# Sausages

Danny Calegari  Department of Mathematics
University of Chicago
Chicago, Illinois, 60637<EMAIL_ADDRESS>
###### Abstract.

The shift locus is the space of normalized polynomials in one complex variable
for which every critical point is in the attracting basin of infinity. The
method of sausages gives a (canonical) decomposition of the shift locus in
each degree into (countably many) codimension 0 submanifolds, each of which is
homeomorphic to a complex algebraic variety. In this paper we explain the
method of sausages, and some of its consequences.

##  1\. Sausages

For each integer $q\geq 2$ the shift locus $\SS_{q}$ is the set of degree $q$
polynomials $f$ in one complex variable of the form

$f(z):=z^{q}+a_{2}z^{q-2}+a_{3}z^{q-3}+\cdots+a_{q}$

for which every critical point of $f$ is in the attracting basin of $\infty$.
One can think of $\SS_{q}$ as a open submanifold of $\mathbb{C}^{q-1}$;
understanding its topology is a fundamental problem in complex dynamics. For
example, when $q=2$ the complement of $\SS_{2}$ in $\mathbb{C}$ is the
Mandelbrot set. Knowing that $\SS_{2}$ is homeomorphic to a cylinder implies
the famous theorem of Douady–Hubbard that the Mandelbrot set is connected.

Although the $\SS_{q}$ are highly transcendental spaces, the method of
sausages (which we explain in this section) shows that each $\SS_{q}$ has a
canonical decomposition into codimension 0 submanifolds whose interiors are
homeomorphic to certain explicit algebraic varieties. From this one can deduce
a considerable amount about the topology of $\SS_{q}$, especially in low
degree.

The construction of sausages has several steps, and goes via an intermediate
construction that associates, to each polynomial $f$ in $\SS_{q}$, a certain
combinatorial object called a dynamical elamination.

###  1.1. Green’s function

Let $K$ be a compact subset of $\mathbb{C}$ with connected complement
$\Omega_{K}:=\mathbb{C}-K$. If $K$ has positive logarithmic capacity (for
example, if the Hausdorff dimension is positive) then there is a canonical
Green’s function $g:\Omega_{K}\to\mathbb{R}^{+}$ satisfying

  1. (1)

$g$ is harmonic;

  2. (2)

$g$ extends continuously to $0$ on $K$; and

  3. (3)

$g$ is asymptotic to $\log{|z|}$ near infinity (in the sense that
$g(z)-\log{|z|}$ is harmonic near infinity).

There is a unique germ near infinity of a holomorphic function $\phi$, tangent
to the identity at $\infty$, for which $g=\log|\phi(z)|$.

###  1.2. Filled Julia Set

Let $f$ be a degree $q$ complex polynomial. After conjugacy by a complex
affine transformation $z\to\alpha z+\beta$ we may assume that $f$ is
normalized; i.e. of the form

$f(z):=z^{q}+a_{2}z^{q-2}+a_{3}z^{q-3}+\cdots+a_{q}$

The filled Julia set $K(f)$ is the set of complex numbers $z$ for which the
iterates $f^{n}(z)$ are (uniformly) bounded. It is a fact that $K(f)$ is
compact, and its complement $\Omega_{f}:=\mathbb{C}-K(f)$ is connected. The
union $\hat{\Omega}_{f}:=\Omega_{f}\cup\infty$ is the attracting basin of
$\infty$.

Böttcher’s Theorem (see e.g. [21] Thm. 9.1) says that $f$ is holomorphically
conjugate near infinity to the map $z\to z^{q}$. For normalized $f$, the germ
of the conjugating map $\phi$ (i.e. $\phi$ so that $\phi(f(z))=\phi(z)^{q}$)
is uniquely determined by requiring that $\phi$ is tangent to the identity at
infinity. The (real-valued) function $g(z):=\log|\phi(z)|$ is harmonic, and
satisfies the functional equation $g(f(z))=q\cdot g(z)$. We may extend $g$ via
this functional equation to all of $\Omega_{f}$ and observe that $g$ so
defined is the Green’s function of $K(f)$.

###  1.3. Maximal domain of $\phi^{-1}$

Let $\overline{\mathbb{D}}\subset\mathbb{C}$ denote the closed unit disk, and
$\mathbb{E}:=\mathbb{C}-\overline{\mathbb{D}}$ the exterior. We will use
logarithmic coordinates $h=\log(|z|)$ and $\theta=\arg(z)$ on $\mathbb{E}$ and
on Riemann surfaces obtained from $\mathbb{E}$ by cut and paste. Note that
$g=h\phi$ where $g$ and $\phi$ are as in § 1.1.

For any $K$ with Green’s function $g$ and associated $\phi$ we can
analytically continue $\phi^{-1}$ from infinity along radial lines of
$\mathbb{E}$. The image of these radial lines under $\phi^{-1}$ are the
descending gradient flowlines of $g$ (i.e. the integral curves of
$-\text{grad}(g)$), and we can analytically continue $\phi^{-1}$ until the
gradient flowlines run into critical points of $g$. Figure 1 shows some
gradient flowlines of $g$ for a Cantor set $K$.

Figure 1. Gradient flowlines of $g$ for a Cantor set $K$

Note that some critical points of $g$ might have multiplicity greater than
one; however because $g$ is harmonic, the multiplicity of every critical point
is finite, and the critical points of $g$ are isolated and can accumulate (in
${\hat{\mathbb{C}}}$) only on $K$. With this proviso about multiplicity, we
want to do a sort of ‘Morse theory’ for the function $g$.

Let $L^{\prime}$ be the union of the segments of the gradient flowlines of $g$
descending from all the critical points of $g$; in Figure 1 these are in red
(gray, for black and white reproduction). Then $\Omega_{K}-L^{\prime}$ is the
image of the maximal (radial) analytic extension of $\phi^{-1}$. The domain of
this maximal extension $\phi^{-1}$ may be described as follows. For
$w\in\mathbb{E}$ define the radial segment $\sigma(w)\subset\mathbb{E}$ to be
the set of points $z$ with $\arg(z)=\arg(w)$ and $|z|\leq|w|$. The height of
$\sigma$, denoted $h(\sigma)$, is $\log(|w|)$. The domain of $\phi^{-1}$ is
$\mathbb{E}-L$ where $L$ is the union of a countable proper (in $\mathbb{E}$)
collection of radial segments.

If $K=K(f)$ for a polynomial $f$, the critical points of $g$ are the critical
points and critical preimages of $f$; i.e. points $z$ for which
$(f^{n})^{\prime}(z)=0$ for some positive $n$. Thus $L^{\prime}$ is nearly
$f$-invariant: the image $f(L^{\prime})$ is equal to
$L^{\prime}\cup\ell^{\prime}$ where $\ell^{\prime}$ is the (finite!) set of
descending flowlines from the critical values of $f$ in $\Omega_{f}$ (which
are themselves not typically critical).

Likewise the map $z\to z^{q}$ on $\mathbb{E}$ takes $L$ to $L\cup\ell$ where
$\ell$ is a finite set of radial segments mapped by $\phi^{-1}$ to
$\ell^{\prime}$.

###  1.4. Cut and Paste

Let $c$ be a critical point of $g$ and let $L^{\prime}_{c}$ be the union of
the gradient flowlines of $g$ descending from $c$ (and for simplicity, here
and in the sequel let’s suppose these flowlines do not run into another
critical point). Then $L^{\prime}_{c}$ is the union of $n+1$ proper embedded
rays from $c$ to $K$ where $n$ is the multiplicity of $c$ as a critical point
(these rays extend continuously to $K$ when the components of $K$ are locally
connected; otherwise they may ‘limit to’ a prime end of a component of $K$).
There is a corresponding collection $L_{c}$ of $n+1$ radial segments
$\sigma_{j}:=\sigma(w_{j})$ all of the same height, where indices are
circularly ordered according to the arguments of the $w_{j}$. The map
$\phi^{-1}$ extends continuously along radial lines from infinity all the way
to the $w_{j}$: the $w_{j}$ all map to $c$. But any ‘extension’ of $\phi^{-1}$
over $L_{c}$ will be multivalued. We can repair this multivaluedness by cut
and paste: cut open $\mathbb{E}$ along the segments $L_{c}$ to create two
copies $\sigma_{j}^{+}$ resp. $\sigma_{j}^{-}$ for each $\sigma_{j}$ on the
‘left’ resp. ‘right’ in the circular order. Then glue each segment
$\sigma_{j}^{-}$ to $\sigma_{j+1}^{+}$ by a homeomorphism respecting absolute
value. Under this operation the collection of segments $L_{c}$ are reassembled
into an ‘asterisk’ which resembles the cone on $n+1$ points; see Figure 2.

Figure 2. Cut and paste over $L_{c}$ of multiplicity 4

The result is a new Riemann surface for which the map $\phi^{-1}$ now extends
(analytically and single-valuedly) over the (cut-open and reglued) image of
$L_{c}$, whose image is exactly $L^{\prime}_{c}$.

If we perform this cut and paste operation simultaneously for all the
different $L_{c}$ making up $L$, the Riemann surface $\mathbb{E}$ is
reassembled into a new Riemann surface $\Omega$ so that $\phi^{-1}$ extends to
an isomorphism $\phi^{-1}:\Omega\to\Omega_{K}$.

If $K=K(f)$ for a polynomial $f$, then the map $z\to z^{q}$ on $\mathbb{E}$
descends to a well-defined degree $q$ holomorphic self-map $F:\Omega\to\Omega$
and $\phi^{-1}$ conjugates $F|\Omega$ to $f|\Omega_{f}$.

###  1.5. Elaminations

It is useful to keep track of the partition of $L^{\prime}$ and $L$ into
finite collections $L^{\prime}_{c}$ and $L_{c}$ associated to the critical
points $c$ of $g$.

For each critical point $c$ of multiplicity $n$ we span the $n+1$ segments of
$L_{c}$ by an ideal hyperbolic $(n+1)$-gon in $\overline{\mathbb{D}}$. The
segments of $L_{c}$ become the tips and the ideal polygon becomes the vein of
a leaf of multiplicity $n$ in an object called an extended lamination — or
elamination for short. When every critical point has multiplicity 1, we say
the elamination is simple. See Figures 3 and 7 for examples of simple
elaminations. The key topological property of elaminations is that the veins
associated to different leaves do not cross. This is equivalent to the fact
that the result $\Omega$ of cut and paste along $L$ is a planar surface
(because it is isomorphic to $\Omega_{X}\subset\mathbb{C}$).

Elaminations are introduced and studied in [11]. The set ${\mathcal{EL}}$ of
elaminations becomes a space with respect to a certain topology (the collision
topology), and can be given the structure of a disjoint union of (countable
dimensional) complex manifolds. For example, the space of elaminations with
$n-1$ leaves (counted with multiplicity) is homeomorphic (but not
biholomorphic) to the space of degree $n$ normalized polynomials with no
multiple roots.

###  1.6. Dynamical Elaminations

Figure 3 depicts the elamination associated to $K(f)$ for a degree 3
polynomial $f$. The critical leaves — i.e. the leaves with tips $L_{c}$
associated to $c$ a critical point of $f$ — are in red. Every other leaf
corresponds to a precritical point of $f$ (which are critical points of the
Green’s function). This elamination is simple: every leaf has exactly two
tips.

Figure 3. Simple dynamical elamination of degree $3$; critical leaves are in
red

Let $\Lambda$ denote the elamination associated to $L$. Note that $\Lambda$
depends not just on $L$ as a set of segments, but also on their partition into
subsets $L_{c}$.

The map $z\to z^{q}$ on $\mathbb{E}$ acts on segments and therefore also on
leaves, with the following exception. If $\ell$ is a leaf whose tips have
arguments that all differ by integer multiples of $2\pi/q$ then these segments
will have the same image under $z\to z^{q}$. Since leaves should have at least
two tips (by convention), if $\ell$ is a leaf all of whose tips have arguments
that differ by integer multiples of $2\pi/q$ then the image of $\ell$ under
$z\to z^{q}$ is undefined.

Suppose $K=K(f)$ for a degree $q$ polynomial. Let $C$ denote the critical
leaves of $L$ (those associated to critical points of $f$). The map $z\to
z^{q}$ takes leaves to leaves in the obvious sense, and takes $\Lambda-C$ to
$\Lambda$.

We say an elamination $\Lambda$ is a degree $q$ dynamical elamination if

  1. (1)

it has finitely many leaves $C$ each of whose arguments differ by integer
multiples of $2\pi/q$ (the critical leaves);

  2. (2)

the map $z\to z^{q}$ takes $\Lambda-C$ to $\Lambda$; and

  3. (3)

every leaf has exactly $q$ preimages.

A degree $q$ dynamical elamination is maximal if there are $q-1$ critical
leaves, counted with multiplicity.

The elamination $\Lambda$ associated to a degree $q$ polynomial $f$ is a
degree $q$ dynamical elamination. It is maximal if and only if all the
critical points of $f$ are in $\Omega_{f}$.

A set of (non-crossing) leaves $C$, each with arguments that differ by integer
multiples of $2\pi/q$ is called a degree $q$ critical set. A critical set is
maximal if there are $q-1$ leaves counted with multiplicity. It turns out that
every maximal degree $q$ critical set $C$ is exactly the set of critical
leaves of a unique (maximal) degree $q$ dynamical elamination $\Lambda$; see
[11] Prop. 5.3. The set of maximal degree $q$ dynamical elaminations is
denoted $\mathcal{DL}_{q}$. As a subset of ${\mathcal{EL}}$ it has the
structure of an open complex manifold of dimension $q-1$ with local
coordinates coming from the (endpoints of) segments of $C$ (at least at a
generic $\Lambda$).

###  1.7. The Shift Locus

For each degree $q$ the Shift Locus $\SS_{q}$ is the space of degree $q$
normalized polynomials $f(z):=z^{q}+a_{2}z^{q-2}+a_{3}z^{q-3}+\cdots+a_{q}$
for which every critical point is in the basin of infinity $\Omega_{f}$. The
coefficients $a_{2},\cdots,a_{q}$ are coordinates on $\SS_{q}$ realizing it as
an open subset of $\mathbb{C}^{q-1}$.

A polynomial $f$ is in $\SS_{q}$ if and only if the Julia set of $f$ is a
Cantor set on which $f$ is uniformly expanding (for some metric). Thus for
such polynomials, $J(f)=K(f)$ and $\Omega_{f}$ is the entire Fatou set (i.e.
the maximal domain of normality of $f$ and its iterates; see e.g. [21]).

Suppose $f\in\SS_{q}$ with associated dynamical elamination $\Lambda$. Since
all critical points of $f$ are in $\Omega_{f}$, it follows that $\Lambda$ is
maximal; thus there is a map $\SS_{q}\to\mathcal{DL}_{q}$ called the Böttcher
map. Conversely, if $\Lambda$ is any maximal degree $q$ dynamical elamination,
and $\Omega$ is obtained from $\mathbb{E}$ by cut and paste along $\Lambda$,
then $F|\Omega$ extends (topologically) over the space of ends of $\Omega$ to
define a degree $q$ self-map $\bar{F}$ of a topological sphere
$\bar{\Omega}\cong S^{2}$. It turns out that there is a canonical conformal
structure on $\bar{\Omega}$ extending that on $\Omega$ so that $\bar{F}$ is
holomorphic. After choosing suitable coordinates on $\bar{\Omega}$ near
$\infty$ the map $\bar{F}$ becomes a degree $q$ normalized polynomial, which
is contained in $\SS_{q}$. The analytic content of this theorem is essentially
due to de Marco–McMullen; see e.g. [16] Thm. 7.1 or [11] Thm. 5.4 for a
different proof.

Thus the Böttcher map $\SS_{q}\to\mathcal{DL}_{q}$ is a homeomorphism (and in
fact an isomorphism of complex manifolds).

###  1.8. Stretching and Spinning

There is a (multiplicative) $\mathbb{R}^{+}$ action on ${\mathcal{EL}}$ called
stretching where $t\in\mathbb{R}^{+}$ acts on $\Lambda$ by multiplying the $h$
coordinate of every leaf by $t$. This action is free and proper. It preserves
$\mathcal{DL}_{q}$ for each $q$, and shows that $\mathcal{DL}_{q}$ (and
therefore also $\SS_{q}$) is homeomorphic to the product of $\mathbb{R}$ with
a manifold of (real) dimension $2q-3$. It is convenient for what follows to
define $\mathcal{DL}_{q}^{\prime}$ to be the open subspace of
$\mathcal{DL}_{q}$ for which the highest critical leaf has
$\log_{q}(h)\in(-1/2,1/2)$. By suitably ‘compressing’ orbits of the
$\mathbb{R}^{+}$ action we see there is a homeomorphism
$\mathcal{DL}_{q}\to\mathcal{DL}_{q}^{\prime}$.

There is also an $\mathbb{R}$ action on ${\mathcal{EL}}$ called spinning where
$t\in\mathbb{R}$ simultaneously rotates the arguments of leaves of height $h$
by $th$. This makes literal sense for the (finitely many) leaves of greatest
height. When leaves of lesser height are collided by those of greater height
the shorter leaf is ‘pushed over’ the taller one; the precise details are
explained in [11] § 3.2. This $\mathbb{R}$ action also preserves each
$\mathcal{DL}_{q}$. The closure of the $\mathbb{R}$-orbits in each
$\mathcal{DL}_{q}$ are real tori, and the $\mathbb{R}$-orbits sit in these
tori as parallel lines of constant slope. A typical $\mathbb{R}$-orbit has
closure which is a torus of real dimension $q-1$, but if some critical leaves
have multiplicity $>1$ or if distinct critical leaves have rationally related
heights, the closure will be a torus of lower dimension.

Stretching and spinning combine to give an action of the (oriented) affine
group $\mathbb{R}\rtimes\mathbb{R}^{+}$ of the line on ${\mathcal{EL}}$ and on
each individual $\mathcal{DL}_{q}$.

###  1.9. Sausages

Suppose $K=K(f)$ for a degree $q$ polynomial. The map $f$ is algebraic, but
the domain $\Omega_{f}$ is transcendental. When we move to the elamination
side, the map $z\to z^{q}$ and the domain $\mathbb{E}$ are (semi)-algebraic,
but the combinatorics of $L$ is hard to understand. Sausages are a way to find
a substitute for $(f,\Omega_{f})$ for which both the map and the domain are
algebraic and more comprehensible.

The idea of sausages is to find a dynamically invariant way to cut up the
domain $\Omega$ into a tree of Riemann spheres, so that $F$ induces polynomial
maps between these spheres. The sausage map is not holomorphic, but it induces
homeomorphisms between certain codimension 0 submanifolds of
$\mathcal{DL}_{q}^{\prime}$ and certain explicit algebraic varieties whose
topology is in some ways much easier to understand.

Now let’s discuss the details of the construction. First consider the map
$z\to z^{q}$ on $\mathbb{E}$ alone. Let $h:=\log(|z|)$ and $\theta=\arg(z)$ be
cylindrical coordinates on $\mathbb{E}$, so that $\mathbb{E}$ becomes the
half-open cylinder $S^{1}\times\mathbb{R}^{+}$ in $(\theta,h)$-coordinates,
and $z\to z^{q}$ becomes the map which is multiplication by $q$ which we
denote $\times q$. For each integer $n$ let $I_{n}$ denote the open interval
$(q^{n-1/2},q^{n+1/2})$ and let $A_{n}$ be the annulus in $\mathbb{E}$ where
$h\in I_{n}$ and let $A=\cup_{n}A_{n}\subset\mathbb{E}$; the complement of $A$
is the countable set of circles with $\log_{q}(h)\in 1/2+\mathbb{Z}$. Then
$\times q$ takes $A_{n}$ to $A_{n+1}$.

This data is holomorphic but not algebraic. So let’s choose (rather
arbitrarily) an orientation-preserving diffeomorphism
$\nu_{0}:I_{0}\to\mathbb{R}$ and for each $n$ define
$\nu_{n}:I_{n}\to\mathbb{R}$ by $\nu_{n}(h)=q^{n}\nu_{0}(q^{-n}h)$ (so that by
induction the $\nu_{n}$ satisfy $\nu_{n+1}(qh)=q\nu_{n}(h)$ for all $n$ and
$h\in I_{n}$), and define $\mu:A\to S^{1}\times\mathbb{R}$ to be the map that
sends $(\theta,h)$ to $(\theta,\nu_{n}(h))$ if $(\theta,h)\in A_{n}$. By
construction, $\mu$ commutes with multiplication by $q$:

$\mu(q\theta,qh)=(q\theta,\nu_{n+1}(qh))=(q\theta,q\nu_{n}(h))=q\mu(\theta,h)$

In other words, $\mu$ semi-conjugates $\times q$ on $A$ to $\times q$ on
$S^{1}\times\mathbb{R}$, which (by exponentiating) becomes the map $z\to
z^{q}$ on $\mathbb{C}^{*}$, an algebraic map on an algebraic domain. Actually,
it is better to keep a separate copy $\mathbb{C}^{*}_{n}:=\mu(A_{n})$ of
$\mathbb{C}^{*}$ for each $n$, so that $\mu$ conjugates $\times q$ on $A$ to
the self-map of $\cup_{n}\mathbb{C}^{*}_{n}$ which sends each
$\mathbb{C}^{*}_{n}$ to $\mathbb{C}^{*}_{n+1}$ by $z\to z^{q}$.

###  1.10. Sausages and Dynamics

Now suppose we have a dynamical elamination $\Lambda$ with critical leaves $C$
invariant under $z\to z^{q}$. For each $A_{n}$ the tips of $\Lambda$ intersect
$A_{n}$ in a finite collection of vertical segments $L_{n}$ (some of which
will pass all the way through $A_{n}$) and we can perform cut-and-paste
separately on each $A_{n}$ to produce a (typically disconnected) surface
$B_{n}$. Furthermore we can perform cut-and-paste on $\mathbb{C}^{*}_{n}$
along the image $\mu(L_{n})$ which, by construction, is compatible with the
Riemann surface structure. The result is to cut and paste $\mathbb{C}^{*}_{n}$
into a finite collection of algebraic Riemann surfaces, each individually
isomorphic to $\mathbb{C}$ minus a finite set of points and which can be
canonically completed to Riemann spheres in such a way that the map $F$ on
$\Omega$ descends to a map $\mathfrak{f}$ from this union of Riemann spheres
to itself; see Figure 4.

Figure 4. $A_{n}$ is cut and paste into $B_{n}$ which in turn maps to a
disjoint union of Riemann spheres

Denote the individual Riemann spheres by $X_{v}$, and by abuse of notation,
write $\mathfrak{f}_{v}:X_{v}\to X_{\mathfrak{f}(v)}$ for the restriction of
$\mathfrak{f}$ to the component $X_{v}$. By the previous discussion, each map
$\mathfrak{f}_{v}$ is holomorphic, so that if we choose suitable coordinates
on $X_{v}$ and $X_{\mathfrak{f}(v)}$ the map $\mathfrak{f}_{v}$ becomes a
polynomial. There is almost a canonical choice of coordinates, which we
explain in the next two sections.

Each $X_{v}$ corresponds to a component $B_{v}$ of some $B_{n}$, and gets a
canonical finite set of marked points $Z_{v}^{\prime}$ which correspond to the
‘boundary circles’ of $B_{v}$. The unique boundary circle with greatest $h$
coordinate picks out a point that we can identify with $\infty\in X_{v}$; we
denote by $Z_{v}$ the set consisting of the rest of the marked points. The
collection of individual Riemann spheres $X_{v}$ can be glued up along their
marked points into an infinite genus zero nodal Riemann surface so that the
indices $v$ are parameterized by the vertices $v$ of the tree of gluings $T$.
This tree is oriented, so that an edge $v$ goes to $w$ if $X_{v}$ is glued
along $\infty$ to one of the (non-infinite) marked points of $X_{w}$. We call
$w$ the parent of $v$ and $v$ one of the children of $w$. If we make the
assumption that no boundary component of any $B_{v}$ contains a critical point
(this is the generic case) then each $\zeta\in Z_{w}\subset X_{w}$ is attached
to a unique $X_{v}$ for $v$ some child of $w$. If $v$ is a child of $w$, and
$X_{v}$ is glued to $X_{w}$ at the point $\zeta\in Z_{w}$, then if $\zeta$ is
a critical point of $\mathfrak{f}_{w}$ of multiplicity $m$, the degree of
$\mathfrak{f}_{v}$ is $m+1$. By abuse of notation, we denote the induced
(simplicial, orientation-preserving) map on $T$ also by $\mathfrak{f}$.

If $\Lambda$ is empty, then $T$ is just a line, and each vertex has a unique
child. If $\Lambda$ is nonempty, then since there are only finitely many
leaves of greatest height, there is a unique highest vertex $v$ of $T$ with
more than one child. Let $w$ be the parent of $v$. The uppermost boundary
components of $B_{v}$ and $B_{w}$ are canonically identified with the unit
circle $S^{1}:=\mathbb{R}/2\pi\mathbb{Z}$. By identifying these circles with
the unit tangent circles at $\infty$ in $X_{v}$ and $X_{w}$ we can choose
coordinates on these Riemann spheres so that the tangent to the positive real
axis corresponds to the angle $0\in S^{1}$. In these coordinates $X_{v}$ and
$X_{w}$ are identified with copies ${\hat{\mathbb{C}}}_{v}$ and
${\hat{\mathbb{C}}}_{w}$ of the Riemann sphere ${\hat{\mathbb{C}}}$, and after
precomposing with a suitable complex affine translation, $\mathfrak{f}_{v}$
becomes a normalized degree $q$ polynomial map $\mathfrak{f}_{v}:z\to
z^{q}+b_{2}z^{q-2}+\cdots b_{q}$, and the (non-infinite) marked points of
$X_{v}$ become the roots of $\mathfrak{f}_{v}$ in ${\hat{\mathbb{C}}}_{v}$.

Vertices of $T$ above $v$ and the maps between their respective Riemann
surfaces do not carry any information. Let $w_{1}:=w$ denote the parent of
$v$, and inductively let $w_{n}$ be the parent of $w_{n-1}$. Then each
$X_{w_{n}}$ has exactly two marked points, which we can canonically identify
with $\infty$ and $0$, and the map
$\mathfrak{f}_{w_{n-1}}:{\hat{\mathbb{C}}}_{w_{n-1}}\to{\hat{\mathbb{C}}}_{w_{n}}$
is canonically normalized as $z\to z^{q}$.

Since these vertices carry no information, we discard them. Thus we make the
convention that $T$ is the rooted tree consisting of $v$ together with its
(iterated) children, and we let $\mathfrak{X}$ denote the nodal Riemann
surface corresponding to the union of $X_{w}$ with $w$ in $T$. We record the
data of the polynomial $\mathfrak{f}_{v}$ associated to the root $v$, though
we do not interpret this any more as a map between Riemann spheres, so that
$\mathfrak{f}$ is now a map from $\mathfrak{X}-X_{v}$ to $\mathfrak{X}$ and
$\mathfrak{f}_{v}$ is a polynomial function on $X_{v}\cong{\hat{\mathbb{C}}}$.

###  1.11. Tags and sausage polynomials

The choice of a distinguished point on a boundary $S^{1}$ component of some
$B_{u}$ is called a tag. Tags are the data we need to choose coordinates on
$\mathfrak{X}$ so that every $\mathfrak{f}_{u}$ becomes a polynomial. We may
identify this boundary circle with the unit tangent circle at a marked point
on $X_{u}$, and think of the tag as data on $X_{u}$. By induction, we can
choose tags on $X_{u}$ in the preimage of the tags of $X_{\mathfrak{f}(u)}$
under the map $\mathfrak{f}_{u}:X_{u}\to X_{\mathfrak{f}(u)}$ and inductively
define coordinates ${\hat{\mathbb{C}}}_{u}$ on $X_{u}$ for which
$\mathfrak{f}_{u}$ is represented by a normalized polynomial map (in general
of degree $\leq q$).

Suppose $u$ has parent $u^{\prime}$, and $\infty$ in ${\hat{\mathbb{C}}}_{u}$
is attached at some point $\zeta\in
Z_{u^{\prime}}\in{\hat{\mathbb{C}}}_{u^{\prime}}$. Suppose $\zeta$ is a
critical point of $\mathfrak{f}_{u^{\prime}}$ with multiplicity $m$. Then
$\mathfrak{f}_{u}$ has degree $m+1$. There are $m+1$ different choices of tag
at $\zeta$ that map to the tag at $\mathfrak{f}_{u^{\prime}}(\zeta)$, and the
different choices affect the normalization of $\mathfrak{f}_{u}$ by
precomposing with multiplication by an $(m+1)$st root of unity.

The endpoint of this discussion is that we can recover
$\mathfrak{X},\mathfrak{f}$ from the data of a rooted tree $T$, and a set of
equivalence classes of pair $(\text{tag},\text{normalized polynomial
}\mathfrak{f}_{u})$. Call this data a (degree $q$) sausage polynomial.

A dynamical elamination $\Lambda$ is generic if the critical points of $F$ are
all contained in $A$; i.e. if no critical (or by induction, precritical) point
has $h$ coordinate with $\log(h)\in 1/2+\mathbb{Z}$. The sausage map is the
map that associates a sausage polynomial to a degree $q$ dynamical
elamination. A sausage polynomial is generic resp. maximal if it is in the
image of a generic resp. maximal dynamical elamination.

A polynomial $\mathfrak{f}_{w}$ associated to a (generic) sausage polynomial
has two kinds of critical points. The genuine critical points are those in
${\hat{\mathbb{C}}}_{w}-Z_{w}^{\prime}$ (recall that $Z_{w}^{\prime}$ is
$Z_{w}\cup\infty$). The fake critical points are those in $Z_{w}^{\prime}$
($=\infty\cup Z_{w}$) which correspond to circle components of $B_{w}$ mapping
with degree $>1$. For a generic dynamical elamination, the genuine critical
points of the associated sausage polynomial are exactly the images of the
critical points of the elamination (i.e. the endpoints of the critical leaves)
under the sausage map. Thus for a generic maximal sausage polynomial of degree
$q$, there are exactly $q-1$ genuine critical points, counted with
multiplicity.

For a generic, maximal sausage polynomial, all but finitely many
$\mathfrak{f}_{v}$ have degree one. A degree one map uniquely pulls back tags,
and has only one possible normalized polynomial representative, namely the
identity map $z\to z$. Thus a generic, maximal sausage polynomial is described
by a finite amount of combinatorial data together with a finite collection of
normalized polynomials. The reader who wants to see some examples should look
ahead to § 2.1 and § 2.3.

Let $\mathcal{X}_{q}$ denote the space of generic maximal degree $q$ sausage
polynomials. Then $\mathcal{X}_{q}$ is the disjoint union of countably
infinitely many components, indexed by the combinatorics of $T$ and the
degrees of the normalized polynomials between the associated Riemann spheres.
Each component of $\mathcal{X}_{q}$ is a quasiprojective complex variety of
complex dimension $q-1$. In fact, each component is an iterated fiber bundle
whose base and fibers are certain affine (complex) varieties called Hurwitz
varieties, which we shall describe in more detail in § 2.6.

###  1.12. Sausage space

Recall that $\mathcal{DL}_{q}^{\prime}\subset\mathcal{DL}_{q}$ denotes the set
of maximal degree $q$ dynamical elaminations for which the highest critical
point has $\log_{q}(h)\in(-1/2,1/2)$. Let
$\mathcal{DL}_{q}^{\prime\prime}\subset\mathcal{DL}_{q}^{\prime}$ denote the
subspace of generic maximal degree $q$ dynamical elaminations. Then the
construction of the previous few sections defines a map
$\mathcal{DL}_{q}^{\prime\prime}\to\mathcal{X}_{q}$.

In fact, this map is invertible. Given a sausage polynomial
$\mathfrak{X},\mathfrak{f}$ over a tree $T$ with root $v$ we can inductively
construct (singular) vertical resp. horizontal foliations on each
${\hat{\mathbb{C}}}_{w}$ as follows. On ${\hat{\mathbb{C}}}_{v}$ we pull back
the foliations of $\mathbb{C}^{*}$ by lines resp. circles of constant argument
resp. absolute value under the polynomial $\mathfrak{f}_{v}$. Then on every
other $w$ we inductively pull back these foliations under
$\mathfrak{f}_{w}:{\hat{\mathbb{C}}}_{w}\to{\hat{\mathbb{C}}}_{\mathfrak{f}(w)}$.
These foliations all carry coordinates pulled back from $\mathbb{C}^{*}$, and
${\hat{\mathbb{C}}}_{w}$ minus infinity and its marked points becomes
isomorphic to a branched Euclidean Riemann surface with ends isomorphic to the
ends of (infinite) Euclidean cylinders. We can reparameterize the vertical
coordinates on each of these Riemann surfaces by the inverses of the maps
$\nu_{n}$, and then glue together the result by matching up boundary circles
using tags. This defines an inverse to the map
$\mathcal{DL}_{q}^{\prime\prime}\to\mathcal{X}_{q}$ and shows that this map is
a homeomorphism. See [11] Thm. 9.20 for details.

###  1.13. Decomposition of the Shift Locus

Putting together the various constructions we have discussed so far we obtain
the following summary:

  1. (1)

§ 1.7: The map that associates to $f\in\SS_{q}$ a maximal degree $q$ dynamical
elamination $\Lambda$ gives an isomorphism of complex manifolds
$\SS_{q}\to\mathcal{DL}_{q}$.

  2. (2)

§ 1.8: By compressing orbits of the free $\mathbb{R}^{+}$ action on
$\mathcal{DL}_{q}$ we obtain a homeomorphism
$\mathcal{DL}_{q}\to\mathcal{DL}_{q}^{\prime}$ to the subspace whose largest
critical leaf has log height $\log_{q}(h)\in(-1/2,1/2)$.

  3. (3)

§ 1.12: The open dense subset
$\mathcal{DL}_{q}^{\prime\prime}\subset\mathcal{DL}_{q}^{\prime}$ of generic
dynamical elaminations maps homeomorphically by the sausage map
$\mathcal{DL}_{q}^{\prime\prime}\to\mathcal{X}_{q}$.

  4. (4)

§ 1.11: The space $\mathcal{X}_{q}$ is the disjoint union of countably many
quasiprojective complex varieties, each of which has the structure of an
iterated bundle of affine (Hurwitz) varieties.

In words: the shift locus $\SS_{q}$ in degree $q$ has a canonical
decomposition into codimension 0 submanifolds whose interiors are homeomorphic
to certain explicit algebraic varieties. It is a fact that we do not explain
here (see [11] § 8 especially Thm. 8.11) that the abstract cell complex which
combinatorially parameterizes the decomposition of $\SS_{q}$ into these pieces
is contractible, so that all the interesting topology of $\SS_{q}$ is
localized in the components of $\mathcal{X}_{q}$.

In the remainder of the paper we give examples, and explore some of the
consequences of this structure.

##  2\. Sausage Moduli

Each component $Y$ of $\mathcal{X}_{q}$ parameterizes sausages of a fixed
combinatorial type. The combinatorial type determines finitely many vertices
$u$ for which the (normalized) polynomial $\mathfrak{f}_{u}$ has degree $>1$.
The combinatorics constrains these polynomials by imposing conditions on their
critical values, for instance that the critical values are required to lie
outside a certain (finite) set. Thus, each component has the structure of an
algebraic variety which is an iterated fiber bundle, and so that the base and
each fiber is something called a Hurwitz Variety.

For this and other reasons, the spaces $\SS_{q}$ and the components $Y$ of
which they are built bear a close family resemblance to the kinds of
discriminant complements that arise in the study of classical braid groups.
The full extent of this resemblance is an open question, partially summarized
in Table 2.

###  2.1. Degree 2

Let $\mathfrak{X},\mathfrak{f}$ be a generic maximal sausage polynomial of
degree $2$. The root polynomial $\mathfrak{f}_{v}$ is quadratic and
normalized. It has one critical point, necessarily genuine. Thus
$\mathfrak{f}_{v}(z):=z^{2}+c$ for some $c\neq 0$. Every other vertex $w$ has
a polynomial $\mathfrak{f}_{w}$ of degree one; since polynomials are
normalized, $\mathfrak{f}_{w}(z):=z$. Thus all the information is contained in
the choice of the (nonzero) constant coefficient $c$ of $\mathfrak{f}_{v}$, so
that $\mathcal{X}_{2}=\mathbb{C}^{*}$. The tree $T$ is an infinite dyadic
rooted tree, where every vertex is attached to its parent at the points
$\pm\sqrt{-c}$; see Figure 5.

Figure 5. A degree 2 sausage; each vertex is attached to its parent at the
points $\pm\sqrt{-c}$.

Furthermore, in this case
$\mathcal{DL}_{2}^{\prime}=\mathcal{DL}_{2}^{\prime\prime}$ so that $\SS_{2}$
is homeomorphic (but not holomorphically isomorphic) to $\mathbb{C}^{*}$. As a
corollary one deduces the famous theorem of Douady–Hubbard [17] that the
Mandelbrot set $\mathcal{M}$ (i.e. $\mathbb{C}-\SS_{2}$) is connected.

###  2.2. Discriminant Locus

In any degree $q$ there is a unique component of $\mathcal{X}_{q}$ for which
all the (genuine) critical points are in the root vertex. Thus
$\mathfrak{f}_{v}$ is a degree $q$ normalized polynomial with no fake critical
points. Since the marked points $Z_{v}$ of the root vertex are exactly the
roots of $\mathfrak{f}_{v}$, this means that $\mathfrak{f}_{v}$ is a
normalized polynomial with no critical roots. Equivalently, $\mathfrak{f}_{v}$
has $q$ distinct roots, so that $\mathfrak{f}_{v}$ is in
$Y_{q}:=\mathbb{C}^{q-1}-\Delta_{q}$ where $\Delta_{q}$ is the discriminant
locus. As is well-known, $Y_{q}$ is a $K(B_{q},1)$ where $B_{q}$ denotes the
braid group on $q$ strands.

###  2.3. Degree 3

Let $\mathfrak{X},\mathfrak{f}$ be a generic maximal sausage polynomial of
degree $3$. If the root polynomial $\mathfrak{f}_{v}$ has two genuine critical
points we are in the case discussed in § 2.2 and the corresponding component
of $\mathcal{X}_{3}$ is a $K(B_{3},1)$. Otherwise, since the root polynomial
must have at least one genuine critical point, if it does not have two it must
have exactly one and $\mathfrak{f}_{v}$ is of the form $z\to(z-c)^{2}(z+2c)$
for some $c\in\mathbb{C}^{*}$.

Figure 6. A degree 3 sausage; the root $v$ has $Z_{v}:=\\{c,-2c\\}$. The child
$w_{1}$ has $Z_{w_{1}}:=\\{\pm\sqrt{c-d},\pm\sqrt{-2c-d}\\}$

The (finite) marked points $Z_{v}$ of ${\hat{\mathbb{C}}}_{v}$ are $c$ and
$-2c$, and the root vertex correspondingly has two children $w_{1},w_{2}$
where ${\hat{\mathbb{C}}}_{w_{1}}$ is attached at $c$ and
${\hat{\mathbb{C}}}_{w_{2}}$ is attached at $-2c$. Because $c$ is a double
root, the polynomial $\mathfrak{f}_{w_{1}}$ has degree $2$; because $-2c$ is a
simple root, the polynomial $\mathfrak{f}_{w_{2}}$ has degree $1$.

Write $\mathfrak{f}_{w_{1}}:z\to z^{2}+d$. If $d\neq c,-2c$ then $Z_{w_{1}}$
has four (non-critical) points (the distinct square roots of $c-d$ and
$-2c-d$) and every other $\mathfrak{f}_{u}$ is degree 1. See Figure 6. Thus
$c$ and $d$ are moduli parameterizing a single component of $\mathcal{X}_{3}$,
and topologically this component is a bundle over $\mathbb{C}^{*}$ whose fiber
is homeomorphic to $\mathbb{C}-\\{c,-2c\\}$.

If $d=c$ or $d=-2c$ then $0$ is a fake critical point for
$\mathfrak{f}_{w_{1}}$, and if $u$ is the child of $w_{1}$ for which
${\hat{\mathbb{C}}}_{u}$ is attached at $0$ then $\mathfrak{f}_{u}$ has degree
$2$. Since $\mathfrak{f}$ is maximal, there is always some vertex $u^{\prime}$
at finite combinatorial distance from the root for which
$\mathfrak{f}_{u^{\prime}}$ has degree $2$ and for which the critical point
$0$ of $\mathfrak{f}_{u^{\prime}}$ is genuine. Thus each component of
$\mathcal{X}_{3}$ is a bundle over $\mathbb{C}^{*}$ with fiber homeomorphic to
$\mathbb{C}$ minus finitely many points.

###  2.4. The Tautological Elamination

The combinatorics of the components of $\mathcal{X}_{3}$ is quite complicated.
Each component of $\mathcal{X}_{3}$ (other than the discriminant complement
c.f. § 2.2) is a punctured plane bundle over the curve $\mathbb{C}^{*}$ with
parameter $c$, and these components glue together in $\SS_{3}$ to form a
bundle over $\mathbb{C}^{*}$ whose fiber $\Omega_{T}$ is homeomorphic to a
plane minus a Cantor set.

Actually, there is another description of $\Omega_{T}$ in terms of
elaminations. For each degree 3 critical leaf $C$ there is a certain
elamination $\Lambda_{T}(C)$ called the tautological elamination which can be
defined as follows. Let’s suppose that we have a maximal degree $3$ dynamical
elamination with two critical leaves $C$ and $C^{\prime}$, and that $C$ has
the greater height. If we fix $C$, then $\Omega_{T}$ parameterizes the space
of configurations of $C^{\prime}$.

The elamination $\Lambda_{T}(C)$ is defined as follows. With $C$ fixed, each
choice of (noncrossing) $C^{\prime}$ determines a dynamical elamination
$\Lambda$. By hypothesis $h(C^{\prime})<h(C)$ and there are only finitely many
(perhaps zero) precritical leaves $P$ of $C$ with $h(P)>h(C^{\prime})$. As we
vary $C^{\prime}$ the laminations $\Lambda$ also vary (in rather a complicated
way) but while $h(P)>h(C^{\prime})$ the leaves $P$ stay fixed under continuous
variations of $C^{\prime}$. It might happen that as we vary the leaf
$C^{\prime}$ it collides with a leaf $P$ with $h(P)>h(C^{\prime})$; the
elamination $\Lambda_{T}(C)$ consists of the cubes $P^{3}$ of all such $P$
(there is a similar, though more complicated construction in higher degrees).
The fact that $\Lambda_{T}(C)$ is an elamination is not obvious from this
definition.

The result of cut and paste (as in § 1.4) on the annulus $1<|z|<|C|$ (thought
of as a subset of $\mathbb{E}$) along $\Lambda_{T}(C)$ is a Riemann surface
$\Omega_{T}(C)$ holomorphically isomorphic to the moduli space of degree 3
maximal dynamical elaminations for which $C$ is the unique critical leaf of
greatest height. Figure 7 depicts the elamination $\Lambda_{T}(C)$ for a
particular value of $C$ whose tips have angles $\pm\pi/3$.

Figure 7. The tautological elamination $\Lambda_{T}(C)$ for $\arg(C)=\pm\pi/6$

These $\Omega_{T}(C)$ are the leaves of a (singular) one complex dimensional
holomorphic foliation of $\SS_{3}$.

Although it is not a dynamical elamination, the tautological elamination
$\Lambda_{T}(C)$ is in a natural way the increasing union of finite
elaminations $\Lambda_{n}$, namely the leaves of the form $P^{3}$ as above
where $P$ is a depth $n$ preimage of $C$. Let $\overline{\mathbb{E}}$ denote
the closure of $\mathbb{E}$ in $\mathbb{C}$ so that
$\overline{\mathbb{E}}=\mathbb{E}\cup S^{1}$, the union of $\mathbb{E}$ with
the unit circle. The result $\Omega_{n}$ of cut and pasting $\mathbb{E}$ along
$\Lambda_{n}$ is partially compactified by a finite set of circles, obtained
from $S^{1}$. By abuse of notation we denote this finite set of circles by
$S^{1}\mod\Lambda_{n}$. It turns out that the the components of
$\mathcal{X}_{3}\cap\Omega_{T}$ corresponding to sausage polynomials with
fixed $c\in\mathbb{C}^{*}$ and for which the second genuine critical point is
in a vertex at depth $n+1$ are in bijection with the set of components of
$S^{1}\mod\Lambda_{n}$. In fact, more is true.

For each combinatorial type $\mathfrak{X},\mathfrak{f}$ let $u$ be the vertex
containing the second genuine critical point (the first, by hypothesis, is
contained in the root). We define the depth $n$ of $\mathfrak{X},\mathfrak{f}$
to be the combinatorial distance of $u$ to the root. There is another
invariant of $\mathfrak{X},\mathfrak{f}$: the $\ell$-value, defined as
follows. Under iteration of $\mathfrak{f}$ (acting on the tree) the vertex $u$
has a length $n$ orbit terminating in the root (note that $\mathfrak{f}(u)$ is
not typically equal to the parent of $u$, but it does have the same depth as
the parent). The point $\infty$ in ${\hat{\mathbb{C}}}_{u}$ is mapped to
$\infty$ in ${\hat{\mathbb{C}}}_{\mathfrak{f}(u)}$ and so on. The product of
the degrees of the polynomials $\mathfrak{f}_{\mathfrak{f}^{i}(u)}$ up to but
not including the root is some power of $2$; by definition, $\ell$ is this
number divided by $2$. The invariants $n$ and $\ell$, taking discrete values,
are really invariants of the components of $\mathcal{X}_{3}$ and ipso facto of
the components of $\mathcal{X}_{3}\cap\Omega_{T}$.

Here is the relation to $\Lambda_{T}(C)$. Components of
$\mathcal{X}_{3}\cap\Omega_{T}$ of depth $n+1$ are in bijective correspondence
with components of $S^{1}\mod\Lambda_{n}$, and a component of
$\mathcal{X}_{3}\cap\Omega_{T}$ with $\ell$-value $\ell$ corresponds to a
component of $S^{1}\mod\Lambda_{n}$ of length $2\pi\ell\cdot 3^{-n}$.

###  2.5. Combinatorics

Let $N_{3}(n,m)$ denote the number of components of $S^{1}$ mod $\Lambda_{n}$
with depth $n+1$ and $\ell=2^{m}$. We do not know a simple closed form for
$N_{3}(n,m)$ and perhaps none exists — one subtle issue is that there are
several combinatorially different ways that a component can have a particular
$\ell$-value. However, an $\ell$-value of $1$ is special, since it corresponds
to an $\mathfrak{f}$ for which $\mathfrak{f}_{\mathfrak{f}^{i}(u)}$ has degree
$1$ for all positive $i$. Correspondingly, there is an explicit formula for
$N_{3}(n,0)$ that we now give; see [10] Thm. 3.6 for a proof.

First of all, $N_{3}(n,0)$ satisfies the recursion $N_{3}(0,0)=1$,
$N_{3}(1,0)=1$ and

$N_{3}(2n,0)=3N_{3}(2n-1,0)\text{ and }N_{3}(2n+1,0)=3N_{3}(2n,0)-2N_{3}(n,0)$

Knowing this, one can write down an explicit generating function for
$N_{3}(n,0)$; the generating function is $(\beta(t)-1)/3t$ where

$\beta(t)=\Bigl{(}\sum_{n=0}^{\infty}h(n)t^{n}\Bigr{)}\prod_{j=0}^{\infty}\frac{1}{(1-3t^{2^{j}})}$

and where the numbers $h(n)$ are defined by

$h(0)=1\text{ and }h(n)=(-3)^{s(n)}(1-(-2)^{k(n)})$

where $2^{k(n)}$ is the biggest power of 2 dividing $n$, and $s(n)$ is the sum
of the binary digits of $n$.

Table 1 gives values of $N_{3}(n,m)$ for $0\leq n,m\leq 12$. Note that
$N_{3}(n,m)=0$ for $n/2<m<n$; see [10] Thm 5.9.

$n\backslash\ell$ | 1 | 2 | $2^{2}$ | $2^{3}$ | $2^{4}$ | $2^{5}$ | $2^{6}$ | $2^{7}$ | $2^{8}$ | $2^{9}$ | $2^{10}$ | $2^{11}$ | $2^{12}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 1 |  |  |  |  |  |  |  |  |  |  |  |
1 | 1 | 1 |  |  |  |  |  |  |  |  |  |  |
2 | 3 | 1 | 1 |  |  |  |  |  |  |  |  |  |
3 | 7 | 6 | 0 | 1 |  |  |  |  |  |  |  |  |
4 | 21 | 16 | 3 | 0 | 1 |  |  |  |  |  |  |  |
5 | 57 | 51 | 13 | 0 | 0 | 1 |  |  |  |  |  |  |
6 | 171 | 149 | 39 | 5 | 0 | 0 | 1 |  |  |  |  |  |
7 | 499 | 454 | 117 | 23 | 0 | 0 | 0 | 1 |  |  |  |  |
8 | 1497 | 1348 | 360 | 66 | 9 | 0 | 0 | 0 | 1 |  |  |  |
9 | 4449 | 4083 | 1061 | 207 | 41 | 0 | 0 | 0 | 0 | 1 |  |  |
10 | 13347 | 12191 | 3252 | 591 | 126 | 17 | 0 | 0 | 0 | 0 | 1 |  |
11 | 39927 | 36658 | 9738 | 1799 | 370 | 81 | 0 | 0 | 0 | 0 | 0 | 1 |
12 | 119781 | 109898 | 29292 | 5351 | 1125 | 240 | 33 | 0 | 0 | 0 | 0 | 0 | 1

Table 1. Number of components of length $\ell/3^{n}$ at depth $n$

###  2.6. Hurwitz Varieties

Let $X$ be a component of $\mathcal{X}_{q}$ parameterizing sausage polynomials
of a fixed combinatorial type. $X$ is an iterated bundle whose base and fibers
$Y$ are all of the following sort. There are specific vertices $u,w$ with
$\mathfrak{f}(u)=w$. The set $Z_{w}\subset{\hat{\mathbb{C}}}_{w}$ is fixed, as
is the degree $p$ of
$\mathfrak{f}_{u}:{\hat{\mathbb{C}}}_{u}\to{\hat{\mathbb{C}}}_{w}$.
Furthermore, for each $\zeta\in Z_{w}$ the ramification data of
$\mathfrak{f}_{u}$ at $\zeta$ is specified; i.e. the monodromy of
$\mathfrak{f}_{u}^{-1}$ in a small loop around each $\zeta$, thought of as a
conjugacy class in the symmetric group on $p$ letters. Then $Y$ is the space
of normalized degree $p$ polynomials with the specified ramification data. We
call $Y$ a Hurwitz Variety, and observe that each $X$ is an iterated bundle
with total (complex) dimension $q-1$ whose base and fibers are all Hurwitz
varieties.

The generic case is that the monodromy of $\mathfrak{f}_{u}^{-1}$ in a small
loop around each $\zeta\in Z_{w}$ is trivial; i.e. that each $\zeta$ is a
regular value. In that case $Y$ is a Zariski open subset of
$\mathbb{C}^{p-1}$. In fact, we can say something more precise. Let
$\Delta_{p}\subset\mathbb{C}^{p-1}$ be the discriminant variety, i.e. the set
of normalized degree $p$ polynomials with a multiple root. For each $\zeta\in
Z_{w}$ let $\Delta_{p,\zeta}:=\Delta_{p}+\zeta$ be the translate of
$\Delta_{p}$ which parameterizes the set of normalized degree $p$ polynomials
$f$ for which $\zeta$ is a critical value. Then

$Y=\mathbb{C}^{p-1}-\bigcup_{\zeta\in Z_{w}}\Delta_{p,\zeta}$

It turns out that the topology of $Y$ depends only on the cardinality of
$Z_{w}$; see [11] Prop. 9.14. This is not obvious, since the
$\Delta_{p,\zeta}$ are singular, and they do not intersect in general
position.

###  2.7. $K(\pi,1)$s

For a finite set $Z\subset\mathbb{C}$ and degree $p$ let $Y_{p}(Z)$ denote the
Hurwitz variety of normalized degree $p$ polynomials for which no element of
$Z$ is a critical value.

As we remarked already in § 2.2, when $|Z|=1$ the space $Y_{p}(Z)$ is a
$K(B_{p},1)$ where $B_{p}$ denotes the braid group on $p$ strands.
Furthermore, when $p=2$ the space $Y_{2}(Z)$ may be identified with
$\mathbb{C}-Z$ in the obvious way, so that $Y_{2}(Z)$ is a $K(F_{n},1)$ where
$F_{n}$ is the free group on $n$ elements, and $n=|Z|$.

It turns out ([11] Thm. 9.17) that $Y_{3}(Z)$ is a $K(\pi,1)$ for any finite
set $Z$. This is proved by exhibiting an explicit $\text{CAT}(0)$ 2-complex
with the homotopy type of each $Y_{3}(Z)$. One component of $\mathcal{X}_{4}$
is a $K(B_{4},1)$ and all the others are nontrivial iterated fibrations where
the fibers are $Y_{2}(Z)$ or $Y_{3}(Z)$s. It follows that every component of
$\mathcal{X}_{4}$ is a $K(\pi,1)$, and in fact so is the shift locus $\SS_{4}$
itself (the same is true for simpler reasons of $\SS_{3}$ and $\SS_{2}$).

One knows few example of algebraic varieties which are $K(\pi,1)$s, and fewer
methods to construct or certify them (one of the few general methods, which
applies to certain complements of hyperplane arrangements, is due to Deligne
[15]). Is $Y_{p}(Z)$ a $K(\pi,1)$ for all $p$ and all $Z$?

###  2.8. Monodromy

For each $p$ and $|Z|$ there is a natural representation (well-defined up to
conjugacy) $\pi_{1}(Y_{p}(Z))\to B_{p|Z|}$ defined by the braiding of the
$p|Z|$ points $f^{-1}(Z)$ in $\mathbb{C}$ as $f$ varies in $Y_{p}(Z)$. This
map is evidently injective when $p=2$ or when $|Z|=1$. Is it injective in any
other case? I do not know the answer even when $p=3$ and $|Z|=2$.

Here is one reason to be interested. There is a monodromy representation of
$\pi_{1}(\SS_{q})$ into the ‘Cantor braid group’ — i.e. the mapping class
group of a disk minus a Cantor set — defined by the braiding of the (Cantor)
Julia set $J_{f}$ in $\mathbb{C}$ as $f$ varies in $\SS_{q}$. A priori this
representation lands in the mapping class group of the plane minus a Cantor
set, but it lifts canonically to the Cantor braid group (which is a central
extension) because every $f\in\SS_{q}$ acts in a standard way at infinity. If
one forgets the braiding and only considers the permutation action on the
Cantor set itself, the image in $\text{Aut}(\text{Cantor Set})$ is known to be
precisely equal to the automorphism group of the full (one-sided) shift on a
$q$ element alphabet, by a celebrated theorem of Blanchard–Devaney–Keen [5].
However, this action of $\pi_{1}(\SS_{q})$ on the Cantor set alone is very far
from faithful.

The automorphism group of the Cantor set is to the Cantor braid group as a
finite symmetric group is to a (finite) braid group. It is natural to ask: is
the monodromy representation from $\pi_{1}(\SS_{q})$ to the Cantor braid group
injective? It turns out that the restriction of the monodromy representation
to the image of $\pi_{1}(Y_{p}(Z))$ in $\pi_{1}(\SS_{q})$ factors through the
representation to $B_{p|Z|}$. So a precondition for the monodromy
representation to the Cantor braid group to be injective is that each
$\pi_{1}(Y_{p}(Z))\to B_{p|Z|}$ should be injective.

When $q=2$ we have $\pi_{1}(\SS_{2})=\mathbb{Z}$ and the monodromy
representation is evidently injective, since the Cantor braid group is
torsion-free. With Yan Mary He and Juliette Bavard we have shown that the
monodromy representation is injective in degree 3 (work in progress).

###  2.9. Big Mapping Class Groups

The Cantor braid group, and the (closely related) mapping class group of the
plane minus a Cantor set, are quintessential examples of what are colloquially
known as big mapping class groups. The study of these groups is an extremely
active area of current research; for an excellent recent survey see
Aramayona–Vlamis [1]. There are connections to the theory of finite type
mapping class groups (particularly to stability and uniformity phenomena in
such groups); to taut foliations of 3-manifolds; to pruning theory and the de-
Carvalho–Hall theory of endomorphisms of planar trees; to Artinizations of
Thompson-like groups and universal algebra; etc. (see [1] for references).

One major goal of this theory — largely unrealized as yet — is to develop new
tools for applications to dynamics in 2 real and 1 complex dimension. Cantor
sets appear in surfaces as attractors of hyperbolic systems (e.g. in
Katok–Pesin theory [19]), and big mapping class groups (and some closely
related objects) are relevant to the study of their moduli. The paper [11] and
the theory of sausages is an explicit attempt to work out some of these
connections in a particular case.

###  2.10. Rays

Let $\Gamma$ denote the mapping class group of the plane (which we identify
with $\mathbb{C}$) minus a Cantor set $K$. The Cantor braid group
$\hat{\Gamma}$ is the universal central extension of $\Gamma$. Some of the
tools discussed in this paper may be used to study $\hat{\Gamma}$ and its
subgroups in some generality; for instance, components of ${\mathcal{EL}}$ are
classifying spaces for subgroups of $\hat{\Gamma}$.

The group $\Gamma$ acts in a natural way on the set $\mathcal{R}$ of isotopy
classes of proper simple rays in $\mathbb{C}-K$ from $\infty$ to a point in
$K$. Associated to this action are two natural geometric actions of $\Gamma$:

  1. (1)

there is a natural circular order on $\mathcal{R}$, so that $\Gamma$ acts
faithfully by order-preserving homeomorphisms on a certain completion of
$\mathcal{R}$, the simple circle; see [3, 7, 12]; and

  2. (2)

the elements of $\mathcal{R}$ are the vertices of a (connected) graph (the ray
graph) whose edges correspond to pairs of rays that may be realized
disjointly; this graph is connected, has infinite diameter, and is
Gromov–hyperbolic; see [2, 9].

(Landing) rays are also a critical tool in complex dynamics, and in the
picture developed in the previous two sections. For $K$ a Cantor Julia set,
nonsingular gradient flowlines of the Green’s function extend continuously to
$K$; the set of distinct isotopy classes of nonsingular flowlines associated
to single $K$ form a clique in the ray graph. Because the ray graph is
Gromov–hyperbolic, there is (up to bounded ambiguity) a canonical path in the
ray graph between any two such cliques; one can ask whether such paths are
coarsely realized by paths in $\SS_{q}$, and if so what geometric properties
such paths have, and how this geometry manifests itself in algebraic
properties of $\pi_{1}(\SS_{q})$. For example: does $\pi_{1}(\SS_{q})$ admit a
(bi-)automatic structure? (to make sense of this one should work with a
locally finite groupoid presentation for $\pi_{1}(\SS_{q})$). One piece of
evidence in favor of this is that $\SS_{3}$ (and, for trivial reasons,
$\SS_{2}$) is homotopy equivalent to a locally $\text{CAT}(0)$ complex, and it
is plausible that the same holds for all $\SS_{q}$. Although there are known
examples of groups which are locally $\text{CAT}(0)$ but not bi-automatic
[20], nevertheless in practice these two properties often go hand in hand.

###  2.11. Left orderability

A group is left-orderable if it admits a total order that is preserved under
left multiplication. The left-orderability of braid groups (see [14]) is key
to some of their most important properties (e.g. faithfulness of the
Lawrence–Kraamer–Bigelow representations [4]). Left-orderability of 3-manifold
groups is also conjecturally ([6]) related to both symplectic topology (via
Heegaard Floer homology) and to big mapping class groups via the theory of
taut foliations and universal circles; see e.g. [13, 8]. The Cantor braid
group is left-orderable (via the faithful action of $\Gamma$ on the simple
circle) so to show that $\pi_{1}(\SS_{q})$ is left-orderable it would suffice
to prove injectivity of the monodromy representation as in § 2.8.

###  2.12. Comparison with finite braids

Define $Y_{q}:=\mathbb{C}^{q-1}-\Delta_{q}$, the space of normalized degree
$q$ polynomials without multiple roots. Our study of $\SS_{q}$ has been guided
by a heuristic that one should think of $\SS_{q}$ as a sort of ‘dynamical
cousin’ to $Y_{q}$, and that they ought to share many key algebraic and
geometric properties. Table 2 compares some of what is known about the
topology of $Y_{q}$ and $\SS_{q}$.

| $Y_{q}$ | $\SS_{2},\SS_{3}$ | $\SS_{4}$ | $\SS_{q},q>4$
---|---|---|---|---
locally $\text{CAT}(0)$ |  yes for $q\leq 6$ | yes | unknown | unknown
$K(\pi,1)$ | yes | yes | yes | unknown
$H_{*}$ vanishes below middle dimension | yes | yes | yes | yes
$\pi_{1}$ is mapping class group | yes | yes | unknown | unknown
$\pi_{1}$ is left-orderable | yes | yes | unknown | unknown
$\pi_{1}$ is biautomatic | yes | yes | yes | unknown

Table 2. Comparison of $\SS_{q}$ with discriminant complements $Y_{q}$

##  3\. Acknowledgements

I would like to thank Lvzhou Chen, Toby Hall, Sarah Koch, Curt McMullen,
Sandra Tilmon, Alden Walker and Amie Wilkinson for useful feedback on early
drafts of this paper.

## References

  * [1] J. Aramayona and N. Vlamis, _Big mapping class groups: an overview_ , preprint https://arxiv.org/abs/2003.07950
  * [2] J. Bavard, _Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire_ , Geom. Top. 20 (2016), 491–535
  * [3] J. Bavard and A. Walker, _The Gromov boundary of the ray graph_ , TAMS, 370 (2018), no. 11, 7647–7678
  * [4] S. Bigelow, _Braid groups are linear_ , JAMS, 14 (2000), no. 2, 471–486
  * [5] P. Blanchard, L. Devaney and L. Keen, _The dynamics of complex polynomials and automorphisms of the shift_ , Invent. Math. 104 (1991), 545–580
  * [6] S. Boyer, C. Gordon and L. Watson, _On $L$-spaces and left-orderable fundamental groups_, Math. Ann. 356 (2013), no. 4, 1213–1245
  * [7] D. Calegari, _Circular groups, planar groups and the Euler class_ , Proceedings of the Casson Fest, 431–491, Geom. Topol. Monogr. 7, Geom. Topol. Publ., Coventry, 2004
  * [8] D. Calegari, _Foliations and the Geometry of 3-Manifolds_ , Oxford Mathematical Monographs. Oxford University Press, Oxford, 2007.
  * [9] D. Calegari, _Big mapping class groups and dynamics_ , blog post, 2009, https://lamington.wordpress.com/2009/06/22/big-mapping-class-groups-and-dynamics/
  * [10] D. Calegari, _Combinatorics of the Tautological Lamination_ , preprint https://arxiv.org/abs/2106.00578
  * [11] D. Calegari, _Sausages and Butcher paper_ , preprint https://arxiv.org/abs/2105.11265
  * [12] D. Calegari and L. Chen, _Big mapping class groups and rigidity of the simple circle_ , Ergodic Thy. Dyn. Sys. 41 (2021), no. 7, 1961–1987
  * [13] D. Calegari and N. Dunfield, _Laminations and groups of homeomorphisms of the circle_ , Invent. Math. 152 (2003), no. 1, 149–204
  * [14] P. Dehornoy, I. Dynnikov, D. Rolfsen and B. Wiest, _Ordering braids_ , Math. Surv. Mon. 148. AMS, Providence, RI, 2008.
  * [15] P. Deligne, _Les immeubles des groupes de tresses généralisés_ , Invent. Math. 17 (1972), 273–302
  * [16] L. DeMarco and C. McMullen, _Trees and the dynamics of polynomials_ , Ann. Sci. Éc. Norm. Sup. (4) 41 (2008), no. 3, 337–382
  * [17] A. Douady and J. Hubbard, _Itération des polynômes quadratiques complexes_ , C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 3, 123–126
  * [18] P. Jones, _On removable sets for Sobolev spaces in the plane_ , in Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), volume 42 of Princeton Math. Ser., pages 250-–267. Princeton Univ. Press, Princeton, NJ, 1995.
  * [19] A. Katok, _Lyapunov exponents, entropy and periodic orbits for diffeomorphisms_ , Publ. Math. IHES 51 (1980), 137–173
  * [20] I. Leary and A. Minasyan, _Commensurating HNN-extensions: non-positive curvature and biautomaticity_ , preprint https://arxiv.org/abs/1907.03515
  * [21] J. Milnor, _Dynamics in one complex variable_ , Third Ed. Ann. Math. Stud. 160, Princeton University Press, Princeton NJ, 2006.


# Phase transitions of the Dicke model: a unified perspective

Pragna Das  Indian Institute of Science Education and Research Bhopal 462066
India  Auditya Sharma  Indian Institute of Science Education and Research
Bhopal 462066 India

###### Abstract

The Dicke model exhibits a variety of phase transitions. The quantum phase
transition from the normal phase to the super-radiant phase is marked by a
dramatic change in the scaling of the participation ratio. We find that the
ground state in the super-radiant phase exhibits multifractality manifest in
the participation ratio scaling as the square root of the full Hilbert space
dimension. The thermal phase transition temperature, for which we obtain an
exact analytical expression, is strikingly captured by the mutual information
between two spins. In the excited state quantum phase transition within the
super-radiant phase, we discover a new upper cut-off energy; the central
energy band between the lower and upper cut-off energies shows distinctly
different behaviour. This finding is corroborated with the aid of several
_eigenvector_ properties: von Neumann entanglement entropy between spins and
bosons, the mean photon number, concurrence between two spins, and
participation ratio. Thus we obtain a unified picture for the three different
kinds of phase transitions.

The Dicke model, which incorporates the interactions of an ensemble of $N$
two-level atoms via dipole coupling with a single bosonic mode Haroche (2007);
Kimble  _et al._ (1996); Mirhosseini  _et al._ (2019) has its origin in
quantum optics, but has found application in a wide range of fields from
quantum chaos to quantum entanglement Furuya  _et al._ (1998); Lakshminarayan
(2001); Bandyopadhyay and Lakshminarayan (2002, 2004); Tanaka  _et al._
(2002); Žnidarič and Prosen (2005); Jacquod (2004); Ghose and Sanders (2004);
Demkowicz-Dobrzański and Kuś (2004); Weinstein and Hellberg (2005);
Lakshminarayan and Subrahmanyam (2003); Lambert  _et al._ (2004); Emary and
Brandes (2003a, b) to scrambling and thermalization Lewis-Swan  _et al._
(2019). Besides possessing an intimate connection to experiments Baumann  _et
al._ (2011); Klinder  _et al._ (2015); Baden  _et al._ (2017), the Dicke model
is a testbed for a variety of phase transitions Kirton  _et al._ (2019).
Although a lot is known about these different transitions, the literature
presents a rather scattered treatment of them Dicke (1954); Kadantseva  _et
al._ (1990); Kirton and Keeling (2018); Kirton  _et al._ (2019); Hepp and Lieb
(1973); Wang and Hioe (1973); Pérez-Fernández and Relaño (2017); Zhu  _et al._
(2019). In this Letter, we provide a transparent unified picture of three
different kinds of phase transitions in the Dicke model.

The nature of the ground state of the Dicke model is dramatically different
depending on the magnitude of the coupling between the atoms and the field.
While for small coupling, in the normal phase, the average photon number in
the ground state is close to zero, when the coupling is greater than a
critical value, in the super-radiant phase, the ground state mean photon
number scales linearly with the number of atoms Kirton  _et al._ (2019); Dicke
(1954); Kadantseva  _et al._ (1990); Kirton and Keeling (2018). Entanglement
properties Lambert  _et al._ (2004); Emary and Brandes (2003a, b) offer clear
signatures of this quantum phase transition S. (1998); Sondhi  _et al._
(1997); Cejnar  _et al._ (2010); Casten and McCutchan (2007); Ma and Wang
(2009); Osterloh  _et al._ (2002) (QPT). Furthermore, a study of level
statistics shows that the system in fact also undergoes a transition from
quasi-integrable to quantum chaotic Emary and Brandes (2003a) at the QPT. In
this Letter, with the aid of a careful study of the participation ratio
Tsukerman (2017) of the ground state, we show how the normal to super-radiant
phase transition is really a localization-to-multifractal transition. We find
that in the super-radiant phase, the ground state participation ratio scales
as the square root of the full Hilbert space dimension.

The Dicke model also exhibits a thermal phase transition (TPT) which was
realized many decades ago Hepp and Lieb (1973); Wang and Hioe (1973). When the
coupling is greater than the critical coupling, as the temperature is
increased, we see a transition back from the super-radiant to the normal phase
Pérez-Fernández and Relaño (2017). In this Letter, we obtain an exact
analytical expression for the transition temperature. Generalizing the
approach of Wang and Hieo Wang and Hioe (1973), we write down the partition
function as a double integral. The transition temperature is identified to be
the point at which the method of steepest descent used to evaluate the
integral in the thermodynamic limit breaks down. Furthermore, just like
entanglement in the ground state marks the quantum phase transition, we show
how the mutual information (MI) between atoms offers a striking signature at
the thermal phase transition.

The Dicke model also exhibits an excited state quantum phase transition
(ESQPT), a term that is used to denote criticality in the excited states of a
quantum system Pérez-Fernández and Relaño (2017); Caprio  _et al._ (2008);
Stránskỳ  _et al._ (2014); Zhu  _et al._ (2019); Cejnar and Stránský (2008);
García-Ramos  _et al._ (2017). The ESQPT, which is a generalization of the
QPT, and is characterized by abrupt variations of the energy and other excited
state properties at a sharp critical value of the energy Cejnar  _et al._
(2021) must be viewed in the backdrop of the tremendous recent interest in the
properties of excited states Pal and Huse (2010); Nandkishore and Huse (2015);
Alet and Laflorencie (2018); Karthik  _et al._ (2007); Caprio  _et al._
(2008); Pérez-Fernández  _et al._ (2011a, b); Beugeling  _et al._ (2015) of
quantum systems. In the present Letter, we uncover how the ESQPT of the Dicke
model affects not only energy levels below a certain lower cut-off, but also
the top-lying energy levels above a second  _upper_ cut-off, a feature that
has apparently been hitherto unnoticed in the literature Pérez-Fernández  _et
al._ (2011a); Lewis-Swan  _et al._ (2019). Strikingly, in contrast to prior
studies, we are able to identify these features with the aid of several
_eigenstate_ properties: von Neumann entanglement entropy (VNEE), the mean
photon number, concurrence and $PR$. Supporting evidence comes from eigenvalue
properties like level statistics Poilblanc  _et al._ (1993) and the
consecutive level spacing ratio Atas  _et al._ (2013) considering both the
whole and different parts of the spectrum.

Figure 1: (a) The inverse participation ratio ($IPR$) of the ground state as a
function of coupling $g$. The inset shows a similar plot for participation
ratio ($PR$). (b) Multifractal dimension $D_{q}$ ($q=1,2,\infty$) of the
ground state as a function of $g$. The inset shows the scaling of $PR$ with
atom number $N$ for the ground state at $g=4.0$. In all the figures, we set
$N=512$, $n_{\text{max}}=32$.

The Hamiltonian of the Dicke model is

$\displaystyle\mathcal{H}$ $\displaystyle=$ $\displaystyle\omega
a^{\dagger}a+\omega_{0}J_{z}+\frac{g}{\sqrt{2j}}(a+a^{\dagger})(J_{+}+J_{-})$
(1)

where $a$ and $a^{\dagger}$ are bosonic operators satisfying the commutation
relation: $[a,a^{\dagger}]=1$ in units where $\hbar=1$. $\omega$ is the
single-mode frequency of the bosonic field while $\omega_{0}$ is the level
splitting of the atoms, and $g$ is the coupling strength of the light-matter
interaction. The angular momentum operators
$J_{\pm,z}=\sum_{i=1}^{2j}\frac{1}{2}\sigma_{\pm,z}^{(i)}$ correspond to a
pseudospin with length $j$, composed of $N=2j$ spin-$\frac{1}{2}$ atoms
described by Pauli matrices $\sigma_{\pm,z}^{(i)}$ acting on site $i$ and
satisfy the commutation relations: $[J_{z},J_{\pm}]=\pm J_{\pm}$,
$[J_{+},J_{-}]=2J_{z}$. The basis of the full Hilbert space of the system is
$\\{|n\rangle\otimes|j,m\rangle\\}$ where $|n\rangle$ are the bosonic basis
states satisfying $a^{\dagger}a|n\rangle=n|n\rangle$ and $|j,m\rangle$ are the
Dicke states satisfying $J_{\pm}|j,m\rangle=\sqrt{j(j+1)-m(m\pm 1)}|j,m\pm
1\rangle$, $J_{z}|j,m\rangle=m|j,m\rangle$. In our work, we take $N$ to be
even, and consider the symmetric subspace which fixes $j=\frac{N}{2}$, and
thus $m$ takes the $(N+1)$ values $(-\frac{N}{2},...,0,...,\frac{N}{2})$. We
also truncate the bosonic mode to take the values $n=0,1,...,n_{\text{max}}$.
Thus the dimension of the Hilbert space is given by
$N_{D}=(N+1)(n_{\text{max}}+1)$. In the thermodynamic limit the system shows a
second-order quantum phase transition from the normal phase (NP) to the super-
radiant phase (SP) at $g=\frac{\sqrt{\omega\omega_{0}}}{2}$ ($=g_{c}$) Emary
and Brandes (2003a). In all our numerical calculations we have set
$\omega=\omega_{0}=1$ and hence $g_{c}=0.5$.

Quantum phase transition: The inverse participation ratio ($IPR$) of an
eigenstate $|\psi\rangle=\sum_{j}^{N_{D}}\psi_{j}|j\rangle$ (where $N_{D}$ is
the Hilbert space dimension) defined as:

$IPR=\sum_{j=1}^{N_{D}}|\psi_{j}|^{4}.$ (2)

It is useful to quantify the degree of delocalization of the eigenstate. Fig.
1(a) shows an exact diagonalization study of the $IPR$ of the ground state as
$g$ is varied; we observe that it is close to one in the NP and close to zero
in the SP. Thus we see that the ground state is localized in the NP whereas it
is extended in nature in the SP. Echoes of these features are also found in
both static and dynamical studies of a variety of other measures of quantum
correlations (see supplementary section). The inset of Fig. 1(a) studies the
participation ratio $PR$ (which is the inverse of $IPR$), as a function of the
coupling $g$ and it shows a phase transition from NP where it takes values
close to zero, to SP with a sharp transition to a non-zero value at the
critical coupling. We plot the $PR$ for different atom number $N$ and notice
that in the SP the value of $PR$ increases with $N$.

A finer understanding of the localization properties may be obtained by
studying the multifractal dimension Macé  _et al._ (2019); Lindinger  _et al._
(2019):

$\displaystyle D_{q}$ $\displaystyle=$ $\displaystyle\frac{S_{q}}{\ln(N_{D})}$
(3)

where $S_{q}=\frac{1}{1-q}\ln\Big{(}\sum_{j=1}^{N_{D}}|\psi_{j}|^{2q}\Big{)}$
is known as the $q$-dependent participation entropy. In the Shannon limit
($q=1$), $S_{1}=\sum_{j}|\psi_{j}|^{2}\ln\Big{(}|\psi_{j}|^{2}\Big{)}$, while
$q=2$ yields the usual $IPR$ with $S_{2}=-\ln(IPR)$. $S_{\infty}$ is
determined by the maximum value of the densities
$p_{\text{max}}=\text{max}_{j}|\psi_{j}|^{2}$ and
$D_{\infty}=-\frac{\ln(p_{\text{max}})}{\ln(N_{D})}$. For a perfectly
delocalized state $S_{q}=\ln(N_{D})$ (when $N_{D}$ is large) and hence
$D_{q}=1$ for all $q$. On the other hand for a localized state
$S_{q}=constant$ and $D_{q}=0$. In an intermediate situation, wave functions
are extended but non-ergodic with $S_{q}=D_{q}\ln(N_{D})$ where $0<D_{q}<1$
and the state is multifractal. In Fig. 1(b) we show $D_{1}$, $D_{2}$ and
$D_{\infty}$ for the ground state as a function of the coupling parameter $g$.
In the NP $D_{q}\approx 0$ hence we can say that the ground state is localized
in the NP. Contrastingly in the SP, $0<D_{q}<1$ with $D_{1}>D_{2}>D_{\infty}$
($D_{1}\approx 0.58$, $D_{2}\approx 0.55$, $D_{\infty}\approx 0.47$), with a
sharp transition at the critical point. In the inset of Fig. 1(b) we show that
at $g=4.0$, $PR$ goes as the square root of the Hilbert space dimension. Hence
we find that the SP is neither perfectly delocalized nor localized, and in
fact displays multifractal character Macé  _et al._ (2019). Given the intense
current interest in multifractal states Lindinger and Rodríguez (2017); Pino
_et al._ (2017); Serbyn  _et al._ (2017), this discovery in a familiar model
is an exciting finding.

Figure 2: (a) Mutual information (MI) of two spins as a function of
temperature at $g=1.0$, with the inset showing the numerical derivative of MI
wrt temperature $\frac{dI_{12}}{dT}$. (b) The red line denotes the critical
temperature as a function of atom number $N$, while the black line is the
theoretical value $(T_{c}=0.850181)$ for $g=1.0$. (c) Mutual information of
two spins as a function of coupling $g$ and temperature $T$. The parameters
are $N=6$, $n_{max}=10$. The black region corresponds to the NP, and the white
region to the SP. The solid line corresponds to $g_{c}$ and the dashed line
denotes the critical temperature theoretically calculated in Eqn. 6. In all
the plots $\omega=\omega_{0}=1$.

Thermal phase transition: To compute the partition function
($Z=Tr(e^{-\frac{\mathcal{H}}{k_{B}T}})$) of the Dicke Hamiltonian it is
useful to first write it in units of $\omega$ as:

$\displaystyle\tilde{\mathcal{H}}=\frac{\mathcal{H}}{\omega}=a^{\dagger}a+\sum_{j=1}^{N}\frac{\epsilon}{2}\sigma_{j}^{z}+\frac{\lambda}{\sqrt{N}}\sum_{j=1}^{N}(a+a^{\dagger})\sigma^{x}$
(4)

where $\epsilon=\frac{\omega_{0}}{\omega}$ and $\lambda=\frac{g}{\omega}$.
Following the method of Wang and Hieo Wang and Hioe (1973) (who studied the
Dicke model within the rotating wave approximation), the computation of the
partition function reduces to the evaluation of a double integral (see
supplementary section):

$\displaystyle
Z(N,T)=\int\frac{d^{2}\alpha}{\pi}e^{-\beta|\alpha|^{2}}\Big{(}2\cosh\Big{[}\frac{\beta\epsilon}{2}\Big{[}1+\frac{16{\lambda}^{2}\alpha^{2}}{\epsilon^{2}N}\Big{]}^{1/2}\Big{]}\Big{)}^{N}$
(5)

which in the thermodynamic limit ($N\to\infty$), may be carried out with the
aid of the method of steepest descent, within the super-radiant phase.
Tracking the point at which the method breaks down (see supplementary
section), we obtain an exact expression for the transition temperature:

$T_{c}=\frac{1}{\beta_{c}}=\Big{(}\frac{\omega_{0}}{2\omega}\Big{)}\frac{1}{\tanh^{-1}\Big{(}\frac{\omega\omega_{0}}{4g^{2}}\Big{)}}.$
(6)

The critical temperature expression is meaningful only when $g>g_{c}$. When
$g<g_{c}$, the system is in the normal phase at all temperatures. When
$g>g_{c}$, it is only above the critical temperature that the system is in the
normal phase, while for $T<T_{c}$ the system is in the super-radiant phase.

Figure 3: (a) Von Neumann entanglement entropy between the atoms and the
bosons as a function of coupling strength $g$ and the energy density $E_{n}/N$
(eigenenergy divided by the atom number $N$) of the DM. (b) Similar plot for
mean photon number, (c) for concurrence between any two atoms and (d)
participation ratio of the eigenstates. The parameters for all the four plots
are $N=512$, $n_{\text{max}}=32$.

The mutual information (MI) between two atoms $I_{12}$ proves to be a very
useful quantity to study. Using the single-spin reduced density matrices
$\rho_{1}$, $\rho_{2}$ and the two-spin reduced density matrix $\rho_{12}$, we
can work out the von Neumann entropies
$S_{1,2}=-Tr(\rho_{1,2}\ln(\rho_{1,2}))$,
$S_{12}=-Tr(\rho_{12}\ln(\rho_{12}))$ from which the mutual information Vedral
(2003); DiVincenzo  _et al._ (2004); Adesso and Datta (2010); Maziero  _et
al._ (2010) is immediately written down:

$I_{12}=S_{1}+S_{2}-S_{12}.$ (7)

In Fig. 2(a) we show $I_{12}$ as a function of temperature at $g=1$
($g>g_{c}$). At low temperatures in the SP, $I_{12}$ takes a value close to
unity while at high temperatures in the NP, $I_{12}$ drops to a value close to
zero, with a dramatic drop happening at a temperature close to the transition
temperature. For a finer understanding of the variation of the mutual
information across the transition, we study in the inset of Fig. 2(a) the
first-order temperature derivative $\frac{d{I_{12}}}{dT}$, for different atom
numbers. We observe that the temperature at which the derivative takes the
minimum value is consistent with the transition temperature $T_{c}$, denoted
by the vertical line. Fig. 2(b) confirms that as the number of atoms is
increased the temperature at the minimum does indeed approach the
theoretically obtained critical temperature. From the surface plot of the MI
as a funtion of $g$ and $T$ in Fig. 2(c), it is clear that for $g<g_{c}$ there
is no phase pransition, but for $g>g_{c}$ there exists a critical temperature
$T_{c}$ at which the system changes from the super-radiant phase ($T<T_{c}$)
to the normal phase ($T>T_{c}$). While it is widely known that entanglement in
the ground state signals the QPT, our work shows that despite also including
classical correlations, the mutual information between atoms offers a striking
signature at the thermal phase transition.

ESQPT: The Dicke model exhibits an excited state quantum phase transition in
the super-radiant phase. When $g>g_{c}$, it has been reported Pérez-Fernández
_et al._ (2011a); Lewis-Swan  _et al._ (2019) that the eigenvalues above a
cut-off energy $E_{c}$ behave in a distinctly different manner in comparison
with the eigenvalues below the cut-off. We find that in fact there is not just
a lower cut-off, but also an upper cut-off. Our data show that we must study
separately the eigenvalues drawn from a central band that comprises of energy
levels between a lower and upper cut-off. The lower and upper energy bands
show different behaviour. While eigenvalue properties like level-statistics
and gap ratio provide supporting evidence (see supplementary section), we
highlight how _eigenstate properties_ offer a striking demonstration of this
picture.

In Fig. 3(a) we show the VNEE Lambert  _et al._ (2004) between spins and
bosons:

$S=-Tr\Big{(}\rho_{\text{boson}}\log_{2}\rho_{\text{boson}}\Big{)},$ (8)

for all the eigenstates of the Dicke model. Here
$\rho_{\text{boson}}=Tr_{\text{atom}}\rho$ is the reduced density matrix of
the bosonic part. We observe two cut-off energies: $(i)$ lower cut-off energy
(corresponding to the ground state energy at $g=g_{c}$) and $(ii)$ upper cut-
off energy (corresponding to the maximum energy for $g=0$). The value of VNEE
is larger in the eigenstates of the central band in comparison with that of
the top and bottom bands. Thus the eigenstates carry a clear signature of the
two excited state quantum phase transitions when $g>g_{c}$. In Fig. 3(b) we
show a similar plot for the mean photon number Emary and Brandes (2003a),
$\langle a^{\dagger}a\rangle$ which is scaled by the pseudospin length $j$ of
the system. It carries information pertaining to the bosonic part of the
eigenstates. In the middle band the value of the mean photon number is
comparatively lower than that of the other two bands. However, we observe that
neither the VNEE between the atoms and the bosons, nor the mean photon number
is able to distinguish between the $g<g_{c}$ and $g>g_{c}$ regions of the
middle band. A study of the entanglement between atoms provides useful further
perspective.

The concurrence Hill and Wootters (1997); Wootters (1998, 2001); Dennison and
Wootters (2001) between (any) two atoms is given by:

$C=max\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\},$ (9)

where $\lambda_{i}$ are the square roots of the eigenvalues of the matrix
product,
$\tilde{{\rho}}_{12}=\rho_{12}(\sigma_{1y}\otimes\sigma_{2y})\rho^{\star}_{12}(\sigma_{1y}\otimes\sigma_{2y})$,
in decending order ($\lambda_{1}>\lambda_{2}>\lambda_{3}>\lambda_{4}$). Here
$\rho^{\star}_{12}$ denotes complex conjugation of $\rho_{12}$, and
$\sigma_{iy}$ are Pauli matrices for two-level systems. In Fig. 3(c) we plot
concurrence between two atoms for the whole spectrum as a function of $g$. We
observe that in addition to showing a signature of the ESQPT in the super-
radiant phase, concurrence is also able to distinguish the eigenstates of the
middle band in the $g<g_{c}$ region and the $g>g_{c}$ region. In the NP, the
concurrence value of the central states is comparatively smaller than that for
the central states of the SP. Again the value of concurrence in the bottom and
top bands of the super-radiant phase is a bit lower than that of the central
band. Fig. 3(d) shows the participation ratio of all the eigenstates as a
function of coupling parameter $g$. We are able to identify the ESQPT, which
divides the whole spectrum into three bands: top, bottom, and central. In the
NP ($g<g_{c}$), the whole region shows a uniform comparatively small value of
$PR$. On the other hand in the SP ($g>g_{c}$) while the central band exhibits
a larger value of $PR$, the top and bottom bands show mixed behaviour,
although they resemble the NP more than the SP.

Summary: We study the phase transitions (QPT, TPT, ESQPT) of the Dicke model,
with the aid of a number of measures of localization, entanglement and mutual
information. Different quantities are more suitable for the different kinds of
phase transitions involved, and a comprehensive look at all of them helps us
obtain a unique overall big-picture of the Dicke model. We are thus able to
provide a unified perspective of three different kinds of phase transitions.
The $IPR$ for the ground state shows a sharp phase transition at $g_{c}$;
while in the NP the ground state behaves like a localized state, the ground
state in the SP is not localized. A careful study of the scaling of $PR$ (for
the ground state in the SP) with the dimension $N_{D}$ of the full Hilbert
space reveals that the $PR$ scales as $\sqrt{N_{D}}$ suggesting multifractral
character. In the $g>g_{c}$ region there exists some critical temperature
$T_{c}$, above which the SP disappears and the system goes into the NP whereas
for $g<g_{c}$ the system remains in the NP for all temperatures. We obtain a
closed-form expression for the transition temperature in the super-radiant
phase, and numerically verify that the mutual information between two atoms
provides a useful signature at the transition. Thus at the temperature
transition, the mutual information proves to be a worthy generalization of
entanglement, which marks the ground state QPT. We find that in the super-
radiant phase, the ESQPT is signalled not just by a lower energy cut-off, but
an upper energy cut-off as well. The ESQPT is studied with the aid of VNEE,
mean photon number, concurrence and $PR$. For the VNEE and mean photon number
the whole central band is uniform, with no distinction between $g<g_{c}$ and
$g>g_{c}$ regions. We find that concurrence and $PR$ reveal more structure. In
addition to showing a signature of the ESQPT in the SP, these quantities are
also able to distinguish the eigenstates of the central band between the
$g<g_{c}$ region and $g>g_{c}$ region. Hence we are able to present various
phase transitions in the DM in terms of several quantities that measure
localization, multifractality, mutual information and entanglement. It would
be interesting to extend the ideas in this study to other spin-boson models,
to open quantum systems that include a bosonic bath, and models with a
periodic drive.

## Acknowledgments

We are thankful to Devendra Singh Bhakuni, Nilanjan Roy, Suhas Gangadharaiah
and Sebastian Wüster for fruitful comments and discussions. P.D. is grateful
to IISERB for the PhD fellowship. A.S acknowledges financial support from SERB
via the grant (File Number: CRG/2019/003447), and from DST via the DST-INSPIRE
Faculty Award [DST/INSPIRE/04/2014/002461].

## References

  * Haroche (2007) S. Haroche, in  _Conference on Coherence and Quantum Optics_ (Optical Society of America, 2007) p. CTuF2.
  * Kimble  _et al._ (1996) H. Kimble, Q. Turchette, N. P. Georgiades, C. Hood, W. Lange, H. Mabuchi, E. Polzik, and D. Vernooy, in  _Coherence and Quantum Optics VII_ (Springer, 1996) pp. 203–210.
  * Mirhosseini  _et al._ (2019) M. Mirhosseini, E. Kim, X. Zhang, A. Sipahigil, P. B. Dieterle, A. J. Keller, A. Asenjo-Garcia, D. E. Chang, and O. Painter, Nature 569, 692 (2019).
  * Furuya  _et al._ (1998) K. Furuya, M. C. Nemes, and G. Q. Pellegrino, Phys. Rev. Lett. 80, 5524 (1998).
  * Lakshminarayan (2001) A. Lakshminarayan, Phys. Rev. E 64, 036207 (2001).
  * Bandyopadhyay and Lakshminarayan (2002) J. N. Bandyopadhyay and A. Lakshminarayan, Phys. Rev. Lett. 89, 060402 (2002).
  * Bandyopadhyay and Lakshminarayan (2004) J. N. Bandyopadhyay and A. Lakshminarayan, Phys. Rev. E 69, 016201 (2004).
  * Tanaka  _et al._ (2002) A. Tanaka, H. Fujisaki, and T. Miyadera, Phys. Rev. E 66, 045201 (2002).
  * Žnidarič and Prosen (2005) M. Žnidarič and T. c. v. Prosen, Phys. Rev. A 71, 032103 (2005).
  * Jacquod (2004) P. Jacquod, Phys. Rev. Lett. 92, 150403 (2004).
  * Ghose and Sanders (2004) S. Ghose and B. C. Sanders, Phys. Rev. A 70, 062315 (2004).
  * Demkowicz-Dobrzański and Kuś (2004) R. Demkowicz-Dobrzański and M. Kuś, Phys. Rev. E 70, 066216 (2004).
  * Weinstein and Hellberg (2005) Y. S. Weinstein and C. S. Hellberg, Phys. Rev. Lett. 95, 030501 (2005).
  * Lakshminarayan and Subrahmanyam (2003) A. Lakshminarayan and V. Subrahmanyam, Phys. Rev. A 67, 052304 (2003).
  * Lambert  _et al._ (2004) N. Lambert, C. Emary, and T. Brandes, Phys. Rev. Lett. 92, 073602 (2004).
  * Emary and Brandes (2003a) C. Emary and T. Brandes, Phys. Rev. E 67, 066203 (2003a).
  * Emary and Brandes (2003b) C. Emary and T. Brandes, Phys. Rev. Lett. 90, 044101 (2003b).
  * Lewis-Swan  _et al._ (2019) R. Lewis-Swan, A. Safavi-Naini, J. J. Bollinger, and A. M. Rey, Nature communications 10, 1 (2019).
  * Baumann  _et al._ (2011) K. Baumann, R. Mottl, F. Brennecke, and T. Esslinger, Phys. Rev. Lett. 107, 140402 (2011).
  * Klinder  _et al._ (2015) J. Klinder, H. Keßler, M. Wolke, L. Mathey, and A. Hemmerich, Proceedings of the National Academy of Sciences 112, 3290 (2015).
  * Baden  _et al._ (2017) M. P. Baden, K. J. Arnold, A. L. Grimsmo, S. Parkins, and M. D. Barrett, Phys. Rev. Lett. 118, 199901 (2017).
  * Kirton  _et al._ (2019) P. Kirton, M. M. Roses, J. Keeling, and E. G. Dalla Torre, Advanced Quantum Technologies 2, 1800043 (2019).
  * Dicke (1954) R. H. Dicke, Physical review 93, 99 (1954).
  * Kadantseva  _et al._ (1990) E. Kadantseva, W. Chmielowski, and A. Shumovsky, in  _Nonlinear Optics in Solids_ (Springer, 1990) pp. 37–41.
  * Kirton and Keeling (2018) P. Kirton and J. Keeling, New Journal of Physics 20, 015009 (2018).
  * Hepp and Lieb (1973) K. Hepp and E. H. Lieb, Annals of Physics 76, 360 (1973).
  * Wang and Hioe (1973) Y. K. Wang and F. T. Hioe, Phys. Rev. A 7, 831 (1973).
  * Pérez-Fernández and Relaño (2017) P. Pérez-Fernández and A. Relaño, Phys. Rev. E 96, 012121 (2017).
  * Zhu  _et al._ (2019) G.-L. Zhu, X.-Y. Lü, S.-W. Bin, C. You, and Y. Wu, Frontiers of Physics 14, 1 (2019).
  * S. (1998) S. S., draft ed. (CUP, 1998).
  * Sondhi  _et al._ (1997) S. L. Sondhi, S. Girvin, J. Carini, and D. Shahar, Reviews of modern physics 69, 315 (1997).
  * Cejnar  _et al._ (2010) P. Cejnar, J. Jolie, and R. F. Casten, Reviews of Modern Physics 82, 2155 (2010).
  * Casten and McCutchan (2007) R. Casten and E. McCutchan, Journal of Physics G: Nuclear and Particle Physics 34, R285 (2007).
  * Ma and Wang (2009) J. Ma and X. Wang, Phys. Rev. A 80, 012318 (2009).
  * Osterloh  _et al._ (2002) A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002).
  * Tsukerman (2017) E. Tsukerman, Phys. Rev. B 95, 115121 (2017).
  * Caprio  _et al._ (2008) M. Caprio, P. Cejnar, and F. Iachello, Annals of Physics 323, 1106 (2008).
  * Stránskỳ  _et al._ (2014) P. Stránskỳ, M. Macek, and P. Cejnar, Annals of Physics 345, 73 (2014).
  * Cejnar and Stránský (2008) P. Cejnar and P. Stránský, Phys. Rev. E 78, 031130 (2008).
  * García-Ramos  _et al._ (2017) J. E. García-Ramos, P. Pérez-Fernández, and J. M. Arias, Phys. Rev. C 95, 054326 (2017).
  * Cejnar  _et al._ (2021) P. Cejnar, P. Stránskỳ, M. Macek, and M. Kloc, Journal of Physics A: Mathematical and Theoretical 54, 133001 (2021).
  * Pal and Huse (2010) A. Pal and D. A. Huse, Phys. Rev. B 82, 174411 (2010).
  * Nandkishore and Huse (2015) R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  * Alet and Laflorencie (2018) F. Alet and N. Laflorencie, Comptes Rendus Physique 19, 498 (2018).
  * Karthik  _et al._ (2007) J. Karthik, A. Sharma, and A. Lakshminarayan, Phys. Rev. A 75, 022304 (2007).
  * Pérez-Fernández  _et al._ (2011a) P. Pérez-Fernández, A. Relaño, J. M. Arias, P. Cejnar, J. Dukelsky, and J. E. García-Ramos, Phys. Rev. E 83, 046208 (2011a).
  * Pérez-Fernández  _et al._ (2011b) P. Pérez-Fernández, P. Cejnar, J. M. Arias, J. Dukelsky, J. E. García-Ramos, and A. Relaño, Phys. Rev. A 83, 033802 (2011b).
  * Beugeling  _et al._ (2015) W. Beugeling, A. Andreanov, and M. Haque, Journal of Statistical Mechanics: Theory and Experiment 2015, P02002 (2015).
  * Poilblanc  _et al._ (1993) D. Poilblanc, T. Ziman, J. Bellissard, F. Mila, and G. Montambaux, EPL (Europhysics Letters) 22, 537 (1993).
  * Atas  _et al._ (2013) Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Phys. Rev. Lett. 110, 084101 (2013).
  * Macé  _et al._ (2019) N. Macé, F. Alet, and N. Laflorencie, Phys. Rev. Lett. 123, 180601 (2019).
  * Lindinger  _et al._ (2019) J. Lindinger, A. Buchleitner, and A. Rodríguez, Phys. Rev. Lett. 122, 106603 (2019).
  * Lindinger and Rodríguez (2017) J. Lindinger and A. Rodríguez, Phys. Rev. B 96, 134202 (2017).
  * Pino  _et al._ (2017) M. Pino, V. E. Kravtsov, B. L. Altshuler, and L. B. Ioffe, Phys. Rev. B 96, 214205 (2017).
  * Serbyn  _et al._ (2017) M. Serbyn, Z. Papić, and D. A. Abanin, Phys. Rev. B 96, 104201 (2017).
  * Vedral (2003) V. Vedral, Phys. Rev. Lett. 90, 050401 (2003).
  * DiVincenzo  _et al._ (2004) D. P. DiVincenzo, M. Horodecki, D. W. Leung, J. A. Smolin, and B. M. Terhal, Phys. Rev. Lett. 92, 067902 (2004).
  * Adesso and Datta (2010) G. Adesso and A. Datta, Phys. Rev. Lett. 105, 030501 (2010).
  * Maziero  _et al._ (2010) J. Maziero, H. Guzman, L. Céleri, M. Sarandy, and R. Serra, Phys. Rev. A 82, 012106 (2010).
  * Hill and Wootters (1997) S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 (1997).
  * Wootters (1998) W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
  * Wootters (2001) W. K. Wootters, Quantum Inf. Comput. 1, 27 (2001).
  * Dennison and Wootters (2001) K. A. Dennison and W. K. Wootters, Phys. Rev. A 65, 010301 (2001).