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. 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