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\vskip 40truemm
\centerline{\bf DYNAMICAL INTEGRAL TRANSFORM ON FRACTAL SETS}
\centerline{\bf AND THE COMPUTATION OF ENTROPY}
\bigskip

\centerline{J.-M. Ghez\footnote{$^1$}{and PHYMAT, D\'epartement de
Math\'ematiques, Universit\'e de Toulon et du Var, B.P. 132, 83957 La Garde
Cedex, France.}, E. Orlandini\footnote{$^2$}{Dipartimento di Fisica,
Universit\`a di Bologna - Italy and INFN - Sezione di Bologna}, M.-C. Tesi$^2$
and S. Vaienti$^{1,}$\footnote{$^3$}{Supported by Contrat CEE n$^{\circ}$
SC1$^{\star}$0281.}}

\vskip 2truecm
\noindent{\bf Abstract} : We introduce an integral transform of wavelet type,
which we call Dynamical Integral Transform, and we show that it can be used to
compute the 2-nd Renyi entropy for a large class of invariant measures. The
method is then generalized to the whole spectrum of the Renyi entropies and
establishes a correspondance between thermodynamic formalism and the Dynamical
Integral Transform of expanding strange sets. Numerical examples are presented.

\bigskip

\vskip 4cm


\n {July 1992}

\noindent CPT-92/P.2663 (revised version)

\noindent To appear in Physica D

\vfill\eject

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\noindent{\bf 1 - Introduction}
\medskip
The wavelet transform and the methods derived from it have revealed to be
powerful tools to analyze fractal sets. In particular they give a complete
description of multifractal measures in two different ways : by a local
analysis of the scaling exponents of the measure [1-3] or through the
computation of the spectrum of the generalized dimensions [4-6].

The same methods have been succesfully extended to extract histograms of
scaling exponents from fully developed turbulence data [7] .In this
article we show that they can be extended to compute the whole spectrum  of
the Renyi entropies. We  recall in section 2 that the Renyi entropies
completely characterize the dynamical properties of an ergodic measure;
we also indicate in  section 5 how to perform a local analysis with a
suitable integral transform in order to detect the entropy at a point $x$ as
given by the Brin-Katok theorem.

The approach we follow is taken from our previous works [4][5]; we defined there
the following integral transform:
$$T_p(a,\mu)=a^{-p}\build{\int\int}_{J\times J\ }^{}g\left ({||x-y||\over
a}\right ) d\mu(x)d\mu(y)\eqno (1)$$
where : $\mu$ is a probability non-atomic measure, $a$ and $p$ are positive
numbers, $J$ is the set supporting $\mu$ and $||.||$ is some distance on $J$. We
called (1) Integrated Wavelet Transform for the close analogy with the usual
wavelet transform. Our principal assumption for the function g [2][4][5] was
that it is in $C^1({\bf R})$ and rapidly decreasing (more than any power) on
the positive semi-axis, in particular : $\build{\lim}_{a\rightarrow 0^+}^{}
a^{-p}g\left ({r\over a}\right )=0$ pointwise for $r>0$ and $p\geq 0$.

When rigorous proofs are carried out on invariant sets $J$, we showed that it
is sufficient to assume $g$ monotone on ${\bf R}^+$; we never need admissible
analyzing  functions $g$ of zero mean (wavelets): we discussed this question
in details in the introduction of [4][5]. There are some other
advantages using our test functions: first, we are able to get some asymptotic
local properties (see section 5 in [5] and 5 in this paper) which are not
immediately recovered with admissible wavelets. Then we showed, in the context
of the generalized dimensions [5], that the monotone analyzing functions work
numerically better than the usual wavelets like the mexican-hat: they are also
used in [8] \footnote{$^{1}$}{``As recently addressed in various
theoretical studies (...), the wavelet analysis of singular measures do not
require the analyzing wavelet $g$ to be of zero mean. In the present study, we
will use a Gaussian function $g(r)=e^{-r^2}$.'' [8], p.4}. Moreover the choice
for monotone wavelets was explicitly assumed by Falconer [9] \footnote{$^{2}$}
{``Choice of a suitable (wavelet) $w$ depends on the purpose for which the
wavelet transform is used. There is  considerable divergence between authors
as to the conditions that $w$ ought to satisfy-for example, some require
certain moments of $w$ to vanish, while others specify rapid decrease at
infinity.''[9], p.781}, in connection with order-two densities of certain
fractal measures. Despite these facts and to  avoid ambiguities, we think that
the name {\it wavelets} could be reserved to the admissible wavelets introduced
in the study of signals [10]. Therefore we call our $g$
{\it analyzing functions} or {\it test functions} and the integral (1) {\it
Fractal Integral Transform} or {\it FIT}.
\medskip
A basic concept in our method is that of adapted analyzing function. We say
that $g$ is $p${\it -adapted} {\it to} $\mu$ {\it for the FIT} if
$\build{\lim}_{a\rightarrow 0^+}^{}\left ({\sup\atop\inf}\right )\left \vert
T_p(a,\mu)\right\vert$ are different from $0$ and $+\infty$.

In this case, we proved in [4] that, for the class of Gibbs measures
$\mu_{\beta}$ on disconnected conformal mixing repellers, $p$ is equal to the
generalized dimension of order $2$ of the measure $\mu_{\beta}$, denoted with
$D_2(\mu_{\beta})$, defined by the usual partition function approach (as a
root of the topological pressure or free energy). This result has then been
generalized to the whole spectrum of the generalized dimensions
$D_q(\mu_{\beta})$ [5]. In [4][5] we checked the adaptedness for smooth sets
in ${\bf R}^n$, for the ternary Cantor set and we presented an argument,
relating the FIT to the correlation integral introduced by Grassberger,
Hentschel and Procaccia [11][12], which formally establishes the adaptedness
of the FIT for a large class of multifractal sets. This last argument is also
illustrated in section 2 for the dynamical integral transform we are going to
introduce. It is useful to remark that if the adaptedness partially fails, in
the sense that at least one, but not both, of the previous limits is zero or
infinity, the equalities in the theorems in [4] and [5] and in those presented
in this paper, must be replaced with bounds, whose direction can be easily
deduced from the proofs.
\medskip
The paper is organized as follows. In section 2 we collect the definition and
some basic properties of the Renyi entropies and we introduce the
Dynamical Integral Transform. In section 3 we show how to use it to compute
the $2$-nd Renyi entropy in the case of mixing repellers  (Theorem 1) and then
we generalize this result to any $q$-Renyi entropy (Theorem 2).
In section 4 we present some examples. In section 5, we suggest a
local analysis to detect the local entropy. The conclusions are in section 6.

\bigskip
\noindent{\bf 2 - The Renyi entropies and the Dynamical Integral Transform }
\medskip
We now briefly recall the definition of the Renyi entropies  for invariant
sets $J$ (see for example [13-16]). Let ${\cal P}$ be a
$\mu$-measurable partition of $J$ and ${\cal P}^{(n)}$ the dynamical partition
obtained intersecting all the sets of the form $P_{\alpha_0}, T^{-1}
P_{\alpha_1},...,T^{-n}P_{\alpha_n}$, where $P_{\alpha_i}$ is an element of
${\cal P}$ and $T^{-i}$ denotes the preimage of order $i$ of the mapping $T$
generating the invariant set $J$. If the diameter of any element $P^{(n)}_i\in
{\cal P}^{(n)}$ goes to zero when $n\rightarrow \infty$, we say that the
partition ${\cal P}$ is generating and we define the $q$-order Renyi entropy
of any invariant probability measure $\mu$ on $J$ as :
$$K_q(\mu)=-\build{\lim}_{n\rightarrow\infty}^{} {1\over n}{1\over (q-1)} \log
\sum_{P^{(n)}_{\alpha}\in {\cal P}^{(n)}} \left [\mu(P^{(n)}_{\alpha})\right
]^q\eqno (2)$$

The Kolmogorov-Sinai metric entropy $K_1(\mu)$ is recovered in the limit
$q\rightarrow 1$, while the topological entropy is recovered in the limit
$q\rightarrow 0$. For the class of Gibbs measures on the invariant sets of
Markov maps and for the hyperbolic iterated function systems (IFS), the
$q$-entropies can be expressed in terms of the topological pressure [14] [17].
For the first class of systems we have [17] :
$$K_q(\mu_{\beta})(q-1) = qP(\beta)-P(q\beta)\eqno (3)$$
where $\mu_{\beta}$ and $P(\beta)$ are respectively the Gibbs measure and the
topological pressure corresponding to the potential : $-\beta\log||DT(x)||$
[18], being DT the derivative of the mapping generating the invariant set. For
the IFS generated by linear mappings with contraction rates
$\lambda_1\dots\lambda_s$ equipped with a balanced measure $\mu$ of weights
$p_1\dots p_s,p_1+\dots+p_s=1$, we have [14] [17] :
$$K_q(\mu)(q-1) = -\log \left (p^q_1+\dots+p^q_s\right )\eqno (4)$$

These relations are important because they are the key of the proofs we give
below; moreover they can be generalized to larger classes of hyperbolic
dynamical systems. The possibility to reconstruct the topological pressure
from the Renyi entropies shows the importance of the latters in the
investigations of the dynamical properties of strange sets. From the pressure
one can extract the Lyapunov exponents and the metric entropy of any Gibbs
measure [18]; moreover the Legendre transform of the pressure is the deviation
function of the sum of the positive Lyapunov exponents. At this regard, the
connection between the Renyi entropies and the large deviations for the
Lyapunov exponents and the Kolmogorov entropy are made explicit in [19]. We
also recall that in [15] a Legendre transform relates the set of $K_q(\mu)$ to
a scaling function $S(\gamma)$ which is the topological entropy of the set of
points for which the (positive) Lyapunov exponents converge to $\gamma$. A
similar Legendre transform has also been proposed by Paladin and Vulpiani in
[19] with a different interpretation. The corresponding function
$\hat{S}(\gamma)$ was in fact related to a set of Local Expansion Parameters
(LEP) which are nothing but the local entropies given by the Brin-Katok
theorem. We will return to this point in section 5.
\medskip
There are at least two formal useful characterizations for the $2$-nd entropy
$K_2(\mu)$. The first is in terms of the so-called correlation integral
introduced by Grassberger and Procaccia in [13], namely :
$$C_d(\ell)=\build{\int\int}_{J\times J}^{}\Theta(\ell
-||x-y||_d)d\mu(x)d\mu(y) $$
where $\Theta$ is the Heaviside function and $||x-y||_d$ is the ''dynamic
norm`` :
$$||x-y||_d=\left[\sum_{j=0}^{d-1}||T^jx-T^jy||^2\right]^{1/2}\eqno (5a)$$

It is also possible to use the norm
$$|||x-y|||_d=\build{\max}_{0\leq j\leq d-1}^{} ||T^jx-T^jy||\eqno (5b)$$
which is equivalent to (5a) and even simpler to handle with in the proofs of
the following theorems. We will use the norm (5a) and we will return to (5b)
in the last section. With $d=0$, we sometimes mean the usual norm.

According to Grassberger and Procaccia [13], if $C_0(\ell)$ scales like
$C_0(\ell)\sim \ell^{\nu}$, where $\nu=D_2(\mu)$ is the correlation dimension,
then $C_d(\ell)$ vanishes like $C_d(\ell)\sim\ell^{\nu}f_d(\ell)$, where
$f_d(0)\sim e^{-dK_2(\mu)}$ for $d\rightarrow +\infty$.
\medskip
The second approach is in term of the $d$-order energy integral
$\phi_d(\alpha)$, defined as :
$$\phi_d(\alpha)=\build{\int\int}_{J\times
J}^{}||x-y||_d^{-\alpha}d\mu(x)d\mu(y)$$

It was proved in [20][21] that, for some expanding sets, $\phi_d(\alpha)$ is
a  meromorphic function with the smallest positive pole located at $D_2(\mu)$;
moreover the residue at this pole behaves like $D_2(\mu)e^{-dK_2(\mu)}$ for
large $d$.
\medskip
We now introduce the {\it dynamical integral transform} (DIT), defined as :
$$T_p(a,\mu,d)=a^{-p}\build{\int\int}_{J\times J}^{} g\left ({||x-y||_d\over
a}\right )d\mu(x)d\mu(y)\eqno (6)$$
where $||x-y||_d$ is the dynamic norm (5) : clearly a dynamic $T$ must be
defined on the set $J$ and $\mu$ is meant to be invariant with respect to $T$.
\medskip
We say that the function $g$ is  $p${\it -adapted to} $\mu$ {\it for the DIT}
if for any $d$ :
$$\build{\lim}_{a\rightarrow 0^+}^{}\left({\sup\atop\inf}\right )\left\vert
T_p(a,\mu,d)\right\vert\not = (0,+\infty)\ .$$

It is easy to relate the DIT to the correlation integral $C_d(\ell)$ defined
above; we follow here the same method as presented in section 4 of [4]. A
direct verification (an integration by parts), shows that :
$$T_p(a,\mu,d)=a^{-p}\left [g\left ({r\over a}\right )C_d(\ell )\right
]^{\Delta}_0-a^{-(p+1)}\int_{0}^{\Delta} C_d(\ell)g'\left ({\ell\over a}\right
)d\ell\eqno (7)$$
where $\Delta$ is the diameter of $J$. We neglect the first term in the right
hand side since it vanishes in the limit $a\rightarrow 0^+$, by the rapid
decay of $g$ at infinity. Assuming for $C_d(\ell)$ the scaling [13] :
$$C_d(\ell)\build{\sim}_{{\ell\rightarrow 0^+\atop d\rightarrow
+\infty}}^{}\ell^{\nu}e^{-dK_2(\mu)}$$
where $\nu=D_2(\mu)$ is the correlation dimension, and substituting in (7), we
immediately get:
$$|T_p(a,\mu,d)|\sim a^{\nu-p}e^{-dK_2(\mu)}\int_{0}^{\Delta/a}\ell^{\nu}\vert
g'(\ell)\vert d\ell$$

Since the integral in the r.h.s. surely converges by the fast decay of $g$ at
infinity, we finally have
$$|T_p(a,\mu,d)|\build{\sim}_{{a\rightarrow 0^+\atop d\rightarrow +\infty}}^{}
a^{\nu-p}\quad e^{-dK_2(\mu)}\eqno (8)$$.
\medskip
This heuristic argument shows that the class of test functions considered in
this paper is adapted for the measures satisfying the preceding scaling for
the correlation integral and therefore gives the correct value for the 2-nd
Renyi entropy. We give now a rigorous proof of the scaling (8).

\bigskip
\noindent{\bf 3 - Rigorous results}
\medskip
In this section we restrict ourselves to disconnected conformal mixing
repellers endowed with Gibbs measures and to the disconneected attractors of
hyperbolic iterated function systems endowed with balanced measures (see [4]
for a review of the properties of these systems). For these dynamical systems,
we can prove the following result :
\bigskip
\noindent{\bf Theorem 1:}
\item{}{\it If the function $g$ is  $p$-adapted to $\mu$ for the FIT, in which
case $p=D_2(\mu)$, then it is  $p$-adapted to $\mu$ for the DIT and moreover
:}
$$\build{\lim}_{d\rightarrow +\infty}^{} -{1\over d} \log
\build{\limsup}_{a\rightarrow 0^+}^{} |T_p(a,\mu,d)|=
\build{\lim}_{d\rightarrow +\infty}^{} -{1\over d} \log
\build{\liminf}_{a\rightarrow 0^+}^{} |T_p(a,\mu,d)|=K_2(\mu)\ .\eqno (9)$$

\noindent{\bf Proof}:

The $p$-adaptedness of $g$ for the DIT follows from the equivalence of the
dynamical norm with the original one.

We prove the rest of the theorem in the particular case of a linear Cantor set
with two scales $\lambda_1$ and $\lambda_2$ equipped with a balanced measure
of different weights $p_1$ and $p_2$; the general proof can be carried out
quite easily using the techniques developped in [4][5] and the characterization
of the Renyi entropies in terms of the pressure. We will indicate at the end
of the proof how to perform these generalizations. Applying the balanced
property of the measure, we can rewrite the DIT as (note that, by the previous
assumptions, $g$ is either positive or negative on the positive semi axis):
\medskip
\noindent$T_p(a,\mu,d) =$
$$a^{-p}\left\{\sum_{k_1,...,k_{d-1}=1}^{2} p^2_{k_1}\dots
p^2_{k_{d-1}}\build{\int\int}_{J\times J}^{} g\left ({\Lambda_{k_1\dots
k_{d-1}}||x-y||\over a}\right )d\mu(x)d\mu(y)\right\}+a^{-p}\phi(a)\eqno (10)$$
where we have put for simplicity:
$$\Lambda_{k_1\dots k_{d-1}}=\left (\lambda^2_{k_1}\dots
\lambda^2_{k_{d-1}}+\lambda^2_{k_2}\dots \lambda^2_{k_{d-1}}+\dots
+\lambda^2_{k_{d-1}}+1\right )^{1/2}\eqno (11)$$
and $\phi (a)$ contains the integrals whose $g$ have an argument strictly
positive since $x$ and $y$ are iterated backward respectively on two sets at a
finite distance (we use here the disconnectedness of $J$). Therefore, when
$a\rightarrow 0^+$, the term $a^{-p}\phi(a)$ goes to zero by the rapid
decrease of $g$ at infinity and we neglect it.
\medskip
Then we can write the DIT as :
$$T_p(a,\mu,d)=\sum^{2}_{k_1,\dots k_{d-1}=1}\left (p^2_{k_1}\dots
p^2_{k_{d-1}}\right )\Lambda^{-p}_{k_1\dots k_{d-1}} T_p(a_{k_1\dots
k_{d-1}},\mu)\eqno (12)$$
where $a_{k_1\dots k_{d-1}}=a\Lambda^{-1}_{k_1\dots k_{d-1}}$.
\medskip
By the adaptedness of $g$ for the FIT for $p=D_2(\mu)$, we have :
$$\build{\lim}_{a\rightarrow 0^+}^{}\left ({\sup\atop\inf}\right )|T_p
(a,\mu)|=\left ({s_1\atop s_2}\right )\not = (0,+\infty)$$
and therefore there are two positive constants $\rho_1$ and $\rho_2$ for which :
$$0<s_2-\rho_2\leq |T_p(a,\mu)|\leq s_1+\rho_1<\infty$$
for $a$ sufficiently small.

Then we can take the $\displaystyle\build{\lim}_{a\rightarrow 0^+}^{}\left
({\sup\atop\inf}\right )$ in (12) and get:
$$\eqalignno{&(s_2-\rho_2)\sum_{k_1\dots k_{d-1}=1}^{2}\left (p^2_{k_1}\dots
p^2_{k_{d-1}}\right )\Lambda^{-p}_ {k_1\dots k_{d-1}}\cr
\leq&\build{\lim}_{a\rightarrow 0^+}^{}\left ({\sup\atop\inf}\right
)|T_p(\mu,a,d)|&(13)\cr \leq& (s_1+\rho_1)\sum_{k_1\dots k_{d-1}=1}^{2}\left
(p^2_{k_1}\dots p^2_{k_{d-1}}\right )\Lambda^{-p}_ {k_1\dots k_{d-1}}\cr}$$

Taking the logarithm, dividing by $d$ and finally sending $d$ to infinity, we
get (note that $p$ is still equal to $D_2(\mu)$):
$$\eqalignno{&\build{\lim}_{d\rightarrow +\infty}^{}-{1\over
d}\log\build{\lim}_{a\rightarrow 0^+}^{}\left ({\sup\atop\inf}\right )\left
\vert T_p(\mu,a,d)\right\vert \cr =&\build{\lim}_{d\rightarrow
+\infty}^{}-{1\over d}\log \sum_{k_1\dots k_{d-1}=1}^{2}\left (p^2_{k_1}\dots
p^2_{k_{d-1}}\right )\Lambda^{-p}_{k_1\dots k_{d-1}}&(14)\cr}$$

Since :
$$1\leq\Lambda\leq d^{1/2}\quad\hbox{and}\quad \sum^{2}_{k_1\dots
k_{d-1}=1}\left (p^2_{k_1}\dots p^2_{k_{d-1}}\right )=\left (p^2_1+
p^2_2\right )^{d-1}\eqno (15)$$
we finally get :
$$\build{\lim}_{d\rightarrow +\infty}^{} -{1\over
d}\log\build{\lim}_{a\rightarrow 0^+}^{}\left({\sup\atop\inf}\right
)T_p(\mu,a,d)=-\log \left (p^2_1+p^2_2\right ),\eqno (16)$$
where the r.h.s. is exactly the $2$-nd Renyi entropy of the balanced measure
of weights $p_1$ and $p_2$ (cf. (4)).
\medskip
To handle with the general case of a disconnected conformal mixing repeller
endowed with a Gibbs measure $\mu_{\beta}$, we have to use the machinery
developed in [4], that allows us to replace the left hand side of (14) with :
$$\build{\lim}_{d\rightarrow \infty}^{} -{1\over d}\log\sum_{k_1\dots
k_{d-1}}{e^{-2(d-1)P(\beta)}\over||DT^{d-1}(\eta_{k_1\dots
k_{d-1}})||^{2\beta}}\Lambda^{-p}_{k_1\dots k_{d-1}}\eqno (17)$$
where $\eta_{k_1\dots k_{d-1}}$ is an arbitrary point in the corresponding
element of the Markov partition of $J$ obtained iterating $J$ backward $(d-1)$
times with the inverse branches of $T$. Besides $\Lambda_{k_1\dots k_{d-1}}$
has the structure
$$\Lambda_{k_1\dots k_{d-1}}=\left[||DT^{d-1}(\eta_{k_1\dots
k_{d-1}})||^{-2}+\dots +||DT(\eta_{k_1})||^{-2}+1\right]^{1/2}$$
and can be bounded uniformly as in (15) by hyperbolicity.
\medskip
Since, for $d\rightarrow +\infty$, the term $d^{-1}\log \sum_{k_1\dots
k_{d-1}}||DT^{d-1} (\eta_{k_1\dots k_{d-1}})||^{-2\beta}$ converges to
$P(2\beta)$ [14], we get from the limit (17) :
$$\build{\lim}_{d\rightarrow +\infty}^{} -{1\over d}\log\build{\lim}_
{a\rightarrow 0^+}^{} \left ({\sup\atop\inf}\right )|T_p(\mu,a,d)|
=2P(\beta)-P(2\beta)$$
which agrees with the $2$-nd Renyi entropy for the Gibbs measure $\mu_{\beta}$
computed with the thermodynamic formalism (cf.(3)). $\diamondsuit$
\medskip
We now generalize the DIT in order to get all the Renyi entropies; we follow
the same idea that leaded us to generalize the FIT to get the generalized
dimensions, that is we define {\it the $q$-Dynamical Integral Transform},
$q-$DIT, as :
$$T_p(a,\mu,d,q)=a^{-p}\left\{\int_{J}^{}d\mu(x)\left\vert\int_{J}^{}
d\mu(y)g\left({||x-y||_d\over a}\right )\right\vert^{q-1}\right\}^{{1\over
q-1}}$$

If the function $g$ is adapted for a certain $p$ for the corresponding
$q-$FIT, $T_p(a,\mu,q)$, (that is obtained setting $d=0$)\footnote{$^{3}$}{In
this case we also say that $g$ is ($p-q$)-adapted to $\mu$}, we proved
that $p$ is equal to the generalized dimension $D_q(\mu)$ when $q>1$ and is a
lower bound to it (eventually equal) for $q<1$. The $q-$DIT behaves in a
similar way; we still formulate the following theorem for the dynamical
systems considered in theorem 1.
\vfill\eject
\noindent{\bf Theorem 2:} \item{}{\it If $g$ is  $p$-adapted to $\mu$
for the q-FIT, then it is $p$-adapted to $\mu$ for the q-DIT and moreover :}
$$\eqalign{&\build{\lim}_{d\rightarrow +\infty}^{}-{1\over
d}\log\build{\lim}_{a\rightarrow 0^+}^{} \left ({\sup\atop\inf}\right
)T_p(a,\mu,d,q)=K_q(\mu)\quad\hbox{for}\quad\ q>1\cr &\hbox{and}\cr
&\build{\limsup}_{d\rightarrow +\infty}^{}-{1\over
d}\log\build{\lim}_{a\rightarrow 0^+}^{} \left ({\sup\atop\inf}\right
)T_p(a,\mu,d,q)\leq K_q(\mu)\quad\hbox{for}\quad\ q<1.\cr} \eqno(18)$$

\noindent{\bf Proof}:

The $p$-adaptedness of $g$ for the $q$-DIT follows from the equivalence of the
dynamical norm with the usual norm.

We specialize again the proof to the linear Cantor set with two scales
$\lambda_1,\lambda_2$ and two weights $p_1\not = p_2$. The generalization is
straightfoward using the same suggestions as at the end of the proof of
theorem 1.
\medskip
Applying the balancement of the measure $(d-1)$ times we get :
$$\eqalignno{T^{q-1}_p(a,\mu,d,q)=&\sum_{k_1\dots k_{d-1}=1}^2p_{k_1}\dots
p_{k_{d-1}}\int_{J}^{}d\mu(x)\left[p_{k_1\dots k_{d-1}}\right.\cr
&\left.a^{-p}\int_{J}^{}|g\left (\Lambda_{k_1\dots k_{d-1}}{||x-y||\over
a}\right )|d\mu(y)+\phi_a(x)\right]^{q-1}&(19)\cr}$$
where $\phi_a(x)$ is a positive function uniformly bounded in $x$ away from
zero and going to $0$ when $a\rightarrow 0^+$; in fact it collects the
integrals which $g$ has the argument bounded away from zero; we also call
$F_{k_1\dots k_{d-1}}(a,x)$ the first term in the square bracket in (19). Note
that we cannot neglect $\phi_a(x)$ at this point since it can be of the same
order as $F_{k_1\dots k_{d-1}}(a,x)$; therefore we need to integrate before.
As in the proof of Theorem 2 in [5], we can bound the integral in the sum in
(19) by Minkowski's inequality from above and below for $q>1$ by an expression
of the type :
$$\int_{J}^{}d\mu(x)F^{q-1}_{k_1\dots
k_{d-1}}(a,x)\left[1+\left\{{\int_{J}^{}d\mu(x)\phi^{q-1}_a(x)
\over\int_{J}^{}d\mu(x)F_{k_1\dots
k_{d-1}}^{q-1}(a,x)}\right\}^{\beta}\right]^{\beta^{-1}}\eqno (20)$$
where $\beta$ is equal to $1$ or ${1\over q-1}$.
\medskip
Whenever $q<1$, we can only keep the upper bound corresponding to the case
$\beta=1$ and this is the reason of the inequality in the statement of the
theorem; in the following we consider the case $q>1$.
\medskip
We observe that :
$$\int_{J}^{}d\mu(x)F_{k_1\dots k_{d-1}}^{q-1}(a,x)=p^{q-1}_{k_1}\dots
p^{q-1}_{k_{d-1}}\ \Lambda^{-p(q-1)}_{k_1\dots k_{d-1}}\ T^{q-1}_p(\mu,a_
{k_1\dots k_{d-1}},q)$$
where $a_{k_1\dots k_{d-1}}=a\Lambda_{k_1\dots k_{d-1}}^{-1}$. The second
factor in (20) goes to 1 when $a\rightarrow 0^+$, since the integral in the
numerator converges to zero and the denominator is bounded away from zero and
infinity by the adaptedness of $g$. Therefore we can bound
$T^{q-1}_p(a,\mu,d,q)$ from  above and below by
$$\sum_{k_1\dots k_{d-1}=1}^{2}p^{q}_{k_1}\dots
p^{q}_{k_{d-1}}\Lambda^{-p(q-1)}_{k_1\dots k_{d-1}}\ T^{q-1}_p(\mu,a_
{k_1\dots k_{d-1}},q)$$

Using the same arguments as in the proof of Theorem 1, we thus get :
$$\build{\lim}_{d\rightarrow +\infty}^{}
-{1\over d}\log\build{\lim}_{a\rightarrow 0^+}^{} \left
({\sup\atop\inf}\right )T_p(a,\mu,d,q) =-{1\over (q-1)}\log (p^q_1+p^q_2)\eqno
(21)$$
which is just the $q$-Renyi entropy $K_q(\mu)$ (cf. (4)). $\diamondsuit$
\bigskip
\bigskip
Under the hypothesis of adaptedness for the $q$-FIT and for more general
systems, we can define:
$${{\overline{K^+_q}(\mu)}\choose {\overline{K^-_q}(\mu)}}
=\build{\limsup}_{d\rightarrow +\infty}^{}-{1\over
d}\log\build{\lim}_{a\rightarrow 0^+}^{} \left ({\sup\atop\inf}\right
)T_p(a,\mu,d,q)$$
where the $\build{\limsup}_{d\rightarrow +\infty}^{}$ is taken in
view of the special but important case of Theorem 2. Apart the obvious bound
${\overline{K^+_q}(\mu)}\leq{\overline{K^-_q}(\mu)}$, we can easily prove the
following monotonicity property of the indices $\overline{K^{\pm}}_q(\mu)$ as
functions of $q$, which is consistent with the analogous property for the
$K_q$ solutions of eqs (3) and (4).
\bigskip
\noindent{\bf Proposition 1:
$${\overline{K^{\pm}_q}(\mu)}\leq{\overline{K^{\pm}_r}(\mu)}\hbox{ for }
r\leq q$$}
\noindent{\bf Proof}:

The proof is similar to that of Proposition 1(iii) in [5] and relies on the
structure of the $q-$DIT as a $L^{q-1}$-norm and on Jensen's inequality.
$\diamondsuit$
\bigskip
\noindent{\bf Remarks:}
\medskip
A] To get the Kolmogorov-Sinai entropy corresponding to $q=1$, we have to take
the logarithm of the $q$-DIT and derive it at the point $q=1$. We are thus led
to conjecture that :
$$\build{\lim}_{d\rightarrow +\infty}^{}-{1\over d}\build{\lim}_{a\rightarrow
0^+}^{}\int_{J}^{}d\mu(x)\log\left\vert a^{-p}\int_{J}^{}g\left
({||x-y||_d\over a}\right)d\mu(y)\right\vert=K_1(\mu)\eqno (22)$$
whenever $p=D_1(\mu)$ (the information dimension): this conjecture is
numerically verified in example 3 below.
\medskip
B] Another interesting question is how the limit (18) approximates the
topological entropy, that is obtained setting $q=0$ in the definition (2) : by
theorem 2, we expect in general this limit to be smaller than the topological
entropy and this is not surprising since the topological entropy is defined
independently of any measure.
\bigskip
\noindent{\bf 4 - Examples}
\medskip
\noindent{\it Example 1: Ternary Cantor set}
\medskip
In the case of the $q$-FIT, we considered in [5] the usual ternary Cantor set
with equal scales $\lambda_1=\lambda_2=\lambda,\quad 0<\lambda\leq 1/3$
endowed with the Gibbs measures $\mu_{\beta},\quad \beta\in {\bf R}$ ;
assuming $g$ $(p-q)$-adapted to $\mu_{\beta}$ for all $q$, we
proved that $p=\log\ 2/\log\ \lambda^{-1}$ for all $q$. A result of the same
type holds for the $q$-DIT. We first observe that the left hand side of (18)
is equal to : $k_q(\mu_{\beta})=\log\ 2$ for $q>1$ and is smaller or equal to
$\log\ 2$ for $q<1$ (this easily follows from the expression for the pressure
: $P(\beta)=\log(2\lambda^{\beta})$). Then, by Proposition 1 and passing to the
limit  for $p=\log\ 2/\log\ \lambda^{-1}$, we have for $r<1<q$ :
$$\eqalignno{ \log\ 2&\geq\build{\limsup}_{d\rightarrow+\infty}^{}-
{1\over
d}\log\lim_{a\rightarrow 0^+}\left({\sup\atop\inf} \right)T_p(a,\mu,d,r)\cr
&\geq\lim_{d\rightarrow+\infty}-{1\over d}\log\lim_{a\rightarrow
0^+}\left({\sup\atop\inf}\right)T_p(a,\mu,d,q)=\log\ 2\cr}$$
The same argument holds by replacing $\limsup$ with
$\lim\quad\hbox{inf}$ in the first limit and this shows that the
$\build\lim_{d\rightarrow+\infty}^{}$ exists for $q<1$ and therefore the limit
in (18) recovers the same entropy for all $q\in{\bf R}$.
\bigskip
\noindent{\it Example 2: Irrational rotation}
\medskip
We showed in [4] that, for the unit hypercube in ${\bf R}^n$, the
adaptedness of $g$ to the Lebesgue measure implies $p=n$ ; moreover we checked
the adaptedness of the function $g(r)=e^{-r}$. We now identify the unit interval
[0,1) with the torus ${\bf T}$ and put on it the irrational rotation :
$T(x)=x+\alpha\ mod\ 1,\quad x\in{\bf T}$ and $\alpha\in{\bf R}\backslash{\bf
Q}\ ;\ T$ leaves invariant the Lebesgue measure. Then we define on ${\bf T}$
the norm : $|| z||=\inf\limits_{k\in {\bf Z}}\mid z+k\mid,\ z\in{\bf R}$,
being $\mid\cdot\mid$ the Euclidean distance on ${\bf R}$. It is easily seen
that :
$$||| x-y|||_d=\left\{\normalbaselineskip=18pt\matrix{ \mid
x-y\mid\hfill&\hbox{for}&\mid x-y\mid\leq 1/2\hfill\cr 1-\mid
x-y\mid\hfill&\hbox{for}&\mid x-y\mid\geq 1/2,\hfill\cr}\right.$$
so that the dynamic norm is independent of $d$. This and the
preceding argument on the adaptedness imply that the 2-nd
Renyi entropy is zero, which is consistent with the well known fact that the
irrational rotations have zero entropy.
\bigskip
\noindent{\it Example 3: Numerical}
\medskip
The relations (9), (18) and (22) give a precise method to compute numerically
the Renyi entropies once the corresponding generalized dimensions are known,
and these can just be computed by means of the $q$-FIT (see [5] for the
examples).
\medskip
We now present a few examples, others being in preparation [22]. First of all,
we compute the 2-nd Renyi entropy by means of formula (9) in two cases : (i)
the ternary Cantor set with scale $\lambda=1/3$ and equal weights
$p_1=p_2=1/2$ ; (ii) the same Cantor set with different weights $p_1=1/4$ and
$p_2=3/4$; these examples will also show the adaptedness of the test function
$g(r)=e^{-r^2}$. In the former  case $D_2=\log\ 2/\log\ 3$ and $K_2=\log\
2=0.6931...$ ; in the latter $D_2=\log(8/5)/\log\ 3$ and
$K_2=\log(16/10)=0.470004...$, as given by (4). The integrals have been
computed by averaging over the predecessors of order $n$ of an arbitrary point
of the unit interval as explained in [4] : the value of $n$ was taken equal to
14. We did not take into account the values of $a$ comparable in size with the
scales of the Cantor set determined by the $n$-th order of iteration, that is
$\left({1\over 3}\right)^{n-d}$ : in fact, for values of $a$ smaller than this
threshold, the DIT goes down abruptly.
\medskip
The values of the DIT for fixed $d$ are given extrapolating the data as a
function of $a$ and are reported in figs. 1 and 2. Morover a non-linear
fitting was performed on these data with a function of type :
$$f(d)=C_1+C_2\ d^{-C_3}$$
where $C_1,\ C_2$ and $C_3$ are constants and $C_1$ is just the
expected value of the DIT in the limit $d\rightarrow +\infty$.

For the Cantor set with equal weights we found : $C_1=0.6914$ and for the
Cantor with different weights : $C_1=0.4669$, in excellent agreement with the
theoretical values. We note that, up to numerical errors, the convergence is
of type ${1\over d}$.
\medskip
In figs. 3 and 4, we report on the computation of the DIT for the same sets but
for different values of $a$. As explained before, when $a$ becomes comparable
with $\left({1\over 3}\right)^{n-d}$, the DIT goes to zero and this is evident
from the figures : this means that we have to keep the structure of the set
enough fine (by taking $n$ sufficiently large) when it is explored by the
integral transform at decreasing values of $a$.
\medskip
The same procedure has then been applied to the computation of the other Renyi
entropies for the Cantor set with weights $p_1=1/4$ and $p_2=3/4$, for which
the entropies are different and given by eq. (4). In fig. 5, we report on the
numerical and analytical curves of the $K_q$ in the range $q\in [-11,11]$; the
precise  values are quoted in table 1. Note that the entropy $K_1$, as given
by formula (22), is in excellent agreement with the theoretical value and
that, when $q<1$, the numerical entropies are smaller than the theoretical
ones, as predicted by Theorem 2. We also report in fig. 6 on the Legendre
 transform
of the quantity  $K_q(q-1)$. The interpretation of the corresponding curve
will be given at the  end of section 5.
\medskip
We conclude this section by computing the entropies for a non-hyperbolic
invariant set, precisely the H\'enon attractor generated by the mapping:
$x'=1-ax^2+y; y'=bx$, with $a=1.4$ and $b=0.3$. The $q$-DIT was computed as
usual by approximating the integral with the ergodic mean with respect to the
physical measure. In figs. 7 and 8, we show the results for the entropies
$K_0$ and $K_2$: the extrapolated values are $0.443049$ for $K_0$ and
$0.298903$ for $K_2$, in agreement with the same entropies given, for example,
in [13] and [24], where $K_0=0.445$ and $K_2=0.325\pm 0.02$.

\bigskip
\noindent{\bf 5 - A ``multientropy'' local analysis}
\medskip
Given the transform at the point $x\in J$ :
$$T_p(a,\mu,x)=a^{-p}\int_{J}^{} g\left ({||x-y||\over a}\right )d\mu(y)$$
and defined the functions of $p:T^{\pm}_p(\mu,x)=\build{\lim}_{a\rightarrow
0^+}^{}\left ({\sup\atop\inf}\right )|T_p(a,\mu,x)|$, we showed in [1][5] that:

i) there are two transition points $p^{\pm}(x)$ such that
$T^{\pm}_p(\mu,x)=0,p<p^{\pm}(x)$ and $T^{\pm}_p(\mu,x)=+\infty,p>p^{\pm}(x)$;
ii) these points are related to the local exponents of the measure $\mu$, that
is :
$$\beta_-(x)\leq p^+(x)\leq p^-(x)\leq \beta_+(x)\eqno (23)$$
where
$$\beta_{\pm}(x)=\build{\lim}_{r\rightarrow 0^+}^{}\left ({\sup\atop
\inf}\right ) {\log\mu(B(x,r))\over\log r}\eqno (24)$$
being $B(x,r)$ a ball of center $x$ and radius $r$ in the metric $||.||$. We
now introduce the {\it local dynamical integral transform} at $x$ defined as :
$$T_p(a,\mu,d,x)=a^{-p}\int_{J}^{}g\left ({|||x-y|||_d\over a}\right
)d\mu(y)$$
where the metric $|||.|||_d$ is defined by (5b). By the equivalence of the
norms, the transition points of the functions :
$$T^{\pm}_p(\mu,d,x)=\build{\lim}_{a\rightarrow 0^+}^{}\left ({\sup\atop
\inf}\right )|T_p(a,\mu,d,x)|$$
will be again $p^{\pm}(x)$. We guess that if the following limits exist (note
that the functions $T^{\pm}_p$ are computed in their transition points):
$$T^+(x)=\build{\lim}_{d\rightarrow +\infty}^{}-{1\over d}\log
T^+_{p^+(x)}(\mu,d,x)$$
$$T^-(x)=\build{\lim}_{d\rightarrow +\infty}^{}-{1\over d}\log
T^-_{p^-(x)}(\mu,d,x)$$
then they coincide respectively with the local entropies $h^{\pm}(x)$ defined
by the Brin-Katok theorem [23] as:
$$\left ({h^+(x)\atop h^-(x)}\right)
=\build{\lim}_{r\rightarrow 0^+}^{}\ \build{\lim}_{d\rightarrow
+\infty}^{}\left ({\sup\atop \inf}\right ) \left (-{1\over
d}\log\mu(B(x,d,r))\right )\eqno (25)$$
where $B(x,d,r)$ is a ball of radius $r$ in the metric $|||.|||_d$. For
$\mu$-almost every $x\in J,h^+(x)=h^-(x)$ and, for an ergodic measure $\mu$,
the local entropies are constant almost everywhere and equal to the metric
entropy $K_1(\mu)$. Our conjecture is suggested by the formal analogy of the
limits (24) and (25) and by a local extension of Theorem 1. A local analysis
could be carried out at a point $x\in J$ by analyzing the behavior of the
function $-{1\over d}\log \vert T_p(a,\mu,d,x)\vert$ in the variables $(a,d)$,
for small values of $a$ and large values of $d$ and keeping $p$ in the
interval given by (23). In this way, the asymptotic oscillations of the
function $-{1\over d}\log\vert T_p(a,\mu,d,x)\vert$ at different points $x\in
J$ should be proportional to the different strength of the local entropies
$h^{\pm}(x)$. An investigation of these questions is in progress, but we
already present here a numerical study of the above conjecture for the ternary
Cantor set and we also give an interpretation of the Legendre transform of
the spectrum of the $K_q$ in terms of the local entropies given by the
Brin-Katok theorem.
\bigskip
\noindent{\bf 5.1 -}
We performed a numerical analysis of the Brin-Katok formula for the ternary
Cantor set endowed with the balanced measure $\mu$ of equal weights and we
observed two facts: first, the limit for $r\rightarrow 0^+$ is inessential,
being the correct value for the entropy already reached when $d\rightarrow
+\infty$, at fixed $r$. This is not surprising if one considers the following
relations among the measures of a ball in the different metrics:
$$\mu(B(x,({\scriptstyle{1\over 3}})^{n+d-1}r))\leq\mu(B(x,d,({\scriptstyle
{1\over 3}})^{n}r))\leq\mu(B(x,({\scriptstyle{1\over 3}})^{n+d-1}))\eqno(26)$$
for $r\leq 1$ and $n\in\bf N$.

By suitably bounding $r$ with power of $1\over 3$ and recalling that the
measure of the intervals generating the Cantor set at the m-th step is
$2^{-m}$, we  immediately recover the entropy $\log 2$ in the limit
$d\rightarrow +\infty$. We show the numerical computation in figs. 9,10 and 11.
\medskip
The second and not yet understood fact appears in figs. 10 and 11 and consists
in the periodic oscillations with affect, for a particular choice of fixed
$r$, the convergence of the limit for $d\rightarrow +\infty$.

In fig. 12, we report on the computation of the local entropy with the local DIT
(the choice of the point $x$ does not change the numerical results),
after having taken the limit for $a\rightarrow 0^+$; what we found is in
agreement with the expected value $\log 2$. We want to point out that a
rigorous proof of this result for the ternary Cantor set seems difficult to
get. For example, the  relation (26), which shifts the problem
from the DIT to the FIT by replacing the measure of the dynamical ball
with the measure of a ball in the ordinary metric, is apparently not
sufficient.
\bigskip
\noindent{\bf 5.2 -}
In [19], Paladin and Vulpiani considered the Legendre transform of the
spectrum of the $K_q(\mu)=K_q$, that is
$$K_q(q-1)=\min_{\gamma}(q\gamma-\hat{S}(\gamma))\eqno(27)$$
and they interpreted $\hat{S}(\gamma)$ as the topological entropy of the set
$\Omega(\gamma)$, where $x\in\Omega(\gamma)$ when, with our notation:
$$\mu(B(T^kx,d,r))\sim e^{-d\gamma}\eqno(28)$$
for $r\rightarrow 0^+$ and $k\geq 0$. They called $\gamma$ Local Expansion
Parameter (LEP): see also Eckmann and Procaccia [19] and [25] for similar
interpretations.

When $k=0$, the scaling (28) is the physical way to write down mathematically
the Brin-Katok formula. We think that the correct way of interpreting the
function $\hat{S}(\gamma)$ is the following: let
$\Omega^{\pm}(\gamma)$ the sets of points $x$ for which respectively
$h^{\pm}(x)=\gamma$. Then  $\hat{S}(\gamma)$ is the common topological entropy
for the sets  $\Omega^{\pm}(\gamma)$. This assertions deserve to be proved
analytically,  probably using the large deviation techniques employed in the
rigorous  derivation of the $\alpha-f(\alpha)$ theory for the generalized
dimensions [26] and the Bowen characterization of the topological entropy in
terms of the  $(n-\epsilon)$ spanning sets [27]. Apparently, the
interpretation of the LEP's as the local Brin-Katok entropies, although
implicit in (28), was at our  knowledge not given before. Note that, according
to this interpretation, the maximum of the concave curve $\hat{S}(\gamma)$ is
the topological entropy $K_0$, while the same curve intersects
the bissectrix at the Kolmogorov entropy $K_1$. This is evident in fig. 6 for
the linear Cantor set and follows also from the Legendre transform of (4).
\bigskip
\noindent{\bf 6 - Conclusions}
\medskip
We showed in this paper that a suitable integral transform of wavelet type,
that we called Dynamical Integral Transform (DIT), allows us to compute the
spectrum of Renyi entropies. For mixing repellers, we gave rigorous results,
that we can extend to non-hyperbolic invariant sets.

Our method can be
numerically implemented quite easily and compared to other techniques, like
the correlation integral and the energy integral, shows some universality in
the choice of the test functions that satisfy all the same asymptotic scalings.
Moreover, our technique is intrinsically dynamic, that is we extract the
entropies by (ergodically) averaging over orbits instead of partitionning the
invariant sets as prescribed by formula (2), which is the most commonly used
for the computation of the Renyi entropies.

Finally, the local version of the DIT allows us to explore the local entropies
of strange sets which topological distribution we claimed is given by the
Legendre transform of the Renyi entropies. This analysis is the natural
extension to entropies of the capability of the integral transform of wavelet
type to capture the local dimensions of fractal measures and should give a
``multientropy'' description of strange sets.
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\noindent{\bf Acknowledgments:} we want to thank S. Siboni for useful
discussions concerning section 5.


\vfill\eject
\centerline{\bf Figures Captions}

\bigskip

\parindent 1.5cm

\item{\hbox to\parindent{\enskip Fig. 1:\hfill}\hfill}Dynamical integral
transform as a function of $d$, after having extrapolated on $a$, for the
ternary Cantor set with equal weights. Fitting the data with the function :
$f(d)=C_1+C_2\ d^{-C_3}$, we found : $C_1=0.6914$ (giving $K_2$)$\ ;\
C_2=-0.5156\ ;\ C_3=1.022$.
\medskip
\item{\hbox to\parindent{\enskip Fig. 2:\hfill}\hfill}Dynamical integral
transform as a function of $d$, after having extrapolated on $a$, for the
ternary Cantor set with weights $p_1=1/4$ and $p_2=3/4$. Fitting the data with
the function : $f(d)=C_1+C_2\ d^{-C_3}$, we found : $C_1=0.4669$ (giving
$K_2$)$\ ;\ C_2=-0.3305\ ;\ C_3=1.060$.
\medskip
\item{\hbox to\parindent{\enskip Fig. 3:\hfill}\hfill}Dynamical integral
transform as a function of $d$ for different values of $a$, for the ternary
Cantor set with equal weights.
\medskip
\item{\hbox to\parindent{\enskip Fig. 4:\hfill}\hfill}Dynamical integral
transform as a function of $d$ for different values of $a$, for the ternary
Cantor set with weights $p_1=1/4\ ;\ p_2=3/4$.
\medskip
\item{\hbox to\parindent{\enskip Fig. 5:\hfill}\hfill}Numerical and analytical
spectra of the Renyi entropies $K_q$ for the ternary Cantor set with weights
$p_1=1/4\ ;\ p_2=3/4$.
\medskip
\item{\hbox to\parindent{\enskip Fig. 6:\hfill}\hfill}Legendre transform
$\hat{S}(\gamma)$ of
$K_q(q-1)$ for the ternary Cantor set with weights $p_1=1/4\ ;\ p_2=3/4$.
The maximum of this curve is the topological entropy $K_0$ and
the curve intersects the bissectrix at the Kolmogorov entropy $K_1$.
\medskip
\item{\hbox to\parindent{\enskip Fig. 7:\hfill}\hfill}0-Dynamical integral
transform as a function of $d$, after having extrapolated on $a$, for the
H\'enon attractor. Fitting the data with the function :
$f(d)=C_1+C_2\ d^{-C_3}$, we found : $C_1=0.4430$ (giving $K_0$)$\ ;
\ C_2=0.7300\ ;\ C_3=1.3885$.
\medskip
\item{\hbox to\parindent{\enskip Fig. 8:\hfill}\hfill}Dynamical integral
transform as a function of $d$, after having extrapolated on $a$, for the
H\'enon attractor. Fitting the data with the function :
$f(d)=C_1+C_2\ d^{-C_3}$, we found : $C_1=0.2989$ (giving $K_2$)$\ ;
\ C_2=0.5773\ ;\ C_3=0.6682$.
\medskip
\item{\hbox to\parindent{\enskip Fig. 9:\hfill}\hfill}Local entropy given by
Brin-Katok formula for the ternary Cantor set with equal weights.
$-{1\over d}\mu(B(x,d,r))$ is plotted vs. $d$ for fixed $r=0.012048$.
Fitting the data with the function :
$f(d)=C_1+C_2\ d^{-C_3}$, we found : $C_1=0.6931$ (giving the local entropy)
$\ ;\ C_2=1.3863\ ;\ C_3=1.0000$.
\medskip
\item{\hbox to\parindent{\enskip Fig. 10:\hfill}\hfill}Local entropy given by
Brin-Katok formula for the ternary Cantor set with equal weights.
$-{1\over d}\mu(B(x,d,r))$ is plotted vs. $d$ for fixed $r=0.162105$. Note the
oscillations which affect the convergence.
\medskip
\item{\hbox to\parindent{\enskip Fig. 11:\hfill}\hfill}Local entropy given by
Brin-Katok formula for the ternary Cantor set with equal weights.
$-{1\over d}\mu(B(x,d,r))$ is plotted vs. $d$ for fixed $r=0.044194$. Note the
oscillations which affect the convergence.
\medskip
\item{\hbox to\parindent{\enskip Fig. 12:\hfill}\hfill}Local dynamical integral
transform as a function of $d$, after having extrapolated on $a$, for the
ternary Cantor set with equal weights. Fitting the data with the function :
$f(d)=C_1+C_2\ d^{-C_3}$, we found : $C_1=0.7001$ (giving the local entropy)
$\ ;\ C_2=-0.5660\ ;\ C_3=0.9122$.

\centerline{\bf Tables Captions}

\bigskip

\parindent 1.5cm

\item{\hbox to\parindent{\enskip Tab. 1:\hfill}\hfill}Spectrum of the Renyi
entropies $K_q$ for the ternary Cantor set with weights $p_1=1/4\ ;\ p_2=3/4$.
Theoretical values are computed according to eq. (4). Numerical values are
computed according to Theorem 2.

\vfill\eject
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\end