# Diophantine Approximation of Anergodic Birkhoff Sums over Rotations

Paul Verschueren
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: Started: 05/2020; Date: This version: 03/2023)

###### Abstract.

We study Birkhoff sums over rotations (series of the form
$\sum_{r=1}^{N}\phi(r\alpha)$), in which the summed function $\phi$ may be
unbounded at the origin. Estimates of these sums have been of significant
interest and application in pure mathematics since the late 1890s, but in
recent years they have also appeared in numerous areas of applied mathematics,
and have enjoyed significant renewed interest. Functions which have been
intensively studied include the reciprocals of number theoretical functions
such as $\phi(x)=1/\\{x\\},1/\\{\\{x\\}\\},1/\left\|x\right\|$ and
trigonometric functions such as $\phi(x)=\cot\pi x$ or $\left|\csc\pi
x\right|$. Classically the Birkhoff sum of each function has been studied in
relative isolation using function specific tools, and the results have
frequently been restricted to Bachmann-Landau estimates. We introduce here a
more general unified theory which is applicable to all of the above functions.
The theory uses only “elementary” tools (no tools of complex analysis), is
capable of giving “effective” results (explicit bounds), and generally matches
or improves on previously available results.

The author wishes to thank Professor Sebastian van Strien of Imperial College,
without whose encouragement and support this paper would never have seen
publication, and also the College itself for providing access to the
literature.
The author also wishes to thank Ben Mestel of the Open University for
introducing him to the subject, for his infectious enthusiasm, and for many
wonderful discussions.
This work was partly supported by ERC AdG RGDD No 339523

###### Contents

  1. 1 Introduction
    1. 1.1 The context of Diophantine Approximation
    2. 1.2 The context of Dynamical Systems
    3. 1.3 Overview of main results
    4. 1.4 Overview of this paper
  2. 2 Preliminaries
    1. 2.1 Basic Terminology & Notation
      1. 2.1.1 Contextual Annotation
      2. 2.1.2 Contextual Annotation Quickstart
      3. 2.1.3 The abuttal operator (Annotation conventions)
    2. 2.2 Logic and Abstract Algebra
    3. 2.3 Dynamical Systems
    4. 2.4 Geometry of the circle
    5. 2.5 Number Theory
  3. 3 Theory of Hom-set Magmas
    1. 3.1 Introduction
    2. 3.2 Pull Ups of EndoMorphisms
    3. 3.3 Pull Ups of EndoRelations
    4. 3.4 Special Cases and Application
  4. 4 Separation of Concerns in Birkhoff Sums
    1. 4.1 Introduction
    2. 4.2 Formal development
    3. 4.3 Proof by Ansatz
  5. 5 Distribution of orbit segments on the circle
    1. 5.1 QuasiPeriod Decomposition
    2. 5.2 Application to irrational rotations
  6. 6 Analysis of spaces of unbounded observables
    1. 6.1 Definitions and Basic Topology
    2. 6.2 Some initial function classes
    3. 6.3 Classification of primitive functions on an interval
    4. 6.4 Representation of functions by primitives
  7. 7 Homogeneous Birkhoff Sums of unbounded observables
    1. 7.1 Introduction
    2. 7.2 Dual Relations for Birkhoff Functionals
    3. 7.3 Partition Sums
    4. 7.4 Birkhoff sums of monotonic functions on $(0,1)$
    5. 7.5 Further bounds for primitive functions
    6. 7.6 Summary of Results
  8. 8 Application to the function family $\theta^{\beta}(x)=x^{-\beta},\beta\geq 1$
    1. 8.1 Some ancillary lemmas
    2. 8.2 Lower bound for $S_{N}\theta^{\beta}$ using the generalised harmonic function $H^{\beta}$
    3. 8.3 Upper bound for $S_{N}\theta^{\beta}$ using the generalised harmonic function $H^{\beta}$
    4. 8.4 Using Estimates for the harmonic function
    5. 8.5 Summary of upper bound results for $S_{N}\theta^{\beta}$
    6. 8.6 Duality results
  9. 9 Comparisons with Prior Art
    1. 9.1 Introduction
    2. 9.2 Summary of our results for $\theta x=\frac{1}{x}$
    3. 9.3 The sum of reciprocals of fractional parts $\sum_{r=1}^{N}\frac{1}{\\{r\alpha\\}}$
    4. 9.4 The sum of reciprocals of the distance to nearest integer function $\sum_{r=1}^{N}\frac{1}{\left\|r\alpha\right\|}$
    5. 9.5 The sum of the anti-symmetrisation of reciprocals $S_{N}(\theta-\overline{\theta})=\sum_{r=1}^{N}\frac{1}{\\{r\alpha\\}}-\frac{1}{1-\\{r\alpha\\}}$
    6. 9.6 The series $\sum_{r=1}^{N}\frac{1}{r^{\gamma}}\phi(r\alpha)$
    7. 9.7 The double exponential sum $Exp2(N)=\sum_{u=0}^{N-1}\sum_{v=0}^{N-1}e^{2\pi iuv\alpha}$
  10. 10 Conclusion
  11. A Context Annotation

##  1\. Introduction

We study estimates for series of the form $\sum_{r=1}^{N}\phi(r\alpha)$ where
$\phi$ is a real function of period 1, and $\alpha$ is an irrational real
number. Such series can be classified as Birkhoff sums over rotations of the
circle (with initial condition $0$). When $\phi$ is Lebesgue integrable, a
first order estimate is easily available via the powerful theorems of Ergodic
Theory. In particular Birkhoff’s Ergodic Theorem gives us
$\frac{1}{N}\sum_{r=1}^{N}\phi(r\alpha)\rightarrow\int\phi$. Series which
cannot be estimated in this way we call anergodic Birkhoff sums. In this paper
we study the case where $\phi$ is not Lebesgue integrable due to being
unbounded.

Such series have been of great historical interest in pure mathematics (eg
Diophantine Approximation, Discrepancy Theory, q-series) but there has been a
recent resurgence of interest as new applications have emerged in a number of
areas including KAM theory, Quantum Field Theory, Quantum Chaos, Quantum
Computing, and String Theory.

These series seem most naturally situated at the intersection of the
disciplines of Diophantine Approximation and Dynamical Systems, although some
sophisticated techniques of Complex Analysis were also famously deployed by
Hardy and Littlewood in their studies. We will briefly position our objects of
study within the two aforementioned disciplines.

One notable difference between the two disciplines is that studies in
Diophantine Approximation have almost always focused on the sums
$\sum_{r=1}^{N}\phi(r\alpha)$ as the natural objects of study, where $\phi$ is
unbounded at the origin. In Dynamical Systems the natural objects of study are
the more general sums $\sum_{r=1}^{N}\phi(x_{0}+r\alpha)$ where $x_{0}$ is an
’initial condition’. This broadening of perspective proves important.

Whilst the general approach of this paper is designed to be applicable to
quite general sums, space will restrict us to developing detailed estimates
only for a single class of functions $\phi$ and for the homogeneous
$(x_{0}=0)$ case. This case does however cover the results of a surprisingly
high proportion of previous studies. We hope to address inhomogeneous sums and
other classes of functions in later papers.

###  1.1. The context of Diophantine Approximation

Particular examples of series $\sum_{r=1}^{N}\phi(r\alpha)$ have long been
studied within the discipline of Diophantine Approximation, particularly
exploiting the theory of Continued Fractions. These studies are typically
challenging, and individual papers have focused on developing results for a
particular given function $\phi$. Perhaps as a result, techniques have often
been dependent upon identities tied to the particular function being studied.
The major contribution of this paper is an approach which separates the study
of the underlying dynamics (an irrational rotation) from the characteristics
of the function $\phi$. We develop new results about the dynamics, and these
are then immediately available for use with any chosen function $\phi$.

The simplest such series is undoubtedly the “sum of remainders”, namely the
series $\sum_{r=1}^{N}\\{r\alpha\\}$ (ie $\phi(x)=\\{x\\}$) in which $\\{x\\}$
is the fractional part of $x$ (or remainder modulo 1). This case is ergodic
and a simple application of ergodic theory immediately gives
$\frac{1}{N}\sum_{r=1}^{N}\\{r\alpha\\}\rightarrow\frac{1}{2}$. However the
rate of convergence is of great interest, and was closely studied by many
illustrious mathematicians in the early 20th century, including Lerch[10],
Sierpinsky[13, 14], Ostrowski[12], Hardy & Littlewood[4, 5], Behnke[1],
Hecke[7] who all contributed insights and techniques. Even after such intense
study, Lang still saw value in developing a new approach as late as the 1960’s
(see [8]), and there have been more papers on the topic in more recent years.

Of all the tools and techniques introduced in these papers, we will make the
greatest use of Ostrowski’s representation of integers developed in [12]
above. The way in which Ostrowski used this technique within his paper is not
easily generalisable, and the technique seems to have been forgotten for many
years (in fact it was independently rediscovered in 1952 by Lekkerkerker [9]
and in 1972 by Zeckendorf[20], but then only for the case of Fibonacci
numbers). However in recent years it has been recognised as a powerful
technique quite independent of its original “sum of remainders” context.

Although the approach in our paper is applicable to the “sum of remainders”
problem, that is not our focus. Rather we are concerned with series in which
$\phi$ is not integrable due to unbounded singularities. In such cases the
Ergodic Theorems do not apply, and and other techniques are necessary to
estimate the sums. A remarkable range of techniques have been applied. Hardy &
Littlewood studied $\phi(x)=\csc\pi x$. using a number of advanced techniques
including double zeta functions, functional equations, and Cesaro means. Lang
[8]studied $\phi(x)=1/\\{x\\}$ and $\left|\csc\pi x\right|$ using an elegant
and elementary recursive technique he developed for the purpose. Sudler[16]
and Wright[19] studied $\phi(x)=\log|\sin\pi x|$ giving perhaps the first
explicit bounds for such a sum. This sum occurs in a remarkable range of
application areas and guises (see the Bibliographies of [11, 18] for a list of
30 papers on the topic prior to 2016). An open question of Erdos-Szekeres[3]
from 1959 was settled positively in 1998 for almost all rotation numbers
$\alpha$ by Lubinsky[11] who felt certain it would prove to apply to all
$\alpha$. It was finally settled negatively in 2016 by Verschueren[17], but
using techniques rather different from those in this paper. Recently (2020)
Beresnevich, Haynes and Velani[2] studied $\phi(x)=1/\left\|x\right\|$ and
explored connections between this particular Birkhoff sum and recent advances
in the long standing Littlewood conjecture, using both elementary techniques
and Minkowski’s Theorem. Sinai & Ulcigrai[15] (2009) studied the case
$\phi(x)=\cot\pi x$ arising out of a problem in Quantum Computing, and used
the “Cut and Stack” technique developed within the Dynamical Systems
discipline.

In this paper we develop instead a single unified approach with which to
tackle these problems. We also show how we can apply it with relatively little
effort to a particular class ($\sum_{r=1}^{N}\phi(r\alpha)$ where $\phi$ is
has a single unbounded point at the origin), where it gives results equivalent
to, or improving upon, those currently existing in the literature.

###  1.2. The context of Dynamical Systems

In the Dynamical Systems context, our series $\sum_{r=1}^{N}\phi(r\alpha)$ is
an example of an additive co-cycle (or equivalently a skew product) and more
specifically, a Birkhoff sum. The base phase space is the circle, and the
space morphism (or evolution function) is an irrational rotation. Birkhoff
sums occur widely in the modelling of physical systems, representing sums of
an observable along an orbit of the phase space. Although we are restricting
here our study of Birkhoff sums to one of the simplest of Dynamical Systems
(the irrational rotation), we recall that many physical systems can be studied
via, or even reduced to, rotations of the circle.

Key tools of study in this area are normally Ergodic theorems (eg Birkhoff,
von Neumann, Oseledets) which establish that averages of the Birkhoff sums
converge (ie the sequence111or the multiplicative analogue
$\left(\prod_{r=1}^{N}\phi(r\alpha)\right)^{1/N}$
$\frac{1}{N}\sum_{r=1}^{N}\phi(r\alpha)$ has a limit) almost everywhere under
suitable constraints (primarily that $\phi$ be $L^{1}-$integrable). We will
call the Birkhoff sum anergodic when ergodic theory cannot be applied.

Such cases already occur in Quantum Field Theory and String Theory (where
tools of renormalisation theory replace those of ergodic theory) and where the
techniques of this paper (seen as a form of renormalisation) may provide the
beginnings of a further alternative approach. Sinai & Ulcigrai report the
occurrence of the case $\phi(x)=\cot\pi x$ in a problem of quantum computing.
They developed their own approach to the problem, but the approach developed
in this paper is also applicable to the problem and provides an alternative
approach (see the application in 9.7).

More recently, the advent of “Big Data” has refocused attention on Koopman
operators and their adjoints, Perron-Frobenius operators, as these seem better
suited to data-driven (rather than equation-driven) analysis, and high
dimensional systems. The original theory was developed in the context of
ergodic theory, and is limited to Banach spaces of observables (usually
Hilbert spaces). Birkhoff sums are an important type of Koopman operator, and
our study of anergodic sums extends the study of Koopman operators into non-
Banach spaces of observables with singularities. It seems entirely possible
that Big Data problems will at some point need such extensions. We therefore
expend some effort in situating our theory within the context of operator
theory (see particularly Section 3).

###  1.3. Overview of main results

  1. (1)

We develop general upper and lower bounds for the Birkhoff sum
$S_{N}\phi=\sum_{r=1}^{N}\phi(r\alpha)$ where $\phi$ is a monotonic period-1
function which may be unbounded at the origin. These bounds are independent of
any other characteristics of $\phi$, and hence widely applicable.

  2. (2)

These bounds are easily extended to the related Birkhoff sums $S_{N}\theta$ in
the cases $\theta(x)=\phi(1-x)$, $\theta(x)=\phi(x)+\phi(1-x)$, and
$\theta(x)=\phi(x)-\phi(1-x)$.

  3. (3)

We compute specific bounds for the particular family $\phi(x)=x^{-\beta}$ for
$\beta\geq 1$. (The techniques can also be used for $\beta<1$ but this case is
$L^{1}$-integrable and is already covered by existing methods.) We show that
that in most cases this leads relatively quickly to results which match or
improve upon results in the literature.

  4. (4)

The underlying theory includes a new analysis of the distribution of the
sequence of fractional parts $\\{r\alpha\\}$ across $[0,1)$ using an extension
of Ostrowski’s representation of integers which seems of interest in its own
right.

  5. (5)

###  1.4. Overview of this paper

The aim of this paper is to develop a general theory of Birkhoff sums of the
form $\sum_{1}^{N}\phi(r\alpha)$ where $\alpha$ is a real irrational and
$\phi$ is a period 1 real function which may be unbounded at the origin. As we
shall use a number of Dynamical Systems concepts in the paper, we note that
the Birkhoff sum is equivalent to an additive co-cycle or skew-product
$\sum_{1}^{N}\phi(R_{\alpha}^{r}0)$ where $R_{\alpha}$ is an irrational
rotation of the circle, $0$ is the initial condition, and $\phi$ is a
observable on the circle which may be unbounded at the initial condition.

In section 2 we introduce a collection of concepts and results which we use in
the rest of the paper. The paper is slightly unusual in that we do not require
any advanced results from any particular mathematical discipline, but we will
however use basic tools and results from a range of disciplines. This gives us
two problems. The first is that finding a level of introduction which will
suit all readers is an impossible task, and we apologise to every reader in
advance for both insulting her intelligence in some sections, and for assuming
too much for him in others. The second is that we have encountered numerous
notation collisions between different disciplines. Rather than embark on the
Sisyphian task of unifying mathematical notation, we have reused the concept
of namespaces from Computer Science, albeit with an informal implementation
which we feel is more suited to mathematical writing.

In section 3 we begin developing a theory of hom-set magmas which provides the
algebraic context for our later sections. This gives us a set of dualities
which save us much duplication of effort later, and also provides a structure
within which to make sense of later sections, particularly in Section 7.

In section 4 we introduce and develop the idea of separation of concerns, a
concept borrowed from the world of software engineering. The sum
$\sum_{1}^{n}\phi(r\alpha)$ mixes values of the observable $\phi$ with the
dynamics of an irrational rotation. Previous approaches to these sums have
developed identities concerning the observable $\phi$ and developed specific
techniques for summing these particular identities. In software engineering
terms the concerns of summation and dynamics are tightly bound. The result is
that each observable with distinct characteristics tends to result in a
different approach. In the approach of this paper, we will consciously aim to
separate the two concerns. This allows us to study the dynamics once only,
independent of any particular observable. The summation process then becomes a
separate concern, and its study is simplified by being unbound from the
dynamics.

The way we achieve this is to use an approach from Category Theory/Operator
Theory to recast the normal Birkhoff sum $\sum_{1}^{n}\phi(r\alpha)$ into the
sum $\phi^{*}\sum_{1}^{*n}(R_{\alpha}^{r}0)$ where $\phi^{*},\sum^{*}$ are now
operators on the space of orbits under the irrational rotation $R_{\alpha}$.
This achieves the desired separation albeit at the cost of a more abstract
approach. We are now left with studying the distribution of the orbit
$\sum_{1}^{*n}(R_{\alpha}^{r}0)$ (independently of $\phi$), and then studying
the problem of applying operators $\phi^{*}$ to such sequences. Like with many
things in basic Category Theory, the approach becomes almost trivial once the
concepts are understood. The value of Category Theory lies in encouraging
thinking which leads to a fruitful recasting.

In section 5 we study our first concern which is the distribution of the orbit
$\sum_{1}^{n}(R_{\alpha}^{r}0)$. The equi-distribution of the points of such
an orbit regarded as a _set_ , is well understood and much studied. However
our concern is rather to study the distribution as a _sequence_ , ie it is to
develop an estimate of the location of the point $r\alpha$ for each $r$. It
turns out we can do this remarkably well using the number theoretic tool of
Ostrowski representation, but lifting this tool to the level of an operator
acting on “quasiperiod” orbit segments - namely the segments of an orbit which
lie between closest returns. The error terms in these positional estimates are
well controlled and effectively obey a renormalisation law.

In section 6 we develop some basic theory on unbounded functions and their
spaces, as this does not appear to be a well studied area.

In section 7 we move to our second concern of applying the derived operator
$\phi^{*}$ to the orbit, and this is now largely the problem of applying
$\phi^{*}$ to the quasiperiod segments identified in Section 5. This results
in upper and lower bounds for the Birkhoff sum $\sum_{1}^{n}\phi(r\alpha)$
which of course involve $\phi$, but which are perfectly general, ie they are
independent of the particular characteristics of the specific observable
$\phi$. The approach also provides a way of identifying important equivalence
classes of observables which share the same higher order growth estimates.
This completes our development of the general theory.

In section 8 we make an application of the general theory to the particular
family of functions $\phi(x)=x^{-\beta}$ for real $\beta\geq 1$. The general
theory can of course be applied to the family $\beta<1$, but as $x^{-\beta}$
is $L^{1}-$integrable on the circle, these Birkhoff sums can be estimated via
existing methods and we shall ignore them here. For $\beta\geq 1$ the fact
that we now have a general theory enables us to develop estimates using a
variety of estimating techniques developed independently in the literature.
The theory enables us to develop the estimates relatively rapidly compared
with earlier papers, perhaps with greater clarity. In addition, as they are
developed within a unifying framework, the the relative merits of the
different techniques can be assessed, something which was not easy to do
previously.

In section 9, we compare the estimates from the general theory to the results
available in the literature. With a few special case exceptions, we show that
the results from the general theory are equivalent to, or improve upon those
in the literature. (It is possible that the exceptions could be improved on
with more careful application of the theory, as we have made no attempt to
achieve best possible results at this stage).

##  2\. Preliminaries

This paper requires little in the way of pre-requisites from any one
mathematical discipline, but we do draw some basic ideas and results from a
range of disciplines. Some of these disciplines have several notations in
common use and in addition there are some notation collisions between
disciplines. In this section, we standardise our notation, summarise the
necessary disciplinary background, and develop from these some simple theory
and results for use in the following sections.

###  2.1. Basic Terminology & Notation

We will adopt the following notation for logical operations: $\&$ (AND), $|$
(OR), $!$ (NOT), $\coloneqq$ (Definition), $\left\llbracket.\right\rrbracket$
(Iverson bracket - see 2).

We will use symbols from the $\mathbb{B}$lackboard font for our main spaces of
interest: we assign $\mathbb{Z},\mathbb{\mathbb{\mathbb{R}}},\mathbb{T}$ their
usual meanings as the ring of integers, the topological field of real numbers
with Lebesgue measure, and the circle (as a Lie Group/real manifold - see 2.4
below). We assign $\mathbb{N}$ the “semi-standard” meaning as an additive
group (and so $0\in\mathbb{N}$ and we define
$\mathbb{N}^{+}\coloneqq\mathbb{N}\backslash\\{0\\}$). Finally we define two
real semi-open intervals $\mathbb{I}\coloneqq[0,1)$ and
$\mathbb{I}^{/}\coloneqq\mathbb{I}-\frac{1}{2}=[-\frac{1}{2},\frac{1}{2})$
which together form the atlas (the covering of coordinate spaces) for our
circle manifold.

We define the set
$\\{x_{r}\\}_{r=1}^{N}\coloneqq\\{x_{1},x_{2},\ldots,x_{N}\\}$ with the
analogous notation $(x_{r})_{r=1}^{N}$ for sequences.

####  2.1.1. Contextual Annotation

Motivation In borrowing from a number of disciplines in this paper, we quickly
hit a problem with homonyms - symbols which have different meanings in
different disciplines. The problem is one of disambiguation - how do we ensure
that the reader understands which meaning is signified by a homonym at any
particular occurrence.

In mathematics we have historically aimed to solve this problem by eliminating
it, stressing the importance of unambiguous notation. In theory we can simply
define sufficient new symbols. However we found two obstacles to this approach
in this paper. First, we are using very basic concepts with very familiar
notation, and this means that introducing new notation may be unhelpful. For
example the notation $\\{x\\}$ signifies different meanings in set theory and
number theory, but it seems undesirable to replace it with new notation in
either discipline. Second, although the supply of new symbols is theoretically
limitless, modern mathematics is in reality constrained by the supply of
defined Unicode symbols. The latter is very finite and can be exhausted
surprisingly quickly.

Both of these problems were encountered in the process of writing down the
theory presented in this paper. This leads us to considering ambiguous
notations, and in practice this proves to be much less of a problem than it
sounds.

Firstly, the meaning of any particular homonym is often easily derived from
the immediate context, and then the danger of ambiguity is theoretical rather
than real. In fact we frequently recognise this in mathematics by saying we
are “abusing notation”. The phrase suggests that we know we are doing
something wrong or lazy, but perhaps feel we can “get away with it”. However
we now argue that, provided that the context does remove ambiguity, we are not
“abusing” so much as “reusing” notation, and this is an effective and laudable
approach to managing a finite resource. As an example, consider the trivial
function $f:\mathbb{N}\rightarrow\mathbb{N}$ defined by $n\mapsto n+1$. We now
extend this to a function $f:\mathbb{C}\rightarrow\mathbb{C}$ defined by
$z\mapsto z+1$. We refer to this as “abuse of notation” since we are using the
symbol $f$ to signify two different functions, but it would serve no good
cause to use different symbol for the two functions, and indeed doing so would
lose sight of the deep connection between the functions.

We will therefore feel free to subject our notation to abusive practices,
subject to an additional authorial responsibility to ensure that ambiguity is
resolved by the context. The reader must judge how well we succeed.

Secondly, the normal way of resolving ambiguity in formulae or equations is to
add explanatory detail to the surrounding text. However this can become
clumsy, and the relevant text can seem somewhat dissociated from the symbols
themselves. We borrow from Computer Science the idea of “inlining” context by
the direct annotation of ambiguous tokens (such as variable names) with
context (“namespace”) tokens which determine the correct disambiguation. In
Computer Science the goal is to provide a formal disambiguation syntax
suitable for processing by machine, and existing implementations we have
considered are probably too heavy handed to suit mathematical writing. We have
instead adopted a lighter and more mathematical notational approach with the
more modest goal of providing sufficient disambiguation to suit an intelligent
mathematical reader, rather than an efficient algorithm. Like any new notation
there is a learning hurdle, but we feel the benefits for this paper make it
worthwhile. A quick-start summary follows here, but see Appendix A for a more
formal explanation.

####  2.1.2. Contextual Annotation Quickstart

Embracing the positive reuse of notation as described above, enables us to
write $X\coloneqq(X,+)$ in order to define the group $X$ with underlying set
$X$ and operator ${}^{\prime}+^{\prime}$. We call ${}^{\prime}X^{\prime}$ here
a homonym because we have given two different mathematical objects (the set
and the group) the same name $X$. Depending upon the document we are writing,
we might judge the intended meaning to be clear from surrounding text. If
there is need for disambiguation, we can make the meaning of each occurrence
of the homonym $X$ explicit by annotating each symbol with a suitable
contextual label. For example we could write $X_{Grp}\coloneqq(X_{Set},+)$. In
a more formal setting we might decide to be even more explicit with something
like $X_{\epsilon Grp}\coloneqq(X_{::Set},+_{AssocOp})$. Whether we needed to
define the context labels $Grp,Set,AssocOp$ would again depend on authorial
judgment. In these examples, the label represents a category or class of
mathematical objects but it is useful not to restrict ourselves to this. Also
given subscripted $X_{i}$ we could write $(X_{i})_{Grp}$. Use $:$ or
$\epsilon$ only when appropriate.

######  Example 1.

Classically in mathematics, the type of a variable is given in accompanying
text, visually separated from the reasoning which follows. The annotation
approach complements the classical approach and the two can be usefully
intermixed. In some situations annotation may be distracting, cluttering an
equation or formula, and in this case the classical approach will be best. In
other situations it may enhance the flow of information by concisely inlining
all necessary context. For example we might write
$z_{\mathbb{C}}=x_{\mathbb{R}}+iy_{\mathbb{R}}$, and this may be either highly
useful or highly distracting depending upon the circumstances.

####  2.1.3. The abuttal operator (Annotation conventions)

Rather than writing $Y^{X}$ we will use $XY$ as a concise notation for the
morphisms from the object $X$ to the object $Y$. In addition to providing more
natural directionality, this notation is particularly convenient for use as an
annotation. For example, writing $\phi_{XY}(a)=b$ tell us that $\phi$ is a
morphism from $X$ to $Y$, and hence we can also deduce that $a\in X,b\in Y$:
it would be redundant to write $\phi_{XY}(a_{X})=b_{Y}$, although it is
permitted. Note that writing $\phi_{XY}$ means the same as $\phi\in XY$ and
$\phi:X\rightarrow Y$, and sometimes it also makes sense to say simply “$\phi$
is $XY$”. If $X=Y$ it often makes sense to abbreviate $\phi_{XX}$ to simply
$\phi_{X}$ but then it needs to be clear from other context that $\phi$ is a
function on $X$, and not an element of $X$. This is an example of why we
regard $X$ here a contextual label rather than the category or class $X$.

We also borrow from mathematical logic the convention of using abuttal to
represent a suitable distinguished operator in a particular context. For
example in the context of an algebraic ring we retain the usual convention
$xy\coloneqq x\times y$, ie the abuttal operator represents the multiplication
operator. However in the context of the function space $XY$, we define abuttal
as evaluation, ie $\phi_{XY}x\coloneqq\phi(x)$. Given also $\theta_{YZ}$ we
will define abuttal to be functional composition, ie
$(\theta\phi)_{XZ}x\coloneqq\theta_{YZ}(\phi_{XY}x)$.

###  2.2. Logic and Abstract Algebra

Functions of Propositions

######  Definition 2.

Let $\mathscr{P}$ be a Boolean algebra of logical propositions, $\mathscr{B}$
the Boolean integers $\\{0,1\\}$, and $\mathscr{B}^{\prime}$ the alternate
Boolean integers $\\{-1,1\\}$. The Iverson bracket
$\left\llbracket.\right\rrbracket_{\mathscr{PB}}$ is a function which maps
logical propositions to $\\{0,1\\}$ as follows: for any proposition $P$ we
define $\left\llbracket P\right\rrbracket:=1$ if $P$ is true, and $0$ if $P$
is false. The alternate Iverson bracket
$\left\llbracket.\right\rrbracket_{\mathscr{PB}^{\prime}}^{\prime}$ is
analogous: we define $\left\llbracket P\right\rrbracket:=1$ if $P$ is true,
and $-1$ if $P$ is false.

Note the useful identities
$\left\llbracket!P\right\rrbracket=1-\left\llbracket P\right\rrbracket$, and
that $\left\llbracket
P\right\rrbracket^{\prime}=(-1)^{\left\llbracket!P\right\rrbracket}=\left\llbracket
P\right\rrbracket-\left\llbracket!P\right\rrbracket$. These functions allows
us to define discontinuous functions very efficiently: for example
$f_{\mathbb{R}\mathbb{R}}(x)\coloneqq\left\llbracket x>0\right\rrbracket x$
gives us the Heaviside function (the real function which takes the value $x$
for $x>0$, and $0$ otherwise.

We will often need to distinguish odd and even cases, and it useful to define
a set of constants for this purpose.

######  Definition 3.

For $r\in\mathbb{Z}$ we define $E_{r}\coloneqq\left\llbracket
r\,\textrm{is\,even}\right\rrbracket,O_{r}\coloneqq\left\llbracket
r\,\textrm{is\,odd}\right\rrbracket$

Note the useful identities $E_{r}+O_{r}=1$ (so $x_{r}=E_{r}x_{r}+O_{r}x_{r}$),
and $E_{r}-O_{r}=(-1)^{r}$.

Abstract Algebra Basic terminology We recall some basic algebraic concepts and
terminology, we fix precisely how we will use them here, and we establish
notation.

Set:

We will not be concerned with foundational issues here, so we content
ourselves with a naive approach and in particular we allow the definition of
sets by use of the naive notation $X\coloneqq\\{x:Px\\}$ where $P$ is a
logical proposition.

Function:

A function $\phi_{XY}$ here always means a single valued relation on the set
$X\times Y$. The annotation $XY$ represents the set of morphisms from $X$ to
$Y$, which in the case of sets is precisely the set of all possible functions
from $X$ to $Y$. The latter is often denoted $Y^{X}$, but we prefer the more
natural direction implied by $XY$. When it seems more appropriate, we will
also use the traditional notation $\phi:X\rightarrow Y$.
A function is defined explicitly using the notation $\phi_{XY}\coloneqq
x_{X}\mapsto y_{Y}$, which also allows the definition of anonymous functions,
eg $x_{X}\mapsto y_{Y}$.
We will also use placeholder notation to allow for function notation involving
decorations (eg $\bar{.}$) or braces (eg $\\{.\\}$ ). Here the character
${}^{\prime}.^{\prime}$ is the placeholder.
We define the $n_{\mathbb{N}}$-ary extension of $\phi_{XY}$ to be
$\phi_{X^{n}Y^{n}}:(x_{i})_{i=1}^{n}\mapsto(\phi x_{i})_{i=1}^{n}$ and the
skew extension $\phi_{X^{n}Y^{n}}^{Op}:(x_{i})\mapsto(\phi x_{i})^{Op}$ where
$\left(.^{Op}\right)_{X^{n}X^{n}}$ is the function which reverses sequences in
$X^{n}$. (This is of course a specialisation of the concept of the $Op$
functor from Category Theory but we wish to avoid deploying the full
generality here. We retain the “skew” terminology as it is more familiar in
the contexts we are using).
If $X=Y$ then $\phi$ is a set endomorphism, and any $x_{X}$ satisfying $\phi
x=x$ is called a fixed point of $\phi$.

Algebraic Operation:

A function $\circ_{X^{n}X}$ is called an $n_{\mathbb{N}}$-ary algebraic
operation on a set $X$, abbreviated to simply an operation on $X$ when $n$ is
unimportant.
A unary operation is also a set endomorphism, which will often be denoted by
the decoration of a symbol, eg $\overline{.}$ denotes the function
$x\mapsto\overline{x}$.
Binary operations have the formal notation $\circ(x_{1},x_{2})$ and this is
occasionally helpful, but we shall of course normally use the more common
notation $x_{1}\circ x_{2}$. Often it is useful to use abuttal (eg $ab$) to
indicate the presence of a well known binary operation - when needed we will
use ′′ to indicate the abuttal operator, eg
${}^{\prime\prime}\coloneqq(a,b)\mapsto a\times b$ means $ab$ is to be
interpreted as $a\times b$.
$n$-ary operations on sequences $(x_{i})_{X^{n}}$ are often constructed by
extending unary or binary operations, eg by
$\overline{(x_{i})}\coloneqq(\overline{x_{i}})$ or
$\circ(x_{i})\coloneqq$$\circ(x_{1},\circ(x_{2},\ldots\circ(x_{n-1},x_{n})\ldots))$
and we will take these these extensions as given unless otherwise specified.
If we also omit specification of $n$ then we intend that the extension may be
made to any $n\geq 1$ (unary) or $n\geq 2$ (binary case). Examples: the unary
operation $-:x\mapsto-x$ gives $-(x_{1},\ldots x_{n})=(-x_{1},\ldots,-x_{n})$,
a commutative binary operation $+$ gives
$,+(x_{1},\ldots,x_{n})=\sum_{1}^{n}x_{i}$ whereas the non-commutative binary
skew operation $\circ^{Op}:(x_{1},x_{2})\mapsto x_{2}\circ x_{1}$ gives
$\circ^{Op}(x_{1},\ldots,x_{n})=\circ(x_{n},\ldots,x_{1})$. (Borrowed from
Universal Algebra).

Algebraic Structures:

An algebraic structure $X_{Struc}$ is a an ordered pair of sets
$(X,\\{\circ_{\alpha}\\}_{\alpha\in A})$ where $X_{Set}$ is called the
underlying set, and $\\{\circ_{\alpha}\\}$is a set of operations on $X$
indexed by a set $A$. We will also call $X$ a $\left|A\right|-Struc$. In the
case of a singleton set $A$ (ie $X$ is a $1-Struc$), we simplify the notation
to $(X,\circ)$ rather than $(X,\\{\circ\\})$. For example, a bare set has
$A=\emptyset$ (meaning there is no additional structure), a bare monoid or
group has $A$ a singleton, a bare ring or field has $|A|=2$. For $|A|\geq 2$
there will usually be relations defining interaction between operators.

Generated Structures:

Let $(U,\circ)$ be a structure with $\circ$ a binary operation. If $U$ has a
unit it is unital. Note that if $U$ is not unital we can always extend it with
a unit, and there can be only one unit for $\circ$. Let $X\subseteq U$ and $X$
is closed under $\circ$, $X$ is a substructure of $U$. The smallest
substructure containing $X$ (which could be $U$ or $X$ or something in
between) is the closure of $X$, designated $<X,\circ>$ (and also called the
structure generated by $X$).
If $\circ$ is not associative (following Bourbaki ##paper) the structure is a
magma and its elements are represented by binary trees with leaves in $X$, eg
$w_{1}\circ w_{2}$ can be regarded as a tree rooted at $\circ$ with
$w_{1},w_{2}$ as left and right sub-trees (which may be simply leaves). When
$\circ$ is associative, the need for brackets disappears, the structure
becomes a semi-group, and elements such as $u_{1}\circ u_{2}\circ u_{3}$ can
be represented as strings $u_{1}u_{2}u_{3}$ or sequences
$(u_{1},u_{2},u_{3})$. If one of these structures lacks a unit, we can always
extend it with a unique unit, but this is more important in the case of a
semi-group - in this case the structure becomes a monoid, and the unit can be
represented as an empty string or sequence, typically designated $\epsilon$.
The structure is called free if there are no relations between elements, ie if
$w_{1}=w_{2}$ means that any representations of $w_{1},w_{2}$ (trees or
strings) are identical. Note that even if $(U,\circ)$ is not free, we can
always introduce a second operator $\circ_{2}$, forget $\circ_{1}$, and
consider $U_{Set}$ as a set of generators for a free magma, semi-group or
monoid $(U^{*},\circ_{2})$. In this case elements of $U_{Set}$ are called the
base elements.
We shall be particularly interested in using free monoids to represent orbits
in a Dynamical System, and in magmas of hom-sets to develop our theory of
induced morphisms (see 3).
We can regard a free magma as a naturally graded structure (ie there is a
function $U^{*}\rightarrow\mathbb{N}_{0}$) in two ways. The first, introduced
by Bourbaki is effectively the width of a term binary tree, measured as the
width of the base of the tree, ie the number of leaf elements in the tree.
This grading also extends to associative free magmas where it reduces to being
simply the length of a string). We introduce a second grading for non-
associative magmas in 3, which we will call the height of the tree, which is
more useful for our purposes. We define base elements to be trees of height
$0$, and inductively a tree of height $n$ is one which has one or both sub-
trees of height $n-1$. Note this means $a\circ b$ has a height of $1$,
$(a\circ b)\circ(c\circ d)$ has a height of $2$, whereas
$(a\circ(b\circ(c\circ d)))$ has a height of 3.

Homomorphisms/Morphisms:

A structure homomorphism $\phi_{XY}$ between two algebraic structures
$X_{Sruc}=(X_{Set},\\{\circ_{\alpha}\\}),Y_{Struc}=(Y_{Set},\\{\circ_{\beta}\\})$
is a function $\phi_{X_{Set}Y_{Set}}$ which preserves (is compatible
with/distributes over) the algebraic structure. This means that for each
$n$-ary ($n>0$) algebraic operation $\circ_{\alpha}$ on $X$ there is a $n$-ary
$\circ_{\beta}$ on $Y$ such that
$\phi\left(\circ_{\alpha}(x_{i})_{n-Seq}\right)=\circ_{\beta}\left(\phi
x_{i}\right)_{n-Seq}$ for all sequences $(x_{i})_{X^{n}}$. It is useful to
define for each $n$ the Cartesian product function
$\phi_{X^{n}Y^{n}}=\prod_{i=1}^{n}\phi_{XY}\coloneqq(x_{i})_{X^{n}}\mapsto(\phi_{XY}x_{i})_{Y^{n}}$,
which gives us
$\phi_{XY}\circ\circ_{\alpha}=\circ_{\beta}\circ\phi_{X^{n}Y^{n}}$. We can
also define the involution
$Op_{X^{n}}\coloneqq(x_{i})_{i=1}^{n}\mapsto(x_{i})_{i=n}^{1}$, which pulls up
to an involution on $X^{n}Y^{n}$ defined by
$Op(\phi_{X^{n}Y^{n}})=\phi_{X^{n}Y^{n}}^{Op}\coloneqq\phi_{X^{n}Y^{n}}\circ
Op_{X^{n}}$. We call $\phi^{Op}$ an anti-structure homomorphism as it reverses
sequences. For example, given groups $(X,\circ_{X}),(Y,\circ_{Y})$, then
$\phi_{XY}$ is an anti-group homomorphism iff $\phi(x_{1}\circ_{X}x_{2})=(\phi
x_{2})\circ_{Y}(\phi x_{1})$.
Note the standard additional terminology for specialised homomorphisms: if the
function $\phi_{X_{Set}Y_{Set}}$ is a bijection then $\phi$ is an isomorphism,
and if $X=Y$ then $\phi_{XX}$ is an endomorphism, and if both conditions hold
then $\phi_{XX}$ is an automorphism. If $\phi_{XX}x=x$, $x$ is a fixed point
of the endomorphism $\phi$.

hom-sets:

Note that $XY$ represents the hom-set (the set of homomorphisms or morphisms)
from $X$ to $Y$. and that the endomorphism hom-set $XX$ of endomorphisms on
$X$ is a subset of the hom-set of unary operators on $X$. We therefore need to
be careful $XY$ can often be equipped with algebraic operations defined by the
operations of the underlying algebraic structures $X$ and $Y$. We call these
new operations derived/induced and $XY$ is then a (second order) algebraic
structure in its own right. We will make significant use of this type of
construction - see section (3) below.
The abuttal map $":=(X,Y)\mapsto XY$ is itself a binary operation on sets, and
so given a set of sets $X=\\{X_{\alpha}\\}$ we will write
$X^{*}=<X,^{\prime\prime}>$ for the set of sets generated from $X$ by ′′. Note
that since ′′ is not associative, $(X^{*},^{\prime\prime})$ is a magma rather
than a semi-group (its elements are represented by binary trees rather than
strings), and we will call it the magma of hom-sets. Also the elements of
$X^{*}$ contain typically morphisms between sets of morphisms rather than
morphisms between morphisms, so that these are distinct from the “higher
order” morphisms encountered in higher order category theory.
second cant use of eg $Y$ group gives $XY$ group, endomorphisms of group form
group (or comp of monids XYZ).

Involution:

An involution $\left(\overline{.}\right)_{XX}$ on an algebraic structure
$X=(X_{Set},\\{\circ\\}_{\alpha})$ is a unary operation of order 2 (ie for
every $x_{X}$ we have $\overline{\overline{x}}=x$), and which is also a skew-
automorphism which is a skew-anti-commutes with each $\circ_{\alpha}$, ie
$\overline{x\circ_{\alpha}y}=\overline{y}\circ_{\alpha}\overline{x}$. The
second constraint is automatically satisfied if $\\{\circ\\}_{\alpha}$ is
empty. Given another set $Y$, an involution on $X$ induces an involution on
$(XY)_{Set}$ by $\overline{\phi_{XY}}x=\phi\overline{x}$. However this is
involution does not generally commute with operators on $XY$ such as
composition or addition, so it is not generally an involution on $XY$ as an
algebraic structure.

We call $\overline{x}$ the involute of $x$, and self-involute when
$x=\overline{x}$. The set of self-involute points is the set of fixed points
of $\overline{.}$ denoted $FP(X,\bar{.}$) or just $FP(X)$ when $\bar{.}$ is
clear. A subset $W\subseteq X$ is fixed point free under $\bar{.}$ if it is
disjoint from $FP(X,\bar{.})$. Note that this means $W,\bar{W}$ are mutually
disjoint, and that $W\neq X$ unless $X=\emptyset$. A partition of $X$ under
$\bar{.}$ is a triple of mutually disjoint subsets $(W,\overline{W},FP(X))$
which cover $X$.

######  Lemma 4.

Let $X,Y$ be algebraic structures with involutions $\bar{.}_{X},\bar{.}_{Y}$,
and let $W\subset X$ be fixed point free under $\bar{.}_{X}$. Suppose further
that $A,B$ are real-valued functions on $X\times Y$ with the property that
$A(x,y)>B(x,y)$ and $A(\overline{x},y)<B(\overline{x},y)$ for any $y$ and any
$x\in W$. Then $\overline{A}(x,y)<\overline{B}(x,y)$ and
$\overline{A}(\overline{x},y)>\overline{B}(\overline{x},y)$ for any $y$ and
any $x\in W$.

###### Proof.

$\overline{A}(x,y)=A(\overline{x},\overline{y})<B(\overline{x},\overline{y})=\overline{B}(x,y)$
and the second result follows similarly. ∎

Involutions

######  Definition 5.

An involution $\overline{.}$ on a set $X$ is a self-inverse bijection, ie
$\overline{\overline{x}}=x$. An involution on an algebraic structure
$(X,\circ)$ is an involution on $X$ which is also an anti-isomorphism, ie
$\overline{x\circ y}=\overline{y}\circ\overline{x}$. If $\circ$ is
commutative, $\overline{.}$ becomes an automorphism.

If $(X,\circ)$ is a group, we call $\overline{x}=x^{-1}$ the inverse
involution, noting $\overline{xy}=\overline{y}\,\overline{x}$. When
$X=\mathbb{R}/\mathbb{Z}$ this gives $\overline{x}=-x$,
$\overline{x+y}=(-y)+(-x)$, and we will use this extensively.

On the free monoid $(X^{*},+),$ reversal is an involution where
$\overline{x_{1}x_{2}\ldots x_{N}}=x_{N}x_{N-1}\ldots x_{1}$ and
$\overline{w_{1}+w_{2}}=\overline{w_{2}}+\overline{w_{1}}$. Note that reversal
restricted to $X$ is the identity.

We will use some useful abstractions inspired by Category Theory in order to
simplify and structure our theory. We start with concrete categories, namely
categories whose objects are mathematical spaces each with an underlying set.
By abuse of notation we will denote the underlying space of $X_{Obj}$ by
$X_{Set}$. Also in a concrete category, the morphism set $XY$ of morphisms
from $X_{Obj}$ to $Y_{Obj}$ is a set of functions
$\\{\phi_{X_{Set}Y_{Set}}\\}$ between the underlying sets. Of course $XY$ will
normally be a strict subset of the full set of functions from $X_{Set}$ to
$Y_{Set}$ (usually denoted $Y^{X}$), and will be chosen to preserve in some
way the additional structure of the spaces $X_{Obj},Y_{Obj}$.

Given a concrete category $C$ we construct the second order category $C^{+}$
whose objects are $Obj(C)\bigcup\\{XY:X,Y\in Obj(C)\\}$, ie we extend $C$ by
making the morphism sets $XY$ of $C$ into objects of $C^{+}$. This allows us
to construct morphisms between morphism sets of $C$, and between morphism sets
of $C$ and objects of $C$.

We use this for separating $S_{N},\phi$ and light use of duality to reduce the
number of cases we need to consider.

Induced/Constructed/Dependent Morphisms General:
$A_{(XY)(VW)}:\phi_{XY}\mapsto\phi_{VW}$ Need to check whether $\phi_{VW}$ is
a morphism.

Sequence extension
$\prod_{(XY)(X^{n}Y^{n})}^{n}:\phi_{XY}\mapsto\phi_{X^{n}Y^{n}}$ where
$\left(\prod^{n}\phi\right)(x_{i})\coloneqq(\phi x_{i})_{Y^{n}}$ and hence to
Kleene Star $\phi_{X^{*}Y^{*}}$ which is $\phi_{X^{n}Y^{n}}$ on seq of length
$n\geq 0$. But note image $\prod_{(XY)(X^{n}Y^{n})}^{n}XY$ is usually a strict
subset of $X^{n}Y^{n}$ consisting of the diagonal entries: general term of
$X^{n}Y^{n}$ is
$(\phi_{j})_{X^{n}Y^{n}}:(x_{j})_{X^{n}}\mapsto(\phi_{j}x_{j})_{Y^{n}}$ \-
diagonal element of $X^{n}Y^{n}$, $X^{n}$ has $\phi_{j}=\phi$, $x_{j}=x$
respectively.

$(XY)^{n}=\left\\{(\phi_{1},\ldots,\phi_{n})\right\\}$ is ambiguous over
domain - is it $XY^{n}$ or $X^{n}Y^{n}$? - don’t use - use either of the
latter which are well defined. But then $\phi_{XY^{n}}\in(XY)_{Set}^{n}$\- but
application of $(XY)^{n}$ as a map is undefined, better to say $XY^{n}$ is
naturally isomorphic to $X^{n}Y^{n}\mid_{Diag(X^{n})}$

Also $(\phi_{i})_{XY^{n}}:x\mapsto(\phi_{i}x)_{Y^{n}}$ \- restriction of
domain of $X^{n}Y^{n}$ from $X^{n}$ to diagonal elements of $X^{n}$,
$Diag(X^{n})$

Pullback/Koopman Operator:
$C_{\psi}:\phi_{XY}\mapsto\left(\phi\psi_{WX}\right)_{WY}$ $C_{\psi}$ is
linear in $\phi$ and has signature $(XY)(WY)$, $C$ is
$\left(WX\right)\left((XY)\left((XY)(WY)\right)\right)$ function to operator
(meta operator, 2nd order operator)

Given $(Y,\circ_{Y^{n}Y}),$
$\circ_{(XY^{n})(XY)}:(\phi_{i})_{XY^{n}}\mapsto\left(\circ_{Y^{n}Y}\cdot(\phi_{i})_{XY^{n}}\right)_{XY}$
eg (n=2)
$(\phi_{1}\circ_{XY}\phi_{2})x\coloneqq\left(\phi_{1}x\right)\circ_{Y}(\phi_{2}x)$
(pushforward of $\circ$ ) Equivalent to $f(g)=f\circ g$ not $g(f)=f\circ g$ \-
simple composition for 1-functions, extended to sequences. What about
$\circ_{Y^{n}Z}$ ?
$(Y,\circ_{Y^{n}Z}),\circ_{(XY^{n})(XZ)}:(\phi_{i})_{XY^{n}}\mapsto\left(\circ_{Y^{n}Z}\cdot(\phi_{i})_{XY^{n}}\right)_{XZ}$
\- this normal comp with $Y$ replaced by $Y^{n}.$ Generalisation of
distribution when $X=Y=Z$.

PushForward/Koopman Adjoint:
$C_{\phi}:\psi_{WX}\mapsto\left(\phi_{XY}\psi_{WX}\right)_{WY}$ is $(WX)(WY)$
instead of $(XY)(WY)$. $C$ is linear in $\phi$, but $C_{\phi}$ is not linear
in $\psi$. So the codomain $X$ of $\phi$ is pushed forward to $Y$, rather than
the domain of $X$ being pulled back to $W$. In the above we have $(W,X,Y)$
replaced by $(X,Y^{n},Y)$.

Let $X,Y$ be two objects of some concrete category $C$, and suppose $Y$ is
equipped with a binary operation $+_{Y}$ (not necessarily commutative). Then
$+_{Y}$ induces a dependent binary operation $+_{XY}$ on the set of functions
$Y^{X}$ as follows: $(\phi_{XY}\,+_{XY}\,\theta_{XY})x\coloneqq\phi
x\,+_{Y}\,\theta x$. Of course the subset $XY$ of $Y^{X}$ may not be closed
under $+_{XY}$. If it is closed, then we say the space
$XY\coloneqq(XY,+_{XY})$ is a dependent space of $Y$.

Now given an endomorphism $T_{XX}$ and a morphism $\phi_{XY}$, then the
composition $\phi_{XY}T_{XX}$ is itself in $XY$. In other words $T$ acts on
$XY$ by $\phi\mapsto\phi T$. When $XY$ is a dependent space on $Y$, the action
is also linear in the sense that
$(\phi_{1}+\phi_{2})(Tx)=\phi_{1}(Tx)+\phi_{2}(Tx)=(\phi_{1}T)(x)+(\phi_{2}T)(x)=(\phi_{1}T+\phi_{2}T)(x)$.

We now wish to be able to reason further about the map $\phi\mapsto\phi T$
which is an endomorphism of $XY$ and consequently does not exist within $C$.
Informally we will just add it in, but we will also describe how to this
formally. We move to the higher order category $C^{+}$ where we can define
morphisms between sets of morphisms - in particular a morphism between $XY$
and $AB$ will lie in the morphism set $(XY)(AB)$. In particular we can now
define the dependent morphism $\mathbb{T}_{(XY)(XY)}(\phi_{XY})\coloneqq\phi
T_{XX}$. Now by the result above, $+_{XY}$ in $XY$ induces a dependent binary
operation $+_{(XY)(XY)}$ in $XY^{(XY)}$ defined by $(T+U)\phi\coloneqq
T\phi+U\phi$. Again $(XY)(XY)$ is a dependent space of $XY$ if closed under
$+$.

The endomorphisms $T_{XX}$ of $X$ form a monoid under composition, and so also
$(XY)(XY)$ and also a semi-group if closed under the operator $+_{XY}$ pulled
up from $+_{Y}$. Also $T^{n}\phi=\phi T^{n}$ for $n\geq 0$.

$X^{*}=(X^{*},+)$ is the free linear monoid on $X$, and we extend $T$ to
$X^{*}$ by $T(\sum_{r=1}^{N}x_{r})\coloneqq\sum_{r=1}^{N}Tx_{r}$ so the
induced $T$ is $X^{*}X^{*}$ and linear. Now the orbit segment of $x_{0}$ with
length $N$ in $(X,T)$ is $(Tx_{0},T^{2}x_{0},\ldots,T^{N}x_{0})$ and now
regard this as the word
$\sum_{r=1}^{N}\left(T^{r}\left(x_{0}\right)_{X^{*}}\right)$ in $X^{*}$ where
$\left(x_{0}\right)_{X^{*}}$ is the single letter word in $X^{*}$
corresponding to $(x_{0})_{X}$. Using the linearity of $T^{r}$ we can rewrite
this as $\left(\sum_{r=1}^{N}T^{r}\right)x_{0}$ where the summation is the
induced operator $+_{(X^{*}X^{*})}$. We now define the linear operator
$S_{N}=\sum_{r=1}^{N}T^{r}$ and we can write the orbit segment as
$S_{N}x_{0}$. Finally we similarly extend $\phi_{XY}$ to $X^{*}$by
$\phi(\sum_{r=1}^{N}x_{r})=\sum_{r=1}^{N}\phi x_{r}$ where now the second
summation uses $+_{Y}$. Hence we can use the rewriting rules of (4)
$\phi(S_{N}x_{0})=\sum_{r=1}^{N}\phi(T^{r}x_{0})=\sum_{r=1}^{N}(T^{r}\phi)x_{0}=\left(\sum_{r=1}^{N}T^{r}\right)\phi
x_{0}=\left(S_{N}\phi\right)x_{0}$ where the second homonym $S_{N}$ is now
$(X^{*}Y)Y$ and $S_{N}\phi$ is $X^{*}Y$ (ie an observable on $X^{*}$) and its
restriction to $X$ is now the Birkhoff sum operator

Free Monoids

######  Definition 6.

Given a set $X$, the free monoid $(X^{*},+)$ on $X$ is defined as the set of
finite words (or strings) $x_{1}x_{2}\ldots x_{N}$ with each letter $x_{i}\in
X$, including the empty word $0$, and with the operation $+$ being the
concatenation of words (which of course is NOT commutative). We write
$\sum_{r=1}^{N}w_{r}$ to represent $w_{1}+w_{2}\ldots+w_{N}$ where $w_{r}$ is
a word of $X^{*}$. We write the reverse sum $w_{N}+w_{N-1}\ldots+w_{1}$ as
$\sum_{r=1}^{N}w_{N+1-r}$.

Duality A duality between two spaces allows us to obtain dual results
automatically in one space from results in the other. We will exploit duality
in a number of places to reduce the number of cases we need to consider.
However the dualities are quite simple, and the theory development required to
make them formal and explicit does not repay the effort. Hence we will simply
identify the dualities informally and use them to structure the cases to be
considered.

We will exploit 2 types of duality: domain dualities which arise through the
existence of useful involutions on underlying sets and groups, and a parity
duality (a duality between odd and even results) which arises in connection
with the study of Continued Fractions.

###  2.3. Dynamical Systems

Although we noted above that Birkhoff sums were studied pre-Birkhoff, they are
normally today associated with Ergodic Theory and Dynamical Systems, and we
will introduce appropriate terminology and notation here.

A Dynamical System $(X,T_{X})$ is simply a space $X$ (a set $X$ equipped with
some suitable algebraic and analytical structure) with an endomorphism
$T:X\rightarrow X$. Given $x_{0}\in X$ (called an initial condition), the
point $x_{N}=T^{N}x_{0}$ is the $N-$th iterate of $x_{0}$, and the sequence
$(x_{1},x_{2},\ldots)$ is the (forward) orbit of $x_{0}$ (NB it is very
convenient NOT to regard $x_{0}$ itself as the first point of the orbit). For
$n,m\geq 0$, the subsequence of the orbit $(x_{r})_{r=m+1}^{m+n}$ is called a
segment of length $n$, and an initial segment if $m=0$.

We will focus in this paper on the Dynamical system $(\mathbb{T},R_{\alpha})$
where $\mathbb{T}=[0,1)$ is the circle (a compact 1 dimensional real manifold)
and $R_{\alpha}$ is a rotation through $\alpha_{\mathbb{R}}$ revolutions. Note
that $R_{\alpha}=R_{\\{\alpha\\}}$

When $X$ has a distance function $d$ we say $x_{N}$ is a point of closest
return to $x_{0}$ if $x_{N}=T^{N}x_{0}$ satisfies
$d(x_{0},x_{N})<d(x_{0},x_{r})$ for $r<N$, ie $x_{N}$ is the first point of
minimum distance from $x_{0}$ in the initial orbit segment of length $N$. We
call $N$ a quasiperiod with quasiperiod error $d(x_{0},x_{N})$. Note that $1$
is trivially a quasiperiod.

Note also that if $x_{N}=x_{0}$ then: the orbit is periodic, and $N$ is a
multiple of the period $p=\min\\{r>0:R^{r}x=x\\}$, and the period is itself a
quasiperiod with quasiperiod error $0$.

We now introduce the concept of an observable of the Dynamical System as a
function $\phi_{XY}$ with values in a semi-group $(Y,\circ_{Y})$. By 2.2 this
induces a natural binary function $\circ_{XY}$ on $XY$. In general
$\circ_{Y},\circ_{XY}$ need not be commutative (as in the theory of co-cycles.
where $(Y,\circ)$ is a space of matrices with matrix multiplication). In this
paper we restrict ourselves to the simple case
$(Y,\circ)=(\mathbb{\mathbb{R}},+)$, and so in the sequel an observable on $X$
will be a real function on $X$, and the set of observables $X\mathbb{R}$ is a
commutative group, namely $(X\mathbb{R},+_{X\mathbb{R}})$.

######  Definition 7 (Birkhoff Sum).

The $N$th Birkhoff sum over the Dynamical System $(X,T)$ of the observable
$\phi_{X\mathbb{R}}$ with initial condition $x$ is
$S_{N}^{T}(\phi,x)\coloneqq\sum_{r=1}^{N}\phi(T^{r}x)$.

We shall find it convenient to define a number of homonymous forms derived
from $S_{N}$. We allow the omission of $T$ or $\phi$ when these are
understood, obtaining the forms $S_{N}(\phi,x)\coloneqq S_{N}^{T}(\phi,x)$,
$S_{N}x\coloneqq S_{N}(\phi,x)$. When $T$ is $R_{\alpha}$, namely a rotation
of the circle through $\alpha_{\mathbb{R}}$ revolutions, we will also write
$S_{N}^{\alpha}\coloneqq S_{N}^{R_{\alpha}}$. Finally we define the
homogeneous form $S_{N}\phi\coloneqq S_{N}(\phi,0)$,

Separation of concerns Let $(X,T)$ be a Dynamical system with a Value system
$(Y,+)$ and Observables system $(XY,+)$.

In (2.2) we noted that given $T:X\rightarrow X$ and $\text{$\phi$}\in(XY,+)$,
that $T$ has a homomorphic action on $T:XY\rightarrow XY$ defined by

$(T\phi)x=(\phi\circ T)x=\phi(Tx)$

For the purposes of this paper, the right hand side is more tractable since it
means we can study a given Dynamical System $(X,T)$ independent of $\phi$.
However our goal is to study Birkhoff sums which have the form
$(S_{N}\phi)x_{0}$. Noting that this has the same form as the left hand side
of (2.2), it would be nice to rewrite $(S_{N}\phi)x_{0}$ as
$\phi(S_{N}x_{0})$. Unfortunately $S_{N}x_{0}$ is undefined, but we can easily
remedy this by extending our study from $X$ to $X^{*}$, the free monoid on $X$
(see 2.2), which provides a richer context in which to work.

Note that we can write any word $w=x_{1}\ldots x_{N}\in X^{*}$ as a
concatenation $\sum_{r=1}^{N}x_{i}$. We can also identify the initial orbit
segment $(T^{r}x_{0})_{r=1}^{N}$ of $(X,T)$ with the concatenation
$\sum_{r=1}^{N}T^{r}x_{0}$.

We now extend the function $\phi:X\rightarrow Y$ to a monoid homomorphism
$\phi:X^{*}\rightarrow Y$ by $\phi(\sum_{r=1}^{N}x_{r})=\sum_{r=1}^{N}\phi
x_{r}$. Similarly we extend each function $T^{r}:X\rightarrow X$ to a monoid
homomorphism $T^{r}:X^{*}\rightarrow X^{*}$ by
$T^{r}(\sum_{r=1}^{N}x_{r})=\sum_{r=1}^{N}T^{r}x_{r}$, noting by (2.2) that
$+$ on $X^{*}$ induces $+$ on $X^{*}X^{*}$ so that
$M=\left((X^{*}X^{*}),+\right)$ is a monoid of monoid endomorphisms. In
particular we can now write $S_{N}=\sum_{r=1}^{N}T^{r}\in M$ with
$S_{N}w=\sum_{r=1}^{N}T^{r}w$ for each word $w\in X^{*}$. We can now use (2.2)
with $U:X^{*}\rightarrow X^{*}\in M$ and $\phi\in(X^{*}Y,+)=M_{2}$ that $U$
has a homomorphic action $U:M_{2}\rightarrow M_{2}$ by $(U\phi)w=\phi(Uw)$. In
particular taking $U=S_{N},w=x_{0}$ gives us (as desired)
$(S_{N}\phi)x_{0}=\phi(S_{N}x_{0})$, and indeed much more.

Noting that we can now identify $S_{N}x_{0}$ with the initial orbit segment
$(T^{r}x_{0})_{r=1}^{N}$ in the Dynamical System $(X,T)$, we use the identity
above to separate the study of Birkhoff sums $S_{N}\phi x_{0}$ into the study
of initial orbit segments, and subsequently the action of $\phi$ on these
segments. We do this in Sections (5,6) respectively.

###  2.4. Geometry of the circle

As usual we will adopt the topological quotient group
$\mathbb{T}=\mathbb{\mathbb{\mathbb{R}}}/\mathbb{Z}$ as our model of the
circle, together with its quotient topology and (pushed forward from
$\mathbb{I}$) Lebesgue measure. Recall that the elements of
$\mathbb{R}/\mathbb{Z}$ are the cosets $x_{\mathbb{\mathbb{R}}}+\mathbb{Z}$.
We write $x_{\mathbb{T}}\coloneqq x_{\mathbb{\mathbb{R}}}+\mathbb{Z}$ and
designate the quotient homomorphism
$\pi_{\mathbb{\mathbb{\mathbb{R}}}\mathbb{T}}$ so that
$\pi(x_{\mathbb{\mathbb{\mathbb{R}}}})=x_{\mathbb{\mathbb{R}}}+\mathbb{Z}=x_{\mathbb{T}}$.
Given $x,x_{0}\in\mathbb{\mathbb{\mathbb{R}}}$, we also regard $\mathbb{T}$ as
a compact manifold with the atlas of charts222Strictly a chart is a be a
homeomorphism between open sets, but we can easily extend these charts to the
full manifold. extension to the full manifold/interval we must first define
$\chi_{x_{0}}:\mathbb{T}\backslash\\{x_{0}+\mathbb{Z}\\}_{Set}\rightarrow\mathbb{I}\backslash\\{x_{0}\\}_{Set}$
and then extend it to include $x_{0}+\mathbb{Z}\mapsto\\{x_{0}\\}.$Similarly
for $\chi_{x_{0}}^{/}$
$\chi_{x_{0}}^{\mathbb{T}\mathbb{I}}:x_{\mathbb{T}}=x+\mathbb{Z}\mapsto\\{x-x_{0}\\}$.
Note that
$\chi_{x_{0}}^{\mathbb{T}\mathbb{I}}\chi^{\mathbb{\mathbb{\mathbb{R}}}\mathbb{T}}=\chi_{x_{0}}^{\mathbb{\mathbb{\mathbb{R}}}\mathbb{I}}:x\mapsto\\{x-x_{0}\\}$.
We also have a similar set of charts
$\chi_{x_{0}}^{\mathbb{T}\mathbb{I}^{/}}:x_{\mathbb{T}}\mapsto\\{\\{x-x_{0}\\}\\}$
with
$\chi_{x_{0}}^{\mathbb{T}\mathbb{I}^{/}}\chi^{\mathbb{\mathbb{\mathbb{R}}}\mathbb{T}}=\chi_{x_{0}}^{\mathbb{\mathbb{\mathbb{R}}}\mathbb{I}^{/}}:x\mapsto\\{x-x_{0}\\}$.
When $x_{0}=0$ we will drop the subscript from each of the maps $\chi$ and
regard these as canonical maps. In particular we take
$\chi^{\mathbb{T}\mathbb{I}},\chi^{\mathbb{T}\mathbb{I}^{/}}$ as a canonical
atlas, and call
$x_{\mathbb{I}}=\chi^{\mathbb{T}\mathbb{I}}(x_{\mathbb{T}}),x_{\mathbb{I}^{/}}=\chi^{\mathbb{T}\mathbb{I}^{/}}(x_{\mathbb{T}})$
the natural coordinate, and signed coordinate of $x_{\mathbb{T}}$
respectively.

Note the transition map or change of coordinates
$\chi_{x_{1}}^{\mathbb{T}\mathbb{I}}\left(\chi_{x_{0}}^{\mathbb{T}\mathbb{I}^{/}}\right)^{-1}x=\\{x-(x_{1}-x_{0})\\}$,
and so for the canonical coordinates the morphism
$\chi_{x_{1}}^{\mathbb{T}\mathbb{I}}\left(\chi_{x_{0}}^{\mathbb{T}\mathbb{I}^{/}}\right)^{-1}$
lies in the hom-set $\mathbb{I}^{/}\mathbb{I}$ and its inverse lies in
$\mathbb{I}\mathbb{I}^{/}$.

The natural order on $\mathbb{\mathbb{\mathbb{R}}}$and $\mathbb{I}$ is
ambiguous on $\mathbb{T}$. Whilst it is normal to define circle order as a
ternary relation (without reference to coordinates), it will suit us better to
use the order induced from the natural coordinate, so that we define
$x_{\mathbb{T}}\leq y_{\mathbb{T}}\,\Leftrightarrow\,0\leq x_{\mathbb{I}}\leq
y_{\mathbb{I}}<1$. We will then say that a sequence
$\left(x_{i}\right)_{i=1}^{n}$ for $n\geq 3$ is in cyclic order if there is a
rotation $R_{\alpha}$ under which the full sequence is placed in natural
order, ie
$\\{x_{i}+\alpha\\}_{\mathbb{I}}\leq\\{x_{i+1}+\alpha\\}_{\mathbb{I}}$ for
$i=1..n-1$. We say $\left(x_{i}\right)_{i=1}^{n}$ for $n\geq 3$ is in anti-
cyclic order if the reverse sequence $\left(x_{i}\right)_{i=n}^{1}$ is in
cyclic order. A sequence which is in cyclic or anti-cyclic order is simply
ordered. A sequence of length $0\leq n\leq 2$ is always both in cyclic and
anti-cyclic order.

The ambiguity of order means that a pair of distinct circle points
$a_{\mathbb{T}}<b_{\mathbb{T}}$ identifies 2 circle intervals defined by the
direction of travel from $a$ to $b$. We will find it useful to have 2 ways of
resolving this ambiguity. First we use the coordinate order: the cyclic
interval $\mathbb{I}$ contains interior points $x$ such that $(a,x,b)$ are in
cyclic order; and the anti-cyclic interval contains $x$ such that $(a,x,b)$
are in anti-cyclic order. We call these directed intervals, and if we do not
specify a direction, we will take cyclic by default. Secondly we define
undirected intervals: the minor interval is the one of smaller measure, the
major interval is the one of larger measure. If the measure of both is
precisely $1/2$, we take the minor interval to be the cyclic directed
interval, although we will not make use of such intervals in this paper. Note
that for directed intervals $(a,b)\neq(b,a)$ whereas for undirected intervals
(of measure not $\frac{1}{2}$) we have $(a,b)=(b,a)$ for both minor and major
intervals.

In addition to open and closed intervals, we are equally concerned with semi-
open intervals. We will use the usual notation of open $()$ and closed $[]$
brackets to indicate whether endpoints are included or not. We will also use
$|$ to avoid specifying which is the case, so that $|a,b|$ represents a
particular but unspecified one of the 4 possible interval configurations of
endpoints.

We will also use $1$ as a synonym for $0$ where this aids readability, so
$[\frac{3}{4},1]$ represents the subset $[\frac{3}{4},1)\cup\\{0\\}$in
$\mathbb{I}$, and the cyclic directed interval $[\frac{3}{4},0]$ in
$\mathbb{T}$.

Any circle interval $J$ is also a topological subspace of $\mathbb{T}$
equipped with the subspace topology. One subtlety to note is that a subspace
may have boundary points in the parent topology, but these are not boundary
points in the subspace topology: for example if $J=[a,b)$ then the interval
$[a,c_{J})$ is _open_ (not _semi-open_) in the subspace topology of $J$. With
this in mind, it becomes simple to extend the well known characterisation
about open sets of $\mathbb{R}$ to arbitrary real or circle intervals. We
include a proof for completeness:

######  Proposition 8.

In the subspace topology of an interval $J$ of either $\mathbb{R}$ or
$\mathbb{T}$, an open set is an at most countable union of disjoint open
intervals

###### Proof.

Let $X$ be an open set in the topology of $J$. For each $x_{X}$ let $I_{x}$ be
the maximal interval in $X$ containing $x$. If $I_{x},I_{y}$ are not disjoint,
their union is an interval, so their maximality requires $I_{x}=I_{x}\cup
I_{y}=I_{y}$. Hence $X$ is a union of disjoint intervals. However $X$ is also
open and so contains no boundary points. Hence each $I_{x}$ is an open
interval. Finally each interval with interior contains a rational, so any
family of disjoint intervals with interior is at most countable. ∎

A partition of the circle of length $n\geq 1$ is a an ordered sequence
$\left(x_{i}\right)_{i=1}^{n}$. A partition of a circle interval $J$ is a
partition of the circle, all of whose points lie in $J$. (Note that this is a
slight generalisation of the usual definition: we do not require $J$ to be
closed, nor do we require the endpoints of $J$ to be included in the
partition).

We introduce the natural involution on $\mathbb{T}$ define by
$\overline{x_{\mathbb{T}}}=-x_{\mathbb{T}}$ which induces coordinate
involutions $\overline{\\{\chi
x_{\mathbb{T}}\\}}=\\{\chi\overline{x_{\mathbb{T}}}\\}=\\{-x_{I}\\}=\\{1-x\\}$
and
$\overline{\\{\\{\chi^{/}x_{\mathbb{T}}\\}\\}}=\\{\\{\chi^{/}\overline{x_{\mathbb{T}}}\\}\\}=\\{\\{-x_{\mathbb{I}^{/}}\\}\\}=-\\{\\{x_{\mathbb{I}^{/}}\\}\\}$.
Note that for $x_{\mathbb{I}}\neq 0$ we have $\\{1-x\\}=1-x$, whereas
$\\{1-0\\}=0$. The duality which results from application of this involution
we call Circle Duality.

We now introduce the real translation
$R_{\alpha}^{\mathbb{\mathbb{\mathbb{RR}}}}:x\mapsto x+\alpha$. This induces
the translations (which we will here call rotations)
$R_{\alpha_{\mathbb{T}}}^{\mathbb{TT}}:x_{\mathbb{T}}\mapsto
x_{\mathbb{T}}+\alpha_{\mathbb{T}}$ and
$R_{\alpha_{\mathbb{I}}}^{\mathbb{I}\mathbb{I}}:x_{\mathbb{I}}\mapsto\\{x_{\mathbb{I}}+\alpha_{\mathbb{I}}\\},R_{\alpha_{\mathbb{I}^{/}}}^{\mathbb{I}^{/}}:x_{\mathbb{I}^{/}}\mapsto\\{\\{x_{\mathbb{I}}+\alpha_{\mathbb{I}}\\}\\}$.
Since each of these has identical structure, we will abuse notation and regard
all of them as $R_{\alpha}$.

Note that $\overline{R_{\alpha}}=R_{\overline{\alpha}}=R_{1-\alpha}$ (and
$R_{\alpha}=R_{\\{\alpha\\}}=R_{\alpha+n_{\mathbb{\mathbb{Z}}}}$)

Note that in these systems we have $R_{\alpha}^{N}x=\\{x+N\alpha\\}$ and
$S_{N}(\phi,x,R_{\alpha})=\sum_{r=1}^{N}\phi(\\{x+r\alpha\\})$. Note we can
always lift $\phi:\mathbb{T}\rightarrow\mathbb{\mathbb{\mathbb{\mathbb{R}}}}$
to
$\phi^{\uparrow}:\mathbb{\mathbb{\mathbb{\mathbb{\mathbb{R}}}\rightarrow}\mathbb{\mathbb{\mathbb{R}}}}$
by defining $\phi^{\uparrow}(x)=\phi(\\{x\\})$, and by abuse of notation we
will simply write $\phi$ for $\phi^{\uparrow}$ so for example we can simply
write $S_{N}(\phi,x,R_{\alpha})=\sum_{r=1}^{N}\phi(x+r\alpha)$. We call
$S_{N}(\phi,x,R_{\alpha})$ a homogeneous sum for $\\{x\\}=0$, and an
inhomogeneous sum for $\\{x\\}\neq 0$.

Duality results:.
$S_{N}^{\overline{T}}\overline{\phi}\overline{x}=S_{N}^{T}\phi x$
$S_{N}(-\phi)x=-S_{N}\phi x$
$S_{N}^{\overline{T}}(-\overline{\phi})\overline{x}=-S_{N}^{T}\phi x$

A partition of the circle of length $q\geq 1$ is a sequence of $q$ circle
points with coordinates satisfying $0\leq x_{1}<x_{2}\ldots<x_{q}<1$. Note
that any sequence of distinct points can be reordered to give a partition. The
intervals of the partition are the images under
$\chi_{\mathbb{T}\mathbb{I}}^{-1}$ of the $q-1$ intervals $(x_{i},x_{i+1})$
for $r<q$, and the interval $(x_{q},1)\bigcup(0,x_{1})$.

The partition is regular if $x_{r+1}-x_{r}=\frac{1}{q}$ for $1\leq r<q$, and
also (if $q>1)$ $\\{x_{1}-x_{q}\\}=\frac{1}{q}$. We will only deal with
regular partitions, so we will drop the word regular. Note that a partition of
length 1 is always regular.

###  2.5. Number Theory

Algebraic Context Given some set $A\subseteq\mathbb{\mathbb{\mathbb{R}}}$ we
define $x_{\mathbb{R}}A=\\{xa:a\in A\\}$. If $A$ is an additive group then so
is $xA$ and there is a quotient group $\mathbb{R}/xA$ which is also an
additive group with elements $y_{(\mathbb{R}/xA)}=y+xA\coloneqq\\{y+xa:a\in
A\\}$. Further $xA$ is also closed under multiplication iff $z\in\mathbb{Z}$.
When $z\in\mathbb{Z}$, $z\mathbb{Z}$ is a commutative ring (without unit for
$|z|>1$), and the quotient $\mathbb{Z}/z\mathbb{Z}$ is a commutative ring with
unit (a field if $z$ is a prime, but with zero divisors if $z$ is composite).

In this paper we will work primarily in the projective closure of the positive
reals, ie the interval $\mathbb{P}=[0,\infty]$ equipped with the involution
$R:x\mapsto\frac{1}{x}$, where we define $\infty\coloneqq R0$ (and hence
$R\infty=0$, and $R[0,1]=[1,\infty]$). We also define $\infty$ to be an
integer, together with the usual operator extensions
$\infty+x_{\mathbb{P}}\coloneqq\infty,\,\infty.x_{\mathbb{P}}\coloneqq\infty,\,x_{\mathbb{P}}\leq\infty$.
Number Functions The most important group of functions for our purposes are
the remainder (residue) and floor functions. Recall $\mathbb{I}\coloneqq[0,1)$
and for $x_{\mathbb{R}}\neq 0$ $x\mathbb{I}=[0,x)$.

For each $z_{\mathbb{R}}\neq 0$ we define the $z-$remainder function
$\\{.\\}_{z}:\mathbb{R}\rightarrow\left|z\right|\mathbb{I}$ by
$\\{x\\}_{z}\coloneqq\min\\{x-k_{\mathbb{Z}}z:kz\leq x\\}$, ie $\\{x\\}_{z}$
is the positive remainder after integer division by $z$. Note as $z\rightarrow
0$, $\\{x\\}_{z}\rightarrow 0$ and so we extend to $\\{x\\}_{0}=0$. Note
$\\{x+nz\\}_{z}=\\{x\\}_{z}$ so there is a natural bijection
$x+z\mathbb{Z}\longleftrightarrow\\{x\\}_{z}$, ie
$\mathbb{\mathbb{\mathbb{R}}}/z\mathbb{Z}\leftrightarrow|z|\mathbb{I}$. This
means we can give $|z|\mathbb{I}$ group structure via
$x+_{z\mathbb{I}}\,y\coloneqq\\{x+y\\}_{z}$. Similarly
$\mathbb{Z}/q\mathbb{Z}\longleftrightarrow|q|\mathbb{I}\bigcap\mathbb{Z}$ and
we can add distributive multiplication via
$r\cdot_{|q|\mathbb{I}}s=\\{rs\\}_{q}$.

The signed remainder is
$\\{\\{x\\}\\}_{z}=\\{x\\}_{z}-\left\llbracket\\{x\\}_{z}\geq\frac{\left|z\right|}{2}\right\rrbracket\left|z\right|\in\left|z\right|\mathbb{I}^{/}$.
The remainder and signed remainder are inverse bijections
$|z|\mathbb{I}^{/}\longleftrightarrow|z|\mathbb{I}$. Together they provide an
involution on $|z|\left(\mathbb{I}\bigcup\mathbb{I}^{/}\right)$ which fixes
$|z|[0,\frac{1}{2})$ and transposes $[-\frac{1}{2},0)$ with $[\frac{1}{2},1)$.

We also define the floor function $\left\lfloor x\right\rfloor_{z}\coloneqq
x-\\{x\\}_{z}$ and the smallest distance from $x$ to $z\mathbb{Z}$ given by
$\left\|x\right\|_{z}=\left|\\{\\{x\\}\\}_{z}\right|\in\frac{|z|}{2}\mathbb{I}$.
Note that the latter is a metric on $\mathbb{I}$ though not a norm (it fails
the requirement of scalar multiplication). It is useful to extend these
functions by defining $\\{\infty\\}=0$ from which we also get
$\\{\\{\infty\\}\\}=0,\left\lfloor\infty\right\rfloor=\infty,\left\|\infty\right\|=0$.

When $z=1$ we will omit it from the notation. The functions
$\\{x\\},\\{\\{x\\}\\}$ are then also called fractional parts of $x$.

We can use $\\{x\\}_{z}$ to derive other common functions:

The highest common factor is the binary operation
$(x_{\mathbb{R}},y_{\mathbb{R}})\coloneqq\max\\{z_{\mathbb{R}}:\\{x\\}_{z}=0\&\\{y\\}_{z}=0\\}$.
Note $(x,x)=(x,0)=\left|x\right|$.

The reflexive & transitive relation ’divides’ is given by
$z|x\coloneqq\;\\{x\\}_{z}=0$ (so we allow $0|x$), and the equivalence
relation ’congruent’ by $x\equiv y\bmod
z\;\coloneqq\;\\{x\\}_{z}=\\{y\\}_{z}$. Now
$\\{x\\}_{z}=\\{y\\}_{z}\Longleftrightarrow x+z\mathbb{Z}=y+z\mathbb{Z}$ so
that there is a natural bijection between $[0,z)$ and
$\mathbb{\mathbb{\mathbb{R}}}/z\mathbb{Z}$ giving $[0,z)$ a group structure by
$x_{[0,z)}\leftrightarrow[x]$, and a ring structure on $\\{r\\}_{r=0}^{q-1}$
induced from $\mathbb{Z}/q\mathbb{Z}$.

The signum function $\operatorname{sgn}$ maps $\alpha_{\mathbb{R}}$ to its
sign:$\operatorname{sgn}(x_{\mathbb{R}})=\left\llbracket
x>0\right\rrbracket-\left\llbracket x<0\right\rrbracket$. (Note
$\operatorname{sgn}(0)=0$). Residues When $z,x$ are natural numbers we call
$\\{x\\}_{z}$ a residue of $z$, noting that there are just $z$ possible
residues, namely the set $\\{i_{\mathbb{N}}:0\leq i<z\\}$. We will also tend
to use We have basic results, if $(p_{\mathbb{N}},q_{\mathbb{N}})=1$ then
$\left\\{\,\\{r_{\mathbb{N}}p\\}_{q}\,\right\\}_{r=1}^{q-1}=\\{r\\}_{r=1}^{q-1}$,
ie the postfix operator $\times p$ induces a permutation on non-zero residues
of $q$ \- in particular for $q>1$ there is a unique $1\leq r_{p}\leq q-1$ with
$\\{r_{p}p\\}_{q}=1$ which we designate $\\{p\\}_{q}^{-1}$.

$\\{.\\}_{q}$ induces $+_{q\mathbb{I}}$ by:
$\\{x\\}_{q}+_{q\mathbb{I}}\\{y\\}_{q}=\\{x+y\\}_{q}$ \- need to show indt of
choice of $x,y$.

$\\{x+y\\}_{q}=\left\\{\\{x\\}_{q}+\\{y\\}_{q}\right\\}_{q}=\\{x\\}_{q}+\\{y\\}_{q}-\left\llbracket\\{x\\}_{q}+\\{y\\}_{q}\geq|z|\right\rrbracket|z|$.
$\\{rs\\}_{q}=\left\\{\\{r\\}_{q}\\{s\\}_{q}\right\\}_{q}$

Diophantine Approximation Given $a_{\mathbb{R}}$, the theory of Continued
Fractions provides a sequence
$\left(\frac{p_{r}}{q_{r}}\right)_{r=0}^{\infty}$ (convergents) of
exponentially improving rational approximations of $a$. These have long
provided a primary tool in our area of study, and there are many good
introductions, eg Hardy & Wright[6]. We will give a very brief summary from
the slightly novel viewpoint of our needs in this paper.

Recall that we can split any $x_{\mathbb{R}}$ into its integer and fractional
parts, ie $x=\left\lfloor x\right\rfloor+\left\\{x\right\\}$. In the sequel it
will be useful to adopt the convention that symbols with $.^{/}$ indicate real
numbers, and symbols without indicate integer numbers. In addition we will
regard $\infty$ as a positive integer reciprocal with $0$, so that
$\frac{1}{0}\coloneqq\infty=\left\lfloor\infty\right\rfloor+\frac{1}{\infty}$.
An integer, real therefore means an element of
$\widehat{\mathbb{Z}}=\mathbb{Z}\bigcup\\{\infty\\}$,$\widehat{\mathbb{R}}=\mathbb{R}\bigcup\\{\infty\\}$
respectively.

We start with a modernisation of the method of anthyphairesis (alternating
subtraction) developed by the early Greeks, probably by the School of
Pythagoras, and later a central component of what we now call Euclid’s
algorithm.

######  Definition 9.

The anthyphairetic relation is the recurrence relation
$a_{n}^{/}=a_{n}+\frac{1}{a_{n+1}^{/}}$ where $a_{n}=\left\lfloor
a_{n}^{/}\right\rfloor$. Given an initial real number $a^{/}$ and setting
$a_{0}^{/}=a^{/}$ the relation defines (for $n\geq 0$) 2 infinite sequences
$(a_{n}^{/}),(a_{n})$ of real numbers and their integer parts respectively.
333Technical note: Classically, expositions distinguish between finite
sequences (for $a_{0}^{/}$ rational) and infinite sequences (for $a_{0}^{/}$
irrational). For our purposes it is simpler to regard all sequences as
infinite, extending finite sequences with the integer $\infty$. In practice
this makes little difference to the discussion. For historical reasons, we
call $(a_{n}^{/})$ the sequence of complete quotients of $a^{/}$, and
$(a_{n})$ the sequence of partial quotients. The integer sequence $(a_{n})$ is
also called the anthyphairesis of the real number $a^{/}.$

Note for $n\geq 0$ that $\frac{1}{a_{n+1}^{/}}$ is the fractional part of
$a_{n}^{/}$, and so $1<a_{n+1}^{/}\leq\infty$ which also means $a_{n+1}\geq
1$. Also by definition $a_{n}\leq a_{n}^{/}<a_{n}+1$ unless
$a_{n}^{/}=a_{n}=\infty$.

In modern terms, the relation defines the anthyphairetic map
$\mathscr{A}:a^{/}\mapsto(a_{n})$ of real numbers to sequences of integers. It
is easy to see that this map is injective, and so it provides an invertible
encoding of reals by integers. The map is not surjective: if $a_{n}=\infty$
then also $a_{n+1}=\infty$ and $a_{n-1}\neq 1$ (with the single exception of
$A(1)=(1,\infty,\infty,\ldots)$). The image of $\mathscr{A}$ is therefore set
of anthyphaireses, a subset of all integer sequences.

The natural question arises as to whether we can construct the inverse map
$A^{-1}$ from the set of anthyphaireses to $[1,\infty]$. Note first that using
the anthyphairetic relation recursively gives us
$a_{0}^{/}=a_{0}+\frac{1}{a_{1}^{/}}=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}^{/}}}=...$.
These expressions are called Continued Fractions, although a better modern
term might be recursive fractions, namely fractions $\frac{a}{b}$ in which
$a,b$ may themselves be recursive fractions. Let us write the $n-$th Continued
Fraction as $CF_{n}^{/}(a_{0},\ldots,a_{n-1},a_{n}^{/})$, noting that if we
simplify the continued fraction back to a normal fraction we will end up with
$a_{0}^{/}=CF_{n}^{/}(a_{0},\ldots,a_{n}^{/})=\frac{P_{n}(a_{0},\ldots,a_{n}^{/})}{Q_{n}(a_{0},\ldots,a_{n}^{/})}$,
a rational polynomial in $a_{0},\ldots,a_{n}^{/}$. If $a_{0}^{/}$ is
irrational, so is $a_{n}^{/}$ and we have simply shifted the problem of
constructing the inverse from $a_{0}^{/}$ to $a_{n}^{/}.$444There is an
important exception: if the anthyphairesis is eventually periodic,
Euler/Lagrange showed that $a_{0}^{/}$ is a root of a quadratic equation which
can be recovered from the sequence.).

However if $a_{n}^{/}$ is itself an integer, $a_{0}^{/}$ is rational and easy
to calculate, and so we the effects of replacing $a_{n}^{/}$ with its integer
part $a_{n}$. This gives us a derived sequence of Continued Fractions
$CF_{n}=CF_{n}^{/}(a_{0}...a_{n})$, each of which now simplifies to some
normal fraction $\frac{p_{n}}{q_{n}}$. For example
$CF_{0}=\frac{p_{0}}{q_{0}}=a_{0}$,
$CF_{1}=\frac{p_{1}}{q_{1}}=a_{0}+\frac{1}{a_{1}}=\frac{a_{0}a_{1}+1}{a_{1}}$,
$CF_{2}=\frac{p_{2}}{q_{2}}=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}}}=\frac{a_{0}a_{1}a_{2}+a_{0}+a_{2}}{a_{1}a_{2}+1}$
etc.

The remarkable results of Continued Fraction theory are now that

(2.1)
$a_{0}^{/}-\frac{p_{n}}{q_{n}}=\frac{(-1)^{n}}{q_{n}\left(a_{n+1}^{/}q_{n}+q_{n-1}\right)}$

This means that $\left(\frac{p_{n}}{q_{n}}\right)$ is a sequence whose limit
is $a_{0}^{/}$, ie
$A^{-1}(a_{r})=\lim_{n\rightarrow\infty}\frac{p_{n}}{q_{n}}$, and we call
$\frac{p_{n}}{q_{n}}$ a convergent of $a_{0}^{/}$. Further, these are the best
possible rational approximations given the size of denominator, meaning that
$\left|a_{0}^{/}-\frac{p}{q}\right|<\left|a_{0}^{/}-\frac{p_{n}}{q_{n}}\right|$
requires $q>q_{n}$. Indeed each approximation is actually stronger: the
(stricter) inequality
$\left|qa_{0}^{/}-p\right|<\left|q_{n}a_{0}^{/}-p_{n}\right|$ in fact requires
$q\geq q_{n+1}$.

We will need two other basic results of Continued Fraction theory. We assume
each $\frac{p_{n}}{q_{n}}$ is in lowest terms. Then first there is an
invariant of the sequence given by the identity:

(2.2) $p_{n+1}q_{n}-p_{n}q_{n+1}=(-1)^{n}$

Second, $(p_{n}),(q_{n})$ both satisfy the continuant recurrence relation
$r_{n}=a_{n}r_{n-1}+r_{n-2},$which means:

$\displaystyle p_{n}$ $\displaystyle=a_{n}p_{n-1}+p_{n-2}$ $\displaystyle
q_{n}$ $\displaystyle=a_{n}q_{n-1}+q_{n-2}$

Note that from $\frac{p_{0}}{q_{0}}=a_{0}$ we get $p_{0}=a_{0},q_{0}=1$ and
from $\frac{p_{1}}{q_{1}}=a_{0}+\frac{1}{a_{1}}=\frac{a_{0}a_{1}+1}{a_{1}}$ we
get $p_{1}=a_{0}a_{1}+1$,$q_{1}=a_{1}$. Noting the recurrence relation can
also be written in descending form as $r_{n-2}=r_{n}-a_{n}r_{n-1}$, we can
then deduce the simpler initial conditions:

$\displaystyle p_{-1}$ $\displaystyle=1,p_{-2}=0$ $\displaystyle q_{-1}$
$\displaystyle=0,q_{-2}=1$

Continued Fraction theory began from the Greeks’ algebraic interest in the
relations between natural numbers, rationals and reals, and this algebraic
focus continued into exploring their connection with the solution of
Diophantine equations (ie solutions in integers or rationals). Our interest by
contrast is in using the results of the theory to study Dynamical Systems
consisting of rotations of the circle. This changes our view of the first
class citizens of the theory. So whereas the first class citizens in classical
theory are rational convergents and approximation errors, ours are periods of
closest return, and the periods of the associated return errors. Let us unpack
this.

Given a Dynamical System $(X,T)$ with $(X,d)$ a metric space, recall that a
period of closest return (or quasiperiod) to $x_{0}$ is an integer $n>0$ such
that $d(T^{m}x_{0},x_{0})<d(T^{n}x_{0},x_{0})$ requires $m>n$. In the system
$(\mathbb{T},R_{a_{0}^{/}})$ (being rotations of the circle through
$a_{0}^{/}$ revolutions), then the “best possible” result of classical
Continued Fraction theory are equivalent to the result that each $q_{n}$ is a
quasiperiod. The approximation error
$q_{n}a_{0}^{/}-p_{n}=\frac{(-1)^{n}}{a_{n+1}^{/}q_{n}+q_{n-1}}$ is equivalent
to saying that $q_{n}$ rotations through $a_{0}^{/}$ returns to a (directed)
distance of $\frac{(-1)^{n}}{a_{n+1}^{/}q_{n}+q_{n-1}}$ revolutions from the
starting point, ie it is the return error, which has an error period of
$a_{n+1}^{/}q_{n}+q_{n-1}$. Since this will be of primary interest we will
give it its own notation: motivated by the continuant relation for
quasiperiods, namely $q_{n+1}=a_{n+1}q_{n}+q_{n-1}$, we will denote the $n-$th
error period $q_{n+1}^{/}\coloneqq a_{n+1}^{/}q_{n}+q_{n-1}$.555Historical
note: the notation $q_{n+1}^{/}$ was in fact used by Hardy & Wright[6], but
purely as a convenient local shorthand for the expression
$a_{n+1}^{/}q_{n}+q_{n-1}$ in the proof of the inequality
$\left|\alpha-\frac{p_{n}}{q_{n}}\right|<\frac{1}{q_{n}q_{n+1}}$. It does not
seem to have been named, or used in further theory development. In fact
Lang[8] does not use any shorthand at all, always writing
$a_{n+1}^{/}q_{n}+q_{n-1}$ in full. We are not aware of distinct names being
introduced for $(q_{n}),(q_{n}^{/})$ previously.

In summary, the sequences $a_{n},a_{n}^{/}$ are first class citizens of both
classical theory and of the theory in this paper. However classical theory
then focuses on the convergents $\frac{p_{n}}{q_{n}}$ and their
numerators/denominators, whereas our focus is on the periods
$q_{n},q_{n}^{/}$. For convenience we give a formal definition:

######  Definition 10.

Given a real number $a_{0}^{/}$ with full quotient sequence $(a_{n}^{/})$ and
partial quotient sequence $(a_{n})$, we also define for $n\geq 0$ the
quasiperiod sequence of $a_{0}^{/}$ as $(q_{n}=a_{n}q_{n-1}+q_{n-1})$ and the
error-period sequence of $a_{0}^{/}$ as $(q_{n}^{/}=a_{n}^{/}q_{n-1}+q_{n-1})$
where $q_{-1}=0,q_{-2}=1$. Although of less interest, we also define
$p_{n},p_{n}^{/}$ analogously.

Finally we remark that a rotation through $a_{0}^{/}\geq 1$ is equivalent to a
rotation through $\alpha=\\{a_{0}^{/}\\}<1$. In this paper we focus on the
dynamics of irrational rotations, and so we will adopt the convention that the
symbol $\alpha$ is used to represent a real number (usually irrational) with
$0\leq\alpha\coloneqq\\{a_{0}^{/}\\}<1$, noting that
$A(\alpha)=CF(0,a_{1},a_{2}\ldots)$, ie $a_{0}=0$ and hence also
$\alpha=a_{0}^{/}(\alpha)=\frac{1}{a_{1}^{/}}$. It also gives
$p_{0}=0,p_{1}=1$.

######  Definition 11 (Diophantine type functions).

In estimating we find that it is useful to establish bounds on the growth of
the sequences $(q_{n}),(q_{n}^{/})$. Since $q_{n}=a_{n}q_{n-1}+q_{n-2}$ we
have for $n\geq 1$ $a_{n}q_{n-1}\leq q_{n}<(a_{n}+1)q_{n-1}$ so that
$\prod_{r=1}^{n}a_{r}\leq q_{n}<\prod_{r=1}^{n}(a_{r}+1)$ with equality only
for $n=1$. Define $a_{n}^{\max}=\max_{r\leq n}a_{r}$ then
$q_{n}<(a_{n}^{\max}+1)^{n}$. Note $a_{n}^{\max}$ is an increasing positive
sequence. Define $a_{n}^{\max},a_{n}^{/\max}$ and $A_{n}=\max_{r\leq
n}\left(\frac{q_{r}}{q_{r-1}}\right)$ and $A_{n}^{/}=\max_{r\leq
n}\left(\frac{q_{r}^{/}}{q_{r-1}}\right)$. Note
$a_{n}^{\max}<a_{n}^{/\max}<a_{n}^{\max}+1$ and $A_{n}<A_{n}^{/}<A_{n}+1$
$a_{n}^{\max}<A_{n}<a_{n}^{\max}+1$ $a_{n}^{/\max}<A_{n}^{/}<a_{n}^{/\max}+1$.
All are $O(a_{n}^{\max})$ and $A_{n}^{/}<a_{n}^{\max}+2$.

Estimates relating $n,q_{n}$ Since $q_{n}=a_{n}q_{n-1}+q_{n-2}$ we get
$q_{n}=\left(a_{n}a_{n-1}+1\right)q_{n-2}+q_{n-3}\geq 2q_{n-2}$ for $n\geq 2$
(equality only for $n=2)$. Hence for $n$ even $q_{n}\geq 2^{n/2}$ for $n\geq
2$ and for $n$ odd $q_{n}\geq 2^{(n-1)/2}q_{1}$ for $n\geq 1$. Hence
$n\leq\frac{2}{\log 2}\log q_{n}$ and $n\leq 1+\frac{2}{\log 2}(\log
q_{n}-\log q_{1})$ for $n$ odd.

Writing $\phi\coloneqq\frac{1}{2}\left(\sqrt{5}+1\right)$ for the golden
ration, we can get a slightly better estimate by observing that $q_{n}$
increases with increasing $a_{n}$ and so the lowest quasiperiods are those of
the golden rotation, amely $\phi-1=\frac{1}{2}\left(\sqrt{5}-1\right)$ with
$a_{i}=1$. Hence $q_{n}\geq
F_{n+1}=\frac{1}{\sqrt{5}}\left(\phi^{n+1}-(-\phi)^{-(n+1)}\right)$ giving
$n\leq\frac{1}{\log\phi}\left(\log
q_{n}+\log\sqrt{5}-\log\left(1-\phi^{-2(n+1)}\right)\right)-1$. Since
$\phi^{-2(n+1)}$ is a reducing function with $n$ we have for $n\geq 1$,
$n\leq\frac{1}{\log\phi}\left(\log
q_{n}+\log\sqrt{5}-\log\left(1-\phi^{-4}\right)\right)-1$. Now
$1-\phi^{-4}=\phi^{-2}(\phi^{2}+\phi^{-2})=\phi^{-2}F_{2}\sqrt{5}$ and
$F_{2}=1$ so that $\log\sqrt{5}-\log\left(1-\phi^{-4}\right)=2\log\phi$. So we
have $n\leq\frac{\log q_{n}}{\log\phi}+1$ for $n\geq 1$ with equality only for
$n=1$ and $q_{1}=1$ (and in fact the strict inequality still holds for $n=0$).

Also
$q_{n}=\prod_{r=1}^{n}\frac{q_{r}}{q_{r-1}}\leq\prod_{r=1}^{n}A_{r}\leq\left(A_{n}\right)^{n}$.
Hence

(2.3) $\frac{\log q_{n}}{\log A_{n}}\leq n\leq\frac{\log q_{n}}{\log\phi}+1$

where the left inequality holds for $q_{n}>1$ (and hence also $A_{n}>1)$, and
the right inequality holds for $n\geq 0$.666In this context it is tempting to
index the $a_{r}$ from 0 so $\alpha=[0;a_{0},a_{1},\ldots]$ as this gives the
nicely corresponding sequences
$(a_{r})_{r=0}^{\infty},(b_{r})_{r=0}^{\infty},(q_{r})_{r=0}^{\infty}$ with
$b_{r}\leq a_{r}$. However this would be a significant departure from a strong
historic convention which works well in a wider context. We will therefore
adhere to the established convention so that $\alpha=[0;a_{1},a_{2},\ldots]$
and $q_{r+1}=a_{r+1}q_{r}+q_{r-1}$$a_{r},q_{r}$, as elsewhere in the
literature.

Continued Fractions Preliminary Results We will need some additional simple
results in following sections. Given the maturity of CF theory, it is likely
that these results are not new, however we have not been able to find them
(perhaps they are hidden by notational differences), and so we will give a
proof here.

######  Lemma 12.

$p_{n}\equiv(-1)^{n+1}q_{n-1}^{-1}\bmod q_{n}$ where $r^{-1}$ signifies the
inverse of $r\bmod q$, ie $0\leq r^{-1}<q$ with $rr^{-1}\equiv 1\bmod q$. For
$\alpha=a_{0}^{/}<1$ this is equivalent to $p_{n}=q_{n-1}^{-1}$ ($n$ odd) and
$p_{n}=q_{n}-q_{n-1}^{-1}$ ($n$ even).

###### Proof.

The first part follows directly from expressing (2.2) $\bmod q_{n}$. For the
second part, note for $\alpha<1$ we have $p_{n}<q_{n}$, and the result
follows. ∎

######  Lemma 13.

We have the equalities $q_{n}^{/}=\prod_{1}^{n}a_{r}^{/}$ and
$\frac{p_{n}^{/}}{q_{n}^{/}}=a_{0}^{/}$

###### Proof.

Using the recurrence relations we have
$q_{n+1}^{/}=a_{n+1}^{/}q_{n}+q_{n-1}=a_{n+1}^{/}\left(a_{n}q_{n-1}+q_{n-2}\right)+q_{n-1}$.
The anthyphairetic relation gives $a_{n+1}^{/}a_{n}+1=a_{n+1}^{/}a_{n}^{/}$
and hence
$q_{n+1}^{/}=a_{n+1}^{/}\left(a_{n}^{/}q_{n-1}+q_{n-2}\right)=a_{n+1}^{/}q_{n}^{/}$.
This gives us $q_{n}^{/}=\left(\prod_{1}^{n}a_{r}^{/}\right)q_{0}^{/}$. Now
$q_{0}^{/}=a_{0}^{/}q_{-1}+q_{-2}=1$, and the first result follows. A similar
argument gives $p_{n}^{/}=\left(\prod_{1}^{n}a_{r}^{/}\right)p_{0}^{/}$ and
$p_{0}^{/}=a_{0}^{/}p_{-1}+p_{-2}=a_{0}^{/}$, and the second result follows. ∎

######  Definition 14.

We define the upper Gauss map $\Gamma:[0,\infty]\rightarrow[1,\infty]$ by
$x\mapsto\frac{1}{x-\left\lfloor x\right\rfloor}$, noting this defines the
left shift on the sequence of full quotients, ie
$\Gamma(a_{n}^{/})=a_{n+1}^{/}$.777It is also the conjugate of the lower Gauss
map $\gamma:[0,1]\rightarrow[0,1]$ by $\Gamma=R\gamma R$ where $R$ is the
involution $x\rightarrow\frac{1}{x}$ on $[0,\infty]$

Recall we define the dual rotation $\overline{\alpha}\coloneqq 1-\alpha$. For
any sequence of observables $x_{r}\coloneqq x_{r}(\alpha)$ we define the dual
observable values $\overline{x}_{r}\coloneqq x_{r}(\overline{\alpha})$

$a_{0}^{/}=\alpha,a_{n}=\left\lfloor
a_{n}^{/}\right\rfloor,a_{n+1}^{/}=\frac{1}{a_{n}^{/}-a_{n}}$ so for
$\alpha<\frac{1}{2}$ $a_{1}=\left\lfloor\frac{1}{\alpha}\right\rfloor\geq
2,a_{0}^{/}=\alpha,a_{1}^{/}=\frac{1}{\alpha},\overline{a}_{0}^{/}=1-\alpha,\overline{a}_{1}^{/}=\frac{1}{1-\alpha}$
and
hence$\overline{a}_{2}^{/}=\frac{1}{\overline{a}_{1}^{/}-\overline{a}_{1}}=\frac{1}{\frac{1}{1-\alpha}-1}=\frac{1-\alpha}{\alpha}=\frac{1}{\alpha}-1=a_{1}^{/}-1$.
Hence $\overline{a}_{2}=a_{1}-1$

######  Lemma 15.

For $0<\alpha<\frac{1}{2}$, the quotients
$\overline{a}_{r}^{/},\overline{a}_{r}$ of $\overline{\alpha}$ satisfy the
identity $\overline{x}_{r+1}=x_{r}$ for $r\geq 2$, and further
$\overline{q}_{r}^{/}$ satisfies it for $r\geq 1$, and $\overline{q}_{r}$ for
$r\geq 0$. In addition the early sequence values are
$\overline{a}_{0}^{/}=1-\alpha,\overline{a}_{1}^{/}=\frac{1}{1-\alpha},\overline{a}_{2}^{/}=\frac{\alpha}{1-\alpha}$,
$\overline{a}_{0}=0,\overline{a}_{1}=1,\overline{a}_{2}=a_{1}-1$,
$\overline{q}_{0}^{/}=1,\overline{q}_{1}^{/}=\frac{1}{1-\alpha}$ and
$\overline{q}_{0}=1$.

###### Proof.

Standard results from earlier in this section immediately give us
$\overline{a}_{0}^{/}=1-\alpha,\overline{q}_{0}=1,\overline{q}_{0}^{/}=1$, and
also since $1-\alpha<1$,
$\overline{a}_{0}=0,\overline{q}_{1}^{/}=\overline{a}_{1}^{/}=\frac{1}{1-\alpha}$
(note that $\overline{q}_{1}^{/}\neq q_{0}^{/}$,$\overline{a}_{1}^{/}\neq
a_{0}^{/}$).

Now we use $0<\alpha<\frac{1}{2}$ so that $\frac{1}{2}<\overline{\alpha}<1$
and hence $\overline{a}_{1}^{/}=\frac{1}{1-\alpha}<2$, giving
$\overline{q}_{1}=\overline{a}_{1}=1=q_{0}$. We can now calculate
$\overline{a}_{2}^{/}=\frac{1}{\overline{a}_{1}^{/}-\overline{a}_{1}}=\frac{1}{\frac{1}{1-\alpha}-1}=\frac{1-\alpha}{\alpha}=\frac{1}{\alpha}-1=a_{1}^{/}-1$
and hence $\overline{a}_{2}=a_{1}-1$. These results then give
$\overline{q}_{2}=(a_{1}-1).1+1=a_{1}=q_{1}$ and similarly
$\overline{q}_{2}^{/}=(a_{1}^{/}-1).1+1=a_{1}^{/}=q_{1}^{/}$. Finally
$\overline{a}_{3}^{/}=\frac{1}{\overline{a}_{2}^{/}-\overline{a}_{2}}=\frac{1}{\left(a_{1}^{/}-1\right)-\left(a_{1}-1\right)}=a_{2}^{/}$
and hence also $\overline{a}_{3}=a_{2}$. This then gives
$\overline{q}_{3}^{/}=\overline{a}_{3}^{/}\overline{q}_{2}^{/}=a_{2}^{/}q_{1}^{/}=q_{2}^{/}$,
and we have established the early sequence values.

Now for $n\geq 2$, we can write
$\overline{a}_{n+1}^{/}=\Gamma^{n-2}\overline{a}_{3}^{/}=\Gamma^{n-2}a_{2}^{/}=a_{n}^{/}$
and hence also $\overline{a}_{n+1}=a_{n}$. The recurrence relations for
$\overline{q}_{n+1}^{/},\overline{q}_{n+1}$ quickly give the results for these
sequences for $n\geq 2$, and the early sequence results extend them to the
claimed starting values. ∎

We now develop some useful identities

######  Lemma 16.

Identities:

  1. (1)

For $n\geq 0$ and $a_{0}^{/}$ irrational we have $1\leq
a_{n+1}<a_{n+1}^{/}=a_{n+1}+\frac{1}{a_{n+2}^{/}}<a_{n+1}+1$

  2. (2)
  3. (3)

For $n>0$,
$q_{n}<q_{n}^{/}<q_{n}+q_{n-1}<\min\left(\left(1+\frac{1}{a_{n}}\right)q_{n},q_{n+1}\right)\leq
2q_{n}$ Hence
$\prod_{r=1}^{n}a_{r}<q_{n}<q_{n}^{/}=\prod_{r=1}^{n}a_{r}^{/}=\prod_{r=1}^{n}(a_{r}+\frac{1}{a_{r+1}^{/}})<\prod_{r=1}^{n}(a_{r}+1)$

  4. (4)

$a_{n+1}^{/}(a_{n}^{/}-a_{n})=1$ (equivalent to
$a_{n+1}^{/}a_{n}^{/}=a_{n+1}^{/}a_{n}+1$ hence $a_{n+1}^{/}a_{n}^{/}>2$ and
$q_{n+1}^{/}>2q_{n-1}^{/}$) and hence
$q_{n}^{/}=q_{n}+\frac{q_{n-1}}{a_{n+1}^{/}}<q_{n}\left(1+\frac{1}{a_{n}a_{n+1}^{/}}\right)$.
This is $<\frac{3}{2}q_{n}$ unless $a_{n+1}=a_{n}=1$

  5. (5)

$\frac{1}{q_{r+2}^{/}}+\frac{a_{r+1}}{q_{r+1}^{/}}=\frac{1}{q_{r}^{./}}\leq\frac{1}{q_{r}^{.}}$
(equality only for $r=0$)

  6. (6)

$\frac{1}{q_{r}}-\frac{1}{q_{r}^{/}}=\frac{q_{r-1}}{q_{r}q_{r+1}^{/}}\leq\min\\{\frac{1}{q_{r+1}^{/}},\frac{1}{2q_{r}}\\}\leq\frac{1}{2q_{r}}$
with equality only possible for $r=1,q_{1}=1$

###### Proof.

$\frac{1}{q_{r+2}^{/}}+\frac{a_{r+1}}{q_{r+1}^{/}}=\frac{q_{r+1}^{/}+a_{r+1}q_{r+2}^{/}}{q_{r+2}^{/}q_{r+1}^{/}}=\frac{\left(1+a_{r+1}a_{r_{+2}}^{/}\right)q_{r+1}^{/}}{q_{r+2}^{/}q_{r+1}^{/}}=\frac{\left(a_{r+1}^{/}a_{r_{+2}}^{/}\right)}{q_{r+2}^{/}}=\frac{1}{q_{r}^{/}}$
and $q_{r}^{/}\geq q_{r}$ with equality only for $r=0$

$\frac{1}{q_{r}}-\frac{1}{q_{r}^{/}}=\frac{q_{r}^{/}-q_{r}}{q_{r}q_{r}^{/}}=\frac{\left(a_{r}^{/}-a_{r}\right)q_{r-1}}{q_{r}q_{r}^{/}}=\frac{\frac{1}{a_{{}_{r+1}}^{/}}q_{r-1}}{q_{r}q_{r}^{/}}=\frac{q_{r-1}}{q_{r}q_{r+1}^{/}}$.
The inequalities follow from $q_{r-1}\leq q_{r}$ (equality only possible for
$r=1,\alpha>\frac{1}{2}$), and $q_{r+1}^{/}>q_{r+1}>2q_{r-1}$ ∎

Ostrowski Representation Ostrowski introduced this representation in his paper
(##paper) studying the sum of remainders (the Birkhoff sum
$\sum_{r=1}^{N}\left\\{r\alpha\right\\}$). The precise application of the tool
in that paper does not seem to have a natural generalisation, but the tool
itself it has found many other applications since, and it will be a primary
tool in our study.

We first recall the idea of a non-standard number system for representing
integers. In a standard number system we use a radix $q>1$ to represent an
integer $N$ by taking the sequence of weights $1=q^{0}<q^{1}<q^{2}...<q^{n}$
defined by $q^{n}\leq N<q^{n+1}$ and finding integer coefficients $b_{r}\geq
0$ such that $N=\sum_{r=0}^{n}b_{r}q^{r}$. In general $N$ has many such
representations, but if we impose the condition that we maximise $b_{n}$, then
$b_{n-1}$ and so on down to $b_{0}$, then the $b_{r}$ are determined and we
have a canonical representation. In a non-standard number system we take an
arbitrary unbounded sequence of weights $1=q_{0}\leq q_{1}\leq q_{2}...$ and
apply precisely the same procedure to arrive at a canonical representation
$N=\sum_{r=0}^{n}b_{r}q_{r}$. (We can allow $q_{r}=q_{r+1}$ because maximising
$b_{r+1}$ before $b_{r}$ will result in $b_{r}=0$).

######  Definition 17.

For $N\in\mathbb{N}$ the Ostrowski representation $O_{\alpha}(N)$ is the
canonical representation $N=\sum_{r=0}^{n}b_{r}q_{r}$ in the non-standard
number system defined by the quasiperiods $q_{r}$ of $\alpha$.

Preliminary results In this section we will build a number of simple results
from the foregoing which will be useful in our main development.

We note the following characteristics of the Ostrowski representation:

######  Lemma 18.

Let $\alpha$ have partial quotient sequence $(a_{n})$ and quasiperiod sequence
$(q_{n}),$and suppose $N$ has Ostrowski Representation
$\sum_{r=0}^{n}b_{r}q_{r}$, then:

  1. (1)

$b_{r}\leq a_{r+1}$

  2. (2)

$b_{0}<a_{1}$

  3. (3)

$b_{r}=a_{r+1}\,\Longrightarrow\,b_{r-1}=0$

  4. (4)
  5. (5)

$q_{1}=1\,\Longrightarrow b_{0}=0$

###### Proof.

The crucial insight is that if $b_{r}q_{r}\geq q_{r+1}$ then $b_{r+1}$ is not
the maximal coefficient of $q_{r+1}$. So in the canonical representation we
must have $b_{r}q_{r}<q_{r+1}$. Since $q_{r+1}=a_{r+1}q_{r}+q_{r-1}$ and
$q_{r}>0$ for $r\geq 0$, this gives $b_{r}<a_{r+1}+\frac{q_{r-1}}{q_{r}}$ for
$r\geq 0$. Recall $q_{-1}=0,q_{0}=1,q_{1}\geq 1,q_{2}>1$ and so for
$r=0,$$b_{0}<a_{1}$ and for $r\geq 1$, $b_{r}\leq a_{r+1}$. Also if $q_{1}=1$
then $a_{1}=1$ and then $b_{0}<a_{1}$ gives $b_{0}=0$. Finally, refining the
initial insight, we note that if $b_{r}=a_{r+1}$ then
$b_{r}q_{r}+b_{r-1}q_{r-1}=q_{r+1}+(b_{r-1}-1)q_{r-1}$ which again means that
$b_{r+1}$ cannot be maximal if $b_{r-1}-1\geq 0$. ∎

##  3\. Theory of Hom-set Magmas

###  3.1. Introduction

Given the Birkhoff sum $S_{N}(\phi,x_{0})=\sum_{r=1}^{N}\phi(x_{0}+r\alpha)$,
we can rewrite this in curried form as $S_{N}^{x_{0}}\phi$, ie as the action
of an operator $S_{N}^{x_{0}}$ with signature $(\mathbb{TR})\mathbb{R}$
(see**) mapping the function space of observables $\\{\phi_{\mathbb{TR}}\\}$
to $\mathbb{R}$. More specifically therefore, $S_{N}^{x_{0}}$ is a functional.
In later sections we will decompose $S_{N}^{x_{0}}$ into a sum of many
functionals of the same signature. This leads to much repetitive work which
can be saved by a structured understanding of a number of related underlying
dualities.

In this section we will develop the algebraic theory which contains these
dualities, which seems of interest in itself. Amongst other things, it
delivers a set of 8 dual relations between functionals of signature $(ST)T$
built from a Source structure $(S,\sigma_{S},R_{S})$ and a Target structure
$(T,\tau_{T},R_{T})$ equipped with endomorphisms $\sigma,\tau$ and
endorelations $R_{S},R_{T}$. If any one of the dual relations hold, we can
immediately deduce the other 7.

We will later (Section 7.2) apply this theory to the environment of
$(\mathbb{T},\overline{.},\leq_{\mathbb{T}}),(\mathbb{R},-_{\mathbb{R}},\leq_{\mathbb{R}})$.
Although this is an extremely simple environment in which to apply the theory,
we have still found this approach has saved duplicated effort, and perhaps
more importantly, confusion, by providing a rigorous and systematic approach
to manipulating these relations. The General Hom-set Magma For a brief summary
of free magmas and the height function $h$, and also of hom-sets, see the
sections on generated structure and on hom-sets in subsection 3.

Let $X=\\{X_{\alpha}\\}$ be a set of sets indexed by another set $A$. We will
call each $X_{\alpha}$ a base set, or height 0 set, and set $M_{0}(X)\coloneqq
X$. Recall that we write $AB$ for the set of functions from $A$ to $B$. For
$n\geq 1$ define $M_{n}(X)=\left\\{AB:A\in M_{n-1}(X)|B\in
M_{n-1}(X)\right\\}$ (ie at least one of $A,B$ is in $M_{n-1}(X)$). Each $m\in
M_{n}(X)$ is a hom-set and we say that it has height $n$ (written $h(m)=n$)
and is of type $(h(A),h(B))$. Note that $h(AB)=\max(h(A),h(B))+1$. Finally we
define $M(X)=\bigcup_{n=0}^{\infty}M_{n}(X)$ called the Magma of hom-sets on
$X$. In the sequel we will find it more convenient to refer to the height of
$AB$ as its Level.

Note that if $A,B$ are sets then $AB$ is an element of $P^{2}\left(A\times
B\right)$ where $P$ is the power set operator on sets, and hence together with
$M(X)$ itself satisfies the requirements of a set in $ZF$ so that there are no
foundational issues.

There is however another technical issue: we cannot easily guarantee that the
generating set $X$ is free. Hence $M(X)$ may not be a free magma, and a given
element $m$ may appear at different levels, with different tree
representations. We avoid this issue by regarding $M(X)$ as a family of sets
indexed by the free magma on the symbols ${}^{\prime}X_{\alpha}^{\prime}$.
Then when we talk about $AB\in M(X)$ we must specify which representation tree
in the free magma we have in mind, and then the height, width and type of $AB$
are well defined.

Further notes on notation and annotation We will adopt the conventions of
operator theory in the sequel, and call functions of Level 2 and above
operators. Further, we will call operators of type $(n,0)$ (which map
functions to a base set) functionals. We will use greek letters for functions
of level 1 (which are necessarily of type $(0,0)$), eg
$\phi_{X_{\alpha}X_{\beta}},$ and capital letters for functionals, eg
$A_{(X_{\alpha}X_{\beta})X_{\beta}}$ (so that we can correctly write
$\left(A_{(X_{\alpha}X_{\beta})X_{\beta}}\phi_{X_{\alpha}X_{\beta}}\right)_{X_{\beta}}$
). We also need to refer to relations and will use the symbol
${}^{\prime}R^{\prime}$ to indicate these. In particular $R_{XX}$ refers to an
endorelation on $X$.

For convenience we will also introduce a condensed endo annotation by writing
the annotation $T^{\circ}$ for the annotation $TT$, and
$T^{\circ\circ}=(TT)^{\circ}=(TT)(TT)$ etc. This applies to both endomorphisms
and endorelations. For example the level 2 endomorphism $\alpha_{(XY)(XY)}$ on
$XY$ can be written $\alpha_{\left(XY\right)^{\circ}}$, and the level 3
endomorphism $\sigma_{\left((ST)(ST)\right)\left((ST)(ST)\right)}$ on
$(ST)^{\circ}$ can be written $\sigma_{(ST)^{\circ\circ}}$. Note that the
presence of $.\text{\textdegree}$ always indicates that the annotated morphism
is an endomorphism. Later when we hope the reader has become comfortable with
which symbols represent endomorphism or endorelations, we will drop the
$.^{\circ}$ annotation altogether, so that the (Level 2) annotated
endomorphism $\sigma_{(ST)^{\circ}}$ will be annotated even more succinctly as
$\alpha_{ST}$. This of course has the risk of confusing it with the (Level 1)
function $\alpha_{ST}$. The quack rule applies: if it looks like an
endomorphism, and behaves like an endomorphism, it probably is a higher level
endomorphism rather than a lower level function.

We will also adopt the convention of using annotation only when there are
changes in the referents, reading left to right. For example in the formula
$\sigma_{(ST)^{\circ}}\phi_{ST}=\phi\circ\sigma_{S^{\circ}}$, the symbol
${}^{\prime}\sigma^{\prime}$ changes referents (from Level $2$ of type $(1,1)$
to Level $1$ of type $(0,0)$) whereas the symbol ${}^{\prime}\phi^{\prime}$
has the same referent in both places.

Finally we use the symbol ${}^{\prime}\circ^{\prime}$ conventionally as the
binary operator which composes morphisms, so that the function composition
$(f_{TZ}\circ g_{ST})_{SZ}$ represents the function $s_{S}\mapsto f(g(s))$,
with the appropriate corresponding definition for relations.

Source-Target Magma In this section we will be working with Levels $0$ to $3$
of the hom-set magma $M(X)$ on a pair of Level $0$ sets $X=\\{S,T\\}$. We will
be focusing on morphisms from $S$ to $T$, and call $S$ the source, and $T$ the
target structure. We will here point out the hom-sets of particular interest.

Level 1 hom-sets contain functions of type $(0,0)$, ie between ordered pairs
of the Level 0 sets $S,T$. In particular the hom-sets $SS,TT$ (or
$S^{\circ},T^{\circ}$) are the endomorphisms of $S$ and $T$ respectively, and
$ST$ is the the hom-set of function from the source to the target. The other
Level $1$ hom-set is $TS$, but we shall not focus on this here.

Level 2 hom-sets contain Level 2 operators (ie involving Level 1) objects, and
we shall focus on:

  1. (1)

$(ST)(ST)$ or $(ST)^{\circ}$, the hom-set of type (1,1) endomorphisms mapping
functions in $ST$ to functions in $ST$ ,

  2. (2)

$(ST)T$ the hom-set of type $(1,0)$ operators (functionals) which map
functions in $ST$ to “values”, ie to elements of the base set $T$.

Level 3 hom-sets contain Level 3 operators (ie involving Level 2 objects), and
we shall focus on $\left((ST)T\right)\left((ST)T\right)$ or
$\left((ST)T\right)^{\circ}$, the hom-set of endomorphisms of type $(2,1)$
mapping functionals to functionals in $(ST)T$.

We will also develop the concept of a Pull Up. As the name suggests, Pull Ups
map objects of one level to a higher level, unlike functionals which map
objects down to the base level. Technically therefore pull ups are operators,
but there is less value in surfacing them explicitly, and we will simply
explore their results. In particular we will introduce pull ups of Level 1
objects to Levels 2 and 3, and of Level 2 objects to Level 3. Objects may be
functions or relations. Pull Ups are specialisations of the well known pull
back and push forward constructions.

###  3.2. Pull Ups of EndoMorphisms

Level 2 Pull Ups of Endomorphisms on $S,T$

######  Definition 19 (Pull Up Endomorphisms).

Let $\sigma_{SS},\tau_{TT}$ be Level 1 endomorphisms of $S,T$ respectively.

We now “pull back” $\sigma_{SS}$ to $ST$ to induce a level 2 endomorphism
$\sigma_{(ST)(ST)}$ which is of type $(1,1)$ on $ST$. Using the condensed
endomorphism annotation, we have:

(3.1)
$\sigma_{(ST)^{\circ}}\coloneqq\phi_{ST}\mapsto(\phi\circ\sigma_{S^{\circ}})_{ST}$

We call $\sigma_{(ST)^{\circ}}$ the (source) pull up of $\sigma_{S^{\circ}}$
from $S$ to $ST$ . Similarly we can “push forward” $\tau_{T^{\circ}}$ to
induce the (target) pull up of $\tau_{T^{\circ}}$ from $T$ to $ST$, which is
again a level 2 endomorphism of type $(1,1)$:

(3.2)
$\tau_{(ST)^{\circ}}\coloneqq\phi_{ST}\mapsto\left(\tau_{T^{\circ}}\circ\phi\right)_{ST}$

Note the reversal of the order of composition between the pull ups of source
and target endomorphisms.

######  Lemma 20.

The pull up of an involution is also an involution.

###### Proof.

Let $\sigma_{\left(ST\right)^{\circ}}$ be the source pull up of
$\sigma_{S^{\circ}}$. Then

(3.3)
$\sigma_{(ST)^{\circ}}^{2}\phi=\sigma(\sigma\phi)=\sigma(\phi\circ\sigma_{S^{\circ}})=(\phi\circ\sigma_{S^{\circ}})\circ\sigma_{S^{\circ}}=\phi\circ\sigma_{S^{\circ}}^{2}$

so that if $\sigma_{S^{\circ}}$ is an involution (ie $\sigma^{2}=Id$) then
$\sigma_{(ST)^{\circ}}$ is also an involution. A similar argument works for
target pull ups. ∎

######  Lemma 21.

Two source (or target) pull ups commute if their underlying endomorphisms
commute. Source and target pull ups always commute.

###### Proof.

Let $\sigma_{S^{\circ}},\varsigma_{S^{\circ}}$ be two commuting endomorphisms.
Using the associativity of composition we get

(3.4)
$\left(\sigma_{(ST)^{\circ}}\circ\varsigma_{(ST)^{\circ}}\right)\phi_{ST}=\sigma(\varsigma(\phi))=(\phi\circ\varsigma_{S^{\circ}})\circ\sigma_{S^{\circ}}=\phi\circ(\varsigma\circ\sigma)=\phi\circ(\sigma\circ\varsigma)=(\varsigma_{(ST)^{\circ}}\circ\sigma_{(ST)^{\circ}})\phi$

A similar argument establishes the result for two target endomorphisms. By
another similar argument:

(3.5)
$\left(\sigma_{(ST)^{\circ}}\circ\tau_{(ST)^{\circ}}\right)\phi_{ST}=\sigma(\tau(\phi))=(\tau_{TT}\circ\phi)\circ\sigma_{SS}=\tau\circ(\phi\circ\sigma)=\tau_{(ST)^{\circ}}(\sigma_{(ST)^{\circ}}(\phi))=(\tau\circ\sigma)\phi$

and so $\sigma_{(ST)^{\circ}},\tau_{(ST)^{\circ}}$ always commute. ∎

######  Remark 22.

If two involutions $\sigma,\tau$ commute then
$\left(\sigma\tau\right)^{2}=\sigma^{2}\tau^{2}=Id$ so that $\sigma\tau$ is
also an involution.

Level 3 Pull Ups of endomorphisms on $S,T$ We can now use the above results
about $ST$ to deduce similar results about $(ST)T$ by replacing $S$ with $ST$
and keeping $T$ unchanged. Note that $(ST)T$ is a level 2 set of morphisms of
type $(1,0)$ and so $\left((ST)T\right)^{\circ}$ is a level 3 set of type
$(2,2)$ morphisms.

Proceeding as in the previous subsection, the pullback of
$\sigma_{\left(ST\right)^{\circ}}$ gives us the source pull up:

(3.6) $\displaystyle\sigma_{\left((ST)T\right)^{\circ}}$
$\displaystyle\coloneqq A_{(ST)T}\mapsto
A\circ\sigma_{\left(ST\right)^{\circ}}$

and the pushforward of $\tau_{T^{\circ}}$ gives us the target pull up:

(3.7) $\tau_{\left((ST)T\right)^{\circ}}\coloneqq
A_{(ST)T}\mapsto\tau_{T^{\circ}}\circ A$

However this time we have an additional endomorphism on the source $ST$,
namely $\tau_{\left(ST\right)^{\circ}}$ which we can again pull back. The
natural labelling of this new pull up would be
$\tau_{\left((ST)T\right)^{\circ}}$ but have just used that symbol and
annotation for the pull up of $\tau_{T^{\circ}}$! Instead, since the new pull
up is a source pull up, we will label it
$\sigma_{\left((ST)T\right)^{\circ}}^{/}$ to get:

(3.8) $\displaystyle\sigma_{\left((ST)T\right)^{\circ}}^{/}$
$\displaystyle\coloneqq A_{(ST)T}\mapsto A\circ\tau_{\left(ST\right)^{\circ}}$

By the second part of Lemma 21 we have immediately that
$\sigma_{\left((ST)T\right)^{\circ}},\tau_{\left((ST)T\right)^{\circ}}$
commute and that
$\sigma_{\left((ST)T\right)^{\circ}}^{/},\tau_{\left((ST)T\right)^{\circ}}$
commute. The same part of the lemma also gives us that
$\sigma_{\left(ST\right)^{\circ}},\tau_{\left(ST\right)^{\circ}}$ commute, and
then we can apply the first part to find that
$\sigma_{\left((ST)T\right)^{\circ}},\sigma_{\left((ST)T\right)^{\circ}}^{/}$
commute.

It follows by repeated application of Lemma 20 that if $\sigma_{S^{\circ}}$ is
an involution, so is $\sigma_{\left((ST)T\right)^{\circ}}$, and if
$\tau_{T^{\circ}}$ is an involution, so are
$\tau_{\left((ST)T\right)^{\circ}}$ and
$\sigma_{\left((ST)T\right)^{\circ}}^{/}$. If both
$\sigma_{S^{\circ}},\tau_{T^{\circ}}$are involutions, so are all the 8
compositions of the three Level 3 . Hence we have shown:

######  Theorem 23.

If $S,T$ are equipped with involutions $\sigma_{S^{\circ}},\tau_{T^{\circ}}$
respectively, then the space of functionals $(ST)T$ is equipped with a
$\left(C_{2}\right)^{3}$ group of involutions, generated by the 3 involutions
$\sigma,\sigma^{/},\tau$ of $\left((ST),T\right)^{\circ}$ which are themselves
induced by $\sigma_{S^{\circ}},\tau_{T^{\circ}}$.

###  3.3. Pull Ups of EndoRelations

The theory we have developed so far could be given purely within the context
of Category Theory (ie in terms of objects and morphisms without defining what
we mean by them). However at this stage our progress is easiest if we
introduce the standard set theoretic notion of Relations as elements of sets.

######  Definition 24.

We define a relation $R_{ST}$ between $S,T$ as a subset of $S\times T$, ie an
element of the power set of the Cartesian product of $S,T$. We write
$s_{S}\,R_{ST}\,t_{T}$ to mean $(s,t)\in R_{ST}$ and we also define the dual
or opposite relation $R_{TS}^{Op}=\left\\{(t,s):(s,t)\in R_{ST}\right\\}$. If
$S=T$, then $R$ and $R^{Op}$ are called endorelations on $S$.

Note that $Op$ is an involution on relations, ie $(R^{Op})^{Op}=R$, and that
$sRt\Leftrightarrow tR^{Op}s$.

We now suppose that $S,T$ are equipped with endorelations
$R_{S^{\circ}}\subseteq S\times S,R_{T^{\circ}}\subseteq T\times T$ in
addition to the endomorphisms $\sigma_{S^{\circ}},\tau_{T^{\circ}}$. As usual
if we have $(s_{1},s_{2})\in R_{S^{\circ}}$ we will write this as
$s_{1}\,R_{S^{\circ}}\,s_{2}$, and we define the opposite relation
$R_{S^{\circ}}^{Op}$ by $s_{1}\,R_{S^{\circ}}\,s_{2}\Leftrightarrow
s_{2}\,R_{S^{\circ}}^{Op}\,s_{1}$, and then $R_{T^{\circ}}^{Op}$ similarly.

######  Definition 25.

A relation homomorphism from $R_{S^{\circ}}$ to $R_{T^{\circ}}$ is a morphism
$\phi_{ST}$ such that $\phi\times\phi$ is an element of
$R_{S^{\circ}}R_{T^{\circ}}$, ie
$s_{1}R_{S^{\circ}}s_{2}\Leftrightarrow(s_{1},s_{2})\in
R_{S^{\circ}}\Rightarrow(\phi s_{1},\phi s_{2})\in
R_{S^{\circ}}\Leftrightarrow\phi s_{1}R_{T^{\circ}}\phi s_{2}$. A relation
anti homomorphism from $R_{S^{\circ}}$ to $R_{T^{\circ}}$ is a relation
homomorphism from $R_{S^{\circ}}$ to $R_{T^{\circ}}^{Op}$, and has the
opposite result, ie $s_{1}R_{S}s_{2}\Rightarrow\phi s_{1}R_{T}^{Op}\phi
s_{2}\Leftrightarrow\phi s_{2}R_{T}\phi s_{1}$. . If
$R_{S^{\circ}},R_{T^{\circ}}$ are both functions we call $\phi_{ST}$ a
function homomorphism.

######  Remark 26.

Note that a relation anti-morphism of $R_{S^{\circ}}R_{T^{\circ}}$ transposes
its images under $R_{T^{\circ}}$, so that a composition of a relation morphism
and anti-morphism is a relation anti-morphism, whilst a composition of two
relation morphisms or two relation anti-morphisms is a relation morphism.
Further if $\phi$ is an $R_{S^{\circ}}R_{T^{\circ}}$ morphism, then we have
$s_{2}R_{S^{\circ}}^{Op}s_{1}\Leftrightarrow
s_{1}R_{S^{\circ}}s_{2}\Rightarrow\phi s_{1}R_{T^{\circ}}\phi
s_{2}\Leftrightarrow\phi s_{2}R_{T^{\circ}}^{Op}\phi s_{1}$ so that $\phi$ is
also an $R_{S^{\circ}}R_{T^{\circ}}^{Op}$, $R_{S^{\circ}}^{Op}R_{T^{\circ}}$
anti morphism, and an $R_{S^{\circ}}^{Op}R_{T^{\circ}}^{Op}$ morphism.

Now suppose that $\sigma_{S^{\circ}}$ is an $R_{S^{\circ}}R_{S^{\circ}}$ anti-
morphism, and that similarly $\tau_{T^{\circ}}$ is an
$R_{T^{\circ}}R_{T^{\circ}}$ anti-morphism. Then if $\phi$ is also a relation
anti-morphism of $R_{S^{\circ}}R_{T^{\circ}}$, then
$\sigma_{ST}(\phi)=\phi\circ\sigma_{S^{\circ}},\tau_{ST}(\phi)=\tau_{T}\circ\phi$
are both relation morphisms of $R_{S^{\circ}}R_{T^{\circ}}$ and
$(\sigma\circ\tau)\circ\phi$ is a relation anti-morphism of
$R_{S^{\circ}}R_{T^{\circ}}$.

—

######  Remark 27.

When $R$ is a function we write $xRy$ as $y=R(x)$, so a function homomorphism
has $s_{2}=R_{s}(s_{1})\Rightarrow\phi s_{2}=R_{T}(\phi s_{1})$ hence
$\phi(R_{S}s_{1})=R_{T}(\phi s_{1})$ and so $\phi\circ R_{S}=R_{T}\circ\phi$.

Level 2 Pull Ups of endorelations from $T$ We define in this section a pull up
of an endorelation on $T$. This pull up is _not_ the same operation as the
pull up of a function: there are many possible pull ups, and we will only
define the operation in one direction, as a pull back from the target.
Nevertheless the term pull up still seems appropriate.

######  Definition 28.

For any subset $Z\subseteq S$, the pull up of the endorelation $R_{T}$ to be
the endorelation $\left(R_{Z}\right)_{(ST)^{\circ}}$ on $ST$ defined by
$\phi_{1}\,R_{Z}\,\phi_{2}\Leftrightarrow(\phi_{1}s\,R_{T^{\circ}}\,\phi_{2}s$
for all $s_{Z})$. If $Z=\emptyset$, $R_{Z}$ is the full relation.

######  Remark 29.

Recall that that we may allow ourselves at any point to change annotation at
the risk of confusion. This seems to be a point where improvement in
legibility outweighs the risk.

We will use $\sigma$ (without annotation) to signify $\sigma_{T^{\circ}}$,
$\sigma_{ST}$ to signify $\sigma_{(ST)^{\circ}}$, and analogously for the
symbol $\tau$. We will also call a morphism of $R_{Z}R_{Z}$ an endomorphism of
$R_{Z}$.

######  Remark 30.

Note that if $Z^{/}\subseteq Z$ then
$\phi_{1}\,R_{Z}\,\phi_{2}\Rightarrow\phi_{1}\,R_{Z^{/}}\,\phi_{2}$, ie
$R_{Z}\subseteq R_{Z^{/}}$.

Given $\phi_{1},\phi_{2}$ such that $\phi_{1}R_{Z}\phi_{2}$ we can now deduce
a number of dual results, ie given a result involving $R_{Z}$ we can deduce a
corresponding result involving $R_{Z}^{Op}$. However using we will find it
possible, and in fact more convenient, to translate these results back into
the alternative form of results in terms of $R_{Z}$.

######  Proposition 31.

If $\tau$ is an anti-endomorphism of $R_{T^{\circ}}$ then $\tau_{ST}$ is an
anti-endomorphism of $R_{Z}$, ie
$\phi_{1}\,R_{Z}\,\phi_{2}\Rightarrow\left(\tau_{ST}\phi_{2}\right)\,R_{Z}\,\left(\tau_{ST}\phi_{1}\right)$.
A similar result holds for endomorphisms.

###### Proof.

If $\tau$ is an anti morphism then for any $s_{S}$,
$\phi_{1}s\,R_{T^{\circ}}\,\phi_{2}s\Rightarrow\tau(\phi_{2}s)\,R_{T^{\circ}}\,\tau(\phi_{1}s)\Leftrightarrow(\tau_{ST}\phi_{2})s\,R_{T^{\circ}}\,(\tau_{ST}\phi_{1})s$.
Since this holds trivially for all $s_{Z}$, the result for anti-morphisms
follows. The morphism case is similar. ∎

Recall that $\sigma Z\coloneqq\\{\sigma z:z\in Z\\}$.

######  Proposition 32.

If $\sigma$ is an involution on $S$, then $\sigma_{ST}$ is an endomorphism of
$R_{\sigma Z}$, ie
$\phi_{1}R_{Z}\phi_{2}\Rightarrow\left(\sigma_{ST}\phi_{1}\right)\,R_{\sigma
Z}\,\left(\sigma_{ST}\phi_{2}\right)$

###### Proof.

Since $\sigma$ is an involution, $\phi
s=\phi(\sigma^{2}s)=(\phi\circ\sigma)(\sigma s)=(\sigma_{ST}\phi)(\sigma s)$,
so $\phi_{1}s\,R_{T^{\circ}}\,\phi_{2}s\Rightarrow(\sigma_{ST}\phi_{1})(\sigma
s)\,R_{T^{\circ}}\,(\sigma_{ST}\phi_{2})(\sigma s)$ which establishes the
result. ∎

######  Proposition 33.

If $\sigma$ is an involution on $S$, and $\tau_{T}$ is an (anti) morphism on
$R_{T}$, then $\sigma_{ST}\circ\tau_{ST}$ is an (anti) morphism of $R_{\sigma
Z}$.

###### Proof.

This follows immediately by applying 31 followed by 32, eg in the anti
morphism case,
$\phi_{1}R_{Z}\phi_{2}\Rightarrow(\sigma_{ST}\circ\tau_{ST})\phi_{2}\,R_{\sigma
Z}\,(\sigma_{ST}\circ\tau_{ST})\phi_{1}$ ∎

Note that we established $\sigma_{ST}\circ\tau_{ST}=\tau_{ST}\circ\sigma_{ST}$
in (3.5), and so the order of application of results above is immaterial.

Level 3 Pull Ups of endorelations from $T$ Recall (23) that if $\sigma,\tau$
are involutions on $S,T$ then $\sigma,\sigma^{/},\tau$ on $(ST)T$ are also
commuting involutions and generate an (Abelian) $(C_{2})^{3}$ group of 8
involutions. We wish to investigate how these behave with respect to induced
relations on $(ST)T$. We first look at results we can take from the previous
section by replacing $S$ by $ST$, $Z\subseteq S$ by $\Psi\subseteq ST$, and
$\phi_{ST}$ with $A_{(ST)T}$. Recall that
$\sigma_{\left((ST)T\right)^{\circ}}^{/}(A)=A\circ\tau_{\left(ST\right)^{\circ}}$.

Note we can apply proposition 31 to $\tau_{(ST)T}$, and proposition 32 to
$\sigma_{(ST)T},\sigma_{(ST)T}^{/}$. 31 requires $\tau_{T}$ to be an anti-
morphism on $R_{T}$, and 32 requires $\sigma_{ST},\sigma_{ST}^{/}$ to be
involutions on $ST$. Now $\sigma_{ST}$ is an involution if $\sigma_{S}$ is,
and $\sigma_{ST}^{/}$ is an involution if $\tau_{T}$ is. If these hold then 33
will also hold for the pairs $\sigma,\tau$ and $\sigma^{/},\tau$. Hence 31-33
give dual results relating to
$Id,\sigma,\sigma^{/},\tau,\sigma\circ\tau,\sigma^{/}\circ\tau$ but we are
missing results for $\sigma\circ\sigma^{/},\sigma\circ\sigma^{/}\circ\tau$
which we will now develop.

######  Proposition 34.

If $\sigma_{ST},\sigma_{ST}^{/}$ are involutions on $ST$, then
$\sigma_{(ST)T}\circ\sigma_{(ST)T}^{/}$ is itself an involution and also an
endomorphism of $R_{\sigma\sigma^{/}\Psi}$, ie
$A_{1}R_{\Psi}A_{2}\Leftrightarrow(\sigma\sigma^{/})A_{1}\,R_{(\sigma\sigma^{/}\Psi)}\,(\sigma\sigma^{/})A_{2}$

###### Proof.

By Theorem 23 $\sigma\circ\sigma^{/}$ is itself an involution on $ST$, and the
result follows by applying Proposition 32 ∎

######  Proposition 35.

If $\sigma_{ST},\sigma_{ST}^{/}$ are involutions on $ST$, and $\tau_{T}$ is an
(anti)morphism of $R_{T}$, then
$\sigma_{(ST)T}\circ\sigma_{(ST)T}^{/}\circ\tau_{(ST)T}$ is an (anti)morphism
of $R_{\sigma\sigma^{/}\Psi}$, ie
$A_{1}R_{\Psi}A_{2}\Leftrightarrow(\sigma\sigma^{/}\tau)A_{2}\,R_{(\sigma\sigma^{/}\Psi)}\,(\sigma\sigma^{/}\tau)A_{1}$

###### Proof.

By the previous proposition $\sigma_{(ST)T}\circ\sigma_{(ST)T}^{/}$ is an
involution, and the result follows by applying proposition 33 ∎

We have now established the following:

######  Theorem 36.

If $\sigma_{S},\tau_{T}$ are involutions on
$(S,\sigma_{S},R_{S}),(T,\tau_{T},R_{T})$ respectively, and $\tau_{T}$ is also
an anti-morphism on $R_{T}$, then we have 8 dual equivalences involving the
induced endorelations
$R_{\Psi},R_{\sigma\Psi},R_{\tau\Psi},R_{\sigma\tau\Psi},$ on $(ST)T$ and the
induced involutions $\sigma_{(ST)T},\sigma_{(ST)T}^{/},\tau_{(ST)T}$, namely

$\displaystyle A_{1}R_{\Psi}A_{2}$ $\displaystyle\Leftrightarrow\sigma
A_{1}R_{\sigma\Psi}\sigma
A_{2}\Leftrightarrow\sigma^{/}A_{1}R_{\tau\Psi}\sigma^{/}A_{2}\Leftrightarrow\sigma\sigma^{/}A_{1}R_{\sigma\tau\Psi}\sigma\sigma^{/}A_{2}$

$\Leftrightarrow\tau A_{1}R_{\Psi}^{Op}\tau A_{2}\Leftrightarrow\sigma\tau
A_{1}R_{\sigma\Psi}^{Op}\sigma\tau A_{2}\Leftrightarrow\sigma^{/}\tau
A_{1}R_{\tau\Psi}^{Op}\sigma^{/}\tau A_{2}\Leftrightarrow\sigma\sigma^{/}\tau
A_{1}R_{\sigma\tau\Psi}^{Op}\sigma\sigma^{/}\tau A_{2}$

###  3.4. Special Cases and Application

We now consider 2 special cases of Theorem 36: the case of self dual $\Psi$,
and the case of function homomorphisms from $\tau_{ST}$ to $\tau_{T}$ in
$\Psi$. When $\Psi$ is self-conjugate under the involution $\sigma\tau$ Given
any $\Phi\subseteq ST$, let $\Psi(\Phi)=\Phi\bigcup\sigma\tau\Phi$. Note then
that $\sigma\tau\Psi=\Psi$ (ie $\Psi$ is self-conjugate under the involution
$\sigma\tau=\sigma\circ\tau$), which immediately gives $\tau\Psi=\sigma\Psi$
since $\sigma,\tau$ are commuting involutions. Hence both $\Psi(\Phi)$ and
$\Psi(\sigma\Phi)$ are self-conjugate under $\sigma\tau$ and we can
interchange $R_{\Psi}$ with $R_{\sigma\tau\Psi}$, and $R_{\sigma\Psi}$ with
$R_{\tau\Psi}$ in each of the above dual forms of Theorem 36.

(3.9) $\displaystyle A_{1}R_{\Psi}A_{2}$ $\displaystyle\Leftrightarrow\sigma
A_{1}R_{\sigma\Psi}\sigma
A_{2}\Leftrightarrow\sigma^{/}A_{1}R_{\sigma\Psi}\sigma^{/}A_{2}\Leftrightarrow\sigma\sigma^{/}A_{1}R_{\Psi}\sigma\sigma^{/}A_{2}$
$\Leftrightarrow\tau A_{1}R_{\Psi}^{Op}\tau A_{2}\Leftrightarrow\sigma\tau
A_{1}R_{\sigma\Psi}^{Op}\sigma\tau A_{2}\Leftrightarrow\sigma^{/}\tau
A_{1}R_{\sigma\Psi}^{Op}\sigma^{/}\tau
A_{2}\Leftrightarrow\sigma\sigma^{/}\tau
A_{1}R_{\Psi}^{Op}\sigma\sigma^{/}\tau A_{2}$

Now suppose $\sigma_{S}$ is a relation anti-homomorphism on $R_{S}$ in
addition to being an involution. If $\Phi$ is a set of relation (anti)
homomorphisms then $\sigma\tau\Phi$ is also a set of relation (anti)
homomorphisms by Remark (26), and so the full set $\Psi(\Phi)$ is also a set
of relation (anti) homomorphisms. Similarly
$\sigma\Phi,\tau\Phi,\Psi(\sigma\Phi)$ are sets of relation anti-homomorphisms
(homomorphisms). Finally note that $\Phi,\sigma\tau\Phi\subseteq\Psi(\Phi)$,
and $\sigma\Phi,\tau\Phi\subseteq\sigma\Psi(\Phi)$$\sigma\tau$ and by remark
30, if any of these relations hold, we can then substitute any of the subsets
for the full set, eg from $A_{1}R_{\Psi}A_{2}$ we can deduce
$\sigma\sigma^{/}A_{1}R_{\Phi}\sigma\sigma^{/}A_{2}$.

When $A_{1},A_{2}$ are function homomorphisms of $\tau_{ST}\tau_{T}$ There is
in general no necessary relationship between
$\sigma_{(ST)T}^{/}A_{(ST)T}=A\circ\tau_{ST}$ and
$\tau_{(ST)T}A_{(ST)T}=\tau_{T}\circ A$. However this changes when $A$ is a
function homomorphism from $\tau_{ST}$ to $\tau_{T}$, ie a homomorphism from
the algebraic structure $(ST,\tau_{ST})$ to $(T,\tau_{T})$, ie
$A\left(\tau_{ST}\phi\right)=\tau_{T}\left(A\phi\right)$

######  Lemma 37.

$\sigma_{(ST)T}^{/}=\tau_{(ST)T}$ on function homomorphisms from $\tau_{ST}$
to $\tau_{T}$

###### Proof.

By remark 27 $A_{(ST)T}$ is a function homomorphism means
$A\circ\tau_{ST}=\tau_{T}\circ A$. Then

$\sigma_{(ST)T}^{/}(A)=A\circ\tau_{ST}=\tau_{T}\circ A=\tau_{(ST)T}(A)$

∎

This simplifies the dual forms of Theorem 36 when $A_{1},A_{2}$ are function
homomorphisms to:

(3.10) $\displaystyle A_{1}R_{\Psi}A_{2}$ $\displaystyle\Leftrightarrow\sigma
A_{1}R_{\sigma\Psi}\sigma A_{2}\Leftrightarrow\tau A_{1}R_{\tau\Psi}\tau
A_{2}\Leftrightarrow\sigma\tau A_{1}R_{\sigma\tau\Psi}\sigma\tau A_{2}$
$\Leftrightarrow\tau A_{1}R_{\Psi}^{Op}\tau A_{2}\Leftrightarrow\sigma\tau
A_{1}R_{\sigma\Psi}^{Op}\sigma\tau A_{2}\Leftrightarrow
A_{1}R_{\tau\Psi}^{Op}A_{2}\Leftrightarrow\sigma
A_{1}R_{\sigma\tau\Psi}^{Op}\sigma A_{2}$

The combined case Finally let us consider both special cases. Then we have the
simplified equivalences of $\eqref{eq:DualHom}$ but with the addition of
$\sigma\tau\Psi=\Psi$ and $\sigma\Psi=\tau\Psi$ gives:

(3.11) $\displaystyle A_{1}R_{\Psi}A_{2}\Leftrightarrow\sigma
A_{1}R_{\tau\Psi}\sigma A_{2}\Leftrightarrow\tau A_{1}R_{\tau\Psi}\tau
A_{2}\Leftrightarrow\sigma\tau A_{1}R_{\Psi}\sigma\tau A_{2}$
$\Leftrightarrow\tau A_{1}R_{\Psi}^{Op}\tau A_{2}\Leftrightarrow\sigma\tau
A_{1}R_{\tau\Psi}^{Op}\sigma\tau A_{2}\Leftrightarrow
A_{1}R_{\tau\Psi}^{Op}A_{2}\Leftrightarrow\sigma A_{1}R_{\Psi}^{Op}\sigma
A_{2}$

Example The theory we have developed is of quite wide application, but we will
give one simple example which will become useful to us in Section 7.

We let $S,T$ be 2 real intervals having relations $R_{S},R_{T}$ both as
appropriate subsets of the ${}^{\prime}\leq_{\mathbb{R}}^{\prime}$ order
relation. Let $\sigma_{S}$ be any involution on $S$ which is an anti-morphism
of $\leq$, ie for $s_{1}\leq s_{2}$ in $S$, $\sigma s_{2}\leq\sigma s_{1}$,
and let $\tau_{T}$ be the same on $T$. Suppose $\Psi\subset ST$ is self-
conjugate under $\sigma\tau$.

Suppose further that $A_{1},A_{2}$ are functionals in $(ST)T$ and that under
the pull up of $R_{T}$ to $R_{\Psi}$ that we have $A_{1}R_{\Psi}A_{2}$, ie for
all $\phi_{\Psi}$ we have $A_{1}\phi\leq_{T}A_{2}\phi$. It makes sense then to
write $R_{\Psi}=R_{\sigma\tau\Psi}=\leq_{\Psi}$ and similarly
$R_{\sigma\Psi}=R_{\tau\Psi}=\leq_{\sigma\Psi}$. This means we write
$A_{1}\leq_{\Psi}A_{2}$ to mean $A_{1}\leq A_{2}$ on $\Psi$, and similarly
$A_{1}\leq_{\sigma\Psi}A_{2}$ to mean $A_{1}\leq A_{2}$ on $\sigma\Psi$. Then
by (3.9) when $A_{1}\leq_{\Psi}A_{2}$ we also have the dual results (recalling
$A_{1}R^{Op}A_{2}\Leftrightarrow A_{2}RA_{1}$):

(3.12) $\sigma A_{1}\leq_{\sigma\Psi}\sigma
A_{2},\;\sigma^{/}A_{1}\leq_{\sigma\Psi}\sigma^{/}A_{2},\;\sigma\sigma^{/}A_{1}\leq_{\Psi}\sigma\sigma^{/}A_{2}$
$\tau A_{2}\leq_{\Psi}\tau A_{1},\;\sigma\tau A_{2}\leq_{\sigma\Psi}\sigma\tau
A_{1},\;\sigma^{/}\tau\leq_{\sigma\Psi}\sigma^{/}\tau
A_{1},\;\sigma\sigma^{/}\tau A_{2}\leq_{\Psi}\sigma\sigma^{/}\tau A_{1}$

##  4\. Separation of Concerns in Birkhoff Sums

###  4.1. Introduction

Recall that given a Dynamical System $(X,T_{XX})$ with a value space
$V_{Monoid}=(V,+_{V})$ we define the $N$th Birkhoff sum of an observable
$\phi_{XV}$ as $S_{N}(\phi,x)\coloneqq\sum_{r=1}^{N}\phi(T^{r}x)$ (where the
summation takes place in $V$). As we have noted, previous studies with the
Birkhoff sum in this form have proceeded with properties of $\phi$ being
tightly bound with the entire reasoning process. Our strategy is to separate
the role of $\phi$ from the underlying dynamics of the system $(X,T)$. The key
result which enables us to do this is:

(4.1) $S_{N}(\phi,x)=(S_{N}\phi)x=\phi(S_{N}x)$

The right side of this identity allows us to separate the study of the
composite Birkhoff sum $S_{N}(\phi,x)$ into two parts. First we study the
effects of the homonymic operator $S_{N}$ on $x$. Unlike the original $S_{N}$
this newly constructed operator is independent of $\phi$ and is purely
dynamical. In Section 5 we use it to develop a theory of the sequential
distribution of orbits on the circle. This needs to done only once, and then
the results are available for use with any observable $\phi$.

In section 6 we will study the general classification of unbounded
observables, and develop some initial estimates for the general Birkhoff sum.
In section 5 we will develop the estimates for the case of irrational
rotations of the circle. Finally in section 8 we apply the developed theory to
obtain more specialised estimates for the Birkhoff sum in the case of some
some important specific classes of observables.

The result (4.1) is probably best positioned as a result in Abstract Algebra.
In essence it captures a general insight into a structural decomposition
which, as with many such results, is “obvious” once seen, and can then be more
quickly introduced by way of an ansatz. However as it is fundamental to our
approach we will provide two proofs \- a relatively formal derivation and a
quick proof by ansatz. The first proof is much longer but provides a deeper
understanding of the mechanisms at work. The ansatz simply delivers the result
as quickly as possible, and will suit those to whom the result is intuitive
and are anxious to proceed to the content of subsequent sections. However the
significant difference between the 2 approaches also helps to show that,
although the result may appear “intuitive” to some, this intuition is built
upon a fair amount of internalised mathematical machinery. Also like much
“abstract nonsense”, the proof itself is relatively simple - the challenge
lies in developing the right set of definitions and cocepts within which the
result becomes natural.

###  4.2. Formal development

In the sequel we will fix $N\geq 0$ and to simplify notation we will write
merely $\sum x_{r}$ to denote the formal sum $x_{1}+x_{2}\ldots+x_{N}$ (where
$+$ is an associative binary operator, not necessarily commutative).

Recall three universal constructions

Given sets $X,Y,Z$ then $XY$ is the set of functions $\phi_{XY}\coloneqq
x_{X}\mapsto(\phi x)_{Y}$ from $X$ to $Y$, and $(X\times Y)Z$ is the set of
bifunctions (functions of two arguments)
$f\coloneqq(x,y)\mapsto\left(f(x,y)\right)_{Z}$.

C1:

From a set $X$ we can construct the Kleene star (or free monoid) $X^{*}$ by
$X^{*}=(X^{*},+_{X^{*}})$ where $X^{*}$ is the set of formal sums
$x_{1}+x_{2}\ldots+x_{k}$ for $k\geq 0$, and $+$ is the concatenation of
formal sums (with the empty sum as unit). We regard $X$ as a subset of $X^{*}$
by regarding $x_{X}$ as a formal sum of length 1. Note that “free” here means
free of relations, ie $\sum_{r=1}^{m}x_{r}=\sum_{r=1}^{n}x_{r}^{/}$ means