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{\title\baselineskip=1.728\normalbaselineskip\rightskip=0pt plus1fil
\noindent Computing Bounds on Critical Indices \par}
\vskip.6in
\noindent
Hans Koch$^{\!}$
\footnote{$^1$}
{
{\small Supported by the
National Science Foundation under Grant No. DMR--85--40879.}
}
\HB
\noindent University of Texas at Austin, Department of Mathematics\HB
\noindent Austin, TX 78712
\HB
\vskip.25in
\noindent
Peter Wittwer$^{\!}$
\footnote{$^2$}
{
{\small Supported by the
National Science Foundation under
Grants No. DMS--85--18622 and DMS--87--03539.}
}
\HB
\noindent Rutgers University, Department of Mathematics
\HB
\noindent New Brunswick, NJ 08903
\vskip .8in

\abstract
We use an extended renormalization group map
and computer assisted techniques
to get sharp bounds on the critical exponent
which describes the scaling behavior of the
free energy near a hierarchical
infrared fixed point.
\vfill
\eject

\section Introduction %===========================================

In this paper we
give an outline of a computer assisted proof
in which we use
an extended version of
the Wilson-Kadanoff
renormalization group scheme
to get rigorous bounds on a critical exponent
that is universal
for a class of one parameter families
$F_\mu$
of hierarchical lattice
systems in $d=3$ dimensions.
The parameter $\mu$ will be referred to as temperature.
Assume that a  given family $F_\mu$ undergoes a second order phase
transition at $\mu=\mu_c$, then we define $\kappa$ to be
the exponent that describes the scaling behavior of the
free energy
as the temperature
approaches the critical value $\mu_c$.
More precisely, let $U_n(\mu)$ be the free energy density
coresponding to the system $F_\mu$ confined to
a cube of volume $2^{dn}$ at temperature
$\mu$,
then
$$
\kappa=\lim_{\mu\to \mu_c}
{1\over\log(|\mu-\mu_c|)}\log\bigl(\lim_{n\to\infty} U_n(\mu)\bigr).
\equation(100)
$$
>From a renormalization group analysis this quantity is hard to compute
since it requires global bounds on the renormalization group
transformation.
A possible strategy is
to show that the double limit
in \equ(100) can be replaced by a scaling limit along
certain trajectories $n\mapsto
\mu_n$, i.e. that
$$
\kappa=\kappa_s\equiv\lim_{n\to\infty}
{1\over\log(|\mu_n-\mu_c|)}\log(U_n(\mu_n)).
\equation(100a)
$$
The limit $\kappa_s$ can be computed from local quantities only.
We assume here the validity of this exchange of limits, 
i.e. we assume that $\kappa=\kappa_s$ for our systems,
and concentrate on the task of getting sharp bounds
for $\kappa_s$.
For comparison with existing results we will use
the more common index $\nu$ [5], assuming the correctness of
the scaling relation
$$
\nu=\kappa/d.
\equation(100b)
$$
We obtain our results
by an application of the method
which has been developed in [2] to the class of hierarchical
models which is discussed in [1].
We refer to [1] for the definition of the models 
and for motivations, and restrict ourselves here
to explain the techniques which 
have been used to get the desired bounds.

Let $C$ be the set of bounded continuous functions on
$\real^+=\real\setminus(-\infty,0)$.
In [1] we have studied
for $f\in C$, $f(0)\ne0$,
the renormalization group map
$\NN_{\alpha\beta}$, which is defined through the
equation
$$
\bigl(\NN_{\alpha\beta} f\bigr)(t^2)=
{1\over c}\I_{-\infty}^\infty ds\exp(-\beta^2 s^2)
[f((\alpha t+s)^2)]^\ell,
\equation(101)
$$
and we have been interested in finding fixed points for this map.
The normalization  constant $c=c(f)$ in \equ(101) is chosen  such that
$\bigl(\NN_{\alpha\beta} f\bigr)(0)=1$, i.e.
$$
c=
\I_{-\infty}^\infty
ds \exp(-\beta^2 s^2)[f(s^2)]^\ell.
\equation(102)
$$
The interpretation of $\alpha$, $\beta$ and $\ell$ is as follows.
Let $L$ be the
side length of blocks in
$\integer^d$ over which
the spins are averaged in order to get the renormalization group
map \equ(101), then
$\ell=L^d$ is the number of spins in a block and
$\alpha$ is the rescaling of the spin variables associated with such
a block spin transformation.
For a field with
canonical dimension  on has
$\alpha=L^{-(d-2)/2}$. We choose here $L=2$, and therefore $\ell=8$ and
$\alpha=1/\sqrt 2$.
$\beta$ is just a normalization constant:
if $f$ is a fixed point of $\NN_{\alpha\beta}$, then
the function $f_\beta=f(\beta\>\cdot)$ is a fixed point of $\NN_{\alpha 1}$.
Without loss of generality we can therefore choose $\beta=1$.
\footnote{$^{1)}$}{\small
The recursion relation \equ(101)
can also be studied for
noninteger values of
$\scriptstyle L$ and for fields with
noncanonical dimension.
A numerical study shows, that
he critical index $\scriptstyle \nu$
depends, as expected,
on
the dimension of the field. More carefull
studies show, that it
also depends on the block size
$\scriptstyle L$, for fixed dimension of the field.
This result seems to contradict the universality dogma at first sight.
However, one can understand this effect
in terms of the formulation which we
have given in [1]. There, the recursion
relation \equ(101) is shown to be exact
for spin systems on a cubic lattice
with a
$\scriptstyle L$--dependent
hierarchical symmetry, and models
with different symetries usually do have different
critical behaviour.

It is interesting to compare our results with those obtained
in [3,4],
where
Wilsons hierarchical model --- which differs from the one we
study here --- has been analyzed.
In these references the value
$\scriptstyle L=2^{1/3}$ is used.  We find for this case
numerically that
$\scriptstyle\nu=0.6495704\ldots$ which is
in very good agreement with the values
$\scriptstyle\nu_S=0.649\ldots$ and
$\scriptstyle\nu_E=0.6492\ldots$ given in [3] and [4].
(As a comparison: the best numerical value for
the case of the Ising model is
$\scriptstyle\nu_I=0.638\ldots$.)
Wilson studied originaly his model for
$\scriptstyle L=2$ [6]. For this
case  we prove here rigorously that
$\scriptstyle\nu=0.6501625\ldots$. This value
differs considerably from the number
$\scriptstyle\nu_W=0.609\ldots$ given by Wilson [6] and by Golner [7].
We have no explanation for such a disagreement.
\vfill
}


It is easy to see that the two functions $f_{tr}(t^2)=1$ and
$f_{ht}(t^2)=\exp(-{3\over8 }t^2)$ are fixed points of $\NN_{\alpha 1}$.
$f_{tr}$ is the trivial fixed point. It corresponds to the case
of a noninteracting  massless lattice field theory.
$f_{ht}$ is a massive
high temperature fixed point. The fact that its mass is finite
and not infinite is a
particularity of the renormalization group map \equ(101)
and is not shared by other maps. The existence
of such an additional massive gaussian fixed point is essential
for our analysis:
we use it to get an a priory bound on the large field behavior of the
nontrivial fixed point
of \equ(101).
Assume  therefore that $f^*=\NN_{\alpha 1}f^*$
and that
$f^*(t)$  decays
faster than $\exp(-\eps t)$ as $t\to+\infty$, for some
$\eps>0$,
then by iterating \equ(101) one gets
for every $\delta>0$
the stability bound
$$
C_1\exp(-({3\over8}+\delta)\>t^2)
\le
f^*(t^2)
\le
C_2\exp(-({3\over8}-\delta)\>t^2),
\equation(103)
$$
with $0<C_1\le C_2<\infty$.

In [1] we have already proved that there is a nontrivial
fixed point $f^*\in C$, and we have shown
that $f^*$
extends to a function which is
entire analytic.
Here, we improve now our analysis and show that
the fixed point $f^*$ is
hyperbolic
in a appropriately chosen Banach  space
of analytic functions,
with  a splitting of the tangent space into
a one dimensional unstable direction and a codimension one stable
direction.
As mentioned above,
we apply
the method which has been
developed in [2] to prove the theorems of this paper.
The idea is to use an extended renormalization group
transformation,
which has a
whole piece of the one dimensional unstable
manifold of $f^*$
as its
fixed point.
One aspect of this particular way of doing the analysis is ---
and this might not have been stressed
enough in [2] --- that very precise bounds for the critical exponent
which is associated with this unstable direction can
be obtained.

By comparing the methods in this paper with those of  [1]
the reader will find that
we have been improving
considerably our method of dealing with the large field problem.
The method is formulated now in such a way that a
generalization to
the physically interesting case of translation invariant models
seems feasable.

Before we define the extended renormalization group map we
recall some more details about the
operator $\NN_{\alpha 1}$.
An important  technical step is to choose coordiantes in function space
which take advantage of the a priory bound \equ(103).
Define $C^+$ do be the space of continuous functions $h$
of the form:
$$
h(t^2)=
\bigl(\HH f\bigr)(t^2)=\exp({3\over8}t^2)f(t^2)\,
\equation(104)
$$
for $f\in C$.
Clearly
$\HH:C\to C^+$
is one--to--one.
$\NN_{\alpha 1}$ induces on $C^+$ the map
$\hat\NN=\HH\NN_{\alpha 1}\HH^{-1}$.
An easy computation shows that, for $h\in C^+$, $h(0)\ne0$,
$\hat\NN$ is given by the equation
$$
\bigl(\hat\NN h\bigr)(t)=
{1\over c}\I_{-\infty}^\infty ds\exp(-L^2 s^2)
[h(({\alpha\over L^2} t+s)^2)]^\ell.
\equation(105)
$$
where  $c=c(h)$ is chosen such that
$\bigl(\hat\NN h\bigr)(0)=1$.
A comparision with \equ(101) shows that
$\hat\NN=\NN_{\alpha/L^2 L}$, i.e.\equ(101) is form invariant under the change
of coordinates \equ(104).
In
particular the replacement of $\beta=1$
by $\beta=L$ is just a change of a normalization as we have shown above,
and is therefore irrelevant.
The change from $\alpha$ in $\NN_{\alpha 1}$
to ${\alpha/L^2}$ in $\hat\NN=\NN_{\alpha/L^2 L}$ is however crucial.
The reason
is, that the maps $\NN_{\theta \beta}$  behave
in a certain sense pritty much like the linear
substitution operator  $L_\theta:\zeta(t)\mapsto \zeta(\theta t)$.
(This can be seen by
replacing in \equ(101) and \equ(105) the convolution with the
exponential function by a convolution with a delta function.) On
functions  $\zeta$ which are analytic near zero $L_\theta$ is analyticity
improving for $\theta<1$.
For $\theta$ small enough
this
property carries over to the linearization of $\NN_{\theta \beta}$,
and makes an application of the contraction
mapping principle possible. We will use that method below
to prove our main theorem.

An other
advantage of the new coordinates is, that the stability bound \equ(103)
becomes more transparent.
Let $h^*=\HH f^*$ bet the nontrivial fixed point of $\hat\NN$, then for
every $\delta>0$ there is a constant $C_0>0$
such that for all $t\in\real$,
$$
h^*(t^2)
\le
C_0\exp(\delta\>t^2).
\equation(106)
$$
This property will guide us in our choice
of function space.



\section Results%=================================================


The extended renormaliztion group map  $\RR$ acts on functions $\Phi$
which are jointly analytic in a spin variable $t$ and in the
parameter $\mu$ which
characterizes the temperature of the system. The choice
of $\RR$
is dictated by the following considerations.
Let $h^*=\hat\NN h^*$ be the nontrivial fixed point of
$\hat\NN=\NN_{\alpha/L^2 L}$ discussed above, and let $h_\mu^*$ be
a parametrization of the unstable manifold of $h^*$,
such that the action of $\hat\NN$ becomes linear, i.e.
$\hat\NN h^*_\mu=h^*_{\delta^*\mu}$, where
$\delta^*=2.904071498\dots$ is the eigenvalue
associated with the unstable direction of $h^*$.  It
is then easy to write down an operator $\RR_0$ which has
the whole  one parameter family $h_\mu^*$ as its fixed point.
Namely, one could define
$\RR_0h^*_\mu=\hat\NN h^*_{\mu/\delta^*}$.
The situation is slighly
more complicated if
the unstable manifold
$h_\mu^*$ is known, but not the fact
that $h_{\mu^*}^*$
is a fixed point for
$\mu^*=0$.
The action of $\hat\NN$ would then be
$\hat\NN h^*_\mu=h^*_{\mu^*+\delta^*(\mu-\mu^*)}$,
with $\mu^*$ do be determined. This motivates to define the operator
$\RR$ by the equation
$$
\bigl(\RR \Phi\bigr)(t,\mu)=
{1\over c(\mu/\delta)}
\I_{-\infty}^{\infty}
ds\exp(-L^2 s^2)\>
[\Phi\bigl(({\alpha\over L^2}t+s)^2,\mu_0+\mu/\delta\bigr)]^\ell.
\equation(201)
$$
Here $c(\mu)=c(\mu,\Phi)$, $\delta=\delta(\Phi)$
and  $\mu_0=\mu_0(\Phi)$ are to be determined from normalization
conditions which will be given below.
For convenience later on we decompose now $\RR$ into
a product of simpler operators.
We define for these matters for a given function $\Psi$,
$\Psi(t,\mu)=\sum_{ij\ge0}\Psi_{ij}t^i\mu^j$, the real function
$m$ by the equation
$$
m(\Psi)\!=\!{1\over2}(\Psi_{10}+\Psi_{20}),
\equation(202)
$$
and for $\mu_0\in\real$, $\mu_0$ small, the operators $S_{\mu_0}$
by the equations
$$
\bigl(\SS_{\mu_0}\Psi\bigr)(t,\mu)=\Psi(t,\mu_0+\mu),
\equation(203)
$$
and finally the extension $\NN$  of the operator $\hat\NN$ by the equation
$$
\bigl(\NN \Psi\bigr)(t,\mu)={1\over c(\mu)}
\I_{-\infty}^{\infty}
ds\exp(-L^2 s^2)\>
[\Psi\bigl(({\alpha\over L^2}t+s)^2,\mu\bigr)]^\ell.
\equation(204)
$$
Here,
$$
c(\mu)=c(\mu,\Psi)
=\I_{-\infty}^{\infty}
ds\exp(-L^2 s^2)\>
[\Psi\bigl(s^2,\mu\bigr)]^\ell,
\equation(205)
$$
so that
$\bigl(\NN \Psi\bigr)(0,\mu)=1$.

\bigskip\noindent
$\RR$ is now given by the following
sequence of steps.
\HB
a) Define
$\mu_0^*$ as the  solution of the equation
$$
m(\NN\SS_{\mu_0^*}\Phi)=m(\SS_{\mu_0^*}\Phi).
\equation(206)
$$
\hfill
\break
b) Write
$\SS^*$ for
$\SS_{\mu_0^*(\>.\>)}$ and
$\NN^*$ for
$\NN\SS^*$.
\hfill
$\tag(207)$
\break
c) Define, for $\Psi=\NN^*\Phi$, $\delta$ by the equation
$\delta=\Psi_{11}$.
\hfill
$\tag(208)$
\break
d) Define
$\bigl(\RR \Phi\bigr)(t,\mu)=\bigl(\NN^*\Phi\bigr)(t,\mu/\delta)$
\hfill
$\tag(209)$
\break

Note, that with the definition of $\delta$
given in \equ(208), and with the normalization  \equ(205)
for $\NN$,
we get
for any $\Phi$ on which $\RR$ is defined,
the following normalizations for
$\Psi=\RR\Phi$.
$$
\Psi(0,\mu)=1,
\equation(210)
$$
$$
\bigl(\partial_1\partial_2\Psi\bigr)(0,0)=1.
\equation(211)
$$

For completeness we also specify here the action of the tangent map
$D\RR_\Phi$ of $\RR$ evaluated at $\Phi$ and acting on $\var\Phi$.
($^\bullet$ denotes the partial
derivative with respect to the second argument.)
$$
\eqalignno{
&
\eqalign{
\bigl(D\RR_\Phi\var \Phi\bigr)(t,\mu)
&=\bigl(D\NN^*_\Phi\var \Phi\bigr)(t,{\mu \over \delta})+
\bigl(\NN^*\Phi\bigr)^\bullet(t,{\mu\over\delta}){-\mu\over\delta^2}\var\delta
\cr
&=\bigl(D\NN^*_\Phi\var \Phi\bigr)(t,{\mu\over\delta})+
\bigl(\RR \Phi\bigr)^\bullet(t,\mu){-\mu\over\delta}\var\delta,
\cr
}
&\tag(212)
\cr
&\bigl(D\NN^*_\Phi\var \Phi\bigr)(t,\mu)=\bigl(D\NN_{\SS^*\Phi}
D\SS^*_\Phi\var \Phi\bigr)(t,\mu),
&\tag(213)
\cr
&
\eqalign{
\bigl(D\SS^*_\Phi\var \Phi\bigr)(t,\mu)
&=\bigl(\SS^*\var \Phi\bigr)(t,\mu)
+\bigl((\partial_{\mu_0}
\SS_{\mu_0})\big|_{\mu_0=\mu^*_0(\Phi)}\Phi\bigr)(t,\mu)\var\mu^*_0
\cr
&=\bigl(\SS^*\var \Phi\bigr)(t,\mu)+\bigl(\SS^*\Phi^\bullet\bigr)
(t,\mu)\var\mu_0^*,
\cr
}
&\tag(214)
\cr
&
\eqalign{
\bigl(D\NN_\Phi\var \Phi\bigr)(t,\mu)=
&
{\ell\over c(\mu)}\I_{-\infty}^\infty
\exp(-L^2s^2)
[\Phi\bigl(({\alpha\over L^2}t+s)^2,\mu\bigr)]^{\ell-1}
\var \Phi\bigl(({\alpha\over L^2}t+s)^2,\mu\bigr)ds
\cr
&-{\var c(\mu)\over c(\mu)}\bigl(\NN \Phi\bigr)(t,\mu).
\cr
}
&\tag(215)
\cr
}
$$
Here,
$\var c=\var c(\mu,\var\Phi)$
and
$\var\delta=\var\delta(\var\Phi)$
are defined such that the
normalizations \equ(210) and \equ(211) remain unchanged
by the variation $\var \Phi$. It follows that
$\varphi=D\RR_\Phi\var\Phi$ is normalized according to
$$
\varphi(0,\mu)=0,
\equation(216)
$$
$$
\bigr(\partial_1\partial_2\varphi\bigr)(0,0)=0.
\equation(217)
$$
Furthermore, in \equ(214), $\var\mu^*_0=\var\mu^*_0(\var\Phi)$ is given
by the equation
$$
\var\mu^*_0=-{
m(D\NN_{\SS^*\Phi}\SS^*\var \Phi-\SS^*\var \Phi)
\over
m(D\NN_{\SS^*\Phi}\SS^*\;\Phi^{\!\bullet}-\SS^*\;\Phi^{\!\bullet})
}.
\equation(218)
$$

\smallskip
In order to complete the definition of the operator $\RR$
we define now the  function spaces with
which we will work below.


\claim Definition(201) Given two positive numbers
$\rho_1$ and $\rho_2$, we define $\AA_{\rho_1\rho_2}$ to
be the Banach space of all functions $\Phi$,
$\Phi(t,\mu)=\sum_{ij\ge0}\Phi_{ij}t^i\mu^j$
which are analytic on
the domain $D_{\rho_2}\subset\complex^2$,
$$
D_{\rho_2}=\complex\times\bigl\{|\mu|<\rho_2\bigr\},
\equation(219)
$$
which take real values for real arguments $t$ and $\mu$,
and for which the norm
$$
\|\Phi\|_{\rho_1\rho_2}=\sum_{ij\ge0}|\Phi_{ij}|
\cdot (i!\cdot \rho_1^i)\cdot\rho_2^j
\equation(220)
$$
is finite.
Furthermore, we define
$\AA=\AA_{\rho_t\rho_\mu}$, where $\rho_t=3$ and $\rho_\mu=0.01$.

Let $\Phi\in\AA_{\rho_1\rho_2}$, then we can write
$\Phi(t,\mu)=\sum_{j\ge0}h_j(t)\mu^j$, where the functions $h_j$
are elements of the spaces $\EE_{\rho_1}$ which we now define.
\smallskip

\claim Definition(202) For each $\rho>0$, we define ${\cal E}_\rho$ to be the
Banach space of all functions $h$, $h(t)=\sum_{i\ge0}h_i t^i$
which are entire analytic ,
which take real values for real arguments $t$,
and for which the norm
$$
\|h\|_\rho^\EE=\sum_{i\ge0}|h_i|\cdot (i!\cdot \rho^i)
\equation(221)
$$
is finite. Furthermore, we define
$\EE=\EE_{\rho_t}$.



To understand the choice of the norms \equ(220) and \equ(221)
consider a function
$h(t)=\sum_{i\ge0}h_i t^i$,
for which
$\|h\|^\EE_\rho$
is finite.  Such a function is entire analytic, and it
satisfies, for  $t\in\complex$ the inequality
$$
|h(t)|\le
\sum_{i\ge0}|h_i|\cdot|t|^i\le
\sum_{i\ge0}(|h_i|\cdot i!\cdot\rho^i)\>
{(|t|/\rho)^i\over i!}
\le
\|h\|^\EE_\rho\cdot
\exp(|t|/\rho).
\equation(223)
$$
This shows that
the growth rate of
functions for which the norm \equ(221) is finite
is bounded by $1/\rho$.
In our definition of $\AA$ we have chosen the inverse growth rate
$\rho_t$ such, that for a function $\Phi\in\AA$, and fixed $\mu$,
the function
$f=\HH^{-1}\Phi(\>.\>,\mu)$
is exponentially decreasing along  the positive real
axis, and is therefore within the class
of functions  for which we have the stability bound \equ(103).




\bigskip
The following theorem is our main result.

\claim Theorem(204) There is a ball $\BB\subset\AA$
on which $\RR$ is defined
and once continuously differentiable.
For each $\Phi\in\BB$,
the Fr\'echet derivative $D\RR_\Phi$ of $\RR$ at $\Phi$
is a compact operator on $\AA$, whose spectrum is contained inside
the unit disk.
$\RR$ maps $\BB$ into
itself and has a fixed point
$\Phi^*\in\BB$. The number
$\delta^*=\delta(\Phi^*)$
satisfies the bound
$\delta^*\in[ 2.9040714905322,2.9040715072204]$.

The proof of this theorem is outlined in the next section.
\bigskip

Consider now the one parameter family
of maps $h^*_\mu=\Phi^*(\>.\>,\mu)$.
For $-\rho_\mu<\mu<\rho_\mu$
we have that $h^*_\mu\in\EE$.
We have the following
corollary.

\claim Corollary(205)
There is a ball
$\hat\BB\subset\EE$
on which the operator $\hat\NN$  is defined
and once continuously differentiable.
For
$|\mu|<\rho_\mu/\delta^*$
the action of
$\hat\NN$ is given by
$\hat\NN h^*_\mu=h^*_{\delta^*\mu}$.
$h^*=h^*_0$ is a fixed point of $\hat\NN$.
The Fr\'echet derivative $D\hat\NN_{h^*}$, of $\hat\NN$ at $h^*$
is a compact operator on $\EE$. Its spectrum is contained inside
the unit disk with the exception of the simple eigenvalue
$\delta^*$.
$h^*$ is hyperbolic with
a splitting of its tangent space into a one
dimensional expanding and a codimension one contracting
direction.
The function $\var h^*=$$\partial_\mu\Phi^*(\>.\>,0)$ is
an eigenvector
of $D\hat\NN_{h^*}$
with eigenvalue $\delta^*$.

\proof All the statements of this corollary are immediate consequences
of the definition of the operator $\RR$
and of \clm(204) with the exception of the fact,
that $h_0^*$ is a fixed point of $\hat\NN$,
i.e.
that $\mu_0^*$, as
defined through \equ(206), is zero at the fixed point $\Phi^*$. To see this
we have to discuss briefly how \equ(206) is solved.
We define a function $\Delta:\real\to\real$
$$
\Delta(\mu_0)=m(\NN\SS_{\mu_0}\Phi)-m(\SS_{\mu_0}\Phi),
\equation(211b)
$$
and we prove that this function is zero for some value of its argument.
This proof is given
by using Newtons method:
we show that the map $F:\real\to\real$
$$
F(\mu_0)=\mu_0-{\Delta(\mu_0)\over\Delta'(\mu_0)}
\equation(211c)
$$
is a contraction of a small intervall into itself, which proves that $F$ has a
unique fixed point there.
Equivalently we conclude, that
the function $\Delta$ has a unique zero in this interval,
or equivalently that
\equ(206) has
a unique solution in this interval.
In our computer proof this interval
contains $\mu_0=0$.
Now, from \equ(206) and the definition \equ(202) of
the function $m(.)$ it follows, that
$$
m(\RR\Phi)=m(\SS^*\Phi),
\equation(211a)
$$
for any function $\Phi$ on which $\RR$ is defined. In particular
at the fixed point $\Phi^*=\RR(\Phi^*)$ of
$\RR$
we get that $m(\Phi^*)=m(\SS^*\Phi^*)$.
This equation is satisfied for
$\mu_0^*=0$. Therefore, since the solution
of \equ(206) is unique, we have that
$\mu_0^*(\Phi^*)=0$.
\bigskip

\claim Corollary(206)
Each of the one parameter families
$h_\mu=\Phi(\>.\>,\mu)$ in the domain
of $\RR$ intersects the stable manifold of
$\Phi^*$ transversally for some critical
value $\mu_c$ of the temperature.
If $\nu$ is the index defined in
\equ(100a) and \equ(100b),
evaluated in the scaling limit
$\mu_n=\mu_c+\mu_c+k\delta^*)^{-n}$,
then
$\nu=$$\log(L)/\log(\delta^*)$$\in[0.65016251767896,0.65016252118341]$.
for $k$ any nonzero constant.

\proof
For
the particular one parameter family
of maps $h^*_\mu=\Phi^*(\>.\>,\mu)$
for which $\mu_c=\mu_0^*(\Phi^*)=0$,
the identity
$\nu=$$\log(L)/\log(\delta^*)$
using the relations
$$
Z_n(\mu)=Z_{n-1}(\mu\delta^*)
\equation(316a)
$$
satisfied by the partition functions $Z_{n}$
associated with cubes of volume $2^{d(n+1)}$.
The equation \equ(316a) has been proved in [1] for $\mu=0$,
but the same argument applies for arbitrary $\mu$ and
n sufficiently large.
In the general case
one uses an equation similar to
\equ(316a). However
if $Z_n$ is a partition function associated with
a family
$h_\mu$ then
$Z_{n-1}$ is associated with
the family
$\widetilde h_\mu=\RR(\Phi(\>.\>,\delta\mu))$.
The convergence of the sequence
$Z_n(\mu_n)$ follows from the fact that
$\RR$ is a contraction
with uniformly bounded Fr\'echet derivative.
The details of this computation are left to the
reader.




\section Method of Proof%=========================================

In this section we give
an outline of the proof of \clm(204). Details of the proof,
including the listing of the computer program will
appear elsewhere [8].
First, because a function  $\Psi$ which is
in the range of $\RR$ is normalized
according to \equ(210) and \equ(211),
we can restrict our search for a fixed point
to such functions.
A normalized function  $\Psi$ is of the form
$$
\Psi(t,\mu)=1+t\mu+\varphi(t,\mu)
\equation(301)
$$
with $\varphi$ satisfying \equ(216) and \equ(217).

\claim Definition(301) We define
$\AA_H$ and $\AA_T$
to be the sets of functions
$\Phi\in\AA$ and $\varphi\in\AA$
which satisfy the normalization conditions \equ(210), \equ(211) and
\equ(216), \equ(217), respectively.

If we want to prove \clm(204) by an
application of the contraction mapping principle,
then we have to find a ball $\BB_H\subset\AA_H$
on which $\RR$ is defined and differentiable.
Then we have to study
for each $\Phi\in\BB_H$ the linearization
$D\RR_\Phi$ as a map from $\AA_T$ to $\AA_T$. A sufficient condition
for $\RR$ to be a contraction is, that the operator norm
$\|D\RR_\Phi\|$
is smaller than one, uniformly
for $\Phi$ in such a ball $\BB_H$.
Numerical studies show, however,
that this is
not the case.
We cure this problem brute force by replacing
the norm \equ(220)
by an norm with respect to a basis in which
$D\RR_\Phi$ is essentially diagonal.

\claim Definition(303)
Let $I_1$ and $I_2$ be the two sets
$$
I_1=\{(1,0),(2,0),\dots,(9,0)\},
\equation(306)
$$
and
$$
I_2=\{(i,j)\in\integer^2|\>i>0,(i,j)\ne(1,1),(i,j)\not\in I_1\},
\equation(307)
$$
then we define $\Lambda$ to be the
Banach space of all real sequences $(\lambda_{ij})$
indexed by $I_1\cup I_2$,
$$
\Lambda=\{(\lambda_{ij})|(i,j)\in I_1\cup I_2\},
\equation(308)
$$
for which
with the norm
$$
{\|(\lambda_{ij})\|}_1=\sum_{I_1\cup I_2}
|\lambda_{ij}|
\equation(309)
$$
is finite.

\bigskip
\noindent
To each element $(\lambda_{ij})\in\Lambda$
we can now associate
a function $\varphi\in\AA_T$,
by the equation
$$
\varphi(t,\mu)=
\sum_{k=1\dots9}\lambda_{k0}\cdot {\bf e}_k(t)+
\sum_{I_2}\lambda_{ij}
\cdot
{t^i\over\rho_t^i\cdot i!}
\cdot
{\mu^j\over \rho_\mu^j},
\equation(310)
$$
where the functions ${\bf e}_k$ are given by the equation
$$
{\bf e}_k(t)=
\sum_{i=1\dots9}\sigma_{ik}
\cdot
{t^i\over\rho_t^i\cdot i!}.
\equation(312)
$$
Here, $\sigma=(\sigma_{ij})$
is a nonsingular
$9\times9$--matrix
with inverse $\sigma^{-1}$. (See below for the choice of $\sigma$.)
Conversely if $\varphi\in\AA_T$,
$\varphi=$
$\sum_{I_1\cup I_2}\varphi_{ij}t^i\mu^j$,
then there is an element
$(\lambda_{ij})\in\Lambda$
associated with it by the equations
$$
\eqalignno{
&\lambda_{i0}=\sum_{j=1\dots9}\sigma^{-1}_{ij}
\cdot\varphi_{j0}\cdot(\rho_t^j \cdot j!),
\>\>\>\>\>\>i=1\dots9,
&\tag(313)
\cr
&\lambda_{ij}=\varphi_{ij}
\cdot(\rho_t^i \cdot i!)\cdot \rho_\mu^j,
\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>(i,j)\in I_2.
&\tag(314)
\cr
}
$$
These relations induce a new norm on $\AA_T$. Namely,
let $\varphi\in\AA_T$
and let $(\lambda_{ij})$ be defined
through \equ(313) and \equ(314), then we define (missusing the
notation $\|\ \|_1$ slightly)
$$
\|\varphi\|_1=\|(\lambda_{ij})\|_1.
\equation(316)
$$

We now have the following theorem.

\claim Theorem(304) There is a function $\Phi_0\in\AA_H$, such that
\item{a)}
$\RR$ is defined and continuously differentiable on the neighborhood
$\BB_\beta$,
\HB
$\BB_\beta=$$\{\Phi\in\AA_H|\>\|\Phi-\Phi_0\|_1<\beta\}$, where
$\beta=5\cdot10^{-9}$.
\item{b)}
For $\Phi\in\BB_\beta$, the tangent map $D\RR_\Phi$ is
bounded in norm by
$\|D\RR_\Phi\|_1\le\Theta\le0.86<1$,
\item{c)}
$\Phi_0$ is a good approximate fixed point of $\RR$, i.e.
$\|\RR\Phi_0-\Phi_0\|_1\le\epsilon\le2\cdot10^{-11}<(1-\Theta)\beta$.

\proof The proof of this Theorem is obtained by running our
computer program. For details on the method
see [1,2,9,10]. For details on the proof see [8].
\bigskip

As a result of a),$\ldots$,c) there is a unique fixed point
$\Phi^*$ of $\RR$ in
$\BB_\beta$.
Our computer program also provides the bound
on $\delta^*$ in \clm(204).
\bigskip
\bigskip


\noindent
{\sub{Acknowledgements}}
\smallskip
\noindent
One of the authors (P.W.) would like to acknowledge
NSF support under Grant No.  DMS--85--18622
for the use of the Pittsburgh
supercomputer center, where part of the numerical work as well as runs
of the complete proof have been carried out
on a CRAY--XMP.
\bigskip
\bigskip

\noindent
{\sub{References}}
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A Complete Proof of the Feigenbaum Conjectures.
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\smallskip
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\smallskip
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\smallskip
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\end
