in 4.18 is actually regular, because $q\neq 0$ on $\mu^{-1}(f)$. The rational
map and its inverse extend (uniquely) to an isomorphism of the completions
$\bar{\Gamma}_{f}$ and $\overline{\mu^{-1}(f)}$, which can be thought of
respectively as an elliptic curve and its Jacobian. It allows us to determine
the number and nature of the points at infinity of $\overline{\mu^{-1}(f)}$,
i.e., the points needed to complete ${\mu^{-1}(f)}$ into
$\overline{\mu^{-1}(f)}$; they are the points corresponding to affine points
$(x,y)$ for which the map is not defined, to wit $(x,y)=(0,\pm 1)$, and to the
point at infinity $\infty$ of $\bar{\Gamma}_{f}$, making a total of three
points, as asserted by Proposition 4.6. More specifically, by using the map we
can determine a local parametrisation around these points from local
parametrisations around the points $(0,\pm 1)$ and $\infty$. For the latter,
we can take $(x,y)=(t,1\pm\frac{c_{1}t}{2}+\mathcal{O}(t^{2}))$ and
$(x,y)=(t^{-2},\sqrt{c_{3}}t^{-3}(1+\frac{c_{2}}{2c_{3}}t^{2}+\mathcal{O}(t^{4}))$
to obtain, again using the map, the following local parametrisations at the
three points at infinity of $\overline{\mu^{-1}(f)}$:

$\infty_{0}:\left(-2t^{2},\frac{\sqrt{c_{3}}}{t}\Big{(}1+\frac{c_{2}}{2c_{3}}t^{2}+\mathcal{O}(t^{3})\Big{)}\right)\;,\quad\infty_{1}:\left(-\frac{2}{t},\frac{c_{1}}{2}+\mathcal{O}(t)\right)\;,\quad\infty_{2}:\left(-\frac{2}{t},-\frac{2}{t}-\frac{c_{1}}{2}+\mathcal{O}(t)\right)\;.$

Again, using the map we can derive that the (hyper-) elliptic involution on
$\Gamma_{f}$, which is given by $(x,y)\mapsto(x,-y)$, is given on
$\mu^{-1}(f)$ by $(q,p)\mapsto(q,q-p)$. It permutes the points $\infty_{1}$
and $\infty_{2}$ while leaving $\infty_{0}$ fixed (together with the points
$(2p,p)$ where $-p$ is a root of $f$). The points $\infty_{0}$, $\infty_{1}$
and $\infty_{2}$ correspond to $\Theta\;,\Theta_{+}$ and $\Theta_{-}$,
respectively.

######  Example 4.8.

We also specialize Proposition 4.6 to the case of $g=2$ and provide some extra
information. The polynomial $f$ is now of degree $5$, taking the form
$f(x)=1+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+c_{4}x^{4}+c_{5}x^{5}$, with no multiple
roots, which is equivalent to the curve (1.8) associated with the map (P.iv).
In this case, the theta divisor and its translates are genus 2 curves,
isomorphic to $\bar{\Gamma}_{f}$. This general fact can also be seen here
directly from the description that
$\Theta=\left[\bar{\Gamma}_{f}-\infty\right]$ and similarly for
$\Theta_{+}=\left[\bar{\Gamma}_{f}+(0,1)-2\infty\right]$ and
$\Theta_{-}=\left[\bar{\Gamma}_{f}+(0,-1)-2\infty\right]$. We show that these
curves in $\overline{\mu^{-1}(f)}$ meet according to the intersection pattern
in Figure 2.

$\Theta_{+}$$\Theta_{-}$$\Theta$$0$$[(0,-1)-\infty]$$[(0,1)-\infty]$ Figure 2:
When $g=2$ the divisor at infinity of $\mu^{-1}(f)$ consist of three copies of
the curve $y^{2}=f(x)$, intersecting according to the indicated pattern.

To do this, we first recall the general fact that two translates of the theta
divisor (also called _theta curves_) intersect in two points which coincide if
and only if the curves are tangent. Consider first a point (divisor class) in
$\Theta_{+}\cap\Theta_{-}$. It must be of the form
$\left[P+(0,1)-2\infty\right]$ and of the form
$\left[Q+(0,-1)-2\infty\right]$, for some $P,Q\in\bar{\Gamma}_{f}$. In
particular, the points $P$ and $Q$ must be such that $P+(0,1)\sim Q+(0,-1)$;
as we already recalled, a linear equivalence of such divisors amounts to
equality, so that $P=(0,-1)$ and $Q=(0,1)$ and there is a unique intersection
point $[(0,1)+(0,-1)-2\infty]$ which is the origin $O$ of
$\operatorname{Jac}(\bar{\Gamma}_{f})$, since $(0,1)+(0,-1)\sim 2\infty$.
Consider next a point in $\Theta\cap\Theta_{\pm}$. It must be both of the form
$[P-\infty]$ and $[Q+(0,\pm 1)-2\infty]$, for some $P,Q\in\bar{\Gamma}_{f}$.
This leads us now to the linear equivalence $P+\infty\sim Q+(0,\pm 1)$, whose
only solutions are $P=\infty,\,Q=(0,\mp 1)$ and $P=(0,\pm 1),\,Q=\infty$; the
first solution corresponds again to the origin $O$ while the other
intersection point is the point $[(0,\pm 1)-\infty]$ (see Figure 2).

###  4.3 Discrete Liouville and algebraic integrability

We are now ready to show that the Volterra map ${\cal V}_{g}$ is Liouville
integrable on $(M_{g},\left\\{\cdot\,,\cdot\right\\}^{\phi})$ when $\phi\neq
0$. Recall that a birational map $R$ on an algebraic Poisson manifold
$(M,\left\\{\cdot\,,\cdot\right\\})$ of dimension $n$ and (Poisson) rank $2r$
is said to be _Liouville integrable_ when the following conditions are
satisfied:

  1. (1)

$R$ is a Poisson map;

  2. (2)

$R$ has $n-r$ functionally independent invariants in involution.

In the case at hand, $M=M_{g}$ so that $n=3g+1$, and $R={\cal V}_{g}$, which
we already know to be a Poisson map (Proposition 4.1). Also, we already have
$2g+1$ invariants in involution (Proposition 4.2). So it will be sufficient to
show, in the proof which follows, that the rank of
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$ is $2g$ and that the invariants
$H_{1},\dots,H_{2g+1}$ are functionally independent.

######  Proposition 4.9.

Let $\phi\in{\mathbb{C}}[x]$ be any non-zero polynomial of degree at most
$g+1$, vanishing at $0$. Then the Volterra map ${\cal V}_{g}$ is Liouville
integrable on $(M_{g},\left\\{\cdot\,,\cdot\right\\}^{\phi})$.

###### Proof.

We first show that the components of $\mu$, which are the $2g+1$ polynomial
functions $H_{i}$, defined by $\mu({\cal P},{\cal Q},{\cal
R})=1+\sum_{i=1}^{2g+1}H_{i}x^{i}$, are independent. According to Proposition
4.6, the generic fiber of $\mu$ (which is the generic fiber of
$H_{1},\dots,H_{2g+1}$) is an open subset of the Jacobian of a curve of genus
$g$, hence has dimension $g$. The dimension of the generic fiber of $\mu$ is
given by $\dim M_{g}-s=3g+1-s$, where $s$ denotes the number of independent
functions in $H_{1},\dots,H_{2g+1}$. Therefore, $s=2g+1$ and the components of
$\mu$ are independent.

It remains to be shown that the rank of
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$ is $2g$. Since $\dim M=3g+1$ and
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$ admits $2g+1$ independent functions in
involution, the rank of $\left\\{\cdot\,,\cdot\right\\}^{\phi}$ is at most
$2g$ (see [53, Prop. 3.4]); to show equality, in a neighborhood of a generic
point $({\cal P},{\cal Q},{\cal R})\in M_{g}$ we take the functions
$x_{1},\dots,x_{n},y_{1},\dots,y_{n}$, which we constructed in the proof of
Proposition 4.6. Then
$\left\\{x_{i},x_{j}\right\\}^{\phi}=\left\\{y_{i},y_{j}\right\\}^{\phi}=0$,
since $\left\\{{\cal P}(x),{\cal P}(y)\right\\}^{\phi}=\left\\{{\cal
Q}(x),{\cal Q}(y)\right\\}^{\phi}=0$. We show that the brackets
$\left\\{x_{i},y_{j}\right\\}^{\phi}$ are non-zero if and only if $i=j$, from
which it follows that the rank of $\left\\{\cdot\,,\cdot\right\\}^{\phi}$ is
indeed $2g$. To do this, we compute for $1\leqslant i\leqslant g$ the Poisson
bracket $\left\\{y_{i},\ln{\cal Q}(y)\right\\}^{\phi}$ in two different ways.
First, using (4.1),

$\left\\{y_{i},\ln{\cal Q}(y)\right\\}^{\phi}=\left\\{{\cal P}(x_{i}),\ln{\cal
Q}(y)\right\\}^{\phi}=\frac{\left\\{{\cal P}(x_{i}),{\cal
Q}(y)\right\\}^{\phi}}{{\cal Q}(y)}=\frac{y\phi(x_{i})-x_{i}\phi(y){\cal
Q}(x_{i})/{\cal Q}(y)}{x_{i}-y}=y\frac{\phi(x_{i})}{x_{i}-y}\;,$ (4.19)

and next, using (4.15),

$\left\\{y_{i},\ln{\cal
Q}(y)\right\\}^{\phi}=\sum_{j=1}^{g}\left\\{y_{i},\ln(1-y/x_{j})\right\\}^{\phi}=y\sum_{j=1}^{g}\frac{\left\\{y_{i},x_{j}\right\\}^{\phi}}{x_{j}(x_{j}-y)}\;,$
(4.20)

so that

$\frac{\phi(x_{i})}{x_{i}-y}=\sum_{j=1}^{g}\frac{\left\\{y_{i},x_{j}\right\\}^{\phi}}{x_{j}(x_{j}-y)}\;.$

Since (generically) all $x_{j}$ are different,
$\left\\{x_{j},y_{i}\right\\}^{\phi}=0$ when $j\neq i$, while
$\left\\{x_{i},y_{i}\right\\}^{\phi}=-x_{i}\phi(x_{i})$, so that

$\left\\{x_{i},y_{j}\right\\}^{\phi}=-x_{i}\phi(x_{i})\delta_{ij}\;.$ (4.21)

Since $\phi\neq 0$, this shows that the rank of
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$ is $2g$. ∎

We now conclude with the main result of this section, namely that the Volterra
map is a _discrete a.c.i. system_. By this we mean that, besides being
Liouville integrable, the generic level sets defined by the invariants are
affine parts of Abelian varieties (complex algebraic tori) and the restriction
of the map to any of these Abelian varieties is a translation.

######  Theorem 4.10.

The Volterra map ${\cal V}_{g}$ is a _discrete a.c.i. system_ on
$(M_{g},\left\\{\cdot\,,\cdot\right\\}^{\phi})$.

###### Proof.

Liouville integrability was already shown in Proposition 4.9. Let
$f\in{\mathbb{C}}[x]$ be a polynomial of degree $2g+1$, without repeated roots
and with $f(0)=1$. Writing $f(x)=1+\sum_{i=0}^{2g+1}c_{i}x^{i}$, the common
level set defined by $H_{i}=c_{i}$, $i=1,2,\dots,2g+1$ is the fiber
$\mu^{-1}(f)$ which was shown in Proposition 4.6 to be an affine part of the
Jacobian of $\bar{\Gamma}_{f}$, and this is indeed an Abelian variety. It
remains to be shown that the restriction of the Volterra map ${\cal V}_{g}$ to
$\overline{\mu^{-1}(f)}\cong\operatorname{Jac}(\bar{\Gamma}_{f})$ is a
translation; more precisely we will show that it is a translation over
$[(0,-1)-\infty]=[\infty-(0,1)]$. Let $({\cal P},{\cal Q},{\cal
R})\in\mu^{-1}(f)$ be a generic point (a regular triplet) so that
$(\tilde{\cal P},\tilde{\cal Q},\tilde{\cal R}):={\cal V}_{g}({\cal P},{\cal
Q},{\cal R})$ also belongs to $\mu^{-1}(f)$. The degree $g$ divisors on
$\bar{\Gamma}_{f}$ corresponding to these two triplets are respectively
denoted by $\mathcal{D}=\sum_{i=1}^{g}(x_{i},y_{i})$ and
$\tilde{\mathcal{D}}=\sum_{i=1}^{g}(\tilde{x}_{i},\tilde{y}_{i})$ (so that the
corresponding divisor classes in $\operatorname{Jac}(\bar{\Gamma}_{f})$ are
$[\mathcal{D}-g\infty]$ and $[\tilde{\mathcal{D}}-g\infty]$, respectively).
Consider the rational function $\tilde{F}$ on $\bar{\Gamma}_{f}$ given by the
action of ${\cal V}_{g}$ on (3.6), which in view of (4.3) and the definition
(4.14) of $f$, can be written in a few different ways:

$\tilde{F}:=\frac{y+\tilde{\cal P}(x)}{\tilde{\cal Q}(x)}=\frac{\tilde{\cal
R}(x)}{y-\tilde{\cal P}(x)}=\frac{-wx{\cal Q}(x)}{y-{\cal Q}(x)+{\cal
P}(x)}\;.$ (4.22)

It is clear from the last, respectively first, expression that $\tilde{F}$ has
a simple zero at the each of the points $(x_{i},y_{i})$ and $(0,-1)$, and a
simple pole at each of the points $(\tilde{x}_{i},\tilde{y}_{i})$ and
$\infty$. To verify the behaviour at $\infty$, one should introduce a local
parameter $z$ such that $x=1/z^{2}$,
$y=\sqrt{c_{2g+1}}z^{-(2g+1)}\big{(}1+\mathcal{O}(z)\big{)}$ there, which
gives $\tilde{F}=\sqrt{c_{2g+1}}q_{g}^{-1}/z+\mathcal{O}(1)$. For the other
zero or pole candidates in $\Gamma_{f}$, coming from places where the
numerators or denominators in (4.22) vanish, one checks using one of the
alternative formulae that $\tilde{F}$ is finite and non-zero at these points.
Thus $\tilde{F}$ has precisely $g+1$ zeros and $g+1$ poles, in accord with the
fact that the degree of the divisor of a rational function is zero. The upshot
is that the divisor of zeros and poles of $\tilde{F}$ is given by

$(\tilde{F})=\sum_{i=1}^{g}(x_{i},y_{i})+(0,-1)-\sum_{i=1}^{g}(\tilde{x}_{i},\tilde{y}_{i})-\infty=\mathcal{D}-\tilde{\mathcal{D}}+(0,-1)-\infty\;,$

which leads to the linear equivalence
$\mathcal{D}+(0,-1)\sim\tilde{\mathcal{D}}+\infty,$ and hence to

$[\tilde{\mathcal{D}}-g\infty]=[\mathcal{D}-g\infty]+[(0,-1)-\infty]\;,$
(4.23)

as was to be shown. ∎

According to Proposition 4.3, when $\phi=\phi_{g}x^{g}+\phi_{g+1}x^{g+1}$ we
can restrict the Volterra map and its Poisson structure to
$M_{g}^{c}=\cap_{i=1}^{g}(H_{i}=c_{i})$, and so by the above theorem the
Volterra map is a discrete a.c.i. system on
$(M_{g}^{c},\left\\{\cdot\,,\cdot\right\\}^{\phi})$. In particular, the
recursion relations obtained by fixing the invariants $H_{1},\dots,H_{g}$ to
generic values $c_{i}$ (and possibly also fixing the other Casimir $C_{\phi}$
to some generic value) are discrete a.c.i. systems.

######  Example 4.11.

In the genus 1 case, it follows that (3.34), (3.37) and (3.39), equipped
respectively with the Poisson structures
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$, $\left\\{\cdot\,,\cdot\right\\}^{x}$
and $\left\\{\cdot\,,\cdot\right\\}^{x^{2}}$, are discrete a.c.i. systems. The
same holds for (3.45), which is a discrete a.c.i. system with respect to the
quadratic Poisson structure (4.12).

######  Example 4.12.

In the genus 2 case, we have that for generic $a,b,\nu$ the map (P.iv) is a
discrete a.c.i. system.

##  5 Continuous flows and the infinite Volterra and Toda lattices

The discrete integrable systems that we have discussed so far are naturally
associated with continuous systems which are equally integrable. More
precisely, Liouville integrability of the Hamiltonian systems associated with
the Volterra maps comes for free, and with some extra work we show that these
continuous systems are also algebraically integrable. We further show that in
the genus $g$ case any solution $w_{i}(t)$ of one of the integrable
Hamiltonian vector fields extends, under the action of the Volterra map, to a
sequence $\big{(}w_{n}(t)\big{)}_{n\in{\mathbb{Z}}}$ that is a solution to the
infinite Volterra lattice; notice that in particular, as discussed in the
introduction, this applies to the map (P.iv). We also discuss the relation
between the infinite Toda and Volterra lattices, which explains in part how
some of the results in this paper are related to the results in [28], and what
motivated us to introduce S-fractions and the corresponding Mumford-like
systems to study the map (P.iv) and its higher genus analogues.

###  5.1 Liouville and algebraic integrability

Recall that on $M_{g}$ we have a family of compatible Poisson brackets
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$ of rank $2g$, as well as a family of
polynomial functions $H_{1},\dots,H_{2g+1}$, where $H_{i}$ is the coefficient
in $x^{i}$ of ${\cal P}^{2}(x)+{\cal Q}(x){\cal R}(x)$; said differently,
$H_{1},\dots,H_{2g+1}$ are the components of the momentum map
$\mu:M_{g}\to{\mathbb{C}}[x]$. For the sake of clarity, and since the choice
of Poisson structure is not important for what follows, we will only consider
$\phi=x^{g+1}$ here, and henceforth write $\left\\{\cdot\,,\cdot\right\\}$ for
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$. With this choice of $\phi$, (4.1) –
(4.8) become

$\displaystyle\left\\{{\cal P}(x),{\cal P}(y)^{2}+{\cal Q}(y){\cal
R}(y)\right\\}$ $\displaystyle=y^{g+1}\frac{y{\cal Q}(y){\cal R}(x)-x{\cal
Q}(x){\cal R}(y)}{x-y}\;,$ $\displaystyle\left\\{{\cal Q}(x),{\cal
P}(y)^{2}+{\cal Q}(y){\cal R}(y)\right\\}$ $\displaystyle=2y^{g+1}\frac{x{\cal
P}(y){\cal Q}(x)-y{\cal P}(x){\cal Q}(y)}{x-y}-y^{g+1}{\cal Q}(x){\cal
Q}(y)\;,$ $\displaystyle\left\\{{\cal R}(x),{\cal P}(y)^{2}+{\cal Q}(y){\cal
R}(y)\right\\}$ $\displaystyle=2xy^{g+1}\frac{{\cal P}(x){\cal R}(y)-{\cal
P}(y){\cal R}(x)}{x-y}+y^{g+1}{\cal Q}(y){\cal R}(x)\;.$

As we have seen, $H_{1},\dots,H_{g}$ are Casimir functions of the Poisson
bracket, as well as $H_{g+1}=C_{\phi}$ (see (4.9)). The vector fields
$\frac{1}{2}\mathrm{X}_{H_{g+i+1}}$ are denoted by $\mathrm{X}_{i}$; we will
mainly be interested in $\mathrm{X}_{1}=\frac{1}{2}\mathrm{X}_{H_{g+2}}$,
which we can compute by dividing the above equations by $2y^{g+2}$ and taking
the limit for $y\to 0$, so that

$\dot{\cal P}(x)=\lim_{y\to 0}\frac{y{\cal Q}(y){\cal R}(x)-x{\cal Q}(x){\cal
R}(y)}{2y(x-y)}=\frac{2{\cal R}(x)-x{\cal Q}(x){\cal
R}^{\prime}(0)}{2x}=\frac{{\cal R}(x)}{x}-\frac{r_{1}}{2}{\cal Q}(x)\;,$ (5.1)

where the dot denotes the derivative $\tfrac{\mathrm{d}}{\mathrm{d}t}$, and
similarly

$\displaystyle\dot{\cal Q}(x)$ $\displaystyle=\frac{2p_{1}-q_{1}}{2}{\cal
Q}(x)-\frac{2{\cal P}(x)-{\cal Q}(x)}{x}\;,$ $\displaystyle\dot{\cal R}(x)$
$\displaystyle=r_{1}{\cal P}(x)+\frac{q_{1}-2p_{1}}{2}{\cal R}(x)-\frac{{\cal
R}(x)}{x}\;.$ (5.2)

Notice that the vector field $\mathrm{X}_{1}$ is (non-homogeneous) quadratic.
From Proposition 4.2, the functions $H_{j}$ are in involution with one another
with respect to $\left\\{\cdot\,,\cdot\right\\}$, which means that the vector
fields $\mathrm{X}_{i}$ all commute.

Note that the Liouville integrability of this continuous system is
incorporated into the Liouville integrability of the discrete system, so the
following statement is an automatic consequence of Proposition 4.9.

######  Proposition 5.1.

The Hamiltonian system $(M_{g},\left\\{\cdot\,,\cdot\right\\},\mu)$ is a
Liouville integrable system.

We now turn to the algebraic integrability of the Mumford-like system, which
is slightly more involved in the continuous case. Recall (for example from [1,
Ch. 6]) that $(M_{g},\left\\{\cdot\,,\cdot\right\\},\mu)$ being an a.c.i.
system means that

  1. (1)

$(M_{g},\left\\{\cdot\,,\cdot\right\\},\mu)$ is a (complex) Liouville
integrable system;

  2. (2)

The generic fiber of $\mu$ is (isomorphic to) an affine part of an Abelian
variety;

  3. (3)

The integrable vector fields are holomorphic (hence constant) on these Abelian
varieties.

Items (1) and (2) have been shown already, in Propositions 5.1 and 4.6,
respectively, so it only remains to address item (3).

######  Proposition 5.2.

The Hamiltonian system $(M_{g},\left\\{\cdot\,,\cdot\right\\},\mu)$ is an
algebraic completely integrable system (a.c.i. system).

###### Proof.

We show (3) for one of the integrable vector fields; then it also holds for
the other integrable vector fields, since the latter are holomorphic on the
fiber and commute with a constant vector field.

The vector field which we consider is the Hamiltonian vector field
$\mathrm{X}_{1}$, given by (5.1) and (5.1). Let $({\cal P}_{0},{\cal
Q}_{0},{\cal R}_{0})$ be a generic point of $M_{g}$ and consider for small
$|t|$ the integral curve $t\mapsto({\cal P}_{t},{\cal Q}_{t},{\cal R}_{t})$ of
$\mathrm{X}_{1}$, starting at (${\cal P}_{0},{\cal Q}_{0},{\cal R}_{0})$. Let
$\mathcal{D}_{t}=\sum_{i=1}^{g}(x_{i}(t),y_{i}(t))-g\infty$ denote the
associated divisor on the algebraic curve $\Gamma_{f}$, defined by it; recall
that $\Gamma_{f}$ is given by $y^{2}=f(x)$ where $f={\cal P}_{0}^{2}+{\cal
Q}_{0}{\cal R}_{0}={\cal P}_{t}^{2}+{\cal Q}_{t}{\cal R}_{t}$. Since the
$x_{i}(t)$ are the roots of ${\cal Q}_{t}(x)$, upon substituting $x=x_{i}(t)$
in the equation (5.1) for $\dot{\cal Q}(x)$ we get

$\dot{\cal Q}(x_{i}(t))=-2\frac{{\cal
P}(x_{i}(t))}{x_{i}(t)}=-2\frac{y_{i}(t)}{x_{i}(t)}\;.$

However, we can also compute $\dot{\cal Q}(x_{i}(t))$ directly from the
explicit formula (4.15) for ${\cal Q}(x)$, to wit

$\dot{\cal Q}(x_{i}(t))=\frac{2\dot{x}_{i}(t)}{x_{i}(t)}\prod_{j\neq
i}\left(1-\frac{x_{i}(t)}{x_{j}(t)}\right)\;.$

Comparing these two expressions gives

$y_{i}(t)=-{\dot{x}_{i}(t)}\prod_{j\neq
i}\left(1-\frac{x_{i}(t)}{x_{j}(t)}\right)\;.$ (5.3)

It follows that for $k=0,\dots,g-1$,

$\sum_{i=1}^{g}\frac{x_{i}^{k}(t)\mathrm{d}x_{i}(t)}{y_{i}(t)}=-\sum_{i=1}^{g}x_{i}^{k}(t)\prod_{j\neq
i}\frac{x_{j}(t)}{x_{j}(t)-x_{i}(t)}\mathrm{d}t=-\delta_{k,0}\,\mathrm{d}t\;.$
(5.4)

Above we have used the following identity which is well-known in the theory of
symmetric functions:

$\sum_{i=1}^{g}x_{i}^{k}\prod_{j\neq
i}\frac{x_{j}}{x_{j}-x_{i}}=\delta_{k,0}\;,\qquad k=0,\dots,g-1;$ (5.5)

the proof of the latter follows easily from the fact that any lowest degree
antisymmetric polynomial in $g$ variables is, up to a factor, the Vandermonde
determinant. Integrating (5.4) from $0$ to $t$ gives

$\int_{\mathcal{D}_{0}}^{\mathcal{D}_{t}}\frac{x^{k}\mathrm{d}x}{y}=-t\,\delta_{k,0}\;.$
(5.6)

The left-hand side of (5.6) contains the differentials $x^{k}\mathrm{d}x/y$
for $k=0,\dots,g-1$, which form a basis for the holomorphic differentials on
$\bar{\Gamma}_{f}$. Thus the left-hand side of (5.6) is the image of the
divisor $\mathcal{D}_{t}-\mathcal{D}_{0}$ under the Abel map, which is (by
Abel’s Theorem) an isomorphism between the algebraic Jacobian of
$\bar{\Gamma}_{f}$, consisting of degree zero divisor classes on
$\bar{\Gamma}_{f}$, and the analytic Jacobian of $\bar{\Gamma}_{f}$, which is
a complex torus, that is

$\operatorname{Jac}(\bar{\Gamma}_{f})\cong
H^{0}\left(\Omega^{1}_{\bar{\Gamma}_{f}}\right)^{*}/H_{1}(\bar{\Gamma}_{f}).$

Formula (5.6) then says that the integral curve of $\mathrm{X}_{1}$ is a
straight line in this complex torus, as was to be shown. ∎

It follows that for, generic initial conditions, the solutions to
$\mathrm{X}_{1}$ are meromorphic functions in $t$.

###  5.2 Genus $g$ solutions to the infinite Volterra and Toda lattices

The infinite Volterra lattice is given by the set of equations

$\dot{w}_{n}=w_{n}(w_{n+1}-w_{n-1})\;,\qquad n\in{\mathbb{Z}}\;.$ (5.7)

It was first considered by Kac and van Moerbeke [31], who also studied the
$N$-periodic case ($w_{N+n}=w_{n}$ for all $n$). We now show that the Volterra
map allows us to define, for any $g$, infinite sequences of meromorphic
functions $(w_{n}(t))_{n\in{\mathbb{Z}}}$ which satisfy (5.7). Since these
sequences are defined from solutions of the genus $g$ Mumford-like system, and
hence can be written in terms of genus $g$ theta functions, we will refer to
these solutions to the Volterra lattice as _genus $g$ solutions_.

Let $g>0$ be fixed and consider the vector field on ${\mathbb{C}}^{3g+1}_{w}$,
corresponding to the vector field $\frac{1}{2}\mathrm{X}_{1}$ on the genus $g$
Mumford-like system, via the birational transformation constructed in Section
3.5. For the sake of brevity, let us call this the w-system (in genus $g$). By
algebraic integrability, the vector field $\frac{1}{2}\mathrm{X}_{1}$ has
globally defined meromorphic solutions $w_{1}(t),\dots,w_{3g+1}(t)$,
corresponding to generic initial conditions. Using the recursion relation
induced by the action of the Volterra map on ${\mathbb{C}}^{3g+1}_{w}$, we get
also globally defined meromorphic functions $w_{n}(t)$ for all $n>3g+1$ and
all $n\leqslant 0$. Algebraic integrability further implies that the recursion
and the flow of the vector field must commute, as they both correspond to
translations on the fibers of the momentum map, which are affine parts of
$g$-dimensional tori. (The fact that the map and the flow commute is already a
consequence of Liouville integrability.) It follows that all formulae only
involving the variables $w_{n}$ remain valid when all indices are shifted by
the same integer. In the proof of the theorem which follows we will make
extensive use of the birational transformation between the w-system and the
Mumford-like system. The triplet corresponding to a meromorphic solution
$(w_{1}(t),\dots,w_{3g+1}(t))$ of the w-system in genus $g$ will be denoted
$({\cal P}_{0}(x;t),{\cal Q}_{0}(x;t),{\cal R}_{0}(x;t))$, the index $0$ being
added because we will also use the triplets $({\cal P}_{n}(x;t),{\cal
Q}_{n}(x;t),{\cal R}_{n}(x;t))$, obtained from it by applying the Volterra map
or its inverse several times. Again, all formulae involving only the
polynomials ${\cal P}_{n},{\cal Q}_{n},{\cal R}_{n}$, $n\in{\mathbb{Z}}$,
remain valid when all indices are shifted by the same integer, and for any
$n\in{\mathbb{Z}}$, $({\cal P}_{n}(x;t),{\cal Q}_{n}(x;t),{\cal R}_{n}(x;t))$
corresponds to $(w_{n+1}(t),\dots,w_{3g+n+1}(t))$ under the birational map.

######  Theorem 5.3.

The sequence of meromorphic functions
$\big{(}w_{n}(t)\big{)}_{n\in{\mathbb{Z}}}$ is a solution to the infinite
Volterra lattice (5.7).

###### Proof.

We first recall the recursion relations (3.9) for the triplets $({\cal
P}_{n},{\cal Q}_{n},{\cal R}_{n})$, which we evaluate at any meromorphic
solution to $\mathrm{X}_{1}=\frac{1}{2}\mathrm{X}_{H_{g+2}}$:

$\displaystyle{\cal P}_{n+1}(x;t)={\cal Q}_{n}(x;t)-{\cal P}_{n}(x;t)\;,\
{\cal Q}_{n+1}(x;t)=\frac{2{\cal P}_{n}(x;t)-{\cal Q}_{n}(x;t)+{\cal
R}_{n}(x;t)}{-w_{n+1}(t)x}\;,$ (5.8) $\displaystyle{\cal
R}_{n+1}(x;t)=-w_{n+1}(t)x{\cal Q}_{n}(x;t)\;.$ (5.9)

From (5.9), since $H_{1}=2(p_{1}+r_{1})$ is a first integral, we have

$w_{n}(t)=-\frac{1}{2}r_{n,1}(t)=\frac{1}{2}p_{n,1}(t)-\frac{c_{1}}{4}\;,\quad\hbox{and}\quad
r_{n,2}(t)=-w_{n}(t)q_{n-1,1}(t)\;,$ (5.10)

where $c_{1}$ is a constant. It follows that

$\displaystyle\dot{w}_{n}(t)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:w_to_p}}}{{=}}\frac{1}{2}\dot{p}_{n,1}(t)\stackrel{{\scriptstyle\eqref{eq:pdot}}}{{=}}\frac{1}{2}r_{n,2}(t)-\frac{1}{4}r_{n,1}(t)q_{n,1}(t)\stackrel{{\scriptstyle\eqref{eq:w_to_p}}}{{=}}\frac{w_{n}(t)}{2}(q_{n,1}(t)-q_{n-1,1}(t))$
$\displaystyle\;\stackrel{{\scriptstyle\eqref{PQrect}}}{{=}}\frac{w_{n}(t)}{2}(p_{n+1,1}(t)-p_{n-1,1}(t))\stackrel{{\scriptstyle\eqref{eq:w_to_p}}}{{=}}w_{n}(t)(w_{n+1}(t)-w_{n-1}(t))\;,$

as was to be shown. ∎

######  Remark 5.4.

It is fairly straightforward to modify the proof of Proposition 5.2, and the
preceding result, to all of the Hamiltonian vector fields $\mathrm{X}_{i}$,
associated with times $t_{i}$, $1\leqslant i\leqslant g$, which correspond to
the first $g$ flows in the Volterra lattice hierarchy. This replaces $t$ by
$t_{i}$ and modifies the Kronecker delta on the right-hand side of (5.6) to
$\delta_{k,i-1}$, hence producing solutions that are meromorphic in
$t=t_{1},t_{2},\ldots,t_{g}$.

We now apply a standard Miura-like formula, to show how a genus $g$ solution
of the Volterra lattice, given by an infinite sequence of meromorphic
functions $w_{n}(t)$, also leads to a corresponding solution to the infinite
Toda lattice, given by

$\frac{\mathrm{d}a_{n}}{\mathrm{d}t}=a_{n}(b_{n-1}-b_{n})\;,\qquad\frac{\mathrm{d}b_{n}}{\mathrm{d}t}=a_{n}-a_{n+1}\;.$
(5.11)

(These are almost the same as the Flaschka variables for the Toda lattice,
except that traditionally $\sqrt{a_{n}}$ is used in place of $a_{n}$; and
similarly, the quantities $\sqrt{w_{n}}$ are used in [37].)

######  Corollary 5.5.

Let $w_{n}(t)$, $n\in{\mathbb{Z}}$ be a genus $g$ meromorphic solution to the
infinite Volterra lattice (5.7). Upon setting, for $j\in{\mathbb{Z}}$,

$a_{j+1}:=w_{2j-1}w_{2j}\;,\qquad b_{j+1}:=-w_{2j}-w_{2j+1}\;,$ (5.12)

the sequence of meromorphic functions $a_{j}(t),b_{j}(t)$ is a solution to the
infinite Toda lattice, while another sequence of meromorphic solutions to
(5.11) is given for $j\in{\mathbb{Z}}$ by

$a^{*}_{j+1}:=w_{2j}w_{2j+1}\;,\qquad b^{*}_{j+1}:=-w_{2j+1}-w_{2j+2}\;.$
(5.13)

###### Proof.

Differentiating (5.12) and using (5.7) one gets immediately (5.11), and
similarly for (5.13). ∎

######  Example 5.6.

Theorem 5.3 and Corollary 5.5 imply that we can obtain elliptic (genus 1)
solutions to the infinite Volterra and Toda lattices, by starting from a
generic solution to (3.34). On a fixed orbit of the latter, any such solution
can be identifed with an orbit of the QRT map (3.45) associated with Somos-5.
Hence this means that the analytic results of [24] can be applied, to write
the tau function explicitly as

$\tau_{n}=A_{\pm}B^{n}\frac{{\sigma}(z_{0}+nz)}{{\sigma}(z)^{n^{2}}},$

where $A_{+},A_{-},B$ are non-zero constants (with $A_{\pm}$ chosen according
to the parity of $n$), and ${\sigma}(z)={\sigma}(z;g_{2},g_{3})$ denotes the
Weierstrass sigma function associated with an elliptic curve
${y}^{2}=4{x}^{3}-g_{2}{x}-g_{3}$, isomorphic to (3.41). The parameters
$z,g_{2},g_{3}$ all depend on $c_{1},c_{2},c_{3}$, while $z_{0}$ also depends
on the initial point on the orbit. Then we can write the solution of the map
explicitly in terms of the Weierstrass zeta function, as

$w_{n}=\frac{{\sigma}\big{(}z_{0}+nz\big{)}{\sigma}\big{(}z_{0}+(n+3)z\big{)}}{{\sigma}(z)^{4}{\sigma}\big{(}z_{0}+(n+1)z\big{)}{\sigma}\big{(}z_{0}+(n+2)z\big{)}}=\hat{c}\Big{(}\zeta\big{(}z_{0}+(n+2)z\big{)}-\zeta\big{(}z_{0}+(n+1)z\big{)}+C\Big{)},$
(5.14)

where $\hat{c}={{\sigma}(2z)}/{{\sigma}(z)^{4}}$, $C=\zeta(z)-\zeta(2z)$. Now
extending this by the flow of the vector field $\mathrm{X}_{1}$, with
parameter $t$, we find that only $z_{0}$ changes, being replaced by
$z_{0}+\hat{c}\,t$ (giving a linear flow on the Jacobian of the elliptic
curve). Hence we arrive at the genus 1 solution of the Volterra lattice, given
for $n\in{\mathbb{Z}}$ by

$w_{n}(t)=\hat{c}\Big{(}\zeta\big{(}z_{0}+\hat{c}\,t+(n+2)z\big{)}-\zeta\big{(}z_{0}+\hat{c}\,t+(n+1)z\big{)}+C\Big{)}$

(equivalent to the travelling waves found in [55]), and from (5.12) we get a
corresponding genus 1 solution of the Toda lattice, that is

$\displaystyle a_{n}(t)$
$\displaystyle=\hat{c}^{4}\Big{(}\wp\big{(}2z\big{)}-\wp\big{(}z_{0}+\hat{c}\,t+(2n-1)z\big{)}\Big{)}\;,$
(5.15) $\displaystyle b_{n}(t)$
$\displaystyle=\hat{c}\Big{(}\zeta\big{(}z_{0}+\hat{c}\,t+(2n-1)z\big{)}-\zeta\big{(}z_{0}+\hat{c}\,t+(2n+1)z\big{)}-2C\Big{)}\;,$

written in terms of the Weierstrass $\wp$ function, with the constants
$\hat{c}$, $C$ as above.

######  Example 5.7.

Similarly, we can produce genus 2 solutions to the infinite Volterra and Toda
lattices, by starting from a generic solution to the map (P.iv), extended to
meromorphic functions $w_{n}(t)$ by the flow of the vector field
$\mathrm{X}_{1}$.

In [37], the transformation (5.12) was used to connect the finite Volterra and
Toda lattices by Moser, who attributed it to Hénon. The same transformation
has further been applied to connect real-valued solutions of the infinite
lattices, subject to suitable (smoothness/boundedness) conditions [19]. Moser
also employed finite continued fractions in [37]. However, it turns out that
the map (5.12) has a much earlier origin in the classical theory of continued
fractions, where it arises from the method of contraction for S-fractions (see
J.3 in [48], and [46]), a fact that has perhaps been overlooked in the
integrable systems literature. In the next subsection, we show how the
Volterra maps, as presented in this paper, are related to the integrable maps
recently constructed by one of us [28]; the key is to apply contraction to the
S-fraction (3.15), which produces a J-fraction, and thereby yields the Miura-
type formula (5.12).

###  5.3 Contraction of continued fractions

We take a fixed set of coefficients $c_{i}$, which are arbitrary except that,
as usual, we assume that the polynomial

$f(x)=1+\sum_{i=1}^{2g+1}c_{i}x^{i}$ (5.16)

is square-free with $c_{2g+1}\neq 0$, so that the hyperelliptic curve
$\Gamma_{f}:\,y^{2}=f(x)$ is smooth and has genus $g$. In order to simplify
the presentation below, initially we make the further assumption that
$c_{1}=0$. Then setting

$X=\frac{1}{x}\;,\qquad Y=\frac{y}{x^{g+1}}\;$ (5.17)

establishes a birational isomorphism between $\Gamma_{f}$ and an algebraic
curve ${\cal C}$ which (by completing the square) can be written in the form

${\cal
C}:\,\,Y^{2}=\hat{\mathrm{F}}(X),\qquad\hat{\mathrm{F}}(X)=A(X)^{2}+4R(X),$
(5.18)

where $A(X)$ is a monic polynomial in $X$ of degree $g+1$ with no term of
degree $g$ (so that the right-hand side of (5.18) has no degree $2g+1$ term),
and $R$ is a polynomial of degree at most $g$ in $X$, not identically zero but
otherwise arbitrary; such curves are exactly the ones which were considered in
[28].

Let $({\cal P},{\cal Q},{\cal R})=({\cal P}_{0},{\cal Q}_{0},{\cal R}_{0})\in
M_{g}$ be a generic triplet satisfying ${\cal P}^{2}+{\cal Q}{\cal R}=f$, in
the sense discussed above (3.15). As we have seen in (3.15), the associated
rational function on $\Gamma_{f}$, denoted $F_{0}$, admits the following
expansion as an S-fraction:

$F_{0}=\frac{y+{\cal P}_{0}(x)}{{\cal
Q}_{0}(x)}=1-\cfrac{w_{1}x}{1-\cfrac{w_{2}x}{1-\cfrac{w_{3}x}{1-\cdots}}}\;.$
(5.19)

Now, upon multiplying (5.19) by $x^{-1}=X$, rewriting the S-fraction in terms
of the new spectral parameter $X$, and then applying contraction, we find

$x^{-1}F_{0}=X-\cfrac{w_{1}}{1-\cfrac{w_{2}}{X-\cfrac{w_{3}}{1-\cfrac{w_{4}}{X-\cdots}}}}=X+b_{1}-\cfrac{a_{2}}{X+b_{2}-\cfrac{a_{3}}{X+b_{3}-\cfrac{a_{4}}{X+b_{4}-\cdots}}}\;,$
(5.20)

where the J-fraction on the right above is the contraction of the S-fraction.
More precisely, the second equality in (5.20) is an identity of continued
fractions, obtained by combining successive pairs of adjacent lines in the
S-fraction into a single sequence of lines in the J-fraction, with
coefficients $a_{j},b_{j}$ related to $w_{j}$ by

$b_{1}=-w_{1}\;,\qquad a_{j+1}=w_{2j-1}w_{2j}\;,\qquad
b_{j+1}=-w_{2j}-w_{2j+1}\;,\qquad\mathrm{for}\quad j\geqslant 1\;.$ (5.21)

In particular, we observe that this reproduces the transformation (5.12)
between the Volterra and Toda lattices, but for indices $j\geqslant 1$ only.

We now briefly recall the construction of integrable maps associated with
J-fractions, as presented in [28]. The starting point is a rational function
$Y_{0}$ on an even hyperelliptic curve ${\cal C}$ of the form (5.18), whose
completion $\bar{{\cal C}}$ includes two points $\infty_{1},\infty_{2}$ at
infinity. This function has $g+1$ simple poles and $g+1$ simple zeros, with
one pole being at the point $\infty_{1}$ (where $Y\sim X^{g+1}\sim A(X)$ as
$X\to\infty$), and one zero being at $\infty_{2}$ (where
$Y\sim-X^{g+1}\sim-A(X)$), taking the form

$Y_{0}=\frac{Y+P_{0}(X)}{Q_{0}(X)}\;,$ (5.22)

for polynomials $P_{0}$, of degree $g+1$ with no term at ${\cal O}(X^{g})$,
and $Q_{0}$, of degree $g$; and there exists another polynomial $Q_{-1}$, of
degree $g$, satisfying $Y^{2}=P_{0}^{2}+Q_{0}Q_{-1}=\hat{\mathrm{F}}$. The
expansion of $Y_{0}$ around the point $\infty_{1}$, with $X^{-1}$ as a local
parameter, can be considered as an element of ${\mathbb{C}}((X^{-1}))$, and it
was shown by van der Poorten (see [40, 43]) that this power series admits a
J-fraction expansion of the form

$Y_{0}={\alpha}_{0}+\cfrac{1}{Y_{1}}={\alpha}_{0}+\cfrac{1}{{\alpha}_{1}+\cfrac{1}{Y_{2}}}=\cdots={\alpha}_{0}+\cfrac{1}{{\alpha}_{1}+\cfrac{1}{{\alpha}_{2}+\cfrac{1}{\ddots}}}$
(5.23)

with ${\alpha}_{n}:=\left\lfloor Y_{n}\right\rfloor$, the polynomial part of
$Y_{n}$. Furthermore, for a generic choice of such $P_{0}$, $Q_{0}$ in (5.22),
the polynomial parts ${\alpha}_{n}$ are of degree 1 in $X$ for any $n$, and
the recursion $Y_{n}={\alpha}_{n}+\frac{1}{Y_{n+1}}$ leads to a sequence of
polynomials $P_{n},Q_{n}$ satisfying the same degree constraints as above,
such that

$Y_{n}=\frac{Y+P_{n}(X)}{Q_{n}(X)}=\frac{Q_{n-1}(X)}{Y-P_{n}(X)}\;\implies
Y^{2}=\hat{\mathrm{F}}(X)=P_{n}(X)^{2}+Q_{n}(X)Q_{n-1}(X),$ (5.24)

where the above relations extend to all $n\in{\mathbb{Z}}$, not just
$n\geqslant 0$, by reversing (5.23) to find $Y_{-1}$ from $Y_{0}$, etc.

The situation for the J-fraction expansion (5.23) is very similar to that for
the expansion (5.19), and allows the construction of a birational dynamical
system that is defined by a recursion for the polynomials $P_{n},Q_{n}$,
analogous to the derivation of the Volterra map from the S-fraction in Section
3. Here we refer to the dynamical system (in dimension $3g+1$) defined by
(5.23) as the J-fraction map in genus $g$, denoted ${\cal J}_{g}$. In order to
explain the very close connection between ${\cal V}_{g}$ and ${\cal J}_{g}$,
and prove Theorem 5.10, we further summarize some features of the latter,
while referring the reader to [28] for a complete description.

To describe the dynamics defined by $P_{n},Q_{n}$, new variables $u_{n}$,
$d_{n}$ and $v_{n}$ are introduced from

$P_{n}(X)=A(X)+2d_{n}X^{g-1}+{\cal O}(X^{g-2})\;,\quad
Q_{n}(X)=u_{n}\big{(}X^{g}-v_{n}X^{g-1}+{\cal O}(X^{g-2})\big{)}\;,$ (5.25)

so that from the terms of ${\cal O}(X^{2g})$ in the equation for
$\hat{\mathrm{F}}$ on the right-hand side of (5.24), the relation

$u_{n}u_{n-1}=-4d_{n}\neq 0\;$ (5.26)

must hold, while calculating the (degree 1) polynomial parts in each line of
(5.23) shows that, for any $n$, we have ${\alpha}_{n}=(X+b_{n})/u_{n}$. Thus,
upon substituting for ${\alpha}_{n}$ and rescaling each line of the continued
fraction using (5.26), we may rewrite the J-fraction (5.23) more explicitly as

${Y_{0}}={\alpha}_{0}+\frac{1}{Y_{1}}={\frac{2(X+v_{0})}{u_{0}}+\cfrac{1}{\frac{2(X+v_{1})}{u_{1}}+\cfrac{1}{\frac{2(X+v_{2})}{u_{2}}-\cdots}}},$

where, by setting $\hat{s}_{0}=\frac{u_{1}}{2}$, we have

$Y_{1}=\frac{1}{\hat{s}_{0}}\left(\,{X+v_{1}-\cfrac{d_{2}}{X+v_{2}-\cfrac{d_{3}}{X+v_{3}-\cdots}}}\;\right)\;.$
(5.27)

Then the J-fraction map ${\cal J}_{g}$ is a dynamical system on an affine
phase space $\hat{M}_{g}^{\hat{c}_{1}=0}$ of dimension $3g+1$, which fibers
over the space of curves ${\cal C}$ of the form (5.18), with each (generic)
fiber being an affine part of the corresponding Jacobian variety
$\operatorname{Jac}(\bar{{\cal C}})$; and on each such fiber, the map
corresponds to a translation by the class of the divisor
$\infty_{2}-\infty_{1}$. It is defined recursively by (5.24), in terms of the
coefficients of the polynomials $P_{n},Q_{n}$, except that the prefactors
$u_{n}$ in front of each $Q_{n}$, as in (5.25), are completely decoupled from
the dynamics. Indeed, the constant $\hat{s}_{0}$ in (5.27) is arbitrary: it
determines the first coefficient in the series expansion of the moment
generating function $1/Y_{1}=\sum_{j\geqslant 0}\hat{s}_{j}X^{-j-1}$, whose
coefficients allow the solutions of ${\cal J}_{g}$ to be written in terms of
tau functions given by Hankel determinants; but $\hat{s}_{0}$ can be removed
by a gauge transformation on the tau functions. Moreover, once $\hat{s}_{0}$
is fixed, then $u_{1}$ and all the other prefactors $u_{n}$ are determined
from $\hat{s}_{0}$ and $d_{n}$, due to (5.26); and the phase space
$\hat{M}_{g}^{\hat{c}_{1}=0}$ (which is an affine space of Lax matrices) does
not include the parameter $\hat{s}_{0}$. After decoupling from $u_{n}$, the
map ${\cal J}_{g}$ can be written equivalently as a recursion for the
remaining coefficients in $P_{n},Q_{n}$, or as coupled recurrences for the
quantities $d_{n},v_{n}$. (See (5.33) and (5.35) in the examples below for the
cases $g=1$ and $g=2$, respectively.)

We would now like to identify (5.27) with the J-fraction appearing on the
right in (5.20), but there are two problems: firstly, the relation (5.21) is
valid only for $j\geqslant 1$, and gives a different formula for $b_{1}$ when
$j=0$; and secondly, we initially made the assumption that $c_{1}=0$, which
does not hold in general. To relax the latter assumption, we must shift the
spectral parameter $X$, as in (5.30), make a compensating shift in $b_{j}$,
and allow a linear relation between $x^{-1}F_{0}$ and $Y_{1}$, yielding a
modification of (5.12), valid for all $j\in{\mathbb{Z}}$.

######  Proposition 5.8.

The odd, genus $g$ spectral curve $\Gamma_{f}:\,y^{2}=f(x)$ with
$f(x)=1+\sum_{i=1}^{2g+1}c_{i}x^{i}$, associated with a generic orbit of the
Volterra map ${\cal V}_{g}$, is isomorphic to ${\cal C}$, the even spectral
curve (5.18) for a corresponding orbit of the J-fraction map ${\cal J}_{g}$,
via the birational equivalence $X=x^{-1}+c_{1}/(2g+2)$, $Y=y/x^{g+1}$. Under
this isomorphism of curves, the function $F_{0}$ on $\Gamma_{f}$ and the
function $Y_{1}$ on ${\cal C}$ are related by

$x^{-1}F_{0}-w_{0}=\hat{s}_{0}Y_{1},$ (5.28)

and the quantities $d_{n},v_{n}$ satisfying the map ${\cal J}_{g}$ are given
in terms of the solution of ${\cal V}_{g}$ by

$d_{j+1}=w_{2j-1}w_{2j}\;,\qquad
v_{j+1}=-w_{2j}-w_{2j+1}-\frac{c_{1}}{2(g+1)}\;,\qquad\mathrm{for}\quad
j\in{\mathbb{Z}}\;.$ (5.29)

Hence each iteration on the orbit of ${\cal J}_{g}$ corresponds to two
iterations on the corresponding orbit of ${\cal V}_{g}$.

###### Proof.

The shift in $X$ in the birational transformation, as in (5.30), ensures that
the equation $Y^{2}=\hat{\mathrm{F}}(X)=A(X)^{2}+4R(X)$ for ${\cal C}$ has no
term at ${\cal O}(X^{2g+1})$ in $\hat{\mathrm{F}}(X)$, so that
$A(X)=X^{g+1}+{\cal O}(X^{g-1})$, as required for a spectral curve of the
J-fraction map. Also, from the explicit form of the function $F_{0}$ in
(5.19), we may rewrite the left-hand side of (5.28) in terms of $Y$ and
$\tilde{X}=X-c_{1}/(2g+2)$, as

$\frac{\tilde{X}\big{(}\tilde{X}^{-(g+1)}Y+{\cal
P}_{0}(\tilde{X}^{-1})\big{)}-w_{0}{\cal Q}_{0}(\tilde{X}^{-1})}{{\cal
Q}_{0}(\tilde{X}^{-1})}=\frac{Y+P_{1}(X)}{Q_{1}(X)/\hat{s}_{0}}$

where we calculate $P_{1}(X)=\tilde{X}^{g+1}{\cal
P}_{0}(\tilde{X}^{-1})-w_{0}\tilde{X}^{g}{\cal
Q}_{0}(\tilde{X}^{-1})=\tilde{X}^{g+1}+(p_{0,1}-2w_{0})\tilde{X}^{g}+{\cal
O}(\tilde{X}^{g-1})$, and then in view of (3.25) we see that
$P_{1}(X)=X^{g+1}+{\cal O}(X^{g-1})$, while
$Q_{1}(X)=\hat{s}_{0}\tilde{X}^{g}{\cal
Q}_{0}(\tilde{X}^{-1})=\tfrac{u_{1}}{2}\big{(}2\tilde{X}^{g}+{\cal
O}(\tilde{X}^{g-1})\big{)}=u_{1}\big{(}X^{g}+{\cal O}(X^{g-1})\big{)}$, so
both of $P_{1}$ and $Q_{1}$ are polynomials in $X$ of the required form for
the J-fraction map. Now from the S-fraction in (5.28), we find that combining
contraction with the shift of spectral parameter modifies (5.20), so that, in
terms of fractions in $X$, $x^{-1}F_{0}-w_{0}$ is equal to

$X-\tfrac{c_{1}}{2g+2}-w_{0}-\cfrac{w_{1}}{1-\cfrac{w_{2}}{X-\tfrac{c_{1}}{2g+2}-\cfrac{w_{3}}{1-\cfrac{w_{4}}{X-\tfrac{c_{1}}{2g+2}-\cdots}}}}=X+v_{1}-\cfrac{d_{2}}{X+v_{2}-\cfrac{d_{3}}{X+v_{3}-\cfrac{d_{4}}{X+v_{4}-\cdots}}}\;,$

where we have inserted the formula for $Y_{1}$ from (5.27), and cancelled the
arbitrary constant $\hat{s}_{0}=\frac{u_{1}}{2}$. Comparing the first line of
the above fractions on each side, we see that
$v_{1}=-w_{0}-w_{1}-c_{1}/(2g+2)$, which is the correct form of the relation
for $v_{1}$ in (5.29) when $j=0$, and contraction of the subsequent lines on
the left give these expressions for $d_{j+1},v_{j+1}$ for all $j\geqslant 1$.
One can also shift the fraction on the left down by two lines, to get a
relation between $F_{2}$ and $Y_{2}$, and the fraction can be inverted to find
a relation between $F_{-2}$ and $Y_{0}$; so, by continuing down or up in this
way, we find that $x^{-1}F_{2j}-w_{2j}=\tfrac{u_{j+1}}{2}Y_{j+1}$ holds for
all $j\in{\mathbb{Z}}$, extending (5.29) to negative $j$ as well. In all
expressions, a shift of indices $j\to j+1$ gives a single iteration of ${\cal
J}_{g}$, but all indices of the Volterra variables increase by 2, giving two
iterations of ${\cal V}_{g}$. ∎

######  Corollary 5.9.

Under the action of the Hamiltonian vector field $\mathrm{X}_{1}$, each
generic solution of ${\cal V}_{g}$ produces a genus $g$ solution of the Toda
lattice equation (1.11) which also satisfies the map ${\cal J}_{g}$, via the
transformation (5.29).

###### Proof.

This follows immediately from the preceding result, by applying Theorem 5.3,
and noting that the result of Corollary 5.5 still stands if we set
$a_{n}=d_{n}$, $b_{n}=v_{n}+c_{1}/(2g+2)$ for all $n$. ∎

The main results of this subsection are collected in the following statement.

######  Theorem 5.10.

By contraction of the S-fraction (5.19) for the associated rational function
$F_{0}$, a generic orbit of the Volterra map ${\cal V}_{g}$ corresponding to a
fixed odd spectral curve $\Gamma_{f}:\,y^{2}=f(x)$ of genus $g$, for square-
free $f(x)$ as in (5.16), is transformed to an orbit of the integrable map
${\cal J}_{g}$ constructed in [28] from the J-fraction (1.10), with the even
spectral curve ${\cal C}$ given by (5.18) with
$\hat{\mathrm{F}}(X)=X^{2g+2}+\sum_{j=2}^{2g+2}\hat{c}_{j}X^{2g+2-j}$. For
coefficients $\hat{c}_{j}$ given suitably in terms of $c_{i}$, there is a
birational equivalence between $\Gamma_{f}$ and ${\cal C}$, given by

$X=x^{-1}+\frac{c_{1}}{2(g+1)},\qquad Y=\frac{y}{x^{g+1}}.$ (5.30)

Moreover, the translation on $\operatorname{Jac}(\bar{{\cal C}})$ associated
with a single iteration of the J-fraction map corresponds to twice the
translation on $\operatorname{Jac}(\bar{\Gamma}_{f})$ associated with each
iteration of ${\cal V}_{g}$. In fact, each generic orbit of ${\cal V}_{g}$ is
related to two different orbits of ${\cal J}_{g}$ in this way.

###### Proof.

The main statements in the theorem were already proved in Proposition 5.8. For
the relation between shifts on complex tori, note that in $M_{g}$, we have
that $\mu^{-1}(f)$, the fiber over a generic curve $\Gamma_{f}$, is an affine
part of $\operatorname{Jac}(\bar{\Gamma}_{f})$, while in
$\hat{M}_{g}^{\hat{c}_{1}=0}$ the fiber over ${\cal C}$ is an affine part of
$\operatorname{Jac}(\bar{{\cal C}})$; but then the isomorphism (5.30) between
these two spectral curves means that
$\operatorname{Jac}(\bar{\Gamma}_{f})\cong\operatorname{Jac}(\bar{{\cal C}})$.
It was shown in [28] that each iteration of the J-fraction map ${\cal J}_{g}$
corresponds to a translation by the class of the divisor
$\infty_{2}-\infty_{1}$ on $\operatorname{Jac}{\bar{{\cal C}}}$, where
$\infty_{1,2}$ are the two points at infinity on ${\bar{{\cal C}}}$, and these
are equivalent to the points $(0,\pm 1)$ on (3.5). So in terms of
$\operatorname{Jac}(\bar{\Gamma}_{f})$, this is a translation by the class of
the divisor $(0,-1)-(0,1)=2(0,-1)-(0,-1)-(0,1)\sim
2\big{(}(0,-1)-\infty\big{)}$, that is by $2[(0,-1)-\infty\big{]}$, twice the
shift corresponding to the Volterra map ${\cal V}_{g}$ (as found in the proof
of Theorem 4.10). Finally, notice that in Corollary 5.5 there is the second,
alternative formula (5.13), with the indices on all $w_{i}$ shifted one step
forwards. This corresponds to the freedom to start the contraction procedure
one line lower in the S-fraction (5.20), beginning with $F_{1}$ rather than
$F_{0}$; so the indices on all Volterra variables must be shifted by the same
amount, and the relation (5.28) with the corresponding rational function on
${\cal C}$ is modified to $x^{-1}F_{1}-w_{1}=\hat{s}_{0}Y_{1}$. Then, in terms
of the J-fraction coefficients, this produces

$d_{j+1}=w_{2j}w_{2j+1}\;,\qquad
v_{j+1}=-w_{2j+1}-w_{2j+2}-\frac{c_{1}}{2(g+1)}\;,\qquad\mathrm{for}\quad
j\in{\mathbb{Z}}\;.$ (5.31)

Thus each orbit of ${\cal V}_{g}$ is transformed to two different orbits of
${\cal J}_{g}$, since the resulting orbit of the latter map remains the same
when the index on the $w_{i}$ in (5.31) is shifted by a multiple of 2. ∎

######  Remark 5.11.

In [28], the phase space $\hat{M}_{g}^{\hat{c}_{1}=0}$ for the J-fraction map
${\cal J}_{g}$ is obtained from $\hat{M}_{g}$, an affine space of dimension
$3g+2$ with a specific Poisson structure $\left\\{\cdot\,,\cdot\right\\}$, by
restricting to a subvariety defined by setting the value of one of the
Casimirs to zero. (This is analogous to the situation described in the
appendix, Section 7.) Although Theorem 5.10 has been stated in terms of a
correspondence between specific orbits of ${\cal V}_{g}$ and ${\cal J}_{g}$,
the considerations in the proof make it clear that, since the coresponding
generic fibers are birationally equivalent, the restriction of ${\cal J}_{g}$
to each fiber is (conjugate to) the square $({\cal V}_{g})^{2}={\cal
V}_{g}\circ{\cal V}_{g}$. This gives a strong hint that
$\hat{M}_{g}^{\hat{c}_{1}=0}$ and $M_{g}$ should also be birationally
equivalent, making this into a global statement about the two maps. While this
global statement is by no means obvious, the first example below shows that it
is correct when $g=1$; but it is not true as a Poisson isomorphism, at least
for the specific Poisson structure introduced in [28]. The problem of finding
alternative Poisson structures on $\hat{M}_{g}$, and making all these
statements precise, is best left for future work.

######  Example 5.12.

When $g=1$, with the cubic
$\Gamma_{f}:\,y^{2}=1+c_{1}x+c_{2}x^{2}+c_{3}x^{3}$, the transformation (5.30)
is

$X=x^{-1}+\frac{c_{1}}{4},\qquad Y=\frac{y}{x^{2}}\quad\implies\quad{\cal
C}:\quad Y^{2}=(X^{2}+\hat{f})^{2}+4\hat{u}X+4\hat{h},$

where the quartic curve ${\cal C}$ is written in terms of the parameters

$\hat{f}=\frac{1}{2}c_{2}-\frac{3}{16}c_{1}^{2},\quad\hat{u}=\frac{1}{4}c_{3}-\frac{1}{8}c_{1}c_{2}+\frac{1}{32}c_{1}^{3},\quad\hat{h}=\frac{1}{16}(c_{1}^{2}c_{2}-c_{1}c_{3}-c_{2}^{2})-\frac{3}{256}c_{1}^{4}.$
(5.32)

Under the transformation (5.29), solutions of the $g=1$ Volterra map ${\cal
V}_{1}$, given by (3.34), or equivalently by (3.37) (with fixed $c_{3}$), or
by (3.39) (with fixed $c_{2}$), are mapped to solutions of the corresponding
J-fraction map, which (according to the results in Example 3.2 in [28]) can be
written as a 2D map defined by

$\displaystyle d_{n+1}$ $\displaystyle=-d_{n}-v_{n}^{2}-\hat{f}\;,$ (5.33)
$\displaystyle v_{n+1}$ $\displaystyle=-v_{n}+\frac{\hat{u}}{d_{n+1}}\;,$

on a reduced phase space with fixed parameters $\hat{f},\hat{u}$, which are
determined from (5.32) in terms of the values of the constants
$c_{1},c_{2},c_{3}$ for the solution of the map ${\cal V}_{1}$. The map (5.33)
has the conserved quantity

$\hat{H}=d_{n}(v_{n}^{2}+d_{n}+\hat{f})-\hat{u}v_{n},$

which, on the orbit corresponding to a fixed solution of ${\cal V}_{1}$, takes
the value $\hat{H}=\hat{h}$ given in (5.32). The Poisson bracket presented for
the J-fraction maps in [28] becomes the canonical bracket

$\\{v_{n},d_{n}\\}=1$ (5.34)

on the 2D phase space of the map (5.33), with coordinates $(d_{n},v_{n})$, and
it can be verified directly that the vector field $\\{\cdot,\hat{H}\\}$
extends to the Toda lattice flow (1.11) for all $n\in{\mathbb{Z}}$ under the
action of this J-fraction map. Upon comparing with Example 5.6, it is clear
that the analytic expressions for $a_{n},b_{n}$ in (5.15) provide explicit
formulae for the solutions of both the map (5.33) and the Toda lattice, by
setting $d_{n}(t)=a_{n}(t)$ and $v_{n}(t)=b_{n}(t)-\tfrac{1}{4}c_{1}$. Upon
comparison of (5.14) with (5.15), it can be seen that each iteration of ${\cal
V}_{1}$ gives a shift by $z$ on the Jacobian of the elliptic curve, while each
iteration of (5.33) produces a shift by $2z$.

However, the bracket (5.34) cannot be a reduction of any of the Poisson
structures in Example 4.4, because the parameters $\hat{f},\hat{u}$ do not
correspond to Casimirs of any of these brackets on the phase space $M_{1}$.
(This is immediately obvious for the pencil of brackets generated by
$\left\\{\cdot\,,\cdot\right\\}^{x}$ and
$\left\\{\cdot\,,\cdot\right\\}^{x^{2}}$, while a short calculation shows this
to be the case for (4.12) as well.) Nevertheless, it is still possible to
interpret (5.29) as a Poisson map in terms of members of the pencil
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$, with $c_{1}$ being the fixed value of
a Casimir. For instance, using the Nambu-Poisson structure, we can construct a
birational transformation between the symplectic leaves of
$\left\\{\cdot\,,\cdot\right\\}^{x^{2}}$, that is
$M_{1}^{c}\cap(H_{2}=c_{2})$, and a 2D phase space for the J-fraction map with
coordinates $(d_{n},v_{n})$, with $c_{1},c_{2}$ viewed as fixed parameters, on
which the bracket reduces to

$\\{v_{n},d_{n}\\}^{x^{2}}=v_{n}-\frac{1}{4}c_{1}.$

Now we take a particular numerical example, with the elliptic curve
$y^{2}=1-4x+4x^{3}$ ($c_{1}=-4$, $c_{2}=0$, $c_{3}=4$), and $w_{1}=1$,
$w_{2}=2$. The sequence $(w_{n})$ extends backwards to $n\leqslant 0$ to give
a singular orbit of (3.34), with the same singularity pattern appearing as was
found for (P.iv) in Section 2:

$\ldots,\tfrac{7}{15},\tfrac{10}{3},-\tfrac{3}{2},-\tfrac{1}{2},2,1,0,\infty,\infty,0,1,2,-\tfrac{1}{2},-\tfrac{3}{2},\tfrac{10}{3},\tfrac{7}{15},\ldots.$

This orbit is symmetrical, in the sense that $w_{-n}=w_{n-3}$ for all
$n\in{\mathbb{Z}}$. By applying the formulae in Example 3.9, the solution is
expressed in terms of Hankel determinants $\Delta_{n}$ constructed from the
moment sequence determined from $s_{1}=1$, $s_{2}=2$,
$s_{j}=\sum_{i=1}^{j-1}s_{i}s_{j-i}-\sum_{i=1}^{j-2}s_{i}s_{j-i-1}$
($j\geqslant 3$), that is $(s_{j})_{j\geqslant
1}:\,1,2,3,6,14,37,105,312,956,2996,9554,\ldots$, which gives

$(\Delta_{n})_{n\geqslant-2}:\quad
1,1,1,1,2,-1,3,-5,7,-4,23,29,59,129,314,\ldots.$

With $\Delta_{-3}=0$, this extends backwards to a sequence of tau functions
$\tau_{n}=\Delta_{n-3}=(-1)^{n+1}\tau_{-n}$ for $n\in{\mathbb{Z}}$; hence from
(3.42) and (3.48) we find $c_{2}^{\prime}=-1$, $c_{3}^{\prime}=1$, so for all
$n$ they satisfy the Somos-5 relation

$\tau_{n+5}\tau_{n}=-\tau_{n+4}\tau_{n+1}+\tau_{n+3}\tau_{n+2}.$

Applying (5.30) in this case produces the quartic curve
$Y^{2}=(X^{2}-3)^{2}-4(X+2)$, the same one as in Example 4.2 from [28], and
the contraction formulae (5.29) and (5.31) with $j=1$ give initial points on
two different orbits of the map (5.33) with parameters $\hat{f}=-3$,
$\hat{u}=-1$, namely $(d_{2},v_{2})=(2,-\tfrac{1}{2})$ and $(-1,3)$,
respectively, which both correspond to the value $\hat{h}=-2$ for the
conserved quantity of this map. For these two different orbits, we find that

$d_{n}=\frac{\hat{\tau}_{n-1}\hat{\tau}_{n+1}}{\hat{\tau}_{n}^{2}},$

where $\hat{\tau}_{n}=\Delta_{2n-4}$ (even index Hankel determinants) for the
first orbit, and $\hat{\tau}_{n}=\Delta_{2n-3}$ (odd index Hankel
determinants) for the second one. It follows from Proposition 5.1 in [28] that
either of these even/odd index subsequences ($1,1,2,3,7,23,59,314,\ldots$ and
$1,1,-1,-5,-4,29,129,\ldots$, respectively) must satisfy the same Somos-4
relation, in this case the original one introduced by Somos [47]:

$\hat{\tau}_{n+4}\hat{\tau}_{n}=\hat{\tau}_{n+3}\hat{\tau}_{n+1}+\hat{\tau}_{n+2}^{2}.$

This connection between Somos-5 and Somos-4 has already been exploited
elsewhere (in [24], and in [8], for example); however, the specific Hankel
determinants for Somos-4 obtained here via contraction differ from the ones
found in [8], and also from the ones derived directly from the J-fraction in
[28].

######  Example 5.13.

An orbit of the map (P.iv), with the spectral curve (1.8), can be transformed
to an orbit of the $g=2$ J-fraction map, as discussed in Example 3.3 of [28],
which has an associated sextic curve, related to it via $X=x^{-1}+\nu/3$,
$Y=y/x^{3}$, that can be taken in the form

${\cal C}:\quad
Y^{2}=(X^{3}+\hat{f}X+\hat{g})^{2}+4(\hat{u}X^{2}+\hat{h}_{1}X+\hat{h}_{2}).$

Defined on a 4D phase space with coordinates $(d_{n-1},d_{n},v_{n-1},v_{n})$
and depending on the 3 parameters $\hat{f},\hat{g},\hat{u}$, the map in [28]
is given by

$\displaystyle
d_{n+1}+d_{n}+d_{n-1}+\hat{u}/d_{n}+v_{n}^{2}+v_{n}v_{n-1}+v_{n-1}^{2}+\hat{f}$
$\displaystyle=0,$ (5.35)
$\displaystyle(2v_{n}+v_{n-1})d_{n}+(2v_{n}+v_{n+1})d_{n+1}+v_{n}^{3}+\hat{f}v_{n}+\hat{g}$
$\displaystyle=0.$

More precisely, there are two different orbits of (5.35) obtained from each
orbit of (P.iv), depending on whether the formula (5.29) or (5.31) is applied.
Then, upon writing each term $w_{j}$ satisfying (P.iv) as a ratio of the
Hankel determinants $\Delta_{j}$ defined in Example 3.11, we see that the
quantities $d_{n}$ that appear in the solution of the J-fraction map, as
above, are given as a ratio of tau functions, in two different ways:

$d_{n}=\frac{\hat{\tau}_{n-1}\hat{\tau}_{n+1}}{\hat{\tau}_{n}^{2}},\qquad\hat{\tau}_{n}=\Delta_{2n-4}\quad\mathrm{or}\quad\Delta_{2n-3},$

for $n\in{\mathbb{Z}}$, where the orbit is determined by the choice of parity
of the index on $\Delta_{j}$. It follows from the proof of Theorem 5.5 in [28]
that (regardless of which choice is made), the tau functions $\hat{\tau}_{n}$
satisfy a Somos-8 relation, which explains why the relation (2.9) appears in
Example 2.2.

##  6 Conclusions and outlook

We have seen how the map (P.iv) obtained by Gubbiotti et al. can naturally be
viewed as the $g=2$ member of a family of algebraically integrable maps,
defined for each $g$, that are naturally related to the infinite Volterra
lattice equation, leading to genus $g$ solutions of the latter. This begs the
question as to what can be said about the other Liouville integrable maps in
4D found in [21], namely (P.v) and (P.vi), which take the same form (1.1) but
for a different rational function $\mathrm{F}$. Note that (P.vi) depends on an
extra parameter compared with (P.v), which we denote here by ${\delta}$
(instead of ${\delta}^{2}$ in [21]). In fact, (P.v) arises from (P.vi) in the
limit ${\delta}\to 0$.

So far we have made the following observations:

  * •

Each solution $\hat{w}_{n}$ of (P.v) is mapped to a solution $w_{n}$ of (P.iv)
via the transformation

$w_{n}=\hat{w}_{n+1}\hat{w}_{n},$ (6.1)

by suitably identifying the parameters $\nu,a,b$ for (P.iv) in terms of the
parameters and first integrals for (P.v).

  * •

Under the flow of the Hamiltonian vector field
$\tfrac{\mathrm{d}}{\mathrm{d}t}$ associated with one of its first integrals,
the sequence $(\hat{w}_{n})_{n\in{\mathbb{Z}}}$ generated by iteration of the
map (P.v) extends to a solution of the modified Volterra lattice in the form

$\frac{\mathrm{d}\hat{w}_{n}}{\mathrm{d}t}=\hat{w}_{n}^{2}(\hat{w}_{n+1}-\hat{w}_{n-1}).$
(6.2)

  * •

Each solution $\hat{w}_{n}$ of (P.vi) is mapped to a pair of solutions
$w_{n}^{(+)}$, $w_{n}^{(-)}$ of (P.iv), via the transformations

$w_{n}^{(\pm)}=(\hat{w}_{n+1}\pm{\delta})(\hat{w}_{n}\mp{\delta}),$ (6.3)

by suitably identifying the parameters $\nu,a,b$ for (P.iv) in terms of the
parameters and first integrals for (P.vi).

  * •

Under the flow of the Hamiltonian vector field
$\tfrac{\mathrm{d}}{\mathrm{d}t}$ associated with one of its first integrals,
the sequence $(\hat{w}_{n})_{n\in{\mathbb{Z}}}$ generated by iteration of the
map (P.vi) extends to a solution of the modified Volterra lattice in the form

$\frac{\mathrm{d}\hat{w}_{n}}{\mathrm{d}t}=(\hat{w}_{n}^{2}-{\delta}^{2})(\hat{w}_{n+1}-\hat{w}_{n-1}).$
(6.4)

The formulae (6.1) and (6.3) are the well-known expressions for the Miura
transformation connecting the two forms of the modified Volterra lattice
equation, given by (6.2) and (6.4), respectively, to the Volterra lattice
(5.7). Thus the above statements about the connections between the maps can be
viewed as restrictions of a Miura transformation to a finite-dimensional phase
space. Preliminary calculations, and the results of [55] on elliptic
solutions, indicate that this picture should extend to arbitrary genus $g$. We
propose that the full description of the above observations, and their
extension to genus $g$ analogues of the maps (P.v) and (P.vi), should be left
as the subject of future work.

It is also worth pointing out that part of the original motivation for the
work in [21] was to consider autonomous versions of the higher order discrete
Painlevé equations from [11], and new applications of the latter have been
found very recently. Non-autonomous analogues of the Volterra maps ${\cal
V}_{g}$ have been considered in the context of Hermitian matrix models [4],
where they arise as string equations, and they also appear as recursion
relations for orthogonal polynomials associated with generalised Freud weights
of higher order [10]. In these applications, the algebro-geometric structure
of the Volterra maps should be relevant to the asymptotic description of the
oscillatory behaviour that is observed in specific parameter regimes.

##  7 Appendix: the relation between the Mumford-like and the even Mumford
system

Here we now show that the Hamiltonian system
$(M_{g},\left\\{\cdot\,,\cdot\right\\}^{\phi},\mu)$ associated with the
Volterra map is birationally isomorphic (as a Poisson isomorphism) to the even
Mumford system, or more precisely to a subsystem thereof, obtained simply by
fixing the value of one of the Casimirs. Recall from [53, Ch. 6] that the even
Mumford system (of genus $g$) is the Hamiltonian system
$(M_{g}^{\prime},\left\\{\cdot\,,\cdot\right\\}^{\prime}{}^{\psi},\mu^{\prime})$,
whose phase space $M_{g}^{\prime}$ is the $(3g+2)$-dimensional affine space

$M_{g}^{\prime}:=\left\\{(U(\xi),V(\xi),W(\xi))\in{\mathbb{C}}[\xi]^{3}~{}\Bigl{|}~{}\begin{array}[]{lll}\deg
U(\xi)=g\;,&U\hbox{ monic}\\\ \deg V(\xi)<g\;,&\\\ \deg W(\xi)=g+1\;,&W\hbox{
monic}\\\ \end{array}\right\\}\;.$ (7.1)

Elements $(U(\xi),V(\xi),W(\xi))$ of $M_{g}^{\prime}$ are written as Lax
matrices

${\bf L}^{\prime}(\xi):=\left(\begin{array}[]{lr}V(\xi)&U(\xi)\\\
W(\xi)&-V(\xi)\end{array}\right)\;,\qquad$

whose polynomial entries have the form

$U(\xi)=x^{g}+\sum_{i=0}^{g-1}U_{i}\xi^{i}\;,\qquad
V(\xi)=\sum_{i=0}^{g-1}V_{i}\xi^{i}\;,\qquad
W(\xi)=x^{g+2}+\sum_{i=0}^{g+1}W_{i}\xi^{i}\;.$

The $3g+2$ coefficients $W_{g+1},W_{g}$ and $U_{i},V_{i},W_{i}$ with
$0\leqslant i<g$ are used as linear coordinates on $M_{g}^{\prime}$. The
momentum map $\mu^{\prime}$ is given by

$\begin{array}[]{lcrcl}\mu^{\prime}&:&M_{g}^{\prime}&\to&{\mathbb{C}}[\xi]\\\
&&{\bf L}^{\prime}(\xi)=\left(\begin{array}[]{lr}V(\xi)&U(\xi)\\\
W(\xi)&-V(\xi)\end{array}\right)&\mapsto&-\det{\bf
L}^{\prime}(\xi)=V(\xi)^{2}+U(\xi)W(\xi)\;.\end{array}$

It is clear that $-\det{\bf L}^{\prime}(\xi)$ is monic of degree $2g+2$, so
$2g+2$ polynomial functions
$H^{\prime}_{0},H^{\prime}_{1},\dots,H^{\prime}_{2g+1}$ on $M_{g}^{\prime}$
are defined by

$V(\xi)^{2}+U(\xi)W(\xi)=\xi^{2g+2}+\sum_{i=0}^{2g+1}H^{\prime}_{i}\xi^{i}\;,$

For any non-zero polynomial $\psi$ of degree at most $g+1$, a Poisson
structure of rank $2g$ on $M_{g}^{\prime}$ is given by

$\displaystyle\left\\{U(\xi),U(\eta)\right\\}^{\prime}{}^{\psi}$
$\displaystyle=\left\\{V(\xi),V(\eta)\right\\}^{\prime}{}^{\psi}=0\;,$
$\displaystyle\left\\{U(\xi),V(\eta)\right\\}^{\prime}{}^{\psi}$
$\displaystyle=\frac{U(\xi)\psi(\eta)-U(\eta)\psi(\xi)}{\xi-\eta}\;,$
$\displaystyle\left\\{U(\xi),W(\eta)\right\\}^{\prime}{}^{\psi}$
$\displaystyle=-2\frac{V(\xi)\psi(\eta)-V(\eta)\psi(\xi)}{\xi-\eta}\;,$
$\displaystyle\left\\{V(\xi),W(\eta)\right\\}^{\prime}{}^{\psi}$
$\displaystyle=\frac{W(\xi)\psi(\eta)-W(\eta)\psi(\xi)}{\xi-\eta}-(\xi+\eta+W_{g+1}-U_{g-1})U(\xi)\psi(\eta)\;,$
$\displaystyle\left\\{W(\xi),W(\eta)\right\\}^{\prime}{}^{\psi}$
$\displaystyle=2(\xi+\eta+W_{g+1}-U_{g-1})\left(V(\xi)\psi(\eta)-V(\eta)\psi(\xi)\right)\;.$

The polynomial functions $H^{\prime}_{i}$ are independent and in involution,
which accounts for the Liouville integrability of the even Mumford system. It
is algebraically integrable, with the fiber of $\mu^{\prime}$ over any monic
polynomial $f^{\prime}(x)$ of degree $2g+2$ and without multiple roots being
an affine part of the Jacobian of the smooth hyperelliptic curve defined by
$\eta^{2}=f^{\prime}(\xi)$.

For the isomorphism, we consider only the Poisson structures
$\left\\{\cdot\,,\cdot\right\\}^{\prime}{}^{\psi}$ for which $\psi(0)=0$. They
admit $H^{\prime}_{0}$ as Casimir function, hence we can restrict
$(M_{g}^{\prime},\left\\{\cdot\,,\cdot\right\\}^{\prime}{}^{\psi},\mu^{\prime})$
to the subvariety, defined by $H^{\prime}_{0}=0$; the resulting system is
denoted by $(M_{g}^{0},\left\\{\cdot\,,\cdot\right\\}^{0,\psi},\mu^{0})$. We
show that this system is birationally equivalent to the Mumford-like system
$(M_{g},\left\\{\cdot\,,\cdot\right\\}^{\phi},\mu)$, where the relation
between $\psi$ and $\phi$ will be spelled out below. To do this, we construct
a biregular map $\tilde{\Psi}$ and a birational map $\Psi$ making the
following diagram commutative:

${M_{g}^{0}}$${M_{g}}$${B_{g}^{0}}$${B_{g}}$$\scriptstyle{\Psi}$$\scriptstyle{\mu^{0}}$$\scriptstyle{\mu}$$\scriptstyle{\tilde{\Psi}}$

In this diagram, $B_{g}$ and $B_{g}^{0}$ are the images of $\mu$ and
$\mu^{0}$, which we view as spaces of curves: $B_{g}$ consists of the
hyperelliptic curves of the form $y^{2}=f(x)$, with $f(0)=1$ and $\deg
f\leqslant 2g+1$, while $B_{g}^{0}$ consists of the hyperelliptic curves of
the form $\eta^{2}=f^{\prime}(\xi)$, with $f^{\prime}$ monic of degree $2g+2$,
vanishing at $0$. The curves of $B_{g}^{0}$ have two points at infinity, which
we denote by $\infty_{1}$ and $\infty_{2}$.

We first establish a natural correspondence between the curves of $B_{g}^{0}$
and the curves of $B_{g}$. Let $y^{2}=f(x)$ be a curve of $B_{g}$ and
substitute $x=\xi^{-1}$ and $y=\eta\xi^{-g-1}$, to get
$\eta^{2}=f^{\prime}(\xi)=\xi^{2g+2}f(\xi^{-1})$, where $f^{\prime}(0)=0$ and
$f^{\prime}$ is monic of degree $2g+2$, so $\eta^{2}=f^{\prime}(\xi)$ is a
curve of $B_{g}^{0}$. From the latter, one gets back $y^{2}=f(x)$ by setting
$\xi=x^{-1}$ and $\eta=yx^{-g-1}$. Notice that when
$f(x)=1+\sum_{i=1}^{2g+1}c_{i}x^{i}$ then
$\eta^{2}=\xi^{2g+2}+\sum_{i=1}^{2g+1}c_{2g+2-i}\xi^{i}$, which yields the
biregular map $\tilde{\Psi}$.

For the construction of $\Psi$, the biregular map $\tilde{\Psi}$ between the
spaces of curves $y^{2}=f(x)$ and $\eta^{2}=f^{\prime}(\xi)$ is extended to
divisors on these curves. To do this, we compare the description of points
$({\cal P}(x),{\cal Q}(x),{\cal R}(x))$ on a generic fiber of $\mu$ in terms
of divisors on the corresponding curve $y^{2}=f(x)$ with the description of
points $(U(\xi),V(\xi),W(\xi))$ on a generic fiber of $\mu^{\prime}$ in terms
of divisors on the corresponding curve $\eta^{2}=f^{\prime}(\xi)$. The first
description was given in the proof of Proposition 4.6, while the second
description, which we quickly recall, can be found in [53, Ch. 6]. Let
$\xi_{1},\dots,\xi_{g}$ denote the roots of $U(\xi)$ and let
$\eta_{i}:=V(\xi_{i})$, for $i=1,\dots,g$. Then the points
$(\xi_{i},\eta_{i})$ belong to the curve $\eta^{2}=\phi(\xi)$ and so the
divisor class $\left[\sum_{i=1}^{g}(\xi_{i},\eta_{i})-g\infty_{1}\right]$ is a
point of its Jacobian. The relation between the polynomials ${\cal Q}(x)$ and
$U(\xi)$ is clearly given by

$\frac{\xi^{g}}{2}{\cal Q}(\xi^{-1})=U(\xi)\;.$

Indeed, both sides of this equality are polynomials of degree $2g+2$ which
vanish for $\xi=\xi_{i}=x_{i}^{-1}$ (which we may assume to be different).
Similarly, ${\cal P}(x)$ is related to $U(\xi)$ and $V(\xi)$ by

$\xi^{g}{\cal P}(\xi^{-1})-U(\xi)=\frac{U_{0}V(\xi)-V_{0}U(\xi)}{\xi
U_{0}}\;,$

because both sides of this equation are the unique polynomial in $\xi$ of
degree less than $g$ which takes for $\xi=\xi_{i}$ the value
$\eta_{i}/\xi_{i}$. A formula for ${\cal R}$ follows from the equations of the
curves, namely

${\cal P}^{2}(\xi^{-1})+{\cal Q}(\xi^{-1}){\cal
R}(\xi^{-1})=f(\xi^{-1})=\xi^{-2g-2}f^{\prime}(\xi)=\xi^{-2g-2}(V^{2}(\xi)+U(\xi)W(\xi))\;.$

It is clear that this defines a birational map between $M_{g}$ and
$M_{g}^{0}$. Under this map, $H_{i}$ corresponds to $H^{\prime}_{2g+2-i}$, for
$i=1,\dots,2g+1$. To see that $\Psi$ is a Poisson map, hence a birational
Poisson isomorphism, we recall that the Poisson bracket
$\left\\{\cdot\,,\cdot\right\\}^{\prime}{}^{\psi}$ is given in terms of the
$\xi_{i}$ and $\eta_{j}$ by
$\left\\{\xi_{i},\eta_{j}\right\\}^{\prime}{}^{\psi}=\psi(\xi_{i})\delta_{ij},$
and hence

$\left\\{x_{i},y_{j}\right\\}^{\prime}{}^{\psi}=\left\\{\xi_{i}^{-1},\eta_{j}\xi^{-g-1}_{j}\right\\}^{\prime}{}^{\psi}=-\xi_{i}^{-2}\xi_{j}^{-g-1}\left\\{\xi_{i},\eta_{j}\right\\}^{\prime}{}^{\psi}=-{\xi_{i}^{-g-3}}\psi(\xi_{i})\delta_{ij}=-{x_{i}^{g+3}}\psi(x_{i}^{-1})\delta_{ij}\;.$

Compared with (4.21), this shows that
$\Psi:(M_{g}^{0},\left\\{\cdot\,,\cdot\right\\}^{\prime}{}^{\psi})\to(M_{g},\left\\{\cdot\,,\cdot\right\\}^{\phi})$
is a birational Poisson isomorphism when taking $\phi(x)=\psi(x^{-1})x^{g+2}$;
notice that $\phi$, defined by this formula, is indeed a polynomial of degree
at most $g+1$, vanishing at $0$ and that we get all such polynomials $\phi$
for some appropriate polynomial $\psi$ of degree at most $g+1$, vanishing at
$0$.

Acknowledgments: The research of ANWH was supported by Fellowship EP/M004333/1
from the Engineering & Physical Sciences Research Council, UK and grant
IEC\R3\193024 from the Royal Society; he is also grateful to the School of
Mathematics and Statistics, University of New South Wales, for hosting him
during 2017-2019 as a Visiting Professorial Fellow with funding from the
Distinguished Researcher Visitor Scheme, and to Wolfgang Schief, who provided
additional support during his time in Sydney.

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