state

q∈QT, then we represent this in the graph of the transducerT, by doubly circling the stateq. We have the following definitions:

Definition 1.3.11. A transducer,Tis said to besynchronousif for every letterx∈XI, and all states

q∈QT, we have|λT(x,q)|=1. Otherwise we say the transducer isasynchronous.

Definition 1.3.12. LetTbe a transducer, and letq0be a state ofT so thatTq0is an initial transducer.

A stateq∈QT is said to beaccessibleif there is a wordw∈XI∗such thatπT(w,q0) =q, and we say thatqisaccessible (inTq0) byw. Ifw∈X+I then we say thatqis strictly accessible (inTq0 byw). If all the states ofTq0 are (strictly) accessible, we say thatTq0 is (strictly) accessible.

Definition 1.3.13. LetTq0 be an initial transducer. A stateq∈QT is callednon-trivially accessibleif there is a wordw∈X∗Isuch thatqis accessible inTq0 bywandλT(w,q0)6=.

Below are examples of synchronous and asynchronous transducers.

q2 q1

0|1,1|2

2|0

2|0 1|1,0|2

(23)

q1

q2 q3

0|10

1|

0|0 1|11

[image:23.595.215.381.86.203.2]

1|1 0|0

Figure 1.5: Example of an asynchronous transducer with initial stateq3

Eventually we shall see that transducers which satisfy certain non-degeneracy conditions induce continuous functions on Cantor space. To this end we begin by introducing Cantor space: we present several different ways we will be viewing Cantor space in this work.

1.4

Cantor Space from different points of view

In this section we introduce different ways of thinking of Cantor space. Subsequently we shall move between these different points of view as necessary.

From henceforth we shall fixn∈N2andr∈N1a number strictly less thann. We shall also fix

Xn:={0, 1, . . . ,n−1}a finite alphabet, and fix the ordering 0<1<2<. . .< n−1 of elements ofXn. Fix also˙r:={˙1, ˙2, . . . , ˙r}and let the elements of˙rbe ordered ˙1< ˙2<. . .<r˙. We begin by establishing some further notation.

Notation 1.4.1. SetX+n,r :=˙rXn,∗ X∗n,r :=X+n,rt{}andXωn,r :=˙rXnω. Fork ∈NletXkn,rbe the subset ofX∗n,r consisting of all elements of lengthk. Observe thatX1n,r = ˙r. We identify, as in Definition 1.1.9,Xωn,rwith the set of mapsf:N˙rtXnsuch thatf(i)∈ ˙rif and only ifi= 0. Therefore given an infinite wordwinXωn,r, we denote bywithe letterw(i), fori∈Nand we call this letter theithletter ofw.

We extend the partial order6of Definition 1.1.14 to a partial order on the setX∗n,rin the natural way and again denote this partial order by6. We retain the symbol ‘⊥’ for two incomparable words inX∗n,r. More specifically, for two wordsν,η∈X∗n,r, we say thatν6ηifνis a prefix ofη; if

ν66ηandη66νthenνis incomparable toηi.eν⊥η. We also extend the notion of antichains and complete antichains (Definition 1.1.19) to subsets ofX∗n,r.

We now define a metric onXωn,XZnandXωn,rwhich makes each of these sets homeomorphic to Cantor space.

First we define a metricdnonXωn.

Definition 1.4.2. Letdn:Xωn ×Xωn →R0be given by:

dn(w1,w2) =

  

1

k+1,kis minimal such thatw1(k)6=w2(k) 0, otherwise.

(1.4)

The metric d on XZn, which we define below, is a natural extension of dn to bi-infinite sequences.

Definition 1.4.3. Letd:XZn×XZnR0be given by:

d(w1,w2) =

  

1

k+1,k∈Nis minimal such thatw1(k)6=w2(k)orw1(−k)6=w2(−k) 0, otherwise.

(24)

Finally the metric onXωn,ris the natural extension ofdntoXωn,rand we again denote this new metric bydnappealing to the context to clarify any ambiguities that may arise.

Note that each of the spacesXωn,r,Xωn andXZnare homeomorphic to each other. This follows from the well known result that any compact, totally disconnected, perfect, metric space is homeomorphic to Cantor space. Moreover, one can also define homeomorphisms from each of these spaces intoXω2 .

Notation 1.4.4. We also use the symbolCn,rfor the Cantor spaceXωn,r, and the symbolCnfor the Cantor spaceXωn

We will be considering groups of self-homeomorphisms of each of the spacesXωn,r,Xωn,XZn. We begin with the spacesXωn,randXωn. Later on we consider groups of self-homeomorphisms ofXZn. Notation 1.4.5. LetH(Cn,r)andH(Cn)denote the group of self-homeomorphisms ofCn,randCn respectively.

We now describe a basis for each of the topologies induced by the metricsdnonXωn andXωn,r. However, we first establish some further notation.

Notation 1.4.6. Letν∈X∗n, setUν :={νδ |δ ∈Cn}. Forν ∈X+n,r, setUν :={νδ| δ∈ Cn}. For

ν=takeUν:= Cn,r. For a subsetI⊂X∗ntX∗n,rwe shall setU(I) ={Uν |ν∈I}.

Remark 1.4.7. Forν∈X∗norν∈X∗n,r, the subset ofCnorCn,rgiven byUνis a clopen set. That

Uνis open follows since for any pointδ∈Uν, and lettingXbe one ofCnorCn,ras appropriate, the open ballB(δ, 1

|ν|+1) :={ρ∈ X |dn(δ,ρ)61/|ν|+1}is a subset ofUν. ThatUνis closed follows since the complement ofUν is either empty ifν =otherwise it is equal to the union

µ∈X|nν|,µ6=νUµ.

Notation 1.4.8. LetBn:=U(X∗n) ={Uν|ν∈X∗n}and letBn,r:=U(X∗n,r) ={Uν|ν∈X∗n,r}. Remark 1.4.9. The setsBnandBn,rform a basis of clopen sets for the topology induced by the metricdnonCnandCn,rrespectively. This follows since, settingXto be one ofCnorCn,r, for any pointδinXand any numbera∈R1, the open ballB(δ,a)coincides withUνforν∈X∗na prefix ofδof appropriate length. For a pointx∈Cn, we shall use the phraseopen neighbourhood ofx(or neighbourhood ofx)for an open setUcontainingx.

Notation 1.4.10. For a subsetUofCnorCn,rand a continuous functionhwith domainCnorCn,r, we shall use the notation(U)hfor the set{(x)h|x∈U}.

We shall also viewCnandCn,ras the boundaries of the rootedn-ary tree (Definition 1.2.17) and ther-rootedn-ary forest (Definition 1.2.32) respectively. We do this by assigning a label to the edges ofTnandTn,r(Section 1.2) such that the concatenation of labels of infinite geodesics beginning at the root inTnandTn,rcorrespond to elements ofCnandCn,rrespectively. We begin withCn.

Letvbe a node ofTn. Label thenedges leavingvto nodes at levell(v) +1, successively with the symbols 0, 1 . . . ,n−1. This gives a bijection between the set of edges fromvto nodes at level

l(v) +1, and the setXn. Recall thatXnis an ordered set, thus we may assume that the edges of the graph are ordered according to the ordering induced byXn.

Notation 1.4.11. Let{v,u}be an edge ofTn, we denote bylab({v,u})the label of the edge. Given an infinite geodesic,v1,v2, . . . , wherev1 is the root ofTn, we may uniquely identify this geodesic with the elementlab({v1,v2})lab({v2,v3}). . . ∈Cn. Moreover each element ofCn corresponds to a unique infinite geodesic ofTnstarting at the root. Thus elements ofCncorrespond to infinite geodesics inTnbeginning at the root.

The labelling on the edges ofTninduces a labelling on the vertices. This is because ifvis an internal vertex ofTn, we may identifyvwith the wordγv∈Xωn labelling the unique geodesic in

(25)

to the vertex and we identify the root with the empty word. This labelling of the vertices ofTn gives a bijection from the nodes ofTn to the setX∗n. Thus, we may assume that the vertices at leveli,i∈N, are ordered according to the lexicographic ordering induced fromX∗n. Figure 1.6 depicts the labelled rooted binary treeT2, the lexicographic ordering of the edges is indicated by drawing smaller edges to the left of larger ones; likewise the ordering of the vertices at each level is indicated by having smaller vertices appearing to the left of larger ones.

1 11 111 .. . .. . 0 1 110 .. . .. . 0 1 0 1 10 101 .. . .. . 0 1 100 .. . .. . 0 1 0 1 0 1 0 01 011 .. . .. . 0 1 010 .. . .. . 0 1 0 1 00 001 .. . .. . 0 1 000 .. . .. . 0 1 0 1 0 1 0 1

Figure 1.6: Labelled binary tree

Observe that a clopen setUν ∈Bn, forν∈ X∗n, corresponds to the set of infinite geodesics beginning at the root which pass through the vertexνofTn.

Now we extend the labelling ofTnto ther-rooted forestTn,r. Letvbe an internal vertex of one of then-ary trees of the forestTn,r. We label, as in the case ofTn, thenedges fromvto vertices on levell(v) +1, bijectively with the symbols 0, 1, . . . ,n−1. Now letra, for 16a6r, be the root of theathn-ary tree in the forestTn,r. Label all the edges leavingrasuccessively with the symbols ˙aj for 06j6n−1. Thus, we have a bijection from the set of edges leaving the rootrato the set ˙aXn. Therefore, as in the case ofTn, an infinite geodesic beginning at the root of an n-ary tree inTn,r, corresponds uniquely to an element ofCn,r. Moreover, any given element ofCn,r corresponds to a unique geodesic beginning at the root of a particularn-ary tree inTn,r. Thus, whenever we depict ther-rootedn-ary forest, as in Figure 1.7, we shall have: (1.) vertices at leveli+1, fori∈N, appearing below vertices at leveli, (2.) vertices at leveliarranged so that a vertex appears to the left of vertices larger than it in the lexicographic ordering ofX∗n,rand, (3.) edges leaving a vertex arranged so that smaller edges are to the left of larger ones.

˙1 ˙12 .. . .. . .. .

0 1 2

˙11 .. . .. . .. .

0 1 2

˙10 .. . .. . .. .

0 1 2

˙10 ˙11 ˙12

˙2 ˙22 .. . .. . .. .

0 1 2

˙21 .. . .. . .. .

0 1 2

˙20 .. . .. . .. .

0 1 2

˙20 ˙21 ˙22

Figure 1.7: Labelled forestT3,2

This last geometric representation of the spacesCnandCn,r, turns out to be very useful for visualising the action of certain subgroups ofH(Cn)andH(Cn,r)onCnandCn,rrespectively as we will see later on.

(26)

1.5

Transducers and continuous functions on Cantor space

In this section, we demonstrate how finite initial transducers satisfying a certain non-degeneracy condition induce continuous functions of Cantor spaceCn andCn,r. Our exposition follows those given in [30] and the author’s article [10]. We begin first withCn and then make slight modifications forCn,r.

1.5.1

Transducers and continuous functions on

C

n

In this subsection, we demonstrate that finite initial transducers over the alphabetXnsatisfying a certain non-degeneracy condition induce continuous functions onCn. We also identify those homeomorphisms ofCn which may be represented by finite initial transducers satisfying the non-degeneracy condition.

All transducers in this subsection shall be over the alphabetXn.

Definition 1.5.1. LetA=hXn,QA,πA,λAibe a transducer, thenAis callednon-degenerateif there is ak∈N1such that for all wordsΓ ∈Xkn, and for anyqofA, we haveλA(Γ,q)6=.

Remark 1.5.2. Givenδ ∈ Cn, a non-degenerate initial transducer Aq0 over the alphabet Xn

necessarily satisfies:λA(δ,q)∈Cnwhereqis any accessible state ofAq0.

Henceforth, unless stated otherwise, all transducers introduced will be assumed non-degenerate and initial transducers are also assumed to be accessible. Therefore, we shall mostly omit these phrases in subsequent discussions; whenever we include them, we choose to do so for emphasis. Notation 1.5.3. LetAbe a transducer over the alphabetXn, and letqbe a state ofA. We denote by

hAq, the map onCndefined byδ7→λA(δ,q). When it is clear thatqis a state ofAthen we shall

use the symbolhqforhAq.

We have the following result:

Proposition 1.5.4. LetAbe a transducer over the alphabetXnandqbe a state ofA, thenhq: Cn→Cn is continuous.

Proof. Letδ∈Cn. Letρ= (δ)hqand letUbe any open neighbourhood ofρ. Letη∈X+n be such thatUηis an open neighbourhood ofρcontained inU. Letjbe minimal such that for anyΓ ∈Xjn, we have|λA(Γ,q)|>|η|(such ajexists sinceAis non-degenerate). Now let∆∈Xjnbe such that

δ=∆δ¯for some ¯δ∈Cn. Observe thatλA(∆,q)has a prefixηsince∆is a prefix ofδand an element ofXjn, andρ∈Uη. Therefore, for any pointψ∈U∆, we have(ψ)hq∈Uη.

Notation 1.5.5. LetAbe a transducer overCnand letqbe a state ofA. We write im(q)for the image of the maphq: Cn→Cn. We will also refer to im(q)asthe image ofq.

Proposition 1.5.4 demonstrates that an initial non-degenerate transducerAq0 induces several continuous functions onCn. In what follows we shall show how to construct an initial transducer

Aq0 from a homeomorphism h : Cn → Cn such thathq0 = h. We need a few definitions beforehand.

Notation 1.5.6. LetU ⊆ Cn. We set(U)rt ∈ X∗ntXωn to be the greatest common prefix of all elements ofU.

Definition 1.5.7. Leth : Cn → Cn be a continuous function. Defineθh : X∗n → X∗ntXωn by

(ν)θh7→((Uν)h)rt. Given a transducerAq0overCnandqa state ofAthen we will writeθqfor the functionθhq.

Remark 1.5.8. Note that forν∈X∗n, and a continuous functionh: Cn→Cn,(ν)θhis the greatest common prefix of the set(Uν)h. Further observe, sincehis continuous, that for a sequenceνiin

(27)

Remark 1.5.9. Ifhis the continuous map sending all ofCnto a pointx∈Cn, then clearly for any

ν∈X∗n(ν)θh=x. Ifh: Cn →Cnis a homeomorphism, we observe that for any pointν∈X∗n, since(Uν)his clopen (homeomorphisms map clopen sets to clopen sets), then(ν)θh∈X∗n. Definition 1.5.10. Leth: Cn →Cnbe a continuous function and letν∈X∗n. Definehν: Cn→Cn as follows:

(x)hν= (νx)h− (ν)θh.

We callhνalocal map ofh.

Remark 1.5.11. Forν∈X∗nandh: Cn →Cncontinuous, the local maphνis the restriction ofh toUνwhere we delete the prefix(ν)θhfrom all outputs. Observe that ifhis a continuous and injective map fromCnto itself, andν∈X+n, thenhνis also injective by definition.

Proposition 1.5.12. Let h : Cn → Cn be a continuous function, and letν ∈ X∗n, then hν is also continuous.

Proof. Let x ∈ Cn and let Ube an open neighbourhood of y := (x)hν. Let µ ∈ X+n such that y ∈ Uµ ⊆ U. Observe that since (νx)h = (ν)θhy it follows thatU(ν)θhµ is an open

neighbourhood of(νx)h. Sincehis continuous, there is an open neighbourhoodV ofνxsuch that (V)h ⊆ U(ν)θ

hµ. Since V is a neighbourhood of νx, there is some ρ ∈ X

+

n such that

Uνρ⊂Vis an open neighbourhood ofνx. ThereforeUρis an open neighbourhood ofxsuch that

(Uρ)hν⊆Uµ⊆U.

We have the following fact about local maps and the functionθhforha homeomorphism. Proposition 1.5.13. Leth: Cn→Cnbe a homeomorphism and letν,µ∈X∗n. The following holds:

(νµ)θh= (ν)θh(µ)θhν.

Proof. Observe that(ν)θhis a prefix of(Uνµ)h. Moreover,(µ)θhνis the greatest common prefix

of the set(Uνµ)h− (ν)θh. It therefore follows that(ν)θh(µ)θhν= (νµ)θh.

We now give the construction in [30] for building an initial transducer Aq0 from a homeomorphismh∈H(Cn)such thathq0 =h.

Construction 1.5.14. Leth : Cn →Cn be a homeomorphism. Construct an infinite transducer

A=hXn,X∗n,πA,λAi. Forνa state ofAandi∈Xn, the transition and output functions ofA are defined as follows:

πA(i,ν) =νiandλA(i,ν) = (νi)θh− (ν)θh. Observe that forνa state ofAandi∈Xn,λA(i,ν) = (i)θhν.

We have the following result:

Theorem 1.5.15. Leth: Cn→Cnbe a homeomorphism and letAbe the initial transducer constructed fromhas in Construction 1.5.14. For any pointx∈Cnwe have,λA(x,) = (x)h.

Proof. Letx∈Cnand suppose thatx=x0x1x2. . . forxi∈Xn. To demonstrate thatλA(x,) = (x)h, it suffices, by Remarks 1.5.8 and 1.5.9, to show that for every non-empty finite prefixw of x,

λA(w,) = (w)θh. We proceed by induction on|w|.

By definitionλ(x0,) = (x0)θhwhich is a (possibly empty) prefix of(x)h. This proves the base case.

Assume, for m ∈ N1 and for w = x0. . .xm, that λA(w,) = (w)θh. Now consider

λA(wxm+1,). By definition of the transition and output function we have

λA(wxm+1,) =λA(w,)λA(xm+1,w).

Notice thatλA(xm+1,w) = (wxm+1)θh− (w)θh. Therefore, by the inductive assumption, we have:

λA(wxm+1,) = (w)θh((wxm+1h− (w)θh) = (wxm+1h

(28)

Remark 1.5.16. A similar proof to that given above demonstrates that forx∈Cnand forν∈X∗n, we haveλA(x,ν) = (x)hν. A consequence of this is that the states ofA correspond to the local maps ofh. Furthermore,Ais non-degenerate by Remarks 1.5.8 and 1.5.2. In particular Theorem 1.5.15 demonstrates thath=h.

We now modify the Construction 1.5.14 for homeomorphisms ofCn,r. To do this, we will need to adjust our definition of transducers slightly so that they induce continuous functions onCn,r. The approach taken follows that given in the author’s article [10].

1.5.2

Transducers and continuous functions on

C

n,r

In order to allow for transducers to model homeomorphisms of Cantor spaceCn,rfor 1< r < n

we shall have to adjust and impose certain additional restrictions on the transducers.

Definition 1.5.17. Aninitial transducerAq0 = h˙r,Xn,RA,SA,πA,λA,q0ionCn,r is a tuple such that:

(a) the input and output alphabets are equal to the disjoint union˙rtXn, (b) the set of statesQAofAis the disjoint unionRAtSAandq0∈RA,

(c) πA: (˙r×{q0})t(Xn×QA\{q0})→QA\{q0}is the transition function andλA: (˙r×{q0})t

(Xn×QA\{q0})→X∗n,rtX∗nis the output function.

The functionsπAandλAalso satisfy the following restrictions:

(R.1) Whenever we transition from a state inRAto another state inRAwe output the empty word: Ifq1,q2∈RAandπA(x,q1) =q2, thenλA(x,q1) =.

(R.2) Whenever we transition from a state inRAto a state inSAwe output a word inX+n,r: Ifq∈RA, andx∈Xnsuch thatπA(x,q)∈SA, thenλA(x,q)∈X+n,r.

(R.3) We always transition from a state inSAinto another state inSA, and the output is a word in

X∗n:

Ifq∈SA, then∀x∈Xn,λA(x,q)∈X∗n, andπA(x,q)∈SA.

(R.4) Whenever we read a word from a stateq∈QAto the same stateqthe output of this transition is non-empty:

Ifq∈QA, andw∈X+ntX+n,rsuch thatπA(w,q) =qthen,λA(w,q)6=.

Remark 1.5.18. Notice that we can only read a letter from˙rfrom the stateq0. Furthermore, by item (c) of Definition 1.5.17, after processing any element of˙rfromq0, we leave the stateq0and never return to it. We usually writeAq0 =h˙r,Xn,RA,SA,πA,λAiwhen it is clear thatq0is the initial state ofAq0.

We extend the domain ofπAandλAto(˙r×{q0})t(X+n×QA\{q0})by the following rules: for

w∈X+n,i∈Xnandq∈QA\{q0}we have,

πA(wi,q) =πA(i,πA(w,q))andλA(wi,q) =λA(w,q)λA(i,π(w,q)); (1.6) forw∈X+n,r,i∈Xnwe have,

(29)

Notation 1.5.19. For a stateq∈QA\{q0}, and a wordρ∈Xωn, we setλA(ρ,q)as in Notation 1.3.6. Thus, fora∈˙randδ∈Xωn, we setλA(aδ,q0) :=λA(a,q0)λA(δ,πA(a,q0)).

Remark 1.5.20.

(a) Conditions (R.1) to (R.4) imply that whenever we read a wordδ∈Cnthrough a state inSA the output is again an infinite word inCn.

(b) Conditions (R.1) to (R.4) mean that we only ever output a symbol from˙ronce. Conditions (R.1) and (R.4) mean that the transducer must exitRAfor long enough inputs. Conditions (R.2) to (R.4) and the preceding sentence imply that whenever we process an element of

Cn,r through the stateq0of Aq0, the output is also an element ofCn,r. In analogy with Definition 1.5.1, a transducer satisfying conditions (R.1) to (R.3) but which does not satisfy condition (R.4) is called adegeneratetransducer. On the other hand, a transducer satisfying conditions (R.1) to (R.4) is callednon-degenerate.

(c) LetAq0 =h˙r,Xn,RA,SA,πA,λA,q0ibe a non-degenerate initial transducer overCn,rand

letq∈Q\{q0}be an accessible state ofAq0. The initial transducerAqinduces a maphAq

fromCndefined by(δ)hAq=λA(δ,q). The range ofhAqisCnifq∈SAandCn,rotherwise.

The initial transducerAq0 induces a maphAq0 fromCn,rto itself.

Notation 1.5.21. LetAq0 =h˙r,Xn,RA,SA,πA,λA,q0ibe a non-degenerate initial transducer over

Cn,r. For a stateq∈QA, we use the notationhqforhAqwhenever it is unambiguous thatqis a

state ofA.

Remark 1.5.22. LetAq0 =h˙r,Xn,RA,SA,πA,λA,q0ibe a non-degenerate initial transducer over

Cn,r. For any stateqofA, the induced maphqis continuous and the argument demonstrating this is almost identical to the proof of Proposition 1.5.4.

Notation 1.5.23. LetAq0 =h˙r,Xn,RA,SA,πA,λA,q0ibe a non-degenerate initial transducer over

Cn,rand letq∈Q)A. As before (Notation 1.5.5), we use im(q)to denote the image of the maphq. We shall refer to im(q)as the image ofq.

Throughout this work we assume, unless otherwise stated, that all transducers overCn,r

are non-degenerate.

In what follows we outline a procedure given in [10] for constructing a transducerAq0over

Cn,rfrom a homeomorphismhofCn,rsuch thathq0 =h. We first extend the definition of the map

rt of Definition 1.5.6 toCn,rand the definition of the mapθg, forga continuous function ofCn, given in Definition 1.5.7 to homeomorphisms ofCn,rand we do so in the natural way. (It is possible to extendθgto continuous functions onCn,r,, however, as we only focus on homeomorphisms of

Cn,rin this work, we have not done so.) Forg∈H(Cn,r)and a sequenceνi ∈X∗n,r,i∈N, such that|νi|tends to infinity asitends to infinity,θgstill retains the property that|(νi)θg|tends to infinity withialso. ForAq0 a transducer overCn,randqa state ofAq0 we again use the notation

θqforθhq. We now adjust the definition of local maps (Definition 1.5.10) for homeomorphisms of

Cnto homeomorphisms ofCn,r.

Notation 1.5.24. Leth ∈ H(Cn,r)be a homeomorphism. Let Ph ⊂ X∗n,r be the maximal set satisfying the following conditions:

(1) for allν∈Phthere is an ˙a∈˙rsuch that(Uν)h⊆Ua;˙

(2) ifµ is a proper prefix of some element ofPhthen there are distinct ˙a1and ˙a2 such that

(Uµ)h∩Ua˙1 6=∅and(Uµ)h∩Ua˙2 6=∅.

Lemma 1.5.25. Leth∈H(Cn,r)be a homeomorphism. The setPhexists and is a complete antichain for

X∗n,r.

Figure

Figure 1.3: An example of an initial automaton.
Figure 1.5: Example of an asynchronous transducer with initial state q3
Figure 1.11: Reading a word through a state in the product transducer.
Figure 1.12: Figure 1.5 revisited
+7

References

Related documents

Field experiments were conducted at Ebonyi State University Research Farm during 2009 and 2010 farming seasons to evaluate the effect of intercropping maize with

The threshold into the stadium is through a series of layers which delaminate from the geometry of the field to the geometry of the city and creates zones of separation,

modest change detected in our sample is similar to Scott, Minichiello, and Browning (1998) who found that younger adults participating in an older adult education program

• Follow up with your employer each reporting period to ensure your hours are reported on a regular basis?. • Discuss your progress with

Eksperimenti su pokazali da je za mutiranu tRNA Trp , koja nosi supstituciju u D-ruci, smanjena točnost procesa translacije na način da su reakcije koje se odvijaju

Furthermore, while symbolic execution systems often avoid reasoning precisely about symbolic memory accesses (e.g., access- ing a symbolic offset in an array), C OMMUTER ’s test

TPU Educational Standards-2010 develop and supplement FSES requirements with those of international certifying and registering organizations (EMF, APEC Engineer

RoCE (RDMA Over Converged Ethernet) OFA Verbs Interface IB Transport Protocol Network Layer InfiniBand Link Layer InfiniBand OFA* Stack InfinBand Management. RDMA Application