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\begin{document}

\title{Derivation of the auto-correlation function for a triadic Cantor set spectrum}

\author{A. Soret\dag, E. Gurevich \ddag\ and E. Akkermans\ddag}


\address{$^{1}$Department of Physics, Technion Israel Institute of Technology, 32000 Haifa, Israel}

\date{\today}


\begin{abstract}
We derive the analytical expression of the time-averaged correlation function of a quantum system in the case of a triadic 
Cantor set energy spectrum. We generalize the method used and discuss the connection between the presence of scaling symmetries 
in physical systems and the typical log-periodic behaviour of some quantities. As examples, we discuss the case of the 
participation ratio and of the RMS displacement for a triadic Cantor set spectrum, as well as the spontaneous emission 
for a two level atom coupled to a self-similar fractal spectrum. We also present some numerical results obtained recently 
for polaritons in a Fibonacci potential with an associated Cantor like spectrum, which support the theoretical results.     


\end{abstract}

\maketitle


%\jl{1}
%\title{Derivation of the auto-correlation function for a triadic Cantor set spectrum}

%\author{A. Soret\dag, E. Gurevich \ddag\ and E. Akkerman\ddag} %\ftnote{3}{To
%whom correspondence should be addressed.}}

%\address{\dag\ \'Ecole Normale Sup\'erieure de Cachan, 94235 Cachan Cedex, France}

%\address{\ddag\ Department of Physics, Technion Israel Institute of Technology, Haifa 32000, Israel}

%\ead{submissions@iop.org}
%\vspace{10pt}
%\begin{indented}
%\item[]February 2014
%\end{indented}



%\nosections
\section{Introduction}

During the past decades, there has been a growing interest for Schroedinger operators $H = -\frac{\hbar^2}{2m}\Delta + V(\textbf{r})$ with a Cantor set like spectrum, e.g. a perfect, nowhere dense set (for a review, see [1], and more recently [2]). 
%This is usually the case for quasi-periodic potentials $V$, which constitute the generic case ([9]). 
In the context of quantum dynamics for systems verifying such a Hamiltonian, a typical behaviour for some physical quantities, such as the auto-correlation function (time-averaged return probability), has been identified.  More precisely, it has been shown ([3],[4]) that, in the case of a Cantor set spectrum, the auto-correlation function $C(t)$ decreases as : $t^{-D_{2}^{\mu}}$, where $D_{2}^{\mu}$ is a correlation dimension with respect to the measure $\mu$ of the spectrum, although a full analytical expression had not been derived so far. A slightly different quantity, the spatial correlation function $C_{s}(l,\mu) = \int\limits_{K \times K}\theta(l-|x-y|)d\mu(x)d\mu(y)$, with $\theta$ the Heaviside function, $K$ the spectrum and $d\mu$ the measure associated to the $K$, has been derived more precisely for a Cantor set spectrum ([5]), showing that this quantity is a power law $t^{d_{K}}$ ($d_{K}$ being the fractal dimension of the Cantor set) modulated by log-periodic oscillations. 

We recently studied numerically the auto-correlation function for a gas of cavity polaritons placed in a Fibonacci potential and with a Cantor set like energy 
spectrum [6]; we found in this case that $C(t)$ has a log-periodic behaviour. 

In this paper, we study the dynamics of a quantum system whose energy spectrum is described by a self-similar measure $\mu$, e.g. such that there exists a set of contractive similarities $\{\phi_j\}$ and probability weights $\{\pi_j\}$ such that for any continuous function $f$, one has : 
$$\int f(x)d\mu(x) = \sum_j \pi_j \int f(\phi_j(x))d\mu(x)$$
We shall focus on the case of a set of two linear similarities, with $\pi_1 = \pi_2 = \frac{1}{2}$ : 
$$\int f(x)d\mu(x) =  \frac{1}{2} \int f(a_1x)+\frac{1}{2} \int f(a_2x+b)$$
Cantor sets are examples of such spectra, and occur in many cases. We will therefore analyse in detail  
the case of a triadic Cantor set ; we shall derive the exact analytical expression of the auto-correlation function $C(t)$, and show that it is of the form : 
\begin{equation*}
C(t) = \left(\frac{t}{\tau}\right)^{-d_K}g(\ln(t/\tau))
\end{equation*} 
where $d_K=\frac{\ln(2)}{\ln(3)}$ is the fractal 
dimension [7] of the triadic Cantor set, $\tau$ a time scale a $g$ a periodic function. 

We first give a detailed calculation of $C(t)$, then discuss its behaviour and the similarity with numerical results obtained with the Fibonacci cavity. Furthermore, we will generalize the method, which is in fact appropriate for the study of systems described by a self-similar measure (fractals), and gives an intuition of how the fractal properties translate through these quantities or functions. We will finally discuss a few examples in quantum dynamics.   
%The method can be generalized to the case of any self-similar fractal, and can be used to study how the fractal feature of the spectrum translates into the dynamical properties of the system.

\section{}

A triadic Cantor set is constructed 
as follows : start with a segment [0,$\epsilon_{max}$], divide it in three equal parts, and remove the central one, e.g. the segment
$[\epsilon_{max}/3,2\epsilon_{max}/3]$, then repeat the process with the two remaining segments, and so on. After an infinite number of iterations, we 
are left with a set of separated points, of zero Lebesgue measure.

In this paper, $K$ will denote a triadic Cantor set constructed from $[0,1]$.

%\medskip

We now consider a one dimensional quantum system described by the state vector $|\psi(t)\rangle$ and evolving with the Hamiltonian $H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)$. We assume that the energy spectrum of $H$ is a triadic Cantor set. Note that, since all Cantor sets are homeomorphic to the triadic Cantor set ([17]), the results and methods can be adapted to any Cantor set spectrum.
%We can even consider, in a more general case, an unbounded triadic Cantor set, embedded in $[0, \infty[$. For this, note that the usual triadic Cantor set $K$ is composed of the real numbers $\{a_{j}\}$ in $[0,1]$ of the form : $a_{j}=\sum_{k \geq 1} \epsilon_{j,k} 3^{-k}$ with $\epsilon_{j,k}\in \{0,2\}$. One can then generalize to $\mathbb{R}$ by taking all the numbers of the form : $a_{j}=\sum_{k \in \mathbb{Z}} \epsilon_{j,k} 3^{-k}$ with $\epsilon_{j,k}\in\{0,2\}$. We obtain a reunion of usual triadic Cantor sets, located in the unit segments $[b_{j},b_{j}+1]$ with $b_{j}=\sum_{k \geq 0} \epsilon_{j,k} 3^{k}$ and $\epsilon_{j,k}\in\{0,2\}$.
%We assume that $H$ is hermitian and that its spectrum is a triadic Cantor set $K_{\epsilon_{max}}$, constructed from $[0,\epsilon_{max}]$, $\epsilon_{max}<\infty$ ; therefore, $0$ and $\epsilon_{max}$ are respectively the lower and upper bounds for the energy. 

The system is initially in a state described by $|\psi(t=0)\rangle = |\psi_{0}\rangle$. After a time $t$, the state vector  $|\psi(t)\rangle$ is given by :
\[|\psi(t)\rangle = e^{-\mbox{i\^{H}t}/\hbar}|\psi_{0}\rangle\]

We shall also use the spectral decomposition of $|\psi(t)\rangle$ over an orthonormal basis of eigenfunctions $\{|\phi(\epsilon)\rangle\}$ of $H$,$|\phi(\epsilon)\rangle$ being the eigenfunction associated with the energy $\epsilon$ :
\[|\psi(t)\rangle= \int_{spectrum} g(\epsilon) e^{-i\epsilon t/\hbar} |\phi(\epsilon)\rangle d\tilde\mu(\epsilon)\]
where $g(\epsilon)$ is the projection of $|\psi_{0}\rangle$ on $|\phi(\epsilon)\rangle$, and $d\tilde\mu$ a measure defined on the spectrum.

%\medskip

To go further, we need to define an appropriate measure on the spectrum. The triadic Cantor set $K$ has a zero Lebesgue measure. However, in our case, the integrated density of states, e.g. the integral of $\rho(\epsilon)$ over the spectrum $K$ (we shall generalize to the case of an infinite Cantor set later), is non zero. Thus, $\rho(\epsilon)$ must be infinite in every point of $K$.

%We assume here that the integrated density of modes is bounded.

%\medskip

We define a measure $d\mu$ on $K$, such that the integrated density of states $\mathcal{N}(\epsilon)$ is given by : $$\mathcal{N}(\epsilon) = \int_K \theta(\epsilon - \epsilon')d\mu(\epsilon')$$ with $\theta$ the Heaviside function. We furthermore suppose that the density of states is uniform.
%taking into account the density of modes (one can think of $d\mu$ as a measure verifying $d\mu(\epsilon)=\rho(\epsilon)d\epsilon$, even though $\rho(\epsilon)$ is not well defined here). The idea is to build a measure which translates the following scenario : suppose that before imposing the potential 
%$V$ which yields the Cantor spectrum, the system is evolving according to a potential $W$ such that the spectrum is 
%continuous, ; for instance, $W$ could be a periodic potential, in which case the spectrum has a countable number of gaps and
%is continuous. Between each gap, the integrated density of modes is bounded, as the integral of a well defined function : the 
%density of states, over a bounded interval. Now, we slightly change the potential into a quasi-periodic one. This will add new gaps to the initial energy 
%bands, but the integrated density of states, e.g. the number of modes, has to remain constant.

%We shall imagine that the potential $V$ is such that its spectrum is a - finite or infinite - union of triadic Cantor sets ; 
%therefore, the total integrated density of states may be unbounded, but the integral over each triadic Cantor set - the "initial 
%energy bands" in the above analogy - is finite. 
       

Without the loss of generality, we suppose that the integral of $d\mu$ over $K$ equals 1 : $\int_{K}d\mu = 1$. We then proceed iteratively : starting from a segment $[0,1]$, we set $\rho(\epsilon) = 1$ everywhere ; the integral thus equals 1. Now, divide the segment in three, remove the central part and define $\rho(\epsilon)$ to be equal to $\frac{3}{2}$ on $[0,\frac{1}{3}]$ and $[\frac{2}{3}, 1]$, and null on $[\frac{1}{3},\frac{2}{3}]$. We still have : $\int_0^1 \rho(x)dx = 1$. We repeat this process infinitely, and thus define $d\mu$ as a limit : for any function $f$ defined on $[0,1]$, the integral of $f$ over $K$ is : 
\[ \int_{K} f(x)d\mu(x) = \lim_{n \to \infty} \left(\frac{3}{2} \right)^{n} \sum_{a_{j_n}\in P_{n}} \int_{a_{j_n}}^{a_{j_n}+3^{-n}} f(x)dx \]
where $\{P_{n}\}$ is the set of the left edges of the remaining segments after $n$ iterations in the construction of the Cantor set.

One can show (see annexe) that $d\mu$ verifies the following important property : 
\begin{equation}
 \int_{K} f(x)d\mu(x) = \frac{1}{2} \int_{K} f\left(\frac{x}{3}\right)d\mu(x) + \frac{1}{2} \int_{K} f\left(\frac{x+2}{3}\right)d\mu(x)
\end{equation}  
  
This property still holds for multiple integrals ; it becomes, for a double integral :

\begin{eqnarray*}
\lefteqn{ \int\limits_{K\times K} f(x,y)d\mu(x)d\mu(y) }\\
& = &\frac{1}{4} \int\limits_{K\times K} f\left(\frac{x}{3},\frac{y}{3}\right)d\mu(x)d\mu(y) \\
& + & \frac{1}{4} \int\limits_{K\times K} f\left(\frac{x+2}{3},\frac{y}{3}\right)d\mu(x)d\mu(y)\\
& + &\frac{1}{4} \int\limits_{K\times K} f\left(\frac{x}{3},\frac{y+2}{3}\right)d\mu(x)d\mu(y)\\
& + & \frac{1}{4} \int\limits_{K\times K} f\left(\frac{x+2}{3},\frac{y+2}{3}\right)d\mu(x)d\mu(y)
\end{eqnarray*}


\begin{figure}
\centering
\includegraphics[scale=0.3]{idos_cantor.eps}
\caption{Integrated density of states (IDOS) in the case of a triadic Cantor set energy spectrum. The energies are expressed in arbitrary unit and the IDOS has been normalised.}
\end{figure}

\section{}

Let us now study the dynamics of the system described in the previous paragraph.
There are several physical quantities which are useful to characterize the evolution of the system, such as the RMS displacement, the participation ratio or the auto-correlation function.

We will focus here on the auto-correlation function, then discuss briefly the participation ratio and the RMS displacement. 
%However, we shall not study thoroughly the RMS displacement ; as a matter of fact, some numerical results, obtained for a system of cavity polaritons placed in a quasi-periodic potential ([4]) and having a Cantor set like spectrum, indicate that the RMS displacement does not seem to display any characteristic  behaviour due to the fractal feature. 

\

The auto-correlation function, or time averaged return probability, $C(t)$, is by definition the time average of the probability $p(t) = |\langle\psi(0)|\psi(t)\rangle|^{2}$ to find the system in its initial state after a time $t$ : 
\[ C(t) = \frac{1}{t}\int_0^t |\langle\psi(0)|\psi(t')\rangle|^{2} dt' \]
We could directly try to calculate $p(t)$ to study the evolution in time of the system ; however this quantity often exhibits fast local oscillations, which could hide the effects of the fractal feature of the spectrum. Its time average, on the other hand, yields interesting results, as we shall see.

%\medskip

We start with the spectral decomposition of $|\psi(t)\rangle$ : 
\begin{equation*}
|\psi(t)\rangle = \int_{spectrum} g(\epsilon) e^{-i\epsilon t/\hbar} |\phi(\epsilon)\rangle d\tilde\mu(\epsilon)
\end{equation*}

The first step is to obtain an integral over $K$, with a dimensionless variable, in order to use the measure introduced 
earlier.

%Since $K_{\epsilon_{max}}$ is obtained from $K$ by applying a dilatation of factor $\epsilon_{max}$, we can make the 
%change of variable : $\lambda = \epsilon/\epsilon_{max}$ with $\lambda \in K$ and rewrite : 
%\begin{equation*}
%\fl |\psi(t)\rangle = \int_{K_{\epsilon_{max}}} g(\epsilon) e^{-i\epsilon t/\hbar} |\phi(\epsilon)\rangle d\mu_{\epsilon_{max}}(\epsilon)=\int_{K} g(\lambda\epsilon_{max}) e^{-i\lambda\epsilon_{max} t/\hbar} |\phi(\lambda\epsilon_{max})\rangle d\mu(\lambda\epsilon_{max})
%\end{equation*}
We impose that $|\psi(t)\rangle$ is normalized to unity : 
\begin{equation*}
\langle\psi(t)|\psi(t)\rangle=\int_{spectrum} |g(\epsilon)|^{2} d\tilde\mu(\epsilon)=1
\end{equation*}

This implies that $|g(\epsilon)|^{2} \rightarrow_{\epsilon \to \infty} 0$. We will therefore assume that $|g(\epsilon)|^{2}$
is non-zero only for a finite number of triadic Cantor sets, and that it is constant on these sets. By construction, a finite union
of triadic Cantor sets is also a triadic Cantor set embedded in a larger segment, say of - Lesbesgue - measure $\epsilon_{max}$, 
which without loss of generality we will assume to originate at zero. Let us note $K_{\epsilon_{max}}$ this new set, 
obtained from $K$ by applying a dilatation of factor $\epsilon_{max}$. We can make the
change of variable : $\lambda = \epsilon/\epsilon_{max}$ with $\lambda \in K$           
and define a new measure : 
\[d\tilde\mu_{\epsilon_{max}}(\lambda) = |g(\epsilon)|^{2} d\tilde\mu(\lambda\epsilon_{max})\]

 Assuming that $|g(\epsilon)|$ is uniform, we can consider that $d\tilde\mu_{\epsilon_{max}}(\lambda)$ is the measure $d\mu$ introduced in the previous paragraph. To alleviate the notations, we shall now write : $d\tilde\mu_{\epsilon_{max}} = d\mu$. 

Thus :
\begin{eqnarray*}
 C(t) &=  \frac{1}{t}\int_0^t |\langle\psi(0)|\psi(t')\rangle|^{2} dt'\\
&= \frac{1}{t} \int_0^t |\int_{K_{\epsilon_{max}}} |g(\epsilon)|^{2}e^{-i\epsilon t'/\hbar}d\mu_{\epsilon_{max}}(\epsilon)|^{2}dt' \\
%&=& \frac{1}{t} \int_0^t |\int_{K} e^{-i\lambda\epsilon_{max} t'/\hbar}d\mu(\lambda)|^{2}dt' 
\end{eqnarray*}
A standard calculation then leads to :
\begin{equation*}
C(t) = \int_{K\times K} \mbox{sinc} ((\lambda' - \lambda)\epsilon_{max} t/\hbar)d\mu(\lambda)d\mu(\lambda')
\end{equation*}
We now make the change of variables : $u=\frac{\epsilon_{max}t}{\hbar}$ :
\begin{equation*}
C(t) = C(u) = \int_{K\times K} \mbox{sinc} ((\lambda' - \lambda)u)d\mu(\lambda)d\mu(\lambda')
\end{equation*}

The idea is now to transform this double integral into a one variable integral, and then to use Mellin transform and the scaling property (1) of $d\mu$.

%\medskip

Let us do the change of variables : $l=|\lambda' - \lambda|$, $0\leq l\leq1$, and let $dS(l,\mu)$ be the surface (in the sense of $d\mu$) of the 2 dimensional Cantor dust\footnote{obtained by iteration of the following process : start with a full square, divide it in 9 equal squares and keep only the four at the corners of the initial square ; see fig.(2)} contained in the two strips of infinitesimal width located at  $\lambda =  \lambda' +l$ and $\lambda =  \lambda' - l$ (fig. 1).

\begin{figure}
\centering
\includegraphics[scale=0.6]{cant-carpet3.PNG}
\caption{2 dimensional Cantor dust (represented here at the second step of construction) ; the infinitesimal surface $dS(l,\mu)$ corresponds to the surface of the Cantor dust, in the sense of $d\mu$, contained in the two red strips.}
\end{figure}

Since $\mbox{sinc} ((\lambda' - \lambda)t)$ can be considered constant on these strips, we obtain : 
\begin{equation}
C(u) = \int_0^1 \mbox{sinc} (lu) dS(l,\mu)
\end{equation}
We now have to determine $dS(l,\mu)$. For this we shall use Mellin transforms.

%\medskip

Let : $M_{dS}(s) = \int_{0}^{1} l^{s-1}dS(l,\mu)\equiv \int_{0}^{1} l^{s-1}\frac{dS(l,\mu)}{dl}dl$ be the Mellin transform of $dS(l,\mu)$, which is sometimes referred to in the literature as the energy integral ([5], [8]) (note that we wrote $\int_{0}^{1}$ and not $\int_{0}^{\infty}$ : $dS(l,\mu)$ is defined on $[0,1]$, but can be extended on $[0,\infty[$ if we define it as equal to $0$ outside $[0,1]$).  

Now, using the definition of $dS(l,\mu)$ and the property (1) : 
\begin{eqnarray*}
 \lefteqn{M_{dS}(s) = \int_0^1 l^{s-1}dS(l,\mu)}\\
& = &\int\limits_{K\times K} |x-y|^{s-1}d\mu(x)d\mu(y) \\
& = &\frac{1}{4}\int_{K\times K} \left(|\frac{x}{3}-\frac{y}{3}|^{s-1}+|\frac{x+2}{3}-\frac{y+2}{3}|^{s-1}\right)d\mu(x)d\mu(y) \\
& + & \frac{1}{4}\int\limits_{K\times K} \left(|\frac{x}{3}-\frac{y+2}{3}|^{s-1}+|\frac{x+2}{3}-\frac{y}{3}|^{s-1}\right)d\mu(x)d\mu(y) \\
\end{eqnarray*}
%Note that the two terms of the sum in the first integral on the right are equal, as well as the terms in the second one ; thus, rearranging the terms and taking the factor $3$ out of the integral :

\begin{eqnarray*}
 \lefteqn{M_{dS}(s)=\frac{3^{1-s}}{2}\int_{K\times K} |x-y|^{s-1}d\mu(x)d\mu(y)}\\
 &+& \underbrace{\frac{3^{1-s}}{2}\int_{K\times K} |x-y+2|^{s-1}d\mu(x)d\mu(y)}_{\gamma(s)}
 \end{eqnarray*}

Thus : \[M_{dS}(s) = \frac{\gamma(s)}{1-3^{1-s}/2}\]
with $\gamma(s) = \frac{3^{1-s}}{2}\int_{K\times K}|x-y+2|^{s-1}d\mu(x)d\mu(y)$.


The inverse Mellin transform then gives $\frac{dS(l,\mu)}{dl}$ :
 
\begin{eqnarray*}
\lefteqn{\frac{dS(l,\mu)}{dl} = \frac{1}{2i\pi}\int_{\gamma -i\infty}^{\gamma +i\infty} M_{dS}(s)l^{-s}ds}\\
%&=&\frac{1}{2i\pi}\int_{\gamma -i\infty}^{\gamma +i\infty} \zeta(1-\alpha)l^{-\alpha}d\alpha\\
 & =  & \frac{1}{2i\pi}\int_{\gamma -i\infty}^{\gamma +i\infty} \frac{\gamma(s)}{1-\frac{3^{1-s}}{2}}l^{-s}ds
%&=& \sum_{n} \left[\frac{\alpha - (1-\alpha_{n})}{1-\frac{3^{1-\alpha}}{2}}l^{-\alpha}\gamma(\alpha)\right]_{\alpha=\alpha_{n}} \mbox{(residue theorem)}
\end{eqnarray*}
It is important to note that $\gamma(s)$ has no poles. It is clear from its definition : 
\[\gamma(s) = \frac{3^{1-s}}{2}\int_{K\times K}|x-y+2|^{s-1}d\mu(x)d\mu(y)\] 
Since for all $x,y\in[0,1]$, the inequality $1\leq2-x+y\leq 3$ holds, $|x-y+2|^{s-1}$ is well defined and bounded on $K\times K$ for any $s\in\mathbb{C}$, and thus $\gamma(s)$ is well defined and has no poles. The only poles of $\frac{\gamma(s)}{1-\frac{3^{1-s}}{2}}l^{-s}$ are the $\{s_{n}\}_{n \in \mathbb{Z}}$ such that :   
 $1-s_{n} = \frac{\ln(2)}{\ln(3)}+\frac{2i\pi n}{\ln(3)}$. We now apply the residue theorem :

\begin{eqnarray*}
\lefteqn{\frac{dS(l,\mu)}{dl}= \sum_{n} \left[\frac{s - s_{n}}{1-\frac{3^{1-s}}{2}}l^{-s}\gamma(s)\right]_{s=s_{n}}}\\
&=& \frac{l^{d_K-1}}{\ln(3)}\sum_{n}l^{\frac{2i\pi n}{\ln(3)}}\underbrace{\gamma\left(1-d_K-\frac{2i\pi n}{\ln(3)}\right)}_{\gamma_{n}}
\end{eqnarray*}
 
where we used : \[\lim_{s \to s_{n}}\frac{s - s_{n}}{1-\frac{3^{1-s}}{2}} = \frac{1}{\ln(3)}\] and
with $d_K=\frac{\ln(2)}{\ln(3)}$ and  
$$\gamma_{n} = \int_{K\times K} |x-y+2|^{-d_K-\frac{2i\pi n}{\ln(3)}}d\mu(x) d\mu(y)$$ 
Inserting this expression in (2), we get :
\begin{eqnarray*}
\lefteqn{C(u) = \frac{1}{\ln(3)}\sum_{n \in \mathbb{Z}}\gamma_n \int_0^1 l^{d_K-1}l^{2i\pi n/\ln(3)}\gamma_{n}\mbox{sinc}(lu)dl}\\
& = &\frac{u^{-d_K}}{\ln(3)}\sum_{n \in \mathbb{Z}}\gamma_n u^{-2i\pi n/\ln(3)} \int_0^u v^{d_K-1+2i\pi n/\ln(3)}\mbox{sinc}(v)dv
\end{eqnarray*}
with the change of variable : $v=ul$.
Since $\int_0^u v^{d+\frac{2i\pi n}{\ln(3)}-1}\mbox{sinc}(v) dv$ quickly converges to $\int_0^{\infty} v^{d_K+\frac{2i\pi n}{\ln(3)}-1}\mbox{sinc}(v)dv$, one can replace $\int_0^u$ by $\int_0^{\infty}$.

(More precisely : 
%$$|\int_0^u v^{d+\frac{2i\pi n}{\ln(3)}-1}\mbox{sinc}(v) dv -  \int_0^{\infty} v^{d+\frac{2i\pi n}{\ln(3)}-1}\mbox{sinc}(v)dv| < \int_u^{\infty}|v^{d-2}|dv = \frac{u^{d-1}}{1-d}$$ 
%therefore :
$|\int_0^u v^{d_K+\frac{2i\pi n}{\ln(3)}-1}\mbox{sinc}(v) dv -  \int_0^{\infty} v^{d_K+\frac{2i\pi n}{\ln(3)}-1}\mbox{sinc}(v)dv| = O(u^{d_K-1})$)

Using :
\begin{eqnarray*}
\lefteqn{\int_0^{\infty} v^{d_{Kn}+\frac{2i\pi n}{\ln(3)}-1}\mbox{sinc} (v)dv}\\ 
& = & \sin\left(\frac{\pi}{2}(d_{K,n}-1)\right)\Gamma\left(d_{K,n}+\frac{2i\pi n}{\ln(3)}-1\right)
\end{eqnarray*}
with $\Gamma$ the Euler gamma function, and $d_{K,n} = d_K +\frac{2i\pi n}{\ln(3)}$ one finally finds, for $u>>1$ (see fig. (3)): 

\[C(u) = \frac{u^{-d_K}}{\ln(3)}\sum_{n}\gamma_{n}u^{-\frac{2i\pi n}{\ln(3)}}\sin\left(\frac{\pi}{2}(d_{K,n}-1)\right)\Gamma(d_{K,n}-1)\]

%or, for $t>>\frac{\hbar}{\epsilon_{max}}$ (see fig. (3)):
%\begin{equation}
 %C(t)=\frac{1}{\ln(3)}\left(\frac{\epsilon_{max}t}{\hbar}\right)^{-d_K}\sum_{n}\gamma_{n}\left(\frac{\epsilon_{max}t}{\hbar}\right)^{-\frac{2i\pi n}{\ln(3)}}\sin\left(\frac{\pi}{2}(d_{K,n}-1)\right)\Gamma(d_{K,n}-1)
%\end{equation}

\begin{figure}[h]
\centering
\includegraphics[scale=0.3]{ct_loglog2-eps-converted-to.pdf}
\caption{Auto-correlation function in the case of a triadic Cantor set spectrum, in log-scale (log base 10).}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[scale=0.3]{ct_per-eps-converted-to.pdf}
\caption{Auto-correlation function multiplied by $u^{d}$ in the case of a triadic Cantor set spectrum, in log-scale (log base 10). This function is log-periodic in $u$, of period $\ln(3)$.}
\end{figure}

We can prove (see annexe) that the coefficients in the expression (3) decay at least like $\frac{1}{n^{3/2-d}}$ :
\[|\gamma_{n}\sin\left(\frac{\pi}{2}(d+\frac{2i\pi n}{\ln(3)}-1)\right)\Gamma(d+\frac{2i\pi n}{\ln(3)}-1)| = O\left(\frac{1}{n^{3/2-d}}\right)\]

\section{}
%Discussion

The analytical expression (4) of the auto-correlation function is the main result of this paper which we shall now discuss. 
%We will also highlight the similarity with numerical results obtained with the above mentioned Fibonacci cavity. Furthermore, we will try to generalize the method and discuss some other relevant quantities having or likely to have a similar behaviour due to the fractal spectrum.

We find that $C(u)$ is of the form : $C(u) = u^{-d_{K}}g(u)$, with $d_{K}=\frac{\ln(2)}{\ln(3)}$ the fractal dimension of the triadic Cantor set $K$ and  
\[g(u)=\sum_{n \in \mathbb{Z}} \gamma_{n} u^{-\frac{2i\pi n}{\ln(3)}}\sin\left(\frac{\pi}{2}(d_n-1)\right)\Gamma\left(d_n-1\right)\]
with $d_n = d+\frac{2i\pi n}{\ln(3)}$,
a log-periodic function. 
The presence of the fractal dimension $d_{K}$ in the exponent is consistent with previous work on the asymptotic behaviour 
of the auto-correlation function for a Cantor spectrum : $C(t)\approx_{t \to \infty} t^{-d_{k}}$([3], [4]). 

The log-periodicity is the signature of the scaling  symmetry of the spectrum. To see this, note that $C(t)$ is in fact 
the spectral average, for the measure $d\mu$, of the wavelet transform sinc : $C(t) = \int_{K\times K} \mbox{sinc} ((\lambda' - \lambda)\epsilon_{max} t/\hbar)d\mu(\lambda)d\mu(\lambda')$. 
Since the spectrum has a scaling symmetry, one would have expected to find a log-periodic behaviour for the sinc-wavelet transform, 
at least locally (e.g. for the quantity : $C_{local}(t,\lambda) = \int_{K} \mbox{sinc} ((\lambda' - \lambda)\epsilon_{max} t/\hbar)d\mu(\lambda)$). 
It was less obvious that the log-periodicity would remain after averaging over the spectrum ; this is due to the fact that 
we chose the density of modes and the distribution $|g(\epsilon)|^{2}$ to be uniform, and that the set has a unique scaling factor. 
In other words, in the case of a superposition of triadic Cantor sets for example, it is not certain that these oscillations
would survive the averaging.    
      

%\medskip

Let us now highlight the similarity between the theoretical graph of $C(t)$  for a triadic Cantor set spectrum (fig. (3), (4)) 
and the numerical results obtained with the Fibonacci cavity (more details are available in [6]) fig. (5). In this experiment, cavity polaritons are confined in wire cavities, consisting of $\lambda/2$ layers, using Bragg mirrors for the confinement in the vertical direction. The lateral dimension of the $200 \mu m $ long wires are modulated quasi-periodically : the modulation consists in two wire sections (letters) $A$ and $B$, of equal length but different width ; these letters are arranged in a finite sequence, $S_{j}$, obtained recursively using the following Fibonacci like algorithm :
$S_{j>2} = [S_{j-1}S_{j-2}] \mbox{ and } S_{1} = B\mbox{, } S_{2} = A$ 
where $[S_{j-1}S_{j-2}]$ is the concatenation of the sequences $S_{j-1}$ and $S_{j-2}$. The sequence $S_{\infty}$ becomes rigorously quasi-periodic as $j$ tends to infinity. We used a finite sized cavity, however, we observed the features of the fractal spectrum predicted by the theory : namely, gaps densely distributed and an integrated density of states $N(\epsilon)$ well described by a scaling form of the type $N(\epsilon) = \epsilon^{\ln a/\ln b}F(\frac{\ln\epsilon}{\ln b})$ and which follows the gap labelling theorem. 
The photon modes are described by a 2D scalar wave equation with vanishing boundary conditions on the boundary of the wire. This 2D problem is then reduced to a 1D Schroedinger equation with an effective quasi-periodic potential, translating the geometry of the cavity. The equation was then solved numerically (using the transfer matrix formalism) and some useful quantities were plotted : the IDOS, the participation ration, the RMS displacement and the time-averaged return probability. The latter is given fig. (5).

We found that the participation ratio and the return probability evidence a log-periodicity, which would be expected as a signature of the fractal spectrum.
 
\begin{figure}
\centering
\includegraphics[scale=1.2]{c_t_fibo.PNG}
\caption{Auto-correlation function of the wave paquet in the Fibonacci cavity ; numerical results for an initial gaussian wave paquet : $\psi(x,t=0) \sim e^{-2(x-x_{0})^{2}/w_{0}^{2}}$. The graphs correspond to different values of $w_{0}$, with identical $x_{0}$.}
\end{figure}  

%\medskip
\section{}

Let us now discuss the method used in a more general way, to highlight the connection between the presence of a scaling symmetry in a physical system and the typical form of certain quantities.
We see from the calculation that the scaling property of the spectrum leads naturally to a specific behaviour of some physical quantities defined as integrals over the spectrum - power law modulated by log-periodic oscillations. 
%To see this, and to identify which kind of quantities have this behaviour, let us state the method used in a more general way.

Consider a system having a fractal spectrum, characterized by a scaling property, which is expressed through a condition of the kind (1) verified by the measure.

One can show (see annexe) that a function $f$ verifying a functional equation of the type : 
\[  f(x) = \frac{1}{b}f(ax)+g(x)\] 
is (under some conditions on $g$) of the form :
\begin{equation}
f(x) = x^{\ln(b)/\ln(a)}G\left(\frac{\ln{x}}{\ln{a}}\right)
\end{equation}
with $G$ a 1-periodic function.

Now, in the case of a fractal $F$ described by a measure $\mu$ such that : 
\[\int_{F} f(x)d\mu(x) = \sum_{n\in\mathbb{N}}\frac{1}{b_n}f(a_{n}x)d\mu(x)\]
for some $a_n\in\mathbb{R}$, $b_n\in\mathbb{R}*$, then any function $\Psi(t)$ defined by an integral :
$\Psi(t) = \int_{F}h(x,t)d\mu(x)$ will satisfy :
\[ \Psi(t) = \sum_{n\in\mathbb{N}} \frac{1}{b_n}\int_{F}h(a_{n}x,t)d\mu(x)\]
If furthermore $h(x,t)$ can be expressed as a function of $x^{\alpha}t^{\beta}$, $\alpha, \beta \in \mathbb{C}$, $\beta\neq 0$ : $h(x,t)=\psi(x^{\alpha}t^{\beta})$, then :
\begin{eqnarray*}
\lefteqn{\Psi(t) = \sum_{n\in\mathbb{N}} \frac{1}{b_n}\int_{F}\psi(a_{n}x^{\alpha}t^{\beta})d\mu(x)}\\
&=& \sum_{n\in\mathbb{N}} \frac{1}{b_n}\int_{F}h(x,a_{n}^{1/\beta}t)d\mu(x) \\
&=& \sum_{n\in\mathbb{N}}\frac{1}{b_n} \Psi(a_{n}^{1/\beta}t)\\
&=& \frac{1}{b_j} \Psi(a_{j}^{1/\beta}t) + \Phi(t)
 \end{eqnarray*}
with  $\Phi(t) = \sum_{n\in\mathbb{N}-{j}}\frac{1}{b_n} \Psi(a_{n}^{1/\beta}t)$. Thus, if the Mellin transforms of $\Phi(t)$ and $\Psi(t)$ exist and have overlapping domains of definition, and if the Mellin transform of $\Phi(t)$ has no poles, then $\Psi(t)$ will be of the form : 
\[\Psi(t)=t^{\ln(b_j)/\ln(a_j^{1/\beta})}G\left(\frac{\ln{t}}{\ln{a_j^{1/\beta}}}\right)\] 
with $G$ 1-periodic.

%\medskip

More generally, this also occurs whenever we can rearrange the integral in order to have :  $\int_{F}h(\alpha x,t)d\mu(x)\propto  \int_{F}h(x,\beta t)d\mu(x)$.

%\medskip

Thus, we see in the case of quantum dynamics that, since the energy $\epsilon$ and the time $t$ are conjugate variables, it is likely that a time dependant quantity defined as an integral over the spectrum (through the spectral decomposition for instance) of a function defined using the evolution operator $e^{iHt/\hbar}$ or of any function which couples $\epsilon$ and $t$, will be of the form (5). This is well verified for $C(t)$.

%\medskip
\section{}

Let us now give an overview of the other physical quantities which do or should evidence such a behaviour ; we also discuss the RMS displacement, commonly studied in quantum dynamics.


We discuss first the time-averaged participation ratio, usually studied to get information on the dynamics of a quantum system. Moreover, this quantity was studied numerically for the Fibonacci cavity and the results support the idea of a log-periodic feature. By definition, the participation ratio is : 
\begin{equation*}
P_{\psi} = \frac{1}{\int_{espace} |\langle x|\psi\rangle
|^{4}dx} 
\end{equation*}
This quantity reflects the localization in space of the wave function $\psi(x,t)$ : if the state is localized, then $P_{\psi}\simeq 0$ ; if it is extended, then $P_{\psi}\simeq 1$ [9]. 
Taking its time-average, to have a time-dependent quantity, we define :
\begin{equation*}
P_{\psi}(t) = \frac{1}{t}\int_0^t \frac{dt'}{\int_{espace} |\langle x|\psi(t')\rangle|^{4}dx} 
\end{equation*}
Now, using Green functions, we can show [10] that : 
\begin{equation*}
 |\psi(x,t)|=\left|\int\limits_{space}{\int\limits_{spectrum} e^{-i\epsilon t} \phi(x,\epsilon)^{*}\phi(x',\epsilon)
\psi(x',0)dx'd\mu(\epsilon)}\right|
\end{equation*}
$|\phi(\epsilon)\rangle$ being the eigenfunction of the Hamiltonian associated with the energy $\epsilon$. The calculation has not yet completely been carried through, since sufficient information on the behaviour of the space correlation term $\phi(x,\epsilon)^{*}\phi(x',\epsilon)$ is still missing. However, two ideas can be considered.

\color{blue}
The first one is to obtain an approximate expression of the participation ratio by finding an upper and lower bound for $\int \phi(x,\epsilon)^{*}\phi(x',\epsilon)d\mu(\epsilon)$. In fact, it has been established ([11], [12]) for a lot of diffusive processes that the non-diagonal Green function $G(x,x',t)$ verifies : 
\begin{equation*}
 F_{t}(c_{1},c_{2},x,x') \leq G(x,x',t) \leq  F_{t}(c_{3},c_{4},x,x')
\end{equation*}

with $$F_{t}(c_{i},c_{j},x,x') = \frac{c_{i}}{t^{d_{s}/2}}e^{-c_{j}\left(\frac{|x-x'|^{d_{w}}}{t}\right)^{\frac{1}{d_{w}-1}}}$$ 
 where $c_{1}, c_{2},c_{3},c_{4}$are positive constants, $d_{w}$ the walk dimension which occurs in the dispersion relation $\langle x^{2}(t)\rangle = t^{2/d_{w}}$, $d_{s}$ the spectral dimension such that $d_{s}=\frac{2d_{f}}{d_{w}}$ and $d_{f}$ the Hausdorff dimension [13] of the space in which the diffusive process takes place. These bounds hold for diffusive processes ; however, it could be possible to use the same logic for the case of Schroedinger's equation, since the difference with the diffusion equation lies in the factor $i$ in the time derivative.
 %, but in both cases the eigenfunctions used in the definition of the Green function are the eigenfunctions of a time independent differential operator, of the form : $\Delta + V(\textbf{r})$ (Laplacian and a potential).

The second idea is the following : it is an empirical numerical result that, in the case of a euclidean manifold, the 
eigenfunctions $\{\phi(\textbf{r},\epsilon)\}$ of the Laplacian verify (see [12] and references therein) : 
$\overline{\phi(\textbf{r},\epsilon)^{*}\phi(\textbf{r'},\epsilon)} = f\left(\frac{|\textbf{r} - 
\textbf{r'}|}{l_{\epsilon}}\right)$, where $\overline{(...)}$ denotes the average over the points $\textbf{r}, \textbf{r'}$ such as $|\textbf{r} - \textbf{r'}| =$constant and $f$ is some function. This has recently been used for a Laplacian defined on a fractal [12]. Assuming furthermore that, by analogy with the euclidean case, the relation between $l_{\epsilon}$ and $\epsilon$ is of the form : $l_{\epsilon} \propto \epsilon^{\alpha}$, we are able to obtain an analytical expression of the participation ratio, with the expected log-periodic feature.
% We could, in order to use the same hypothesis, modify slightly the problem and consider that the Hamiltonian is reduced to a Laplacian, defined on a fractal, in such a way that the spectrum would also be a fractal. 


A numerical simulation would therefore be interesting in order to verify this conjecture. This requires, in our case, a system such that the Laplacian has a triadic Cantor set spectrum, or any self-similar fractal spectrum.
\color{black}
%\medskip

We shall now briefly discuss the RMS displacement, another quantity commonly studied both in quantum dynamics - spreading of 
the wave packet - and in diffusive processes. First note that, since the above method (which led to the expression of the 
auto-correlation function), uses the linearity of the integral, it is obvious that it is unlikely to work for the RMS
\[\Delta x = \sqrt{\langle(x(t)-x_0)^{2}\rangle} = \sqrt{\int_{space}(x(t)-x_0)^{2}|\psi(t,x)|^{2}dx}\]
because of the square root. However, we should obtain interesting results for the moment of order 2 :
%it even seems difficult to express the squared RMS displacement
\[(\Delta x)^2 = \langle(x(t)-x_0)^{2}\rangle = \int_{space}(x(t)-x_0)^{2}|\psi(t,x)|^{2}dx\]
Although a numerical study, in the case of the Fibonacci cavity, did not give encouraging results, note that this quantity is 
very close to 
\begin{eqnarray*}
\lefteqn{\int_{space}(x(t)-x_0)^{2}|\psi(t,x)|^{2}dx}\\
& = &\left|\int\limits_{space}{\int\limits_{spectrum} e^{-i\epsilon t} \phi(x,\epsilon)^{*}\phi(x',\epsilon)\psi(x',0)dx'd\mu(\epsilon)}\right|^{2}
\end{eqnarray*}
and that using the assumptions discussed for the participation ratio about 
the behaviour of $\overline{\phi(\textbf{r},\epsilon)^{*}\phi(\textbf{r'},\epsilon)}$ should also lead to interesting results.

Furthermore, the case of a Hamiltonian with a one-scale Cantor set spectrum has been solved numerically in the tight-binding 
approximation [14], and it was found that the moments $\langle(x(t)-x_0)^{\alpha}\rangle$ have a log-periodic behaviour. 
      

%\medskip


One can also use this method to derive the decay probability of a two level atom coupled to a fractal spectrum ([15]). 
We show in this case that Fermi's golden rule does not hold, and that the decay probability of the atom from an energy 
level $E_i$ to an energy $E_f\pm \Delta E$ is of the form (5). In this case, the log-periodic oscillations coud be interpretated as a signature of the topological properties of the 
spectrum : a closed, nowhere dense set with no isolated point.  
 
%\medskip

To finish, let us discuss briefly the question of turbulent diffusion, which was found to be related to fractals. A 
powerful tool for the study of turbulent diffusion is the method of breakdown coefficients, bdc, (see [16] and references therein).
If a turbulent system happens to have a physically distinguished scale factor (for instance, unstable eddies breaking up in 
typically always the same number of smaller eddies), then the moments of the bdc's have a log-periodic behaviour (namelly, 
a power law modulated by log periodic oscillations) [16]. It is interesting to note that these log-periodic oscillations were
derived with a very different approach - starting from a functionnal relation obtained in the frame of statistical physics. 
Thus, two very different physical problems (turbulence and quantum dynamics) led to a similar observation : there is a deep 
connection between the topological properties of self-similar sets and the occurence of anomalous log-periodic behaviour
of statistical and dynamical quantities.      


%\medskip
%\begin{center}
%\textbf{Conclusion}
%\end{center}
%\medskip

\section{Conclusion}

In summary, we derived the analytical expression of the auto-correlation function $C(t)$ in the case of a triadic Cantor 
set energy spectrum, and found that it is of the form : $ C(t) = \left(\frac{t}{\tau}\right)^{-d_K}g(\ln(t/\tau))$
 where $d_K=\frac{\ln(2)}{\ln(3)}$ is the fractal 
dimension of the triadic Cantor set, $\tau$ a time scale a $g$ a periodic function. 
We generalize the method and discuss the connection between the presence of self-similar structures or features in a physical system and the 
typical log-periodic behaviour of certain quantities.
In particular, the participation ratio and the moment of order 2 are likely to have such a behaviour for a one-scale Cantor 
set spectrum ; we are at the moment researching on these topics. The method developped also led to some first theoretical results for 
the spontaneous emission of a two level atom coupled to a fractal spectrum ; numerical and experimental work should be pursued in the near future.      
  
  
\medskip
\begin{center}
\textbf{References}
\end{center}
\medskip

%[1] for a review, E. Macia Barber, "Aperiodic Structures in Condensed Matter - Fundamentals and Applications", chap. 5, CRC Press, 2009 

%[2] L. A. Smith, J.-D. Fournier, E. A. Spiegel, 1986, Physics Letter, Vol. 114A, number 8,9

%[3] T. Nakayama, K. Yakubo, "Fractal concepts in condensed matter physics", Springer Science \& Business Media, 11 juil. 2003 

%[4] E. Akkerman, G. V. Dunne, A. Teplyaev, PRL 105, 230407 (2010)  

%[5] E. Akkermans, O. Benichou, G. Dunne, A. Teplyaev and
%R. Voituriez, Phys. Rev. E 86, 061125 (2012).

%[6] C. T. Hill, Fractal theory space: Spacetime of noninteger dimensionality, Phys. Rev. D 67,
%(2003), 085004.

%[7] E. Akkermans, Contemporary Mathematics 601, 1-22 (2013), arXiv:1210.6763
[1] B. Simon, ADVANCES IN APPLIED MATHEMATICS 3, 463--490 (1982)

\medskip

[2] D. Damanik and A. Gorodetski, Commun. Math. Phys. 305,
221 (2011)

\medskip

[3] R. Ketzmerick, K. Kruse, S. Kraut, T. Geisel, arXiv:cond-mat/9611006v2 (1997)

\medskip

[4] R. Ketzmerick, G. Petschel, and T. Geisel, Phys. Rev.Lett. 69, 695 (1992)

\medskip

[5] D. Bessis, J. D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, Phys. Rev. A, Vol. 36, number 2, July  15, 1987

\medskip

[6] Tanese, E. Gurevich, A. Lemaıtre, E. Galopin, I. Sagnes, A. Amo, J. Bloch, E. Akkermans, Physical Review Letters 112, 146404 (2014) 

\medskip

[7] J. Feder, "Fractals", Plenum Press - New York and London, 1988

\medskip

[8] E. ORLANDINIM, . C. TESI and G. TURCHETTI, Europhys. Lett., 21 (7), pp. 719-722 (1993)

\medskip

[9] F. Wegner, Z. Physik B 36, 209-214 (1980)

\medskip

%[9] M. Schreiber and H. Grussbach, Phys. Rev. Lett. 67, 607
%(1991)
[10] E. Akkermans, G. Montambaux, \textit{Mesoscopic Physics of Electrons and Photons}, (Cambridge University Press, 2007)

\medskip



%[13] E. Akkermans, O. Benichou, G. Dunne, A. Teplyaev and
%R. Voituriez, Phys. Rev. E 86, 061125 (2012)

%[14] A. A. Kilbas, M. Saigo, "A remark on asymptotics of the gamma function at infinity" in \textit{Study on Applications for Fractional Calculus Operators in Univalent Function Theory}, RIMS Kokyuroku, 1363 . pp. 33-36

[11] M. T. Barlow and E. A. Perkins, Prob. Theor. Rel. Fields 79, 543
(1988); M. T. Barlow, Lect. Notes Math. 1690, 1 (1998)

\medskip

[12] E. Akkermans, O. Benichou, G. V. Dunne,3 A. Teplyaev, and R. Voituriez, 
"Spatial log-periodic oscillations of first-passage observables in fractals", PHYSICAL REVIEW E 86, 061125 (2012)

\medskip

[13] C. P. Dettmann, N. E. Frankel, J. Phys A: Math. Gen. 26 (1993) 1009-1022

\medskip
 
[14] I. Guarneri, G. Mantica, Phys. Rev. Lett., Vol. 73, Num. 25 (1994)

\medskip

[15] E. Akkerman, E. Gurevich, EPL, 103 (2013) 30009

\medskip

[16] E. A. Novikov, Phys. Fluids A 2, 814 (1990) ; doi: 10.1063/1.857629

\medskip

[17] B. Hasselbatt, A. Katok, Chap. 4, "A First Course in Dynamics, with a Panorama of Recent Developments", Cambridge University Press, 2003

%[9] B. Simon, "Operators with singular continuous spectrum: I. General operators", Annals of Mathematics, 141 (1995), 131-145









  



%[14] W. Arendt, R. Nittka, W. Peter, and F. Steiner, in Mathematical
%Analysis of Evolution, Information, and Complexity,
%edited by W. Arendt and W. P. Schleich, (Wiley, New York,
%2009).

\medskip
\begin{center}
\textbf{Annexe}
\end{center}

\medskip
\textbf{\textit{Scaling property of the measure $d\mu$}}
\medskip

Let $f$ be an integrable function defined on $[0,1]$, with real or complex values.

We derive here the following property :
\[\int_{K} f(x)d\mu(x) = \frac{1}{2} \int_{K} f\left(\frac{x}{3}\right)d\mu(x) + \frac{1}{2} \int_{K} f\left(\frac{x+2}{3}\right)d\mu(x)\]

By definition :
\[\int_{K} f(x)d\mu(x)=\lim_{n \to \infty}\underbrace{\left(\frac{2}{3}\right)^{n}\sum_{a_{j_{n}}\in P_{n}}\int_{a_{j_{n}}}^{a_{j_{n}}+3^{-n}}f(x)dx}_{I_{n}(f(x))}\]

with  $P_{n}$ the set of the left edges of the remaining segments at the $n^{th}$ step of the construction of the triadic Cantor set.
One can directly check that the sequence $(P_{n})_{n}$ follows the recurrence property :
$$P_{n} = \frac{1}{3}P_{n-1}\cup\{\frac{2}{3}+\frac{1}{3}P_{n-1}\}$$
Therefore :
\begin{eqnarray*}
\lefteqn{I_{n}(f(x)) =\left(\frac{2}{3}\right)^{n}\sum_{a_{j_{n}}\in P_{n}}\int_{a_{j_{n}}}^{a_{j_{n}}+3^{-n}}f(x)dx} \\
& = &\left(\frac{2}{3}\right)^{n}\sum_{a_{j_{n-1}}\in P_{n-1}}\int_{a_{j_{n-1}}/3}^{a_{j_{n-1}}/3+3^{-n}}f(x)dx \\
& + & \left(\frac{2}{3}\right)^{n}\sum_{a_{j_{n-1}}\in P_{n-1}}\int_{\frac{2}{3}+a_{j_{n-1}}/3}^{\frac{2}{3}+a_{j_{n-1}}/3+3^{-n}}f(x)dx\\
%&\mbox{change of variable : $u=3x$, $v=3x-2$}&\\
& = &\left(\frac{2}{3}\right)^{n}\sum_{a_{j_{n-1}}\in P_{n-1}}\int_{a_{j_{n-1}}}^{a_{j_{n-1}}+3^{-n+1}}f\left(\frac{u}{3}\right)\frac{du}{3} \\
& + & \left(\frac{2}{3}\right)^{n}\sum_{a_{j_{n-1}}\in P_{n-1}}\int_{a_{j_{n-1}}}^{a_{j_{n-1}}+3^{-n+1}}f\left(\frac{v+2}{3}\right)\frac{dv}{3} \\ 
& = & \frac{3}{2}\frac{1}{3}\left[I_{n-1}\left(f\left(\frac{u}{3}\right)\right)+I_{n-1}\left(f\left(\frac{2+v}{3}\right)\right)\right]\\
\end{eqnarray*}
%\begin{eqnarray*}
   % I_{n}(f(x)) &=\left(\frac{2}{3}\right)^{n}\sum_{a_{j_{n}}\in P_{n}}\int_{a_{j_{n}}}^{a_{j_{n}}+3^{-n}}f(x)dx \\
    %&= \left(\frac{2}{3}\right)^{n}\left[\sum_{a_{j_{n-1}}\in P_{n-1}}\int_{a_{j_{n-1}}/3}^{a_{j_{n-1}}/3+3^{-n}}f(x)dx + \sum_{a_{j_{n-1}}\in P_{n-1}}\int_{\frac{2}{3}+a_{j_{n-1}}/3}^{\frac{2}{3}+a_{j_{n-1}}/3+3^{-n}}f(x)dx\right]\\
    %&\mbox{change of variables : $u=3x$, $v=3x-2$}&\\
    %&=\left(\frac{2}{3}\right)^{n}\left[\sum_{a_{j_{n-1}}\in P_{n-1}}\int_{a_{j_{n-1}}}^{a_{j_{n-1}}+3^{-n+1}}f\left(\frac{u}{3}\right)\frac{du}{3} + \sum_{a_{j_{n-1}}\in P_{n-1}}\int_{a_{j_{n-1}}}^{a_{j_{n-1}}+3^{-n+1}}f\left(\frac{v+2}{3}\right)\frac{dv}{3}\right] \\ 
   %&= \frac{3}{2}\frac{1}{3}\left[I_{n-1}\left(f\left(\frac{u}{3}\right)\right)+I_{n-1}\left(f\left(\frac{2+v}{3}\right)\right)\right]\\
%\end{eqnarray*}

But : $\lim_{n \to \infty}I_{n-1}\left(f\left(\frac{u}{3}\right)\right) = \int_{K}f\left(\frac{u}{3}\right)d\mu(u)$ and 
$\lim_{n \to \infty}I_{n-1}\left(f\left(\frac{2+v}{3}\right)\right) = \int_{K}f\left(\frac{2+v}{3}\right)d\mu(u)$, so finally :
\[\int_{K} f(x)d\mu(x) = \frac{1}{2} \int_{K} f\left(\frac{x}{3}\right)d\mu(x) + \frac{1}{2} \int_{K} f\left(\frac{x+2}{3}\right)d\mu(x)\]
This completes the proof.

\medskip

\textbf{\textit{Log-periodicity of functions verifying :  $f(x) = \frac{1}{b}f(ax)+g(x)$}}
\medskip

Let $f:\mathbb{R} \rightarrow \mathbb{C}$ verifying a functional equality of the kind :
\begin{equation}
f(x) = \frac{1}{b}f(ax)+g(x)
\end{equation}
Let's show that $f$ is of the form :
\[ f(x) = x^{\ln(b)/\ln(a)}G\left(\frac{\ln{x}}{\ln{a}}\right)\]
with $G$ $1$-periodic.

\medskip

We begin by taking the Mellin transform of (3) (assuming that the domain of definition of the Mellin transforms of $f$ and $g$ have a non empty intersection) :
\begin{eqnarray*}
\lefteqn{M_{f} = \int_0^{\infty}f(x)x^{s-1}dx}\\
&=& \frac{1}{b}\int_0^{\infty}f(ax)x^{s-1}dx +\int_0^{\infty}g(x)x^{s-1}dx\\
&=&\frac{1}{b}\int_0^{\infty}f(u)\left(\frac{u}{a}\right)^{s-1}\frac{du}{a} + M_{g}(s)
\end{eqnarray*}

So :
\[ M_{f}(s) = \frac{M_{g}(s)}{1-\frac{a^{-s}}{b}}\]
and by taking the inverse Mellin transform :
\begin{eqnarray*}
f(x)&=\frac{1}{2i\pi}\int_{c-i\infty}^{c+i\infty}M_{f}(s)x^{-s}ds\\
&=\frac{1}{2i\pi}\int_{c-i\infty}^{c+i\infty}\frac{M_{g}(s)}{1-\frac{a^{-s}}{b}}x^{-s}ds
\end{eqnarray*}

Assuming now that $M_{g}$ has no poles, the residue formula finally yields :
\begin{eqnarray*}
\lefteqn{f(x)=\sum_{n\in\mathbb{Z}}\left[\frac{(s-s_{n})M_{g}(s)x^{-s}}{1-\frac{a^{-s}}{b}}\right]_{s=s_{n}}}\\ 
& ( &\mbox{$s_{n}=-\frac{\ln{a}}{\ln{b}}-\frac{2i\pi n}{\ln{b}}$, poles of $\frac{M_{g}(s)}{1-\frac{a^{-s}}{b}}$})\\
&=& x^{\ln(b)/\ln(a)}\underbrace{\sum_{n\in\mathbb{Z}}c_{n}x^{2i\pi n/\ln{b}}}_{G(x)}
\end{eqnarray*}

which is what we wanted to prove.


\medskip

\textbf{\textit{Asymptotic behaviour of the coefficients}}
\medskip




We prove here that the coefficients in the expression (??) decay at least like $\frac{1}{n^{3/2-d}}$ :
\[|\gamma_{n}\sin\left(\frac{\pi}{2}(d+\frac{2i\pi n}{\ln(3)}-1)\right)\Gamma(d+\frac{2i\pi n}{\ln(3)}-1)| = O\left(\frac{1}{n^{3/2-d}}\right)\] 

To see this, let us start with $|\sin(\frac{\pi}{2}\left(d+\frac{2i\pi n}{\ln(3)}-1)\right)\Gamma(d+\frac{2i\pi n}{\ln(3)}-1)|$ ; we have ([14]) :
\[\lim_{n \to \infty} |\Gamma(d+\frac{2i\pi n}{\ln(3)}-1)| = \sqrt{2\pi}e^{-(d-1)}e^{-\frac{2\pi^{2} n}{2ln(3)}}\left(\frac{2\pi n}{\ln(3)}\right)^{d-\frac{3}{2}}\]
Furthermore, since $|\sin(\frac{\pi}{2}(d+\frac{2i\pi n}{\ln(3)}-1))| \sim_{n\rightarrow \infty} e^{\frac{2\pi^{2} n}{2\ln(3)}}e^{1-d}/2$, one has : 
\[|\sin(\frac{\pi}{2}(d+\frac{2i\pi n}{\ln(3)}-1))\Gamma(d+\frac{2i\pi n}{\ln(3)}-1)| = O\left(\frac{1}{n^{3/2-d}}\right)\]

Let us now study $\gamma_n$. By definition :
\begin{eqnarray*}
 \gamma_{n} &=& \int_{K \times K} \frac{d\mu(x)d\mu(y)}{|2-x+y|^{d_{n}}}\\ 
 &= &\lim_{k \to \infty}\left(\frac{3}{2}\right)^{2k}\sum_{a_{j_{k}},a_{j'{k}}\in P_{k}}\int_{a_{j_{k}}}^{a_{j_{k}}+3^{-k}}{\int_{a_{j_{k}'}}^{a_{j_{k}'}+3^{-k}}\frac{dxdy}{|2-x+y|^{d_{n}}}}
\end{eqnarray*}
with $d_{n} = \frac{\ln{2}}{\ln{3}}+\frac{2i\pi n}{\ln{3}}$ and $P_{k}=\{a_{j_{k}}\}$ the set of the left edges of the remaining segments after $k$ iterations in the construction of the triadic Cantor set $K$. Since $K$ is embedded in $[0,1]$, and since for all $x,y \in [0,1]$ the inequality $2 > x-y$ holds, one can suppress the absolute value in the integral. Now, for all $x,y \in [0,1]$, one has : $1\leq2-x+y \leq3$, so $|\frac{1}{|2-x+y|^{d_{n}}}| = \frac{1}{|2-x+y|^{d}} < 1$. 

Thus : 
\[|\gamma_{n}| < \int_{K \times K} \frac{d\mu(x)d\mu(y)}{|(2-x+y)^{d_{n}}|} < \int_{K \times K} d\mu(x)d\mu(y)=1\]
and so : $|\gamma_{n}| = O(1)$.
Numerically, it seems that we even have : $|\gamma_{n}| = O\left(\frac{1}{n}\right)$, but this has not been proven rigorously.

We therefore obtain finally that, at least :
\[|\gamma_{n}\sin\left(\frac{\pi}{2}(d+\frac{2i\pi n}{\ln(3)}-1)\right)\Gamma(d+\frac{2i\pi n}{\ln(3)}-1)| = O\left(\frac{1}{n^{3/2-d}}\right)\] 






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