# Invariance principle for biased Boostrap Random Walk.

A. Collevecchio, K. Hamza, Yunxuan Liu

School of Mathematical Science
Monash University

###### Abstract

Our main goal is to study a class of processes whose increments are generated
via a cellular automata rule. Given the increments of a simple biased random
walk, a new sequence of (dependent) Bernoulli random variables is produced. It
is built, from the original sequence, according to a cellular automata rule.
Equipped with these two sequences, we construct two more according to the same
cellular automata rule. The construction is repeated a fixed number of times
yielding an infinite array ($\\{-K,\ldots,K\\}\times\mathbb{N}$) of
(dependent) Bernoulli random variables. Taking partial sums of these
sequences, we obtain a $(2K+1)$-dimensional process whose increments belong to
the state space $\\{-1,1\\}^{2K+1}$.

The aim of the paper is to study the long term behaviour of this process. In
particular, we establish transience/recurrence properties and prove an
invariance principle. The limiting behaviour of these processes depends
strongly on the direction of the iteration, and exhibits few surprising
features. This work is motivated by an earlier investigation (see [2]), in
which the starting sequence is symmetric, and by the related work [4].

##  1 Introduction

Our main goal is to study a class of processes whose increments are generated
via cellular automata rule. We consider a generalization of a model introduced
in [2] which can be described as follows. Consider a sequence of i.i.d. random
variables taking values in $\\{-1,1\\}$, with mean $2p-1$, where $p\in(0,1)$.
We place these values in a row of an infinite array. We label this row, row 0.
The other rows are built recursively as follows. The $n$-th element of row $k$
is the partial product of the first $n$ elements of row $k-1$. These
“downward” iterations create an array that obeys the cellular automata rule
that the $(k,n)$ entry is simply the product of the entries in $(k-1,n)$ and
$(k,n-1)$, for $k\geq 1,n\geq 2$. The rule can be reversed to create, by
“upward” iteration, rows of negative order. This model is also akin to the
three-dot system introduced by Ledrappier [6] in that both satisfy the same
three-dot identity

$x_{n}(k)x_{n}(k-1)x_{n-1}(k)=1.$

However the Bootstrap Random Walk (BRW) is restricted by design to the half-
lattice as we impose the boundary condition that for any $k\in\mathbb{Z}$,
$x_{1}(k)=x_{1}(0)$.

We emphasize that the randomness in this model is present only through row 0.
In other words, given this row, the array is simply deterministic. In this
regard, the result obtained in [2], studying the case $p=1/2$, was quite
striking, and can be summarized as follows. When $p=1/2$, if we consider the
elements of the row $k$, with fixed $k\in\mathbb{Z}$, they are i.i.d. random
variables taking values in $\\{-1,1\\}$ with mean 0. Let
$S_{n}^{{{\scriptscriptstyle{({k}})}}}$ be the sum of the first $n$ elements
of row $k$. The process
$\big{(}S_{n}^{{{\scriptscriptstyle{({k}})}}}\big{)}_{n}$, for each
$k\in\mathbb{Z}$, is a simple symmetric random walk. Hence, properly re-
scaled, it obeys a functional central limit theorem, i.e. weakly converges to
Brownian motion. What could be surprising is that for any $K$, the process
$S_{n}=(S_{n}^{{{\scriptscriptstyle{({-K}})}}},\ldots,S_{n}^{{{\scriptscriptstyle{({K}})}}})$,
again properly re-scaled, converges to a $(2K+1)$-dimensional standard
Brownian motion#1#1#1We call standard Brownian motion, a multidimensional
process whose components are independent Brownian motions.. In other words,
the strong dependence vanishes in the limit.

In this paper we generalize the above result to any $p\in(0,1)$. Here
$S_{n}^{{{\scriptscriptstyle{({1}})}}}$ is no longer a random walk and the
question of whether or not a properly re-scaled
$S_{n}^{{{\scriptscriptstyle{({1}})}}}$ converges to a Brownian motion
requires attention. It is remarkable that not only do we respond in the
affirmative to this question but we do so collectively for
$S_{n}^{{{\scriptscriptstyle{({-K}})}}},\ldots,S_{n}^{{{\scriptscriptstyle{({K}})}}}$.
In fact, we reveal a dichotomy between the upward and downward processes. For
the latter, we prove that the strong dependence between the components of
$(S_{n}^{{{\scriptscriptstyle{({0}})}}},\ldots,S_{n}^{{{\scriptscriptstyle{({K}})}}})$
still vanishes in the limit. In other words, even in this general setting a
properly re-scaled
$(S_{n}^{{{\scriptscriptstyle{({0}})}}},\ldots,S_{n}^{{{\scriptscriptstyle{({K}})}}})$
converges to a $(K+1)$-dimensional standard Brownian motion. On the other
hand, a limiting upward process retains some dependence beween the components
of
$(S_{n}^{{{\scriptscriptstyle{({-K}})}}},\ldots,S_{n}^{{{\scriptscriptstyle{({0}})}}})$.
A properly re-scaled
$(S_{n}^{{{\scriptscriptstyle{({-K}})}}},\ldots,S_{n}^{{{\scriptscriptstyle{({0}})}}})$
converges to a correlated $(K+1)$-dimensional Brownian motion.

Furthermore, in the general setting of $p\neq 1/2$, the recurrence analysis of
$S_{n}$ is even more interesting as the drift, present in
$(S_{n}^{{{\scriptscriptstyle{({0}})}}})_{n}$, decays exponentially after only
the first downward iteration. In fact, although
$(S_{n}^{{{\scriptscriptstyle{({0}})}}})_{n}$ is a random walk with drift
$2p-1$, the process $(S_{n}^{{{\scriptscriptstyle{({k}})}}})_{n}$ with $k\geq
1$, visits the origin infinitely often. However, this is not the case for the
upward process whose components,
$(S_{n}^{{{\scriptscriptstyle{({k}})}}})_{n}$, $k\leq-1$, are all transient.

In the original model introduced in [2], the increments of
$(S_{n}^{{{\scriptscriptstyle{({k}})}}})_{n}$ are allowed to take values in a
finite set $\mathcal{U}=\\{u_{0},u_{1},...,u_{q-1}\\}\subset\mathbb{R}$
equipped with an operation $\bigotimes$ such that $(\mathcal{U},\bigotimes)$
is an Abelian group. Here $q$ is assumed to be a prime number. The same
generalisation can be undertaken here. However, many of the analytic
expressions we rely on become significantly more cumbersome and significantly
less intuitive. For this reason, we choose to present the results in the
special case $\mathcal{U}=\\{-1,1\\}$ equipped with the usual multiplication.
Different features of a similar model were studied in [4].

The paper is structured as follows. In Section 2, we introduce the model,
state the main results, give an account of the existing literature and
describe the importance of our results within this context. Section 3 sets the
model within a Markov chain setting. It concludes with a study of the
recurrence properties of $(S_{n})_{n}$. Section 4 proves the invariance
principle giving an explicit representation for the normalizing constants. The
paper concludes with an Appendix that contains three combinatorial lemmas.

##  2 Model Set-up and main results.

###  2.1 The model

Denote by $\mathbb{N}$ the set of positive integers, and set
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. Consider the following model. Let
$(\xi_{i})_{i}$ be a sequence of independent random variables having the
Bernoulli$(p)$ distribution on $\\{-1,1\\}$:

$\rho(x)=p^{\frac{1+x}{2}}(1-p)^{\frac{1-x}{2}},$ (2.1)

where $p\in(0,1)$. We build an array
$(\eta_{k,n})_{k\in\mathbb{Z},n\in\mathbb{N}}$ as follows. For
$k\in\mathbb{N}_{0},n\in\mathbb{N}$, we set

$\eta_{0,n}=\xi_{n},\qquad\eta_{k+1,n}=\prod_{m=1}^{n}\eta_{k,m}\qquad\mbox{and}\qquad\eta_{-k,n}=\eta_{-k+1,n-1}\eta_{-k+1,n}.$

Figure 1: (a) Upward process: the white cell is the product of the two grey
cells immediately below and to the left of it. (b) Downward process: the white
cell is the product of all grey cells immediately above and to the left of it.
(c) $\eta_{k,n}$ array: “-” for $-1$ and “+” for $1$.

These rules are represented graphically in Figure 1. As seen in [2],
$\eta_{k,n}$ can then be written in terms of the original sequence
$(\xi_{n})_{n}$:

$\eta_{k,n}=\prod_{m=1}^{n}\xi_{n-m+1}^{\nu_{k,m}},$

where $(\nu_{k,n})_{k\in\mathbb{Z},n\in\mathbb{N}}$ is defined as the array
such that, for any $k\in\mathbb{Z},n\in\mathbb{N}$,

$\nu_{0,1}=1,\quad\nu_{0,n+1}=0,\quad\nu_{k,1}=1,\quad\nu_{k,n}={n+k-2\choose
n-1}=\frac{(n+k-2)\ldots k}{(n-1)!}\mod 2.$

As already mentioned, unlike the symmetric case for which all rows play
identical roles, in the non-symmetric setting ($p\neq 1/2$), which we focus on
in this paper, row 0 plays a unique role and the behaviours of the upward and
downward processes are very different, in many ways, contrasted.

###  2.2 The main results

The main purpose of the paper is to describe the long term behaviour of the
$(2K+1)$-dimensional process
$S_{n}=(S_{n}^{{{\scriptscriptstyle{({-K}})}}},\ldots,S_{n}^{{{\scriptscriptstyle{({K}})}}})$,
where $K\in\mathbb{N}$ is fixed throughout the paper.

We start with a recurrence analysis of the process $S_{n}$.

######  Theorem 2.1.

  1. 1.

$d=1$. The process $(S_{n}^{{{\scriptscriptstyle{({k}})}}})_{n}$ visits 0
finitely many times, for any $k\in[\\![-K,0]\\!]$#2#2#2We write
$[\\![k,\ell]\\!]$ for the set $[k,\ell]\cap\mathbb{Z}$., and infinitely
often, for any $k\in[\\![1,K]\\!]$.

  2. 2.

$d=2$. For any $k,\ell\in[\\![1,K]\\!]$, the 2-dimensional process
$\big{(}(S_{n}^{{{\scriptscriptstyle{({k}})}}},S_{n}^{{{\scriptscriptstyle{({\ell}})}}})\big{)}_{n}$
visits the origin $(0,0)$ infinitely often.

  3. 3.

$d\geq 3$ ($d\leq K$). Any $d$-dimensional process projection of
$(S_{n}^{{{\scriptscriptstyle{({1}})}}},\ldots,S_{n}^{{{\scriptscriptstyle{({K}})}}})$
visits the origin finitely many times.

Next, we state an invariance principle for $S_{n}$. To this end we introduce
the following notations.

For $k,\ell\geq 0$ and $n\in\mathbb{N}$, $\beta_{k,\ell}={k\choose\ell}\mod
2$, $\theta(k)$ is the number of 1’s in the binary representation of $k$,
$2^{\kappa(n)}$ is the largest power of 2 that divides $n$,
$B(k,\ell)=\\#\\{j\in[\\![1,k\vee\ell]\\!]:\
\beta_{k,j}+\beta_{\ell,j}=1\\}$#3#3#3We use the convention that ${n\choose
m}=0$ whenever $n<m$., $\omega_{k,n}=\\#\\{m\in[\\![1,n]\\!]:\nu_{k,m}=1\\}$
and

$\mathcal{B}(k,\ell,n)=2^{\theta(k)}+2^{\theta(\ell)}-2\sum_{j=0}^{k\wedge(-n+\ell)}\beta_{k,j}\beta_{\ell,n+j}.$

As usual we adopt the convention that, for $m<n$, $\sum_{i=n}^{m}x(i)=0$ and
$\prod_{i=n}^{m}x(i)=1$.

We establish in Lemma 5.2 the identity

$B(k,\ell)=2^{\theta(k)}+2^{\theta(\ell)}-2\sum_{j=0}^{k\wedge\ell}\beta_{k,j}\beta_{\ell,j}=\mathcal{B}(k,\ell,0).$

Let

$\mu(k)=\left\\{\begin{array}[]{ll}(2p-1)^{2^{\theta(-k)}}&k\leq 0\\\ 0&k\geq
1\end{array}\right.$

and

$\Sigma(k,\ell)=\left\\{\begin{array}[]{ll}\sum_{n=1}^{-\ell}(2p-1)^{\mathcal{B}(-k,-\ell,n)}+\sum_{n=1}^{-k}(2p-1)^{\mathcal{B}(-\ell,-k,n)}&\\\
\quad+\
(2p-1)^{B(-k,-\ell)}-(-k-\ell+1)(2p-1)^{2^{\theta(-k)}+2^{\theta(-\ell)}}&k,\ell\leq
0\\\ 1+2\sum^{\infty}_{n=1,\ k\leq
2^{\kappa(n)}}(2p-1)^{\omega_{k,n}}&k=\ell\geq 1\\\
0&\mbox{otherwise}\end{array}\right.$

For any $k\in\mathbb{Z}$, $n\in\mathbb{N}$ and $t\in[0,1]$, define

$S_{n}^{{{\scriptscriptstyle{({k}})}}}(t)=\sum_{m=1}^{\left\lfloor
tn\right\rfloor}\eta_{k,m},\quad\bar{S}^{{{\scriptscriptstyle{({k}})}}}_{n}(t):=\big{(}S^{{{\scriptscriptstyle{({k}})}}}_{n}(t)-\mu(k)\lfloor
nt\rfloor\big{)}+\left(nt-\lfloor nt\rfloor\right)\big{(}\eta_{k,\lfloor
nt\rfloor+1}-\mu(k)\big{)}$

and

$U_{n}(t)=\frac{1}{\sqrt{n}}\left(\overline{S}^{{{\scriptscriptstyle{({-K}})}}}_{n}(t),\ldots,\overline{S}^{{{\scriptscriptstyle{({K}})}}}_{n}(t)\right),$

where $\lfloor x\rfloor$ is the integer part of $x$. We shall also write
$\widecheck{U}_{n}(t)$ for the projection of $U_{n}(t)$ on the components
$(0,\ldots,K)$:

$\widecheck{U}_{n}(t)=\frac{1}{\sqrt{n}}\left(\overline{S}^{{{\scriptscriptstyle{({0}})}}}_{n}(t),\ldots,\overline{S}^{{{\scriptscriptstyle{({K}})}}}_{n}(t)\right).$

######  Theorem 2.2.

$(U_{n}(t))_{t\in[0,1]}$ converges weakly to a $(2K+1)$-dimensional Brownian
motion whose covariance matrix is given by the matrix $\Sigma(k,\ell)$. In
particular, $(\widecheck{U}_{n}(t))_{t\in[0,1]}$ converges weakly to a
$(K+1)$-dimensional standard Brownian motion.

###  2.3 Literature Review

####  2.3.1 Elephant Random Walk

The class of processes that we are studying in this paper were introduced and
studied in [2]. As we mentioned before, each of the
$(S^{{{\scriptscriptstyle{({k}})}}}_{n})_{n}$ with $k\geq 1$, is an example of
process with long memory. The process $S_{n}^{{{\scriptscriptstyle{({1}})}}}$
has many similarities, in terms of construction, with the Elephant Walk, which
can be described as follows. Elephant random walks captured a lot of attention
in the literature in recent years, as they are long-memory processes connected
with random recursive trees, and exhibit anomalous diffusion properties. This
family of processes have a parameter $p$ and are built as follows. The first
row of the array is built in the same way the BRW is assembled, i.e.
$\mathcal{E}_{1,i}=\xi_{i}$ where the $\xi$ are i.i.d. in $\\{-1,1\\}$ with
mean $2p-1$. Let $(U_{k})_{k}$ be a sequence of independent random variables,
where $U_{k}$ is discrete uniform over the set $[\\![1,k]\\!]$. Fix
$\mathcal{E}_{2,1}=\mathcal{E}_{1,1}$ and suppose defined $\mathcal{E}_{2,n}$
with $n\in\mathbb{N}$. We define $\mathcal{E}_{2,n+1}$ as
$\xi_{n+1}\mathcal{E}_{2,U_{n}}$. Define
$E_{n}=\sum_{j=1}^{n}\mathcal{E}_{2,j}$. The process $(E_{n})_{n}$ is known in
the literature as an Elephant Random Walk and was introduced by Schütz and
Trimper [10]. It was studied, among others, by Baur and Bertoin [1] where
central limit theorems are proved for a certain range of values of $p$, and a
convergence to non-Gaussian limits for other values. See also Gonzales,
González-Navarrete and Lambert [5].

####  2.3.2 Dependent percolation

Here we focus on the array $(\eta_{\ell,n})_{\ell,n}$. If we color blue the
entries that are equal to $-1$ and orange the ones that equal $+1$ we observe
a typical structure of cellular automata. The orange sizes are organized as
triangles, we call islands, while the blue cells percolate in the manner of a
river until they either hit an orange triangle or reach the extremities of a
finite array. When we change the values of $p$ the triangles have the tendency
of becoming bigger as $p$ approaches 1 or 0. Theorem 2.2 implies that the
difference between the number of $+1$ and $-1$ in any horizontal strip which
does not include the first row, and has $K\times n$ sites, re-scales as
$\sqrt{n}$ and converges to a normal distribution with mean 0. Percolation
properties of the unrestricted (on the entire plane) three-dot system are
studied by Quint [9].

##  3 A Markov chain set-up

Consider the state space $E=\\{-1,1\\}^{2K+1}$ on which we define a process
$X$ as follows. For each vector $e\in E$, we denote by
$e(-K),\ldots,e(-1),e(0),e(1),\ldots,e(K)$ its coordinates, and by $e^{+}$ and
$e^{-}$ be the two “forward” vectors satisfying, for $k\in[\\![1,K]\\!]$,

$e^{\pm}(0)=\pm 1,\quad e^{\pm}(k)=e^{\pm}(k-1)e(k)\quad\mbox{and}\quad
e^{\pm}(-k)=e^{\pm}(-k+1)e(-k+1).$ (3.1)

We shall also need the “backward” vectors ${}^{+}e$ and ${}^{-}e$ satisfying

$^{\pm}e(-K)=\pm 1\quad\mbox{and}\quad^{\pm}e(k)=e(k-1)e(k),$ (3.2)

$\ k\in[\\![-K+1,K]\\!]$. Then the transition probabilities defined, for any
$n\in\mathbb{N}$ and any $e\in E$, by

$\mathbb{P}(X_{n}=e^{\pm}|X_{n-1}=e)=\rho(\pm 1),$ (3.3)

specify a time homogeneous Markov Chain $X=(X_{n})_{n}$ on $E$. We shall
denote its transition probability matrix $P$ and, as usual, index the
probability measure $\mathbb{P}$ and the corresponding expectation
$\mathbb{E}$ with its initial distribution.

We observe that $X_{n}(0)$ is independent of $\sigma(X_{0},\ldots,X_{n-1})$
and has the Bernoulli distribution $\rho$. Indeed,

$\mathbb{P}(X_{n}(0)=\pm
1|X_{n-1}=e)=\mathbb{P}(X_{n}=e^{\pm}|X_{n-1}=e)=\rho(\pm 1).$

This enables us to see the process $(\eta_{n})_{n}$ as the Markov chain
$(X_{n})_{n}$ started at $\mathbf{1}$, and the sequence $(\xi_{n})_{n}$ as the
innovation $(X_{n}(0))_{n}$ in the chain $(X_{n})_{n}$.

In fact, for any integers $k\in[\\![0,K]\\!]$ and $n\in\mathbb{N}$,

$\displaystyle X_{n}(-k)$ $\displaystyle=$ $\displaystyle
X_{n-1}(-k+1)X_{n}(-k+1)\ =\ \prod_{j=0}^{k\wedge n}X_{n-j}(k\wedge
n-k)^{\beta_{k\wedge n,j}}$ (3.4) $\displaystyle X_{n}(k)$ $\displaystyle=$
$\displaystyle X_{0}(k)\prod_{m=1}^{n}X_{m}(k-1)\ =\
\left(\prod_{\ell=0}^{k-1}X_{0}(k-\ell)^{\alpha_{\ell,n}}\right)\left(\prod_{m=1}^{n}X_{n-m+1}(0)^{\nu_{k,m}}\right)$

where $\displaystyle\nu_{k,n}={n+k-2\choose k-1}\mod 2$,
$\alpha_{k,n}=\nu_{k+1,n}$ and $\displaystyle\beta_{k,\ell}={k\choose\ell}\mod
2$.#4#4#4$\alpha_{k,n}=\left(\\!{n\choose k}\\!\right)$ is the number of ways
to choose a multiset of cardinality $k$ from a set of cardinality $n$.

In particular, $X_{n}(-k)$ is a function of
$(X_{0}(-k),\ldots,X_{0}(0),\ldots,X_{n}(0))$. The dependence can be
elaborated on depending on the relative values of $k$ and $n$. If $k<n$, then
the dependence is only on $X_{n-k}(0),\ldots,X_{n}(0)$. While, if $k\geq n$,
then the dependence is on
$(X_{0}(-k+n),\ldots,X_{0}(0),X_{1}(0),\ldots,X_{n}(0))$. Similarly,
$X_{n}(k)$ is a function of
$(X_{0}(k),\ldots,X_{0}(1),X_{1}(0),\ldots,X_{n}(0))$, but not of $X_{0}(0)$.

Two other processes are of interest and will be discussed in the sequel. They
are the two projections on the coordinates $(-K,\ldots,0)$ and the coordinates
$(0,\ldots,K)$. We call the former $(\widehat{X}_{n})_{n}$ and the latter
$(\widecheck{X}_{n})_{n}$, and retain the original numbering of the
components. In other words, the components of $\widehat{X}_{n}$ are numbered
$-K,\ldots,0$, and those of $\widecheck{X}_{n}$ as $0,\ldots,K$. In all, for
$k\in[\\![0,K]\\!]$,

$\widehat{X}_{n}(-k)=X_{n}(-k)\quad\mbox{and}\quad\widecheck{X}_{n}(k)=X_{n}(k).$

It is easy to see that $(\widehat{X}_{n})_{n}$ is a Markov chain with the
following transition probabilities. Let, for $\widehat{e}\in
E^{\prime}=\\{-1,1\\}^{K+1}$, $\widehat{e}^{\pm}$ be the projection of
$e^{\pm}$ (see (3.1)) on the coordinates $(-K,\ldots,0)$. Then

$\mathbb{P}(\widehat{X}_{n}=\widehat{e}^{\pm}|\widehat{X}_{n-1}=\widehat{e})=\rho(\pm
1).$

Similarly, $(\widecheck{X}_{n})_{n}$ is a Markov chain with transition
probabilities:

$\mathbb{P}(\widecheck{X}_{n}=\widecheck{e}^{\pm}|\widecheck{X}_{n-1}=\widecheck{e})=\rho(\pm
1),$

where $\widecheck{e}\in E^{\prime}$ and $\widecheck{e}^{\pm}$ is the
projection of $e^{\pm}$ on the coordinates $(0,\ldots,K)$.

######  Proposition 3.1.

For any $e_{0},e_{2K+1}\in E$ there exists a unique chain
$(e_{1},\ldots,e_{2K})$ such that

$\displaystyle\mathbb{P}(X_{2K+1}=e_{2K+1}|X_{0}=e_{0})$ $\displaystyle=$
$\displaystyle\mathbb{P}(X_{1}=e_{1},\ldots,X_{2K+1}=e_{2K+1}|X_{0}=e_{0})$
$\displaystyle=$ $\displaystyle\prod_{k=1}^{2K+1}\rho(e_{k}(0)).$

And for any $\widehat{e}_{K+1}\in E^{\prime}$, there exists a unique sequence
$(x_{1},\ldots,x_{K})$ such that for any $\widehat{e}_{0}\in E^{\prime}$,

$\mathbb{P}(\widehat{X}_{K+1}=\widehat{e}_{K+1}|X_{0}=\widehat{e}_{0})=\rho(\widehat{e}_{K+1}(0))\prod_{k=1}^{K}\rho(x_{k}).$

In other words, $(\widehat{X}_{n})_{n}$ reaches stationarity after $K+1$
steps.

###### Proof.

The proof relies on Figure 2.

Figure 2: The three-dot system ($K=4$). The white cells are uniquely
determined from the knowledge of the grey cells.

(a) describes the three-point system:

$X_{n}(i)X_{n}(i-1)X_{n-1}(i)=1$ (3.5)

and illustrates the fact that setting any two cells fixes the value of the
third.

Using (a), (b) shows how, for a given value of $e_{2K+1}$, we can uniquely
complete the half-array of size $(2K+1)\times(2K+1)$.

Again using (a), (c) shows that, given $e_{0}$ and
$(e_{1}(K),e_{2}(K-1),\ldots,e_{2K+1}(-K))$, the remaining cells, and in
particular those on row 0, are uniquely determined. Therefore, one can find a
unique sequence $(e_{1},\ldots,e_{2K})$ such that

$\displaystyle\mathbb{P}(X_{0}=e_{0},X_{2K+1}=e_{2K+1})$ $\displaystyle=$
$\displaystyle\mathbb{P}(X_{0}=e_{0},X_{1}(0)=e_{1}(0),\ldots,X_{2K+1}(0)=e_{2K+1}(0))$
$\displaystyle=$
$\displaystyle\left(\prod_{k=1}^{2K+1}\rho(e_{k}(0))\right)\mathbb{P}(X_{0}=e_{0}).$

The second part of the statement is shown in a similar way. Note that
$\mathbb{P}(\widehat{X}_{K+1}=\widehat{e}_{K+1}|X_{0}=\widehat{e}_{0})$ does
not depend on $\widehat{e}_{0}$. ∎

######  Remark 3.1.

From [2], we know that, if $p=\frac{1}{2}$, then the Markov chain reaches its
stationary distribution after exactly $2K+1$ steps regardless of the initial
state. In other words, when $p=1/2$, $(X_{n})_{n}$ is a $(2K+1)$-dependent
process that mixes perfectly after $2K+1$ steps.

If $p\neq\frac{1}{2}$, this is no longer the case. In fact, while the
knowledge of $X_{2K+1}$ is equivalent to that of $X_{1}(K),\ldots,X_{2K+1}(K)$
and does not involve $X_{0}$, to recover the vectors $X_{1},\ldots,X_{2K}$ in
their entirety, one needs $X_{0}$ (as well as $X_{2K+1}$). Therefore, the
sequence $e_{1}(0),\ldots,e_{2K+1}(0)$ in Proposition 3.1, will in general
depend on $e_{0}$.

In fact, while $(\widecheck{X}_{n})_{n}$ fails to mix perfectly after a fixed
number of steps, $(\widehat{X}_{n})_{n}$ reaches stationarity after precisely
$K+1$ steps. This highlights again the difference between the case $p=1/2$,
where downward and upward processes are indistinguishable, and the case $p\neq
1/2$ for which the same two processes exhibit very different behaviours.

######  Proposition 3.2.

The Markov chains $(X_{n})_{n}$, $(\widehat{X}_{n})_{n}$ and
$(\widecheck{X}_{n})_{n}$ are irreducible and aperiodic.

Moreover, their unique stationary distributions $\pi$, $\widehat{\pi}$ and
$\widecheck{\pi}$ are: for any $e\in E$ and $\widehat{e},\widecheck{e}\in
E^{\prime}$,

$\displaystyle\pi(e)$ $\displaystyle=$ $\displaystyle
2^{-K}\prod_{k=0}^{K}\rho\left(\prod_{\ell=0}^{k}e(-\ell)^{\beta_{k,\ell}}\right)$
(3.6) $\displaystyle\widehat{\pi}(\widehat{e})$ $\displaystyle=$
$\displaystyle\prod_{k=0}^{K}\rho\left(\prod_{\ell=0}^{k}\widehat{e}(-\ell)^{\beta_{k,\ell}}\right)$
(3.7) $\displaystyle\widecheck{\pi}(\widecheck{e})$ $\displaystyle=$
$\displaystyle 2^{-K}\rho\left(\widecheck{e}(0)\right)$ (3.8)

###### Proof.

Irreducibility is a direct consequence of Proposition 3.1. Aperiodicity now
follows from the fact that if $e=\mathbf{1}$, then $e^{+}=e$ and $P(e,e)=p>0$.
Similarly for $\widehat{\pi}$ and $\widecheck{\pi}$.

Next we show that if $X_{0}$ has distribution $\pi$ as given in (3.6), then so
does $X_{1}$. The same reasoning extends to $(\widehat{X}_{n})_{n}$ and
$(\widecheck{X}_{n})_{n}$.

Let $e\in E$. The two vectors ${}^{-}e$ and ${}^{+}e$ are the only vectors in
$E$ such that $P(^{\pm}e,e)>0$. Now, for $k<K$,

$\displaystyle\prod_{\ell=0}^{k}{{}^{\pm}e}(-\ell)^{\beta_{k,\ell}}$
$\displaystyle=$
$\displaystyle\prod_{\ell=0}^{k}\big{(}e(-\ell)e(-\ell-1)\big{)}^{\beta_{k,\ell}}\
=\
\left(\prod_{\ell=0}^{k}e(-\ell)^{\beta_{k,\ell}}\right)\left(\prod_{\ell=1}^{k+1}e(-\ell)^{\beta_{k,\ell-1}}\right)$
$\displaystyle=$
$\displaystyle\prod_{\ell=0}^{k+1}e(-\ell)^{\beta_{k+1,\ell}}$

and, for $k=K$,

$\prod_{\ell=0}^{K}{{}^{\pm}e}(-\ell)^{\beta_{K,\ell}}=\pm\prod_{\ell=0}^{K-1}\big{(}e(-\ell)e(-\ell-1)\big{)}^{\beta_{K,\ell}},$

so that

$\rho\left(\prod_{\ell=0}^{K}{{}^{-}e}(-\ell)^{\beta_{K,\ell}}\right)+\rho\left(\prod_{\ell=0}^{K}{{}^{+}e}(-\ell)^{\beta_{K,\ell}}\right)=1.$

It follows that

$\displaystyle\mathbb{P}_{\pi}(X_{1}=e)$ $\displaystyle=$
$\displaystyle\mathbb{P}_{\pi}\big{(}X_{1}=e,X_{0}={{}^{-}e}\big{)}+\mathbb{P}_{\pi}\big{(}X_{1}=e,X_{0}={{}^{+}e}\big{)}$
$\displaystyle=$
$\displaystyle\mathbb{P}_{\pi}\big{(}X_{0}={{}^{-}e},X_{1}(0)=e(0)\big{)}+\mathbb{P}_{\pi}\big{(}X_{0}={{}^{+}e},X_{1}(0)=e(0)\big{)}$
$\displaystyle=$
$\displaystyle\mathbb{P}_{\pi}\big{(}X_{0}={{}^{-}e}\big{)}\mathbb{P}_{\pi}\big{(}X_{1}(0)=e(0)\big{)}+\mathbb{P}_{\pi}\big{(}X_{0}={{}^{+}e}\big{)}\mathbb{P}_{\pi}\big{(}X_{1}(0)=e(0)\big{)}$
$\displaystyle=$ $\displaystyle
2^{-K}\rho(e(0))\prod_{k=0}^{K-1}\rho\left(\prod_{\ell=0}^{k+1}e(-\ell)^{\beta_{k+1,\ell}}\right)$
$\displaystyle=$ $\displaystyle
2^{-K}\rho(e(0))\prod_{k=1}^{K}\rho\left(\prod_{\ell=0}^{k}e(-\ell)^{\beta_{k,\ell}}\right)\
=\ \pi(e).$

∎

In the next proposition we present results on the marginal distributions of
$\pi$. Let $\pi_{k}$ be the $k$-marginal of $\pi$; i.e. the projection of
$\pi$ on component $k$ ($k\in[\\![-K,K]\\!]$):

$\pi_{k}(x)=\sum_{\begin{subarray}{c}e\in E\\\ e(k)=x\end{subarray}}\pi(e).$

The following proposition describes these marginal distributions and how the
corresponding random variables relate to each other. It makes use of the
celebrated Lucas Theorem [7].

######  Theorem 3.1 (Lucas).

A binomial coefficient ${n\choose m}$ is divisible by a prime $p$ if and only
if at least one of the base $p$ digits of $m$ is greater than the
corresponding digit of $n$.

######  Proposition 3.3.

Let $\Xi$ be a random vector whose distribution is $\pi$. Then

  1. 1.

$\widehat{\Xi},\Xi(1),\ldots,\Xi(K)$ are independent and, $\widehat{\Xi}$ has
distribution $\widehat{\pi}$, $\Xi(0)$ has distribution $\rho$ and
$\Xi(1),\ldots,\Xi(K)$ are symmetric Bernoulli random variables;

  2. 2.

for $1\leq k\leq K$ and $x_{0},\ldots,x_{k-1}\in\\{-1,1\\}$,

$\mathbb{P}\left(\Xi(-k)=1|\Xi(-k+1)=x_{k-1},\ldots,\Xi(0)=x_{0}\right)=\rho\left(\prod_{\ell=0}^{k-1}x_{\ell}^{\beta_{k,\ell}}\right);$

  3. 3.

for $0\leq k\leq K$,

$\mathbb{E}\big{[}\Xi(-k)\big{]}=(2p-1)^{2^{\theta(k)}}=\mu(-k);$

  4. 4.

for $0\leq k\leq\ell\leq K$,

$\mathbb{E}\big{[}\Xi(-k)\Xi(-\ell)\big{]}=(2p-1)^{B(k,\ell)},$

and, for $p\neq 1/2$,

$\mbox{cov}(\Xi(-k),\Xi(-\ell))\geq 4p(1-p)(2p-1)^{B(k,\ell)}>0.$

######  Remark 3.2.

In particular, if $k$ is a power of 2, then $\theta(k)=1$ and

$\mathbb{E}\big{[}\Xi(-k)\big{]}=2\mathbb{E}\big{[}\rho(\Xi(0))\big{]}-1=(2p-1)^{2}.$

###### Proof.

  1. 1.

This is a direct consequence of (3.6).

  2. 2.

We prove the second statement by repeating the following argument started at
$K$.

$\displaystyle\mathbb{P}(\Xi(-K+1)=x_{K-1},\ldots,\Xi(0)=x_{0})$
$\displaystyle=$ $\displaystyle\sum_{x_{K}=\pm
1}\prod_{\ell=0}^{K}\rho\left(\prod_{j=0}^{\ell}x_{j}^{\beta_{\ell,j}}\right)\
=\
\prod_{\ell=0}^{K-1}\rho\left(\prod_{j=0}^{\ell}x_{j}^{\beta_{\ell,j}}\right)\sum_{x_{K}=\pm
1}\rho\left(x_{K}\prod_{j=0}^{K-1}x_{j}^{\beta_{K,j}}\right),$

where the last sum takes the form $\rho(y)+\rho(-y)$ which equates to 1.

  3. 3.

To prove the last two statements, we use Proposition 3.1 and more specifically
the fact that $(\widehat{X}_{n})_{n}$ mixes perfectly after $K+1$ steps so
that $\widehat{X}_{K+1}$ has distribution $\widehat{\pi}$. Using (3.4), we can
write

$\mathbb{E}\big{[}\Xi(-k)\big{]}=\mathbb{E}\big{[}\widehat{X}_{K+1}(-k)\big{]}=\mathbb{E}\left[\prod_{j=0}^{k}\widehat{X}_{K+1-j}(0)^{\beta_{k,j}}\right]=\prod_{j=0}^{k}\mathbb{E}\left[\widehat{X}_{K+1-j}(0)^{\beta_{k,j}}\right].$

Now

$\mathbb{E}\left[\widehat{X}_{K+1-j}(0)^{\beta_{k,j}}\right]=\left\\{\begin{array}[]{ll}1&\beta_{k,j}=0\\\
2p-1&\beta_{k,j}=1\end{array}\right.$

Using Lucas Theorem, we know that exactly $2^{\theta(k)}$ $j$’s are such that
$\beta_{k,j}=1$. See Lemma 5.1 of the Appendix. The result follows.

  4. 4.

Again using (3.4) and Lemma 5.2 of the Appendix, we get

$\displaystyle\mathbb{E}\big{[}\Xi(-k)\Xi(-\ell)\big{]}$ $\displaystyle=$
$\displaystyle\mathbb{E}\big{[}\widehat{X}_{K+1}(-k)\widehat{X}_{K+1}(-\ell)\big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\left(\prod_{j=0}^{k}\widehat{X}_{K+1-j}(0)^{\beta_{k,j}}\right)\left(\prod_{j=0}^{\ell}\widehat{X}_{K+1-j}(0)^{\beta_{\ell,j}}\right)\right]$
$\displaystyle=$
$\displaystyle\prod_{j=0}^{k\vee\ell}\mathbb{E}\left[\widehat{X}_{K+1-j}(0)^{\beta_{k,j}+\beta_{\ell,j}}\right]\
=\ (2p-1)^{B(k,\ell)}.$

∎

We conclude the section with a proof of Theorem 2.1 which states that, in
terms of recurrence and transience, the downward process
$(X_{n}(1),\ldots,X_{n}(K))$ behaves like a classical $K$-dimensional random
walk. To this end we recall some facts for (finite-state) Markov chains.

For $e\in E$, we write $\tau_{m}(e)$ for the successive hitting times of state
$e$: $\tau_{0}(e)=0$ and $\tau_{m}(e)=\inf\\{k>\tau_{m-1}(e):X_{k}=e\\}$.
Then, $(\tau_{m}(e)-\tau_{m-1}(e))_{m}$ is a sequence of independent and
identically distributed random variables with finite exponential moments. And
more generally for any function $g$, the random variables

$Y_{m}=\sum_{k=\tau_{m-1}(e)+1}^{\tau_{m}(e)}g(X_{k})$

are independent and identically distributed and have (at least) a finite mean.
It now follows, by application of the classical Law of Large Numbers for
i.i.d. sequences, that

$\frac{1}{n}\sum_{k=1}^{\tau_{n}(e)}g(X_{k})=\frac{1}{n}\sum_{m=1}^{n}\sum_{k=\tau_{m-1}(e)+1}^{\tau_{m}(e)}g(X_{k})=\frac{1}{n}\sum_{m=1}^{n}Y_{m}\xrightarrow{\mbox{\footnotesize{
a.s. }}}\mathbb{E}_{e}\left[\sum_{k=1}^{\tau_{1}(e)}g(X_{k})\right].$

Writing $N_{\tau_{n}(e)}(u)$ for the number of visits to $u$ by time
$\tau_{n}(e)$ (that is the sum of $Y_{1},\ldots,Y_{n}$ when the function $g$
is the indicator function $1_{\\{u\\}}$), we also get that

$\frac{1}{n}\sum_{k=1}^{\tau_{n}(e)}g(X_{k})=\frac{1}{n}\sum_{u\in
E}N_{\tau_{n}(e)}(u)g(u)\xrightarrow{\mbox{\footnotesize{ a.s.
}}}\mathbb{E}_{e}[\tau_{1}(e)]\sum_{u\in
E}\pi(e)g(u)=\mathbb{E}_{e}[\tau_{1}(e)]\mathbb{E}[g(\Xi)],$

where $\Xi$ is a random variable with distribution $\pi$. In total,

$\frac{1}{n}\sum_{k=1}^{\tau_{n}(e)}g(X_{k})\xrightarrow{\mbox{\footnotesize{
a.s.
}}}\mathbb{E}_{e}\left[\sum_{n=1}^{\tau_{1}(e)}g(X_{n})\right]=\mathbb{E}_{e}[\tau_{1}(e)]\mathbb{E}[g(\Xi)].$
(3.9)

######  Proposition 3.4.

Suppose $g$ takes values in $\mathbb{R}^{d}$. Then the process
$Z_{n}=\sum_{m=1}^{n}g(X_{m})$, with $Z_{0}=0$, visits any neighbourhood of 0
infinitely often if and only if so does the process $Z_{\tau_{n}(e)}$.

###### Proof.

The necessity is trivial. To show sufficiency, we first observe that, since
the chain takes finitely many values, so do the increments of the process
$Z_{n}$. It follows that $\max_{n\leq 2K+1}\|Z_{n}\|\leq(2K+1)\max_{e\in
E}\|g(e)\|$. Call this bound $M$.

Next, we fix a state, say $\mathbf{1}$, and call upon Proposition 3.1 to show
that, uniformly in the initial state, $\tau_{1}(\mathbf{1})\leq 2K+1$ on an
event of positive probability.

We reason by contradiction and assume that $Z_{n}$ visits an
$\varepsilon$-ball of 0, $V_{\varepsilon}$, infinitely often but that
$Z_{\tau_{n}(e)}$ only visits the $2M$-ball of 0, $V_{2M}$, finitely many
times. We choose $\varepsilon<M$. Then after the last visit to $V_{2M}$, the
process $Z_{n}$ must return to $V_{\varepsilon}$, and from there, in no more
than $(2K+1)$ steps, and with positive probability, the chain returns to
$\mathbf{1}$. At that time, $\tau$, we have the contradicting statements that
$\|Z_{\tau}\|\leq M+\varepsilon$ (this is the farthest that the chain can
travel on the aforementioned event of positive probability) and
$\|Z_{\tau}\|>2M$ ($Z_{\tau_{n}(e)}$ has exhausted its visits to $V_{2M}$). ∎

### Proof of Theorem 2.1

The proof immediately follows from Proposition 3.4 and Equation (3.9).

  1. 1.

$d=1$. Let $g(e)=e(k)$. Then
$\mathbb{E}_{e}[Y_{1}]=\mathbb{E}_{e}[\tau_{1}(e)]\mu(k)$. If
$k\in[\\![-K,0]\\!]$, then the biased random walk
$Z_{\tau_{n}(e)}=\sum_{m=1}^{n}Y_{m}$ is transient (visits a neighbourhood of
0 finitely many times), from which we deduce that
$S_{n}^{{{\scriptscriptstyle{({k}})}}}=Z_{n}$ itself is transient. If
$k\in[\\![1,K]\\!]$, then $\mathbb{E}[Y_{1}]=0$ and, by the Chung-Fucks
Theorem (see for example [3]), the random walk $Z_{\tau_{n}(e)}$ is recurrent.
It follows from Proposition 3.4 that $S_{n}^{{{\scriptscriptstyle{({k}})}}}$
visits 0 infinitely often.

  2. 2.

$d=2$. Let $g(e)=(e(k),e(\ell))$, $k,\ell\in[\\![1,K]\\!]$. Here again
$\mathbb{E}[Y_{1}]=0$ and, by the Chung-Fucks Theorem, the 2-dimensional
random walk $Z_{\tau_{n}(e)}$ is recurrent, and, by Proposition 3.4, so is
$(S_{n}^{{{\scriptscriptstyle{({k}})}}},S_{n}^{{{\scriptscriptstyle{({\ell}})}}})$.

  3. 3.

$d\geq 3$. Let $g$ be the projection on a given $d$-dimensional space. Then,
the transience of $Z_{n}$ follows from the transience of the $d$-dimensional
random walk $Z_{\tau_{n}(e)}=\sum_{m=1}^{n}Y_{m}$.

##  4 The invariance principle

To show that $U_{n}$ converges weakly to a (correlated) $(2K+1)$-dimensional
Brownian motion, we show that for any
$a=(a_{-K},\ldots,a_{K})\in\mathbb{R}^{2K+1}$,
$\sum_{k=-K}^{K}a_{k}\overline{S}^{{{\scriptscriptstyle{({k}})}}}_{n}(t)$
converges to a (non-)standard Brownian motion.

The argument is standard, follows from the classical theory of the ‘stability’
of Markov chains, as is for example described in [8], and is briefly described
below.

Fix such $a$ and let, for $e\in E$,

$f(e)=\sum^{K}_{k=-K}a_{k}e(k)\quad\mbox{and}\quad\bar{f}(e)=f(e)-\mathbb{E}_{\pi}[f(X_{0})]=\sum_{k=-K}^{0}a_{k}\big{(}e(k)-\mu(k)\big{)}+\sum_{k=1}^{K}a_{k}e(k).$

Now, define the stochastic process $\big{(}S_{n}(t)\big{)}_{t\in[0,1]}$ as

$S_{n}(t)=\sum_{m=1}^{\lfloor
nt\rfloor}\bar{f}(X_{m})=\sum_{k=-K}^{0}a_{k}\big{(}S_{n}^{(k)}(t)-\mu(k)\lfloor
nt\rfloor\big{)}+\sum_{k=1}^{K}a_{k}S_{n}^{(k)}(t)$

and consider the linear interpolation

$\bar{S}_{n}(t):=S_{n}(t)+\left(nt-\lfloor
nt\rfloor\right)\bar{f}\big{(}X_{\lfloor nt\rfloor+1}\big{)}.$

$\bar{S}_{n}(t)$ satisfies the following functional central limit theorem.

A direct application of Theorem 17.4.4 and Section 17.4.3 of [8] yields the
following result.

######  Theorem 4.1.

Let

$\sigma^{2}=\mbox{var}_{\pi}(f(X_{0}))+2\sum_{n=1}^{\infty}\mbox{cov}_{\pi}\big{(}f(X_{0}),f(X_{n})\big{)}.$
(4.1)

If $\sigma^{2}>0$, then, for any initial point $e\in E$,

$\frac{1}{\sigma\sqrt{n}}\bar{S}_{n}(t)\xrightarrow{\text{ d
}}(W_{t})_{t\in[0,1]},$

where $(W_{t})_{t\in[0,1]}$ is a standard Brownian motion.

The next step if to check that $\sigma^{2}>0$. To this end we call upon the
following representation of $\sigma^{2}$ (see Section 17.4.3 of [8]). Writing,
for simplitcity, $\tau(e)$ for $\tau_{1}(e)$,

$\sigma^{2}=\pi(e)\mathbb{E}_{e}\left[\left(\sum_{n=1}^{\tau(e)}\bar{f}(X_{n})\right)^{2}\right].$
(4.2)

The proof of Proposition 4.2 requires the following proposition and notation.
For any $e\in E$ and any $k$, we call $e_{\star k}$ the vector whose
components coincide with those of $e$ except for component $k$ which is
reversed:

$e_{\star k}(\ell)=\left\\{\begin{array}[]{ll}e(\ell)&\ell\neq k\\\
-e(\ell)&\ell=k\end{array}\right.$

######  Proposition 4.1.

Let $e\in E$ be such that for any $k\in[\\![-K,0]\\!]$, $e(k)=1$. Then,

$\mathbb{P}_{e}(\tau(e)<+\infty,X_{1}(0)=\ldots=X_{\tau(e)}(0)=1)>0.$

###### Proof.

We proceed by induction on $K$. To this end we index the chain itself by $K$
and write $X^{(K)}$ etc. The statement is clearly true for $K=1$ in which case
$\mathbb{P}(\tau(e)=1)=p$. Assume it true up to $K-1$. Let $e\in E$ and
$e^{\prime}$ be its projection on components $(-K+1,\ldots,K-1)$. Using the
induction assumption, we know that there is a finite number of steps with
$X_{n}^{(K-1)}(k)=1$, for any $k\in[\\![-K+1,0]\\!]$, that take the reduced
chain $X^{(K-1)}$ from $e^{\prime}$ to itself. For the same transitions, the
original chain $X^{(K)}$ started from $e$ must end up at either $e$ or
$e_{\star K}$. In the latter case, applying the same sequence of transition to
$X^{(K)}$ now at $e_{\star K}$ must lead to $e$, which completes the proof. ∎

######  Proposition 4.2.

For $a\neq 0$, $\sigma^{2}>0$.

###### Proof.

Note that the term on the right hand side of (4.2) is in fact independent of
the choice of the state $e$. We proceed by contradiction. That is, let $f$ be
such that $\sigma^{2}=0$, we show that $f$ must be identically nil. Indeed in
this case the random variable $\sum_{n=1}^{\tau_{e}}\bar{f}(X_{n})$ must be
$\mathbb{P}_{e}$ almost surely nil. Taking $e=\mathbf{1}$, for which
$\mathbb{P}_{e}(\tau(e)=1)=p>0$, yields that $\bar{f}(\mathbf{1})=0$ or
equivalently that
$\sum_{\ell=-K}^{0}a_{k}\big{(}1-\mu(k)\big{)}+\sum_{\ell=1}^{K}a_{k}=0$.
Repeating the previous argument with $e=\mathbf{1}_{\star K}$, for which we
have again $\mathbb{P}_{e}(\tau(e)=1)=p>0$, yields that
$\sum_{\ell=-K}^{0}a_{k}\big{(}1-\mu(k)\big{)}+\sum_{\ell=1}^{K-1}a_{k}-a_{K}=0$.
Combining both equations gives that $a_{K}$ must be nil. We proceed using an
induction argument. Suppose we have established that $a_{\ell}=0$ for
$\ell\in\\{k+1,\ldots,K\\}$. Choose $e=\mathbf{1}_{\star k}$. Then Proposition
4.1 proves the existence of a finite sequence of transitions that take the
chain from $e$ (with $e(k)=1$, for $k\in[\\![-K,0]\\!]$) back to $e$ using
ones on the first component. This is represented in Figure 3.

Figure 3: From $\mathbf{1}_{\star k}$ to $\mathbf{1}_{\star k}$ using 1’s. “-”
for $-1$, “+” for $1$ and grey cells can be either.

Writing $\sum_{n=1}^{\tau_{e}}\bar{f}(X_{n})$ is nil on the set of positive
probability
$\Lambda_{e}=\\{\tau_{e}<+\infty,X_{1}(0)=\ldots=X_{\tau_{e}}(0)=1\\}$, yields
$\sum_{\ell=-K}^{0}a_{k}\big{(}1-\mu(k)\big{)}+\sum_{\ell=1}^{k-1}a_{\ell}-a_{k}=0$,
where we have used the induction assumption that $a_{k+1}=\ldots=a_{K}=0$ and
the fact that $\bar{f}(X_{1})=\ldots=\bar{f}(X_{\tau_{e}})$ on $\Lambda_{e}$.
Combining this equation with that obtained from applying the reasoning to
$e=\mathbf{1}$ gives $a_{k}=0$. This can be repeated for any
$k\in[\\![0,K]\\!]$.

The next step is to look at $k\in[\\![-K,-1]\\!]$. To this end we write a
variation of (3.4) that takes row $k$ as reference and proceeds downward:

$X_{n}(k+j)=\left(\prod_{\ell=0}^{j-1}X_{0}(k-\ell+j)^{\alpha_{\ell,n}}\right)\left(\prod_{m=1}^{n}X_{n-m+1}(k)^{\nu_{j,m}}\right),\quad
1\leq j\leq-k.$

This enables to show that, with $e=\mathbf{1}_{\star k}$ and $h\in\mathbb{N}$
such that $2^{h-1}\leq-k\leq 2^{h}-1$,

$\mathbb{P}_{e}(\widehat{X}_{2^{h}}=\widehat{e},\widehat{X}_{1}(k)=-1,\ldots,\widehat{X}_{2^{h}-1}(k)=-1)>0.$
(4.3)

For example, for $k=-1$, taking $\widehat{X}_{1}(0)=-1$ and
$\widehat{X}_{2}(0)=1$ yields a return to $\widehat{\mathbf{1}_{\star k}}$
after exactly 2 steps such that
$\widehat{X}_{0}(-1)=\widehat{X}_{1}(-1)=\widehat{X}_{2}(-1)=-1$.

To show (4.3), all we need to do is show that the $(\widehat{X}_{n})_{n}$
returns to $\widehat{\mathbf{1}_{\star k}}$ after precisely $2^{h}$ steps and
conditional on
$\\{\widehat{X}_{1}(k)=-1,\ldots,\widehat{X}_{2^{h}-1}(k)=-1\\}$. Indeed, for
$1\leq j\leq-k$,

$\widehat{X}_{2^{h}}(k+j)=\left(\prod_{\ell=0}^{j-1}{\underbrace{\widehat{X}_{0}(k-\ell+j)}_{=1}}^{\alpha_{\ell,2^{h}}}\right)\left(\prod_{m=1}^{2^{h}}{\underbrace{\widehat{X}_{n-m+1}(k)}_{=-1}}^{\nu_{j,m}}\right)=(-1)^{\nu_{j+1,2^{h}}}=1,$

where the last equality follows from the fact, for any
$\kappa\in[\\![0,2^{h}]\\!]$, $\nu_{\kappa,2^{h}}=0$ (see Proposition 18 of
[2]).

As for the case $k\geq 0$, we proceed by induction. Recall that $a_{k}=0$ for
any $k\geq 0$. Taking $e=\mathbf{1}$ and $e=\mathbf{1}_{\star(-1)}$, we get
that $\sum_{\ell=-K}^{-1}a_{k}\big{(}1-\mu(k)\big{)}=0$ and
$\sum_{\ell=-K}^{-2}a_{k}\big{(}1-\mu(k)\big{)}+a_{-1}(-1-\mu(-1))=0$, from
which we deduce that $a_{-1}=0$. This is then repeated to show that $a_{k}=0$
for any $k\in[\\![-K,K]\\!]$, proving the result. ∎

The final step in proving Theorem 2.2 is to untangle the one-dimensional
convergence of $\bar{S}_{n}(t)/(\sigma\sqrt{n})$ into a $(2K+1)$-dimensional
convergence of $U_{n}(t)$. This requires a closer look at, $\sigma^{2}$, the
asymptotic variance of $\bar{S}_{n}(t)/\sqrt{n}$.

######  Proposition 4.3.

  1. 1.

$\mathbb{E}_{\pi}[f(X_{0})]=\sum_{k=0}^{K}a_{-k}(2p-1)^{2^{\theta(k)}}$ and

$\displaystyle\mbox{var}_{\pi}(f(X_{0}))$ $\displaystyle=$
$\displaystyle\sum_{k=-K}^{0}\left(1-(2p-1)^{2^{\theta(-k)+1}}\right)a_{k}^{2}+\sum_{k=1}^{K}a_{k}^{2}$
$\displaystyle\quad+\
2\sum_{k=-K+1}^{0}\sum_{\ell=-K}^{k-1}\left((2p-1)^{B(-k,-\ell)}-(2p-1)^{2^{\theta(-k)}+2^{\theta(-\ell)}}\right)a_{k}a_{\ell}.$

  2. 2.

Let $n\in\mathbb{N}$ and suppose $2^{\kappa(n)}$ is the largest power of 2
that divides it. Let, for $k,\ell\geq 0$ and $n\leq\ell$,

$\mathcal{B}(k,\ell,n)=2^{\theta(k)}+2^{\theta(\ell)}-2\sum_{j=0}^{k\wedge(-n+\ell)}\beta_{k,j}\beta_{\ell,n+j}.$

Then,

$\displaystyle\mbox{cov}_{\pi}(f(X_{0}),f(X_{n}))$ $\displaystyle=$
$\displaystyle\sum_{k=-K}^{0}\sum_{\ell=-K}^{-n}\left((2p-1)^{\mathcal{B}(-k,-\ell,n)}-(2p-1)^{2^{\theta(-k)}+2^{\theta(-\ell)}}\right)a_{k}a_{\ell}$
$\displaystyle\quad+\ \sum_{k=1}^{K\wedge
2^{\kappa(n)}}(2p-1)^{\omega_{k,n}}a_{k}^{2},$

where the first term on the right hand side (double sum) is nil if $n\geq
K+1$.

  3. 3.

$\displaystyle\sigma^{2}=\sum_{k,\ell=-K}^{K}\Sigma(k,\ell)a_{k}a_{\ell}$.

  4. 4.

For any $k\in[\\![-K,K]\\!]$, $\Sigma(k,k)>0$.

###### Proof.

  1. 1.

This is a direct consequence of Proposition 3.3.

  2. 2.

The result immediately follows from the computation of
$\mbox{cov}_{\pi}(X_{0}(k),X_{n}(\ell))$, for any $k,\ell\in[\\![-K,K]\\!]$:

$\displaystyle\mbox{cov}_{\pi}(f(X_{0}),f(X_{n}))$ $\displaystyle=$
$\displaystyle\sum_{k,\ell=-K}^{K}a_{k}a_{\ell}\mbox{cov}_{\pi}\big{(}X_{0}(k),X_{n}(\ell)\big{)}$
$\displaystyle=$
$\displaystyle\sum_{k,\ell=-K}^{K}a_{k}a_{\ell}\Big{(}\mathbb{E}_{\pi}[X_{0}(k)X_{n}(\ell)]-\mu_{k}\mu_{\ell}\Big{)}.$

We start with the case $k\in[\\![1,K]\\!]$ and $\ell\in[\\![-K,0]\\!]$. Using
representation (3.4), we see that $X_{n}(\ell)$ is a function of
$(X_{0}(-\ell),\ldots,X_{0}(0),X_{1}(0),\ldots,X_{n}(0))$. Therefore,
$X_{0}(k)$ and $X_{n}(\ell)$ are independent.

The case $k\in[\\![-K,0]\\!]$ and $\ell\in[\\![1,K]\\!]$ is similar. First, we
observe that $X_{n}(\ell)$ is a function of
$(X_{0}(1),\ldots,X_{0}(\ell),X_{1}(0),\ldots,X_{n}(0))$, and is therefore
again independent of $X_{0}(k)$.

In both of these cases, $\mbox{cov}_{\pi}(X_{0}(k),X_{n}(\ell))=0$.

Next, we look at the case $k,\ell\in[\\![1,K]\\!]$. Using representation (3.4)
and the independence of $X_{0},X_{1}(0),X_{2}(0),\ldots$, we write

$\displaystyle\mathbb{E}_{\pi}[X_{0}(k)X_{n}(\ell)]$ $\displaystyle=$
$\displaystyle\mathbb{E}_{\pi}\left[X_{0}(k)\left(\prod_{j=0}^{\ell-1}X_{0}(\ell-j)^{\alpha_{j,n}}\right)\left(\prod_{m=1}^{n}X_{n-m+1}(0)^{\nu_{\ell,m}}\right)\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\pi}\left[X_{0}(k)\left(\prod_{j=0}^{\ell-1}X_{0}(\ell-j)^{\alpha_{j,n}}\right)\right]\left(\prod_{m=1}^{n}\mathbb{E}_{\pi}[X_{n-m+1}(0)^{\nu_{\ell,m}}]\right)$

which, in the case $k<\ell$, yields to
$\mathbb{E}_{\pi}[X_{0}(k)X_{n}(\ell)]=0$ in view of the fact $\alpha_{0,n}=1$
and, for $\ell\geq 1$, $X_{0}(\ell)$ is symmetric. The case $k>\ell$ is dealt
with in exactly the same way yielding again
$\mathbb{E}_{\pi}[X_{0}(k)X_{n}(\ell)]=0$. Finally, we look at the case
$k=\ell\geq 1$:

$\mathbb{E}_{\pi}[X_{0}(k)X_{n}(k)]=\left(\prod_{j=1}^{k-1}\mathbb{E}_{\pi}[X_{0}(k-j)^{\alpha_{j,n}}]\right)\left(\prod_{m=1}^{n}\mathbb{E}_{\pi}[X_{n-m+1}(0)^{\nu_{k,m}}]\right).$

Since the $X_{0}(k-j)$ are symmetric, the first product takes the value 1 if
$\alpha_{j,n}=0$ for any $j\in[\\![1,k-1]\\!]$, and takes the value 0
otherwise. Using Lemma 5.3 of the Appendix, we see that, letting $2^{\kappa}$
be the largest power of 2 that divides $n$, if $k\leq 2^{\kappa}$, then the
aforesaid product is 1, and if $k>2^{\kappa}$, then the product is 0. The
second product equals $(2p-1)^{\omega_{k,n}}$.

The last case is $k,\ell\in[\\![-K,0]\\!]$. Here, our focus is the upward
process $(\widehat{X}_{n})_{n}$, which reaches stationarity after $K+1$ steps
(see Proposition 3.1):

$\displaystyle\mathbb{E}_{\pi}[X_{0}(k)X_{n}(\ell)]\ =\
\mathbb{E}_{\widehat{\pi}}[\widehat{X}_{0}(k)\widehat{X}_{n}(\ell)]\ =\
\mathbb{E}[\widehat{X}_{K+1}(k)\widehat{X}_{n+K+1}(\ell)]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\left(\prod_{j=0}^{-k}X_{K+1-j}(0)^{\beta_{-k,j}}\right)\left(\prod_{j=0}^{-\ell}X_{n+K+1-j}(0)^{\beta_{-\ell,j}}\right)\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\left(\prod_{m=K+1+k}^{K+1}X_{m}(0)^{\beta_{-k,K+1-m}}\right)\left(\prod_{m=n+K+1+\ell}^{n+K+1}X_{m}(0)^{\beta_{-\ell,n+K+1-m}}\right)\right].$

Clearly, in the case $n+\ell\geq 1$, the two products are independent and we
have once more $\mbox{cov}_{\pi}(X_{0}(k),X_{n}(\ell))=0$. If $n\leq-\ell$,
then we can rearrange the above products in the 3 independent products the
means of which are: $\prod_{j\in I_{1}\setminus I_{2}}(2p-1)^{\beta_{-k,j}}$,
$\prod_{j\in I_{1}\cap
I_{2}}(2p-1)^{\beta_{-k,j}+\beta_{-\ell,n+j}-2\beta_{-k,j}\beta_{-\ell,n+j}}$
and $\prod_{j\in I_{2}\setminus I_{1}}(2p-1)^{\beta_{-\ell,n+j}}$, where
$I_{1}=[\\![0,-k]\\!]$ and $I_{2}=[\\![-n,-n-\ell]\\!]$. However, for
$-k\leq-n-\ell$,

$\displaystyle\sum_{j\in I_{1}\setminus I_{2}}\beta_{-k,j}+\sum_{j\in
I_{1}\cap
I_{2}}\big{(}\beta_{-k,j}+\beta_{-\ell,n+j}-2\beta_{-k,j}\beta_{-\ell,n+j}\big{)}+\sum_{j\in
I_{2}\setminus I_{1}}\beta_{-\ell,n+j}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{-k}\beta_{-k,j}+\sum_{j=0}^{-k}\beta_{-\ell,n+j}-2\sum_{j=0}^{-k}\beta_{-k,j}\beta_{-\ell,n+j}+\sum_{j=-n}^{-1}\beta_{-\ell,n+j}+\sum_{j=-k+1}^{-n-\ell}\beta_{-\ell,n+j}$
$\displaystyle=$ $\displaystyle
2^{\theta(-k)}+2^{\theta(-\ell)}-2\sum_{j=0}^{-k}\beta_{-k,j}\beta_{-\ell,n+j}.$

Similarly, if $-k>-n-\ell$,

$\displaystyle\sum_{j\in I_{1}\setminus I_{2}}\beta_{-k,j}+\sum_{j\in
I_{1}\cap
I_{2}}\big{(}\beta_{-k,j}+\beta_{-\ell,n+j}-2\beta_{-k,j}\beta_{-\ell,n+j}\big{)}+\sum_{j\in
I_{2}\setminus I_{1}}\beta_{-\ell,n+j}$ $\displaystyle=$
$\displaystyle\sum_{j=-n-\ell+1}^{-k}\beta_{-k,j}+\sum_{j=0}^{-n-\ell}\beta_{-k,j}+\sum_{j=0}^{-n-\ell}\beta_{-\ell,n+j}-2\sum_{j=0}^{-n-\ell}\beta_{-k,j}\beta_{-\ell,n+j}+\sum_{j=-n}^{-1}\beta_{-\ell,n+j}$
$\displaystyle=$ $\displaystyle
2^{\theta(-k)}+2^{\theta(-\ell)}-2\sum_{j=0}^{-n-\ell}\beta_{-k,j}\beta_{-\ell,n+j}.$

We immediately get that

$\mathbb{E}_{\pi}[X_{0}(k)X_{n}(\ell)]=(2p-1)^{\mathcal{B}(-k,-\ell,n)}.$

  3. 3.

Theorem 17.0.1 of [8] states that

$\sigma^{2}=\mbox{var}_{\pi}(f(X_{0}))+2\sum_{n=1}^{\infty}\mbox{cov}_{\pi}(f(X_{0}),f(X_{n})).$

Substituting the expressions from 1. and 2. leads to the required
representation.

  4. 4.

Finally, the positivity of $\Sigma(k,k)$ is a direct consequence of
Proposition 4.2. Simply take $a_{k}=1$ and $a_{\ell}=0$ for $\ell\neq k$.

∎

##  5 Appendix

######  Lemma 5.1.

Let, for $k=\sum_{i\geq 0}k_{i}2^{i}$, $\theta(k)=\sum_{i\geq 0}k_{i}$ be the
number of the base 2 digits of $k$ that are equal to 1. Then,

$\\#\\{j\in[\\![0,k]\\!]:\beta_{k,j}=1\\}=2^{\theta(k)}.$

###### Proof.

By Lucas Theorem, $\beta_{k,j}=1$ if and only if, for every $i\geq 0$,
$k_{i}\geq j_{i}$. This leads to $j_{i}=0$ whenever $k_{i}=0$. On the other
hand, if $k_{i}=1$, then $j_{i}$ can be either 0 or 1. The result immediately
follows. ∎

######  Lemma 5.2.

Let, for $k=\sum_{i\geq 0}k_{i}2^{i}$ and $\ell=\sum_{i\geq 0}\ell_{i}2^{i}$,
$\left<k,\ell\right>=\sum_{i\geq 0}k_{i}\ell_{i}2^{i}$. Then,

$B(k,\ell)=2^{\theta(k)}+2^{\theta(\ell)}-2\sum_{j=0}^{k\wedge\ell}\beta_{k,j}\beta_{\ell,j}=2^{\theta(k)}+2^{\theta(\ell)}-2^{\theta(\left<k,\ell\right>)+1}.$

$\theta(\left<k,\ell\right>)$ is the number of the base 2 digits of $k$ and
$\ell$ that are concurrently equal to 1. In particular, $B(k,k)=0$.

###### Proof.

Since $\beta_{k,j}+\beta_{\ell,j}=1$ if and only if
$\beta_{k,j}+\beta_{\ell,j}-2\beta_{k,j}\beta_{\ell,j}=1$ and the second
expression is always in $\\{0,1\\}$,

$\displaystyle B(k,\ell)$ $\displaystyle=$
$\displaystyle\\#\\{j\in[\\![1,k\vee\ell]\\!]:\beta_{k,j}+\beta_{\ell,j}=1\\}\
=\
\sum_{j=1}^{k\vee\ell}\left(\beta_{k,j}+\beta_{\ell,j}-2\beta_{k,j}\beta_{\ell,j}\right)$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{k}\beta_{k,j}+\sum_{j=1}^{\ell}\beta_{\ell,j}-2\sum_{j=1}^{k\wedge\ell}\beta_{k,j}\beta_{\ell,j}$
$\displaystyle=$
$\displaystyle\big{(}2^{\theta(k)}-1\big{)}+\big{(}2^{\theta(k)}-1\big{)}-2\big{(}2^{\theta(\left<k,\ell\right>)}-1\big{)}.$

In the last step, we use the fact that $\beta_{k,j}=\beta_{\ell,j}=1$ if and
only if $\beta_{\left<k,\ell\right>,j}=1$. ∎

######  Lemma 5.3.

If $2^{\kappa}$ is the largest power of 2 that divides $n$ (i.e.
$n=m2^{\kappa}$ where $m$ is odd), then $\min\\{j\geq 1:\
\alpha_{j,n}=1\\}=2^{\kappa}$.

###### Proof.

Let $n=\sum_{\ell\geq 0}n_{\ell}2^{\ell}$ be the binary representation of $n$.
If $n=m2^{\kappa}$, then $n_{\ell}=0$ for $\ell<\kappa$ and $n_{\kappa}=1$,
and

$n-1=\sum_{\ell=0}^{\kappa-1}2^{\ell}+\sum_{\ell\geq\kappa+1}n_{\ell}2^{\ell}.$

Let $j\in[\\![1,2^{\kappa}-1]\\!]$,
$j=\sum_{\ell=0}^{\kappa-1}j_{\ell}2^{\ell}$ be its binary representation and
$\ell^{\star}=\max\\{\ell\in[\\![0,\kappa-1]\\!]:\ j_{\ell}=1\\}$. Then the
$\ell^{\star}$ digit of $n+j-1$ is 0 while the same digit of $j$ is 1. By
Lucas Theorem, $\alpha_{j,n}=0$. On the other hand, all digits of
$n+2^{\kappa}-1$ are at least as large as the same digits of $2^{\kappa}$.
Again, by Lucas Theorem, $\alpha_{2^{\kappa},n}=1$. ∎

## References

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# Dynamics near the $p:-q$ Resonance

Sven Schmidt1,3 and Holger R. Dullin4

1School of Mathematics Loughborough University LE11 3TU, Loughborough, UK
3Schlumberger Oilfield UK Plc Abingdon Technology Centre OX14 1UJ, Abingdon,
UK<EMAIL_ADDRESS>4School of Mathematics and Statistics
The University of Sydney Sydney NSW 2006, Australia<EMAIL_ADDRESS>
###### Abstract

We study the dynamics near the truncated $p:\pm q$ resonant Hamiltonian
equilibrium for $p$, $q$ coprime. The critical values of the momentum map of
the Liouville integrable system are found. The three basic objects reduced
period, rotation number, and non-trivial action for the leading order dynamics
are computed in terms of complete hyperelliptic integrals. A relation between
the three functions that can be interpreted as a decomposition of the rotation
number into geometric and dynamic phase is found. Using this relation we show
that the $p:-q$ resonance has fractional monodromy. Finally we prove that near
the origin of the $1:-q$ resonance the twist vanishes.

Key words: fractional monodromy, resonant oscillator, vanishing twist,
singular reduction

MSC2000 numbers: 37J15, 37J20, 37J35, 37J40, 70K30

##  1 Introduction

The phase space of an integrable system is foliated into invariant tori almost
everywhere. Exceptions occur when the integrals are not independent, i.e. when
the energy-momentum map has critical values. Monodromy is a manifestation of
the singular nature of the fibre over a critical value of the energy–momentum
mapping. The term was introduced in Duistermaat’s 1980 paper [7] as the
simplest obstruction to the existence of global and smooth action–angle
coordinates. Consider a closed, non–contractible loop $\Gamma$ in the image of
the energy–momentum mapping consisting entirely of regular values. Assume the
loop is traversed from $\Gamma(0)$ to $\Gamma(1)$ where $\Gamma(0)=\Gamma(1)$.
For each regular value $\Gamma(s)$ the period lattice $P_{\Gamma(s)}$ gives
the periods of the flows generated by energy and momentum, and thus encodes
the transformation to action-angle variables. The period lattice
$P_{\Gamma(s)}$ at $\Gamma(0)$ and at $\Gamma(1)$ are related by a unimodular
transformation. If this transformation is non-trivial the system is said to
have monodromy. Cushman and Bates [6] use the ratios of periods, i.e. the
rotation number, to compute monodromy. Geometrically monodromy means that the
torus bundle over the loop $\Gamma$ is non-trivial.

If the loop of regular values is contractible through regular values there is
no monodromy. Thus there must be at least one critical value inside the loop
$\Gamma$ for monodromy to occur, and the nature of the critical fibre in
principle determines the monodromy. For non–degenerate focus–focus points the
critical fibre is a pinched torus with $p$ pinches, and the monodromy matrix
of a loop around the critical value of the focus-focus point is
$\begin{pmatrix}1&0\\\ p&1\end{pmatrix}$ [12, 20].

Recently Nekhoroshev et al. [15, 16] found fractional monodromy in certain
resonant oscillators. In fractional monodromy the change of basis of the
period lattice is not an element of $SL(2,\mathbb{Z})$ but of
$SL(2,\mathbb{Q})$ instead. This would be incompatible with the Liouville-
Arnold theorem which gives the actions uniquely up to transformations in
$SL(2,\mathbb{Z})$. Thus for fractional monodromy the loop $\Gamma$
necessarily has to cross critical values. The critical fibre has to be such
that it is possible to continuously pass a sub-lattice of the period lattice
through it [16, 11], also see [10]. The first example was the $1:-2$ resonant
Hamiltonian, with an appropriate compactification [15, 16, 11]. In [14] and
[19] this has been extended to the compactified $p:-q$ resonance.

The present paper is based on the thesis [18]. It is a generalization from the
$1:-2$ resonance as treated in [5] to the general $p:-q$ resonance. We
independently obtained results similar to those presented in [19], but our
approach is complementary. There are two main differences. The first main
difference is that we base our approach on the computation of the action
variables. We derive the remarkable formula relating the action variables
$I_{1}$ and $I_{2}$ by

$I_{2}=\frac{p+q}{4\pi}\Delta hT-I_{1}W$ (1.1)

where $\Delta h$ is the value of the non-linear part of the Hamiltonian, $T$
is the period, and $W$ is the corresponding rotation number. This formula can
be interpreted as decomposing the rotation number into a dynamical phase
proportional to $T$ and a geometric phase proportional to the action $I_{2}$
[13]. Then we show that $W$ changes by $1/(pq)$ upon traversing a loop around
the equilibrium, and thus by (1.1) the action $I_{2}$ changes by $I_{1}/(pq)$
and the system has fractional monodromy. The second main difference is that we
analyze the dynamics near the equilibrium point in detail. The period,
rotation number, and action are dominated by certain singular contributions,
and we show that sufficiently close to the equilibrium point these are the
leading order contributions. This allows us to perform the computation without
any compactification, thereby proving the conjecture made in [16], that
fractional monodromy is independent of the details of the compactification.
Finally we obtain the new result that the isoenergetic non-degeneracy
condition of the KAM theorem is violated near the $1:-q$ resonance, i.e. the
system has vanishing twist near the equilibrium point.

The paper is structured as follows. First we derive the resonant Hamiltonian
normal form near an elliptic–elliptic equilibrium point in $p:-q$ resonance
which is then studied in the reduced phase space after applying singular
reduction. After discussing the structure of the reduced phase space we derive
the set of critical values of the energy–momentum mapping. The period of the
reduced flow, the rotation number, and the non-trivial action of the system
are computed in the next section. We then prove fractional monodromy for the
$p:-q$ resonance. Finally we prove that the twist vanishes in $1:-q$ resonant
systems close to the equilibrium point.

##  2 Resonant Hamiltonain Equilibria

Near a non-resonant elliptic equilibrium point a Hamiltonian can be
transformed to Birkhoff normal form up to arbitrary high order. The truncated
Birkhoff normal form depends on the actions only, and to lowest order these
actions are those of the two harmonic oscillators of the diagonalized
quadratic part of the Hamiltonian. When the frequencies are $p:\pm q$
resonant, the normal form contains additional so called resonant terms that
depend on resonant linear combinations of the angles. Even though the non-
linear non-resonant terms are generically present, we will assume that all the
non-linear non-resonant terms up to the order $p+q$ are vanishing Without this
assumption the dynamics near the equilibrium would be dominated by the non-
resonant terms when $p+q\geq 5$. For the low order resonant cases no such
assumption is necessary, and therefore the results presented in [5] for the
$1:-2$ resonance are relevant for generic Hamiltonian systems. By contrast,
the integrable systems studied in this paper have higher codimension
(increasing with $p+q$) in the class of Hamiltonian systems.

Denote a point in phase space $T^{*}\mathbb{R}^{2}$ with coordinates
$(p_{1},q_{1},p_{2},q_{2})$ and let the symplectic structure be
$\omega=dq_{1}\wedge dp_{1}+dq_{2}\wedge dp_{2}$. Assume the origin
$(p_{1},q_{1},p_{2},q_{2})=(0,0,0,0)\in T^{*}\mathbb{R}^{2}$ is an elliptic
equilibrium point of the system whose eigenvalues
$\pm\operatorname{i}\omega_{1}$, $\pm\operatorname{i}\omega_{2}$ have ratio
$p/q$, where $p$ and $q$ coprime positive integers. Then the quadratic part of
the Hamiltonian $H$ near the equilibrium point can be brought into the form

$H_{2}=\frac{p}{2}\left(p_{1}^{2}+q_{1}^{2}\right)+\sigma\frac{q}{2}\left(p_{2}^{2}+q_{2}^{2}\right)\,,$
(2.2)

by scaling $\omega_{1}$ to $p$ by a linear change of time. For $\sigma=+1$ the
quadratic Hamiltonian $H_{2}$ is definite, while for $\sigma=-1$ it is
indefinite. The system is said to be in $p:\pm q$ resonance, and mostly we are
interested in the case $\sigma=-1$.

The classical treatment of resonant equilibria, see e.g. [2], is as follows.
Denote $(A_{i},\phi_{i})$, $i=1,2$, the canonical polar coordinates
corresponding to $(p_{i},q_{i})$. The resonant Birkhoff normal form depends on
$A_{1}$, $A_{2}$ and the resonant combination (the so called secular term)
$-\sigma q\phi_{1}+p\phi_{2}$. The normal form Hamiltonian truncated at order
$p+q$ is

$H=pA_{1}+\sigma
qA_{2}+\sum\mu_{ij}A_{1}^{i}A_{2}^{j}+\mu\sqrt{A_{1}^{q}A_{2}^{p}}\cos(-\sigma
q\phi_{1}+p\phi_{2}+\varphi)\,,$

where the phase $\varphi$ can be set to zero by a shift of the angles. In
order to reduce the number of variables a linear symplectic transformation to
$(J_{i},{{}\psi_{i}})$ is performed. The resonant combination is introduced as
a new angle ${{}\psi_{2}}$, while the cyclic angle ${{}\psi_{1}}$ is conjugate
to $J_{1}=pA_{1}+\sigma qA_{2}=H_{2}$. The complete transformation is given by
$J=M^{-t}A$, ${{}\psi}=M\phi$, where $M$ contains two arbitrary integers $a$
and $b$ restricted by $\det M=bp-\sigma aq=1$. The new Hamiltonian depends on
$J_{1},J_{2},{{}\psi_{2}}$ only, and is therefore integrable. The lowest order
term in ${{}\psi_{2}}$ has coefficients proportional to
$\sqrt{A_{1}^{q}A_{2}^{p}}$. Setting the non-linear non-resonant terms
$\mu_{ij}$ equal to zero the Hamiltonian becomes

$H=J_{1}+\mu\sqrt{(bJ_{1}-\sigma
qJ_{2})^{q}(-aJ_{1}+pJ_{2})^{p}}\cos{{}\psi_{2}}\,.$

Reverting back to the original Euclidean coordinates the resonant term reads

$\Delta
H=\Re\left[(p_{1}+\operatorname{i}q_{1})^{q}(p_{2}-\sigma\operatorname{i}q_{2})^{p}\right]\,,$

and the Hamiltonian simply is $H=H_{2}+\mu\Delta H$. The functions $H_{2}$ and
$\Delta H$ are in involution and independent almost everywhere, and thus the
system is Liouville integrable. This is the integrable system we are going to
analyze in this paper. Note that unlike previous work [16, 11, 19], we do not
add higher order terms to $\Delta H$ to compactify the system. Moreover, we
will study the integrable system defined by the Hamiltonian $H$, instead of
the Hamiltonian $\Delta H$. For the discussion of monodromy the two are
equivalent, since monodromy is a feature of the Liouville foliation, that does
not depend on the dynamics on the individual tori. Also, they share the same
singularly reduced system. However, we are also interested in the physically
relevant dynamics of the Birkhoff normal form in the context of KAM theory,
and therefore we analyse the Hamiltonian $H$ and not the Hamiltonian $\Delta
H$. In particular, when considering the rotation number of the full system the
difference is crucial. Nevertheless, we will study the momentum map $F=(\Delta
H,H_{2})$ (which is not the energy-momentum map for the Hamiltonian $H$), and
the Hamiltonian $H=H_{2}+\mu\Delta H$ is a (linear) function of the momenta.

##  3 Reduction

In this section we review the steps necessary to reduce to a single degree of
freedom using the symmetry $H_{2}$. We state only the results and refer the
interested reader to the standard literature, for example Cushman [6], Abraham
and Marsden [1] and Broer et. al. [4], for the $p:-q$ resonance in particular
also see [10]. These steps retrace the derivation reviewed in the previous
section in a more geometric way.

The flow of the resonant quadratic part $H_{2}$ is the $S^{1}$–group action

$\displaystyle\Phi^{H_{2}}:S^{1}\times\mathbb{C}^{2}$
$\displaystyle\longrightarrow\mathbb{C}^{2}$
$\displaystyle\left(t,z_{1},z_{2}\right)$
$\displaystyle\mapsto\left(z_{1}\exp\left(p\operatorname{i}t\right),z_{2}\exp\left(\sigma
q\operatorname{i}t\right)\right),\quad z_{1},\,z_{2}\in\mathbb{C}$

where $z_{i}=\frac{1}{\sqrt{2}}\left(p_{i}+\operatorname{i}q_{i}\right)$,
$i=1$, $2$. This action is non–degenerate except on the axis $z_{1}=0$ and
$z_{2}=0$ on which points have isotropy subgroup $\mathbb{Z}_{q}\subset S^{1}$
and $\mathbb{Z}_{p}\subset S^{1}$, respectively (if $q,p>1$). The invariants
of this group action are

$\pi_{1}=z_{1}\bar{z}_{1},\quad\pi_{2}=z_{2}\bar{z}_{2},\quad\pi_{3}=\Re(z_{1}^{q}z_{2}^{p}),\quad\pi_{4}=\Im(z_{1}^{q}z_{2}^{p})$
(3.3)

for $\sigma=-1$. For $\sigma=+1$ instead
$\pi_{3}=\Re(z_{1}^{q}\bar{z}_{2}^{p}),\pi_{4}=\Im(z_{1}^{q}\bar{z}_{2}^{p})$
are the invariants.

Because of the non-trivial isotropy we employ singular reduction in order to
reduce the system to one degree of freedom. The reduced phase space

$P_{h_{2}}=H_{2}^{-1}(h_{2})/S^{1}$

is given by the relation

$\pi_{3}^{2}+\pi_{4}^{2}=\pi_{1}^{q}\pi_{2}^{p},\quad\pi_{1}\geq
0,\;\pi_{2}\geq 0,$ (3.4)

hence it is a semi–algebraic variety in the invariants. The ambient Poisson
space is endowed with the Poisson structure

$\displaystyle\left\\{\pi_{1},\pi_{2}\right\\}$ $\displaystyle=0,\quad$
$\displaystyle\left\\{\pi_{1},\pi_{3}\right\\}$
$\displaystyle=q\mspace{2.0mu}\pi_{4}$
$\displaystyle\left\\{\pi_{1},\pi_{4}\right\\}$
$\displaystyle=-q\mspace{2.0mu}\pi_{3},\quad$
$\displaystyle\left\\{\pi_{2},\pi_{3}\right\\}$ $\displaystyle=-\sigma
p\mspace{2.0mu}\pi_{4}$ $\displaystyle\left\\{\pi_{2},\pi_{4}\right\\}$
$\displaystyle=p\mspace{2.0mu}\sigma\pi_{3},\quad$
$\displaystyle\left\\{\pi_{3},\pi_{4}\right\\}$
$\displaystyle={\textstyle\frac{1}{2}}{\pi_{1}^{q-1}\pi_{2}^{p-1}}\left(\sigma
p^{2}\pi_{1}-q^{2}\pi_{2}\right)$

with Casimir $\mathcal{H}_{2}=p\pi_{1}+q\sigma\pi_{2}$ and symplectic leaves
the reduced phase spaces $P_{h_{2}}$.

Fixing the Casimir $\mathcal{H}_{2}=h_{2}$ we choose to eliminate $\pi_{2}$
from (3.4) and thus get an equation for the reduced phase space in the form

$\pi_{3}^{2}+\pi_{4}^{2}=\pi_{1}^{q}\left(\frac{h_{2}-p\pi_{1}}{\sigma
q}\right)^{p}.$ (3.5)

The interval of valid $\pi_{1}$ is determined by the requirements $\pi_{1}\geq
0$ and $\pi_{2}\geq 0$, thus

$\displaystyle\sigma$ $\displaystyle=+1:$ $\displaystyle\pi_{1}$
$\displaystyle\in[0,h_{2}/p]$ (3.6) $\displaystyle\sigma$
$\displaystyle=-1,\;h_{2}>0:\;$ $\displaystyle\pi_{1}$
$\displaystyle\in[h_{2}/p,\infty)$ $\displaystyle\sigma$
$\displaystyle=-1,\;h_{2}\leq 0:\;$ $\displaystyle\pi_{1}$
$\displaystyle\in[0,\infty).$

Figure 1: The section $\pi_{4}=0$ of the reduced phase space $P_{h_{2}}$ for
the $2:\pm 3$ resonance. Upper left: $\sigma=+1$ and $h_{2}=1.1$. Upper right:
$\sigma=-1$, $h_{2}=0.5$. Lower left: $\sigma=-1$, $h_{2}=-0.5$. Lower right:
$\sigma=-1$, $h_{2}=0$. For $h_{2}\not=0$ points $\pi_{1}=0$ are cusp
singularities, and points $\pi_{1}=h_{2}/2$ are conical singularities, both
due to the non-trivial isotropy of $\Phi^{H_{2}}$.

Fig. 1 shows sections $\pi_{4}=0$ of the rotationally symmetric reduced phase
space $P_{h_{2}}$ for the $2:\pm 3$ resonance for all four relevant cases
$\sigma=+1$, $\sigma=-1$ and $h_{2}>0$, $h_{2}<0$ and $h_{2}=0$. For
$\sigma=+1$ the reduced phase space is compact with a cusp–singularity ($q=3$)
at the origin $\pi_{1}=0$ and a conical singularity ($p=2$) at
$\pi_{1}=h_{2}/2$ (where $\pi_{2}=0$). These singularities in the reduced
phase space are a result of the non–trivial isotropy of the group action
$\Phi^{H_{2}}$ at $z_{1}=\pi_{1}=0$ and $z_{2}=\pi_{2}=0$.

For $\sigma=-1$ the reduced phase space is non-compact. The singular points
are of the same type as in the compact case, but they exist separately for
positive or negative $h_{2}$. For $h_{2}>0$ there is a conical singularity
($p=2$) at $\pi_{2}=0$ and for $h_{2}<0$ there is a cusp singularity ($q=3$)
at $\pi_{1}=0$. For $h_{2}$ there is a singularity of order $p+q=5$ at
$\pi_{1}=0$ (in the compact case $\sigma=1$ the reduced phase space is merely
a point for $h_{2}=0$).

It follows from equation (3.5) that in the general $p:\pm q$ resonance there
is a singularity of order $q$ at $\pi_{1}=0$ and of order $p$ at $\pi_{2}=0$,
assuming $h_{2}\not=0$. If $h_{2}=0$ the order of the singularity is $p+q$.
Note that in the above statement a singularity of order 1 means no
singularity.

Expressed in the invariant the integral $\Delta H$ simply becomes $\pi_{3}$,
so that the reduced Hamiltonian is

$\mathcal{H}(\pi_{1},\pi_{2},\pi_{3})=p\pi_{1}+\sigma
q\pi_{2}+\mu\pi_{3}=h_{2}+\mu\pi_{3}\,.$ (3.7)

As mentioned earlier, the truncated resonant Birkhoff normal form of a generic
resonant Hamiltonian system would also contain terms $\pi_{1}^{i}\pi_{2}^{j}$,
with $i+j\leq(p+q)/2$, but in order to maximize the effect of the resonant
term these are assumed to be zero.

##  4 The Energy–Momentum Mapping

Let

$F:({{}p_{1}},{{}q_{1}},{{}p_{2}},{{}q_{2}})\mapsto\left(H_{2}({{}p_{1}},{{}q_{1}},{{}p_{2}},{{}q_{2}}),\Delta
H({{}p_{1}},{{}q_{1}},{{}p_{2}},{{}q_{2}})\right)$

be the momentum map of the integrable system, and denoted its value by
$(h_{2},\Delta h)$. The elliptic equilibrium at the origin is a critical point
of the momentum map $F$ since both integrals are of order $\geq 2$ in
$(p_{i},q_{i})$, so that $\operatorname{rank}DF(0)=0$. Furthermore, this point
is degenerate in the sense of the momentum map [3]. For this we need to show
that the Hessians of $H_{2}$ and $\Delta H$ are not linear independent at
$x=0$. This follows immediately from the fact that $\Delta H$ is of order
$p+q\geq 3$ in $(p_{i},q_{i})$ so that its Hessian vanishes identically at the
origin.

Figure 2: The critical values $(h_{2},\Delta h)$ of the energy–momentum
mapping are shown as red branches. The top left picture corresponds to the
$1:-q$ resonance, the top right one to the $p:-q$ resonance with $p>1$. The
degenerate equilibrium point is marked by a disk at the origin for
$\sigma=-1$. The lower left picture corresponds to the $2:3$ resonance with
$\sigma=+1$.

The critical values $(h_{2},\Delta h)$ of the momentum mapping for the $p:\pm
q$ resonance (i.e. the bifurcation diagram) are shown in fig. 2, and are
described in the following lemma.

######  Lemma 1.

The entire line $\Delta h=0$ is critical for $p>1$, $q>2$ and $\sigma=-1$. For
$p=1$, the critical values are given by $\Delta h=0$ and $h_{2}\leq 0$ and for
$\sigma=+1$ they are given by $\Delta h=0$ and $h_{2}\geq 0$. Moreover, there
are two additional branches emanating from the origin of the bifurcation
diagram for $\sigma=+1$.

###### Proof.

A point in reduced phase space is critical if 1) it is a singular point of the
reduced phase space or 2) if it is a point of tangency of the surface
$\Delta\mathcal{H}=\Delta h$ (which is simply a horizontal plane) and the
reduced phase space. The first condition together with (3.6) gives that
$\pi_{1}=\pi_{3}=\pi_{4}=0$ is critical for $\sigma h_{2}\geq 0$ and that
$\pi_{1}=h_{2}/p,\pi_{3}=\pi_{4}=0$ is critical for $h_{2}\geq 0$. The
corresponding critical values of the momentum map are $(h_{2},0)$. The second
condition requires the gradients of the two integrals to be parallel, thus
$\pi_{4}=0$, $\pi_{3}\not=0$, and the derivative of the right hand side of
(3.5), $\pi_{1}^{q}\pi_{2}^{p}$, with respect to $\pi_{1}$ vanishes. This
gives $p^{2}\pi_{1}=\sigma q^{2}\pi_{2}$. Since $\pi_{1},\pi_{2}$ are non-
negative there is no tangency for $\sigma=-1$ (appart from the singular point
with $h_{2}=0$). For $\sigma=1$ there are two additional families of critical
values emerging from the origin in a cusp of order $p+q$. ∎

##  5 Dynamics near the Degenerate Equilibrium

In this section we derive equations for the period $T$ of the reduced flow,
the rotation number $W$ of the full system, and the second action $I_{2}$. A
prominent feature of the reduced period $T$ is its algebraic (rather than
logarithmic) divergence when approaching the degenerate equilibrium point.
This is especially easy to see once we introduce weighted polar coordinates
$(\rho,\theta)$ in the bifurcation diagram. In these coordinates, $T$ and $W$
separate into $\rho$– and $\theta$–dependent contributions which considerably
simplify the computations.

###  5.1 The Reduced Period

An equation for $T$ is derived by separation of variables from

$\dot{\pi}_{1}=\left\\{\pi_{1},\mathcal{H}\right\\}=\mu\left\\{\pi_{1},\pi_{3}\right\\}=\mu
q\pi_{4}.$ (5.8)

Using the equation for the reduced phase space (3.5) together with $\Delta
h=\mu\pi_{3}$, $\pi_{4}^{2}$ can be written as a polynomial in $\pi_{1}$. It
follows that the reduced period is defined on the hyperelliptic curve

$\Gamma=\left\\{(\pi_{1},w)\in\mathbb{C}^{2}\mid
w^{2}=Q(\pi_{1})\right\\}\quad\text{where}\quad
Q(z)=\mu^{2}q^{2-p}z^{q}\left(\sigma(h_{2}-pz)\right)^{p}-(q\Delta h)^{2}$

Separating the variables in equation (5.8) and integrating yields

$T(h_{2},\Delta h)=\oint\frac{d\pi_{1}}{w}.$ (5.9)

Our main focus is the $p:-q$ resonance with non–compact fibration. Thus the
integral along a closed loop as it stands makes no sense. We define the
reduced period by dividing the dynamics into two parts: Dynamics close to the
equilibrium point, and dynamics far away from it. If the system is
compactified by an appropriate higher order term (as in [16, 11]) we may
assume that the dynamics far away from the equilibrium will eventually return
to the neighbourhood of the equilibrium. The time spent on this return loop is
a smooth function of initial conditions if we assume that there are no
additional critical points in it. Specifically we consider a Poincaré section
at some small but finite value of $\pi_{1}$ intersecting stable and unstable
invariant manifolds, similarly to the analysis done for symplectic invariants
in [17, 9] and the $1:-2$ resonance in [5]. The contribution of the near-
dynamics (from the section of the stable manifold to the section of the
unstable manifold) is divergent when approaching the equilibrium point, while
the contribution of the far-dynamics (from the section of the unstable
manifold to the section of the stable manifold) remains a smooth and bounded
function.

For convenience we modify the truncated period thus defined one more time.
Notice that the integral of $d\pi_{1}/w$ from any finite positive $\pi_{1}$ to
$\infty$ is finite, and smoothly depends on the parameters. Thus changing the
truncated period integral to an integral over the non-compact domain only
changes it by a smooth function, and the same argument applies. As a result we
can treat the closed loop integral of the non-compact system as our leading
order period. In particular it correctly describes the leading order divergent
terms when approaching the equilibrium point.

An alternative point of view that combines the last two steps (first
restriction to the near-dynamics, then the extension to $\infty$) is to
consider the integral in a compactification of the complex plane into a
projective space. In this space the integral for $T$ is compact, and this also
explains why it is bounded in the first place. This approach was first used in
[5].

In [19] two approaches are followed. The first one generalises the treatment
of the $1:-2$ resonance [11, 10] to the $p:-q$ resonance. There is a
privileged compactification which prevents the rotation number from diverging
when approaching the critical values. Using this compactification the period
lattice, i.e. reduced period and rotation number, is computed. Then the
authors use the Newton polygon to find the leading order terms of these
integrals in the limit approaching the origin of the bifurcation diagram. The
analogue of our polynomial $Q$ appears in their work as the Newton polygon
approximation. What our approach shows is that the resulting hyperelliptic
integrals have direct dynamical meaning as explained above. In the second
approach in [19] the problem is complexified and monodromy is found as Gauss-
Manin monodromy of a loop with complex $(h,\Delta h)$ that avoids critical
values. As the authors point out the singularity at the origin is not of Morse
type, and this is related to the fact the singularity is degenerate in the
sense of the momentum map. As a result the period diverges algebraically
(instead of as usually logarithmically).

######  Lemma 2.

The reduced period $T$ diverges algebraically with exponent $\left|\Delta
h\right|^{\frac{2}{p+q}-1}$ upon approaching the degenerate equilibrium point
on a curve with non–vanishing derivative at the origin. For $p=1$, the period
diverges like $\left|h_{2}\right|^{-\frac{p+q}{2}+1}$ on the line $\Delta h=0$
when $h_{2}\rightarrow 0$.

###### Proof.

Consider the polynomial $Q$ as a polynomial in $\pi_{1}$, $h_{2}$, and $\Delta
h$. It is weighted homogeneous, where $h_{2}$ and $\pi_{1}$ must have the same
weight, so that their weight is $2$ while that of $\Delta h$ is $p+q$.
Therefore we introduce weighted polar coordinates $(\rho,\theta)$ in the image
of the momentum map by

$\displaystyle\Delta h$ $\displaystyle=\rho^{p+q}\sin\theta$ (5.10a)
$\displaystyle h_{2}$ $\displaystyle=\rho^{2}\cos\theta$ (5.10b)

Together with $\pi_{1}=\rho^{2}x$ it follows that

$T(h_{2},\Delta
h)=\rho^{-(p+q)+2}\oint\frac{dx}{\tilde{w}}=:\rho^{-(p+q)+2}A(\theta)$ (5.11)

where

$\tilde{w}^{2}=\mu^{2}q^{2-p}x^{q}\left(\sigma(\cos\theta-
px)\right)^{p}-(q\sin\theta)^{2}$

is independent of $\rho$. Thus $T$ factors into a radial and an angular
contribution $A(\theta)$. The transformation for $(h_{2},\Delta h)$ to
$(\rho,\theta)$ is $C^{0}$ at the origin but $C^{\infty}$ everywhere else.

The period $T$ diverges with $\rho^{2-p-q}$. When approaching the origin on a
line with non-vanishing slope (in the original variables $(h_{2},\Delta h)$
the contribution in $\Delta h\sim\rho^{p+q}$ is of leading order. Thus
$T\sim|\Delta h|^{2/(p+q)-1}$. When $p=1$ the line segment $\Delta h=0$ with
$h_{2}>0$ is not critical. Approaching the origin along this line there is no
contribution from $\Delta h$, and thus only $h_{2}\sim\rho^{2}$ is relevant,
and the result follows. ∎

###  5.2 The Rotation Number

Recall from the introduction the symplectic coordinate system with
$\\{{{}\psi_{i}},J_{i}\\}=1$. The reduced period gives the period of
${{}\psi_{2}}$, while the rotation number gives the advance of
${{}\psi_{1}}/(2\pi)$ during that period. The ODE for ${{}\psi_{1}}$ is

$\dot{{{}\psi}}_{1}=\left\\{\theta_{1},H\right\\}\,.$ (5.12)

Integration of this ODE for time $T$ gives the rotation number. Note that the
angle ${{}\psi_{1}}$ is not be globally defined, but all we need is the
derivative of ${{}\psi_{1}}$, see the comments in [5].

The interpretation of the rotation number is similar to the interpretation of
the period: The true (compactified) rotation number will differ from $R$ by a
smooth function. The leading order singular part of the rotation number is
contained in $R$.

######  Proposition 3.

The rotation number $R$ of the $p:\pm q$ resonance is given by

$R(h_{2},\Delta
h)=\frac{-\sigma}{2\pi}\oint\left(1+\mu\frac{\pi_{3}}{2}\left(\frac{qb}{\pi_{1}}-\frac{ap}{\pi_{2}}\right)\right)\frac{d\pi_{1}}{w}.$

###### Proof.

The angle ${{}\psi_{1}}$ satisfies the following Poisson brackets:

$\displaystyle\left\\{{{}\psi_{1}},\pi_{1}\right\\}$ $\displaystyle=b$
$\displaystyle\left\\{{{}\psi_{1}},\pi_{2}\right\\}$ $\displaystyle=-a$
$\displaystyle\left\\{{{}\psi_{1}},\pi_{3}\right\\}$
$\displaystyle=\frac{\pi_{3}}{2}\left(\frac{qb}{\pi_{1}}-\frac{ap}{\pi_{2}}\right).$

These brackets follow from expressing $\pi_{i}$ in terms of the canonical
variables $(J_{i},{{}\psi_{i}})$, noting $\pi_{1}=A_{1}$, $\pi_{2}=A_{2}$,
$\pi_{3}=A_{1}^{q/2}A_{2}^{p/2}\cos{{}\psi_{2}}$, and then computing the
canonical brackets, using

$\frac{\partial\pi_{1}}{\partial J_{1}}=b,\quad\frac{\partial\pi_{2}}{\partial
J_{1}}=-a\,.$

Thus, the differential equation for ${{}\psi_{1}}$ (5.12) becomes

$\dot{{{}\psi}}_{1}=1+\mu\frac{\pi_{3}}{2}\left(\frac{qb}{\pi_{1}}-\frac{ap}{\pi_{2}}\right).$

Changing the integration variable from $t$ to $\pi_{1}$ using (5.8) gives
${{}\psi_{1}}$ as an Abelian integral. Comparison of the limiting behaviour of
the rotation number and insisting on the relations $R=\omega_{1}/\omega_{2}$
and $\partial H/\partial I_{i}=\omega_{i}$ gives the overall sign $-\sigma$ in
$R$. ∎

We refer to the three integrals $R$ is composed of as

$\displaystyle R(h_{2},\Delta h)$
$\displaystyle=-\sigma\left(\frac{1}{2\pi}T(h_{2},\Delta h)+W(h_{2},\Delta
h)\right),$ $\displaystyle W(h_{2},\Delta h)$
$\displaystyle=W_{1}(h_{2},\Delta h)+W_{2}(h_{2},\Delta h)$

where

$\displaystyle W_{1}(h_{2},\Delta h)$
$\displaystyle=\phantom{-}\frac{qb}{4\pi}\Delta
h\oint\frac{d\pi_{1}}{\pi_{1}w},$ $\displaystyle W_{2}(h_{2},\Delta h)$
$\displaystyle=-\frac{ap}{4\pi}\Delta h\oint\frac{d\pi_{1}}{\pi_{2}w}.$

Expressing these functions in the weighted polar coordinates $(\rho,\theta)$
gives

$\displaystyle B_{1}(\theta)$
$\displaystyle=\frac{qb}{4\pi}\sin\theta\oint\frac{dx}{x\tilde{w}}$ (5.13)
$\displaystyle B_{2}(\theta)$
$\displaystyle=-\frac{apq}{4\pi}\sin\theta\oint\frac{dx}{(px-\cos\theta)\tilde{w}}.$
(5.14)

Note that $B_{1}$ and $B_{2}$ do not depend on $\rho$, which is the main
virtue of the weighted polar coordinates. Nevertheless, the original functions
$W_{1}$ and $W_{2}$ are not continuous at $(h_{2},\Delta h)=(0,0)$. The reason
is that they take different values when approaching the origin along different
lines $\theta=const$.

Figure 3 shows a plot of $B_{1}(\theta)$ on the left and $B_{2}(\theta)$ on
the right on a loop $\Gamma(\theta)$ around the origin of the bifurcation
diagram fig. 2 top right with constant $\rho$ for the $2:-3$ resonance.

Figure 3: Plot of $B_{1}(\theta)$ (left) and $B_{2}(\theta)$ (right) for the
$2:-3$ resonance. $a=-1$, $b=2$. The corresponding bifurcation diagram is
shown in fig. 2 top right. $B_{1}$ is discontinuous on the branch of critical
values with $\Delta h=0$ and $h_{2}<0$, i.e. $\theta=\pm\pi$, with total jump
equal to ${{}-b/q}$. $B_{2}$ is discontinuous on the branch of critical values
with $\Delta h=0$ and $h_{2}>0$, i.e. $\theta=0$, with total jump equal to
${{}-a/p}$, see Lemma 5.

The functions $B_{1}$ and $B_{2}$ are periodic, but discontinuous. The
discontinuity occurs where the critical points are crossed. The functions can
be made continuous by an appropriate shift, but then they will not be periodic
any more. They cannot be made periodic and continuous at the same time. It
turns out that this behaviour is the reason fractional monodromy occurs in the
$p:-q$ resonance, see section 6. From the derivation of $B_{1}$ and $B_{2}$ it
is clear that they are not really independent functions, but can be obtained
from each other by exchanging $p$ and $q$, and shifting $\theta$ by $\pi$.

Note that if we would have considered $\Delta H$ as our Hamiltonian instead of
$H$ the rotation number would not have the diverging contribution from $T$; it
would be $W=W_{1}+W_{2}$ instead of $R=T/2\pi+W$. For the vanishing twist
described later on this contribution is crucial.

###  5.3 The Non–Trivial Action

The flow of $H_{2}$ is periodic with period $2\pi$, and therefore
$I_{1}=H_{2}$ is one action of the system. The other action $I_{2}$ is a non-
trivial function of $\Delta h$ (or $h$) and $I_{1}$. From the local canonical
coordinates mentioned in the introduction we know that $J_{2}$ and
${{}\psi_{2}}$ are conjugate variables. Translating this to the reduced system
gives

######  Proposition 4.

The non–trivial action $I_{2}$ of the $p:\pm q$ resonance is

$I_{2}(I_{1},\Delta h)=-\sigma\left(\frac{p+q}{4\pi}\Delta
h\mspace{2.0mu}T(I_{1},\Delta h)-I_{1}W(I_{1},\Delta h)\right).$ (5.15)

###### Proof.

Using the reduced Poisson bracket we find

$\\{\cos^{-1}\frac{\pi_{3}}{\sqrt{\pi_{1}^{q}\pi_{2}^{p}}},a\pi_{1}+b\pi_{2}\\}=-\sigma\,.$

The action $I_{2}$ is therefore given by

$I_{2}(h_{2},\Delta
h)=\frac{\sigma}{2\pi}\oint\arccos\frac{\pi_{3}}{\sqrt{\pi_{1}^{q}\pi_{2}^{p}}}\mspace{2.0mu}(a\mspace{1.0mu}d\mspace{1.0mu}\pi_{1}+b\mspace{1.0mu}d\mspace{1.0mu}\pi_{2}).$

Integration by parts gives

$I_{2}=\frac{-\sigma}{2\pi}\oint\frac{\pi_{3}}{\sqrt{\pi_{1}^{q}\pi_{2}^{p}-\pi_{3}^{2}}}\left(\left(\frac{aq}{2}+\frac{ap^{2}}{2q}\frac{\pi_{1}}{\pi_{2}}\right)d\pi_{1}+\left(\frac{bp}{2}+\frac{bq^{2}}{2p}\frac{\pi_{2}}{\pi_{1}}\right)d\pi_{2}\right).$

Now $\pi_{1}$ and $\pi_{2}$ are related by the Casimir
$I_{1}=h_{2}=p\pi_{1}+\sigma q\pi_{2}$. Thus
$d\pi_{2}=\sigma\frac{p}{q}d\pi_{1}$ and with $\Delta h=\mu\pi_{3}$ this
equation becomes

$I_{2}=\frac{-\sigma}{2\pi}\left(\frac{\Delta h}{2}(p+q)\left(bp-\sigma
aq\right)\oint\frac{d\pi_{1}}{w}+\frac{\Delta
h\mspace{2.0mu}h_{2}\mspace{2.0mu}a\mspace{2.0mu}p}{2}\oint\frac{d\pi_{1}}{\pi_{2}w}-\frac{\Delta
h\mspace{2.0mu}h_{2}\mspace{2.0mu}b\mspace{2.0mu}q}{2}\oint\frac{d\pi_{1}}{\pi_{1}w}\right).$

Using $\det M=bp-\sigma aq=1$, $h_{2}=I_{1}$, and recalling that
$W=W_{1}+W_{2}$ gives the result. Notice that the final answer is independent
of the choice of integers $a$ and $b$ in $M$, as long as $\det M=1$. ∎

This expression for $I_{2}$ reduces to the one found for the $1:-2$ resonance
[5], where $p+q=3$, $a=0$ and $b=1$. In the next section on fractional
monodromy we shall prove that the terms $W_{1}$ and $W_{2}$ are discontinuous
(see fig. 3) at $\theta=\pm\pi$ and $\theta=\pm 0$ respectively. They can be
made non-periodic and continuous, thus causing the action $I_{2}$ to be
globally multivalued.

Even though $T$ diverges algebraically like $\Delta h^{\frac{2}{p+q}-1}$
(Lemma 2), the action $I_{2}$ goes to zero like $\rho^{2}\sim\Delta
h^{\frac{2}{p+q}}$ when approaching the equilibrium point. The action $I_{2}$
does have the interpretation of a phase space volume. Even though the system
is non-compact the action does not diverge when approaching the equilibrium
point. The geometric reason is that $I_{2}$ measures the volume relative to
the (unbounded) separatrix, which is finite. The boundary terms from the
partial integration cancel, see [18] for the details.

Another interpretation of this formula is obtained by solving it for the
rotation number. This gives a decomposition of the rotation number into a
dynamical phase proportional to $T$ and a geometric phase proportional to the
action $I_{2}$, compare e.g. [13].

##  6 Fractional Monodromy

In this section we establish the fact that the ${{}p:-q}$ resonance has
fractional monodromy. We explicitly calculate the monodromy matrix $M$ that
gives the transformation of the actions $I_{1}$ and $I_{2}$ after one full
anticlockwise cycle around a loop $\Gamma$ enclosing the degenerate
equilibrium point at the origin of the bifurcation diagram fig. 2 top row. We
then describe the singular fibres corresponding to the critical values the
loop $\Gamma$ crosses.

######  Lemma 5.

The function $B_{1}(\theta)$ satisfies

$\lim_{\theta\rightarrow\pm\pi^{\mp}}B_{1}(\theta)=\pm\frac{b}{2q}.$ (6.16)

The function $B_{2}(\theta)$ satisfies

$\lim_{\theta\rightarrow 0^{\pm}}B_{2}(\theta)=\mp\frac{a}{2p}.$ (6.17)

This is the main technical result, but instead of duplicating the proof here,
we refer to [19]. The main addition to their work at this point is the
interpretation of these integrals. In [19] they appeared as the leading order
Newton-polygon approximation of the compactified rotation number integrals. In
our approach they appear directly as the integrals of the rotation number of
the non-compact system interpreted as explained before.

Using this technical result we are now giving another proof of fractional
monodromy in the $p:-q$ resonance which is based on the explicit expression of
the action obtained earlier. Recall that $I_{1}$ is the globally smooth action
$H_{2}$ of the system as introduced in the introduction.

######  Theorem 6.

The $p:-q$ resonance has fractional monodromy near the degenerate elliptic
equilibrium point. The actions change according to

$\begin{pmatrix}I_{1}^{\prime}\\\\[5.0pt]
I_{2}^{\prime}\end{pmatrix}=\begin{pmatrix}1&0\\\\[5.0pt]
{{}-}\frac{1}{pq}&1\end{pmatrix}\begin{pmatrix}I_{1}\\\\[5.0pt]
I_{2}\end{pmatrix}$

after one full anticlockwise cycle on a loop $\Gamma$ around the degenerate
equilibrium point in the bifurcation diagram.

###### Proof.

Assume the loop $\Gamma$ with fixed $\rho=\rho_{0}$ is traversed in the
mathematical positive sense. When crossing the line $\Delta h=0$, $h_{2}<0$,
$\frac{B_{1}(\pi^{-})}{2\pi}=\frac{b}{2q}$, and
$\frac{B_{1}(-\pi^{+})}{2\pi}=-\frac{b}{2q}$. Hence, the effective jump of
$W_{1}$ becomes ${{}-\frac{b}{q}}$. A similar argument holds for $W_{2}$ at
$\Delta h=0$, $h_{2}>0$. Although the loop is traversed such that $\theta$
crosses the line $\Delta h=0$, $h_{2}>0$ from below, $W_{2}$ has the opposite
sign of $W_{1}$. Thus, the effective jump of $W_{2}$ is ${{}-\frac{a}{p}}$.
Using the form of the action variable from proposition 4, it follows that the
action $I_{2}$ changes like

${{}I_{2}\rightarrow I_{2}-I_{1}\left(\frac{a}{p}+\frac{b}{q}\right)}$

upon completing a full cycle. Due to

$\frac{a}{p}+\frac{b}{q}=\frac{aq+bp}{pq}=\frac{1}{pq},$

the monodromy is (independently of the integers $a$ and $b$) given by

${{}I_{2}\rightarrow I_{2}-\frac{1}{pq}I_{1}}.$

∎

The singular fibre of the critical values with $h_{2}<0$ depends on the value
of $q$. Consider the Poincaré section $p_{2}=0$ and $\dot{p}_{2}<0$. The
critical level has $\Delta h=\pi_{3}=0$. Thus we need to study the level set
determined by the three equations $p|z_{1}|^{2}+\sigma q|z_{2}|^{2}=h_{2}<0$,
$\Re(z_{1}^{q}z_{2}^{p})=0$, and $p_{2}=0$ near the critical point $z_{1}=0$.
From the first and last equation we obtain
$q_{2}^{2}=(-h_{2}+p|z_{1}|^{2})/(-\sigma q)$. For small $|z_{1}|$ therefore
$q_{2}$, and hence $z_{2}$, is approximately constant. The remaining equation
thus becomes $\Re(z_{1}^{q})=0$. Writing $z_{1}=r\exp\operatorname{i}\varphi$
gives $r_{1}^{q}\cos q\varphi=0$, so that the level sets of the intersection
of the critical fibre with the Poincaré section are given by the $2q$ rays
with $q\varphi=\pi(n+1/2)$, $n=0,1,\dots,2q-1$. Together they form $q$
straight lines which are the stable and unstable manifolds in alternation, all
passing through the origin.

If the level set is compactified, e.g. by considering
$\Re(z_{1}^{q})+|z_{1}|^{q+1}$, it becomes a flower with $q$ petals. The
action of the flow of $H_{2}$ on the level set is as follows. Start with a
point in the section $p_{2}=0$, hence $z_{2}=q_{2}$. It returns to the section
when the imaginary part of its image vanishes again with positive derivative,
hence for $\Im(q_{2}\exp(\sigma qit))=q_{2}\sin(\sigma qt)=0$, with $\sigma
q\,q_{2}\cos(\sigma qt)<0$. The smallest positive $t$ that solves this is
$t=2\pi/q$. The action of the flow after the return time therefore is

$\Phi^{H_{2}}(2\pi/q,z_{1},z_{2})=(z_{1}\exp(2\operatorname{i}\pi
p/q),z_{2})\,.$

This is simply a rotation by $2\pi p/q$ in the $z_{1}$ plane. As a result the
petals of the flower are mapped into each other, and $p$ is the number of
petals that the rotation advances by. Since $p$ and $q$ are coprime, all
petals are visited before the orbit returns back to the initial one. The
action is the same for either sign of $\sigma$, the difference is that for
$\sigma=-1$ instead of petals we have sectors delineated by stable and
unstable manifolds.

Considering the level set as a whole, and not only in the Poincaré section,
gives a curled torus whose transversal cross section is the petal, and whose
tubes rotate by $2\pi p/q$ when they complete one longitudinal cycle. The fact
that we discussed the flow of $H_{2}$ instead of the flow of $H$ (or $\Delta
H$) is immaterial for the topology of the level set since the flows commute.

For $h_{2}>0$ in a similar way a flower with $p$ petals appears in the section
$p_{1}=0$.

All critical values are degenerate when $q>2$. This is a crucial difference
between the $1:-2$ resonance and the higher order resonances. In particular
the terms $\mu_{ij}$ that were set to zero in the normal form could completely
change the bifurcation diagram. How to envisage the singularity at the
equilibrium point with $h_{2}=\Delta h=0$ is unclear. Both curled tori limit
to this unstable degenerate equilibrium point, but the above argument breaks
down since $|h_{2}|$ cannot be assumed large compared to $|z_{i}|$ any more.

##  7 Vanishing Twist in the $1:-q$ Resonance

For the $p:-q$ resonance with $p>1$ the bifurcation diagram is divided into
two halves by the critical line $\Delta h=0$. We suspect that in this case the
twist does not vanish for regular values, but we have not been able to find a
proof of this. However, when $p=1$ the line of critical values stops at the
origin, see fig. 2. This is therefore a typical case where one can expect
vanishing twist to occur, see [8] for a general topological proof. In this
particular case using the weighted polar coordinates allows for a simple
analytical proof of vanishing twist.

######  Theorem 7.

For the $1:-q$ resonance, $q\geq 2$, the twist vanishes near the degenerate
equilibrium point on the curve $\Delta h=0$, $h_{2}>0$.

###### Proof.

By definition the twist vanishes if the rotation number $R$ has a critical
point on the energy surface, i.e. when $R$ does not change from torus to
torus. On the energy surface this is written as $\partial R/\partial
I_{1}|_{H=const}$. In the image of the energy momentum map the condition is
satisfied when the lines of constant rotation number and the lines of constant
energy are tangent to each other. This implies that $\nabla R$ and $\nabla H$
are parallel. The gradients can be computed in any coordinate system, and we
choose the weighted polar coordinates. Thus the condition for vanishing twist
is

$\frac{\partial R}{\partial\rho}\frac{\partial
H}{\partial\theta}-\frac{\partial R}{\partial\theta}\frac{\partial
H}{\partial\rho}=0\,.$

Now $H=H_{2}+\mu\Delta H$ where $H_{2}$ is order 2 in $\rho$ and $\Delta H$ is
order $p+q$ in $\rho$. For small $\rho$ we can therefore neglect $\Delta H$,
and find $\partial H/\partial\rho\approx 2\rho\cos\theta$, $\partial
H/\partial\theta\approx-\rho^{2}\sin\theta$. Recalling the factorisation
$T=\rho^{-(p+q)+2}A(\theta)$ from (5.11) we find

$\displaystyle-\sigma 2\pi\frac{\partial R}{\partial\rho}$
$\displaystyle=\frac{\partial
T}{\partial\rho}=\left(-(p+q)+2\right)\rho^{-(p+q)+1}A(\theta),$
$\displaystyle-\sigma 2\pi\frac{\partial R}{\partial\theta}$
$\displaystyle=\frac{\partial
T}{\partial\theta}+{B_{1}^{\prime}}(\theta)+{B_{2}^{\prime}}(\theta)\approx\rho^{-(p+q)+2}A^{\prime}(\theta)\,.$

The leading order contribution comes from $T$ alone. Altogether this gives

$-\sigma 2\pi\left(\frac{\partial R}{\partial\rho}\frac{\partial
H}{\partial\theta}-\frac{\partial R}{\partial\theta}\frac{\partial
H}{\partial\rho}\right)=\rho^{-(p+q)+3}\left((p+q-2)A(\theta)\sin\theta-2A^{\prime}(\theta)\cos\theta+O(\rho)\right)$

up to lowest order in $\rho$. The solution $\theta=0$ follows from symmetry:
$A(\theta)$ is even in $\theta$ thus $A^{\prime}(\theta)$ is odd in $\theta$,
and thus both terms are odd in $\theta$. For $p=1$ the line $\theta=0$ is a
line of regular values, and therefore we have shown that the twist vanishes
along $\theta=0$ asymptotically near the origin. ∎

### Acknowledgements

The authors would like to thank the Department of Applied Mathematics at the
University of Colorado, Boulder for its hospitality in 2006/7 where this work
was completed. HRD was supported in part by a Leverhulme Research Fellowship.

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