text:
\section{Introduction }
The topic of space-filling curves has a very long history. Recently,
Rao and Zhang \cite{RaoZhangS16} as well as Dai, Rao, and Zhang \cite{DaiRaoZhang15} found a systematic method
to construct space-filling curves for connected self-similar sets satisfying the open set condition.
This method generalizes almost all known results in this field.

The self-affine sets have more complex structures than self-similar due to the different contraction ratios in different directions. There are almost no systematic works on the space-filling curves of self-affine sets except some examples provided by
Dekking \cite{Dekking82b},  Sirvent's study under some special conditions \cite{Sirvent2008,Sirvent2012}, boundary parametrizations of self-affine tiles by Akiyama and Loridant \cite{AkiyamaLoridant10, AkiyamaLoridant11}, and boundary parametrizations of a class of cubic Rauzy fractals by Loridant \cite{Loridant2016}.
The purpose of the present paper is to carry out studies in this direction.
In particular, we study the space-filling curves (or optimal parameterizations) of single-matrix graph-directed systems.

\subsection{Single-matrix GIFS}
Let $(\mathcal A,\Gamma)$ be a directed graph with vertex set $\mathcal A$ and edge set $\Gamma$.
Let
$$\mathcal{G}=\{g_e: \mathbb R^d\rightarrow \mathbb R^d;~e\in \Gamma\}$$ be a collection of contractions.
The triple $(V,\Gamma,\mathcal{G})$, or simply $\mathcal{G}$, is called \emph{a graph-directed iterated function system }(GIFS). Usually, we set $\mathcal A=\{1,2,\dots,N\}$.
Let $\Gamma_{ij}$ be the set of edges from vertex $i$ to $j$, then there exist unique non-empty compact sets $\{E_i\}_{i=1}^N$ satisfying
\begin{equation}\label{GIFS1}
E_i=\bigcup_{j=1}^N \bigcup_{e\in\Gamma_{ij}} g_e(E_j),\quad\quad 1\leq i \leq N.
\end{equation}
The family $\{E_i\}_{i=1}^N$ is called the family of \emph{invariant sets} of the GIFS (cf. \cite{MauldinWilliams88}).
When the graph has only one vertex with self-edges, then the GIFS will degenerate into an iterated function system (IFS).

We say that the system $\mathcal{G}$ satisfies the \emph{open set condition} (OSC) if there exist non-empty open sets $U_1,\dots, U_N$ such that
$$U_i\subset \bigcup_{j=1}^N \bigcup_{e\in\Gamma_{ij}} g_e(U_j),\quad\quad 1\leq i \leq N,
$$
and the right hand sets union are disjoint.

Denote $M=(m_{ij})_{1\leq i,j\leq N}$ the \emph{associated matrix} of the directed graph $(\mathcal A,\Gamma)$, that is, $m_{ij}=\sharp\Gamma_{ji}$ counts the number of edges from $j$ to $i$. We say a directed graph $({\mathcal A}, \Gamma)$ is \emph{primitive}, if the associated matrix is primitive, i.e., $M^n$ is a positive matrix for some $n$.
(See \cite{MauldinWilliams88}, \cite{Falconer90}.)  In this paper, we always assume that $({\mathcal A}, \Gamma)$ is primitive.

According to Luo and Yang \cite{LuoYang10}, ${\mathcal G}$ is called a \emph{single-matrix GIFS} if there is a $d\times d$ expanding matrix $A$ such that all functions related to $e\in \Gamma$ have the form
\begin{equation}\label{GIFS2}
g_e(x)=A^{-1}(x+d_e),
\end{equation}
where $d_e \in \mathbb{R}^d$.

\subsection{Optimal parametrization}
Let $E\subset \mathbb R^d$ be a compact set and ${\mathcal H}^s(E)$ denote the Hausdorff measure with respect to Euclidean norm of $E$. Basically, if
$\psi:[0,1]\to E$ is a continuous onto mapping, then $\psi$ is a \emph{parametrization} of $E$.
If $E$ is a self-similar set satisfying the open set condition, then
$0<{\mathcal H}^s(E)<\infty$, where $s$ is the Hausdorff dimension of $E$.
In this case, we may expect that $E$ has a better parametrization.
The following concept is first given by Dai and Wang \cite{DaiWang2010}:

\begin{definition}\label{optimal} {\rm
 A surjective mapping   $\psi:[0,1]\rightarrow E$ is called an
\emph{optimal parametrization} of $E$ if the following conditions are fulfilled.
\begin{enumerate}
  \item[($i$)]
   $\psi$ is a \emph{measure isomorphism} between $([0,1],$$ {\mathcal  B}([0,1]),$$ {\mathcal L})$ and
   $(E, {\mathcal  B}(E),{\mathcal H}^s)$, that is,  there exist $E'\subset E$ and $I'\subset [0,1]$ with full measure such that
   $\psi:~I'\to E'$ is a bijection and it is  measure-preserving in the sense that
  $$
  {\mathcal H}^s(\psi(F))=c{\mathcal L}(F) \text{ and } {\mathcal L}(\psi^{-1}(B))=c^{-1}{\mathcal H}^s(B),
  $$
  for any Borel set $F\subset [0,1]$ and any Borel set $B\subset E$, where $c={\mathcal H}^s(E)$. (See for instance, Walters \cite{Walters1982}.)
  \item[($ii$)] $\psi$ is \emph{$1/s$-H\"older continuous}, that is, there is a constant $c'>0$ such that
  $$
  \|\psi(x)-\psi(y)\|\leq c' \|x-y\|^{\frac{1}{s}} \  \text{ for all } x,y\in [0,1].
  $$
 We call $1/s$ the \emph{H\"older exponent}.
  \end{enumerate}

}
\end{definition}

For a self-affine set $K$, the Hausdorff measure may be $0$ or $\infty$, and hence
we cannot require an optimal parametrization satisfying (i) of the above. Also, the $1/s$-H\"older continuity may fail.
So we are forced to define the optimal parametrization in some other way.

To this end, we choose a pseudo-norm $\|\cdot\|_\omega$ instead of the Euclidean norm on $K$.
This pseudo-norm is first introduced by
Lemari{\'e}-Rieusset \cite{LemarieRieusset94} to deal with problems in the theory of wavelets.
   He and Lau \cite{HeLau08} developed the Hausdorff dimension (denoted by $\text{dim}_\omega$) and Hausdorff measure (denoted by $\mathcal H_{\omega}^s$) with respect to pseudo-norm (see Section \ref{Preli} for details). The advantage of the pseudo-norm is that we can regard the expanding matrix $A$ as a `similitude'.
   By replacing the norm, dimension and measure by their counterpart w.r.t.the pseudo-norm, we can define an optimal parametrization similar to Definition \ref{optimal}; details will be given in Section \ref{Preli}.

\subsection{Linear GIFS and our main result}
We equip a GIFS with an order structure (and
call it an \emph{ordered GIFS}), which induces a lexicographical order of the associated symbolic
space. An ordered GIFS is called a \emph{linear GIFS} if every two consecutive cylinders have nonempty intersections (see section \ref{Preli} for precise definitions).

Rao and Zhang \cite{RaoZhangS16} proved that as soon as we find a linear GIFS structure of a self-similar set, then
a space-filling curve can be constructed accordingly. Dai, Rao, and Zhang \cite{DaiRaoZhang15}
develop a very general method to explore linear GIFS structures of a given self-similar set.
In this paper, we only deal with the first problem for self-affine sets. Our main result is

\begin{theorem} \label{Main1}
Let $(\mathcal{A}, \Gamma, \mathcal{G},\prec)$ be  a linear single-matrix graph-directed IFS with expanding matrix $A$ satisfying the open set condition and assume that the associated matrix $M$ of the graph is primitive. Then there exists a parametrization $\psi_j$ of the invariant $E_j$ for all $j\in \mathcal{A}$ such that
\begin{enumerate}
  \item[($i$)]
   $\psi_j$ is a \emph{measure isomorphism} between $$([0,1], {\mathcal  B}([0,1]), {\mathcal L}) \text{ and } (E_j, {\mathcal  B}(E_j),{\mathcal H}_{\omega}^s).$$
  \item[($ii$)] There is a constant $c>0$ such that
  $$
  \|\psi_j(x)-\psi_j(y)\|_\omega\leq c\|x-y\|^{\frac{1}{\alpha}} \  \text{ for all } x,y\in [0,1],
  $$
where $\alpha=\dim_{\omega} E_j$.
 \end{enumerate}

\end{theorem}

According to the relation between Euclidean norm and pseudo-norm, we have the following result for the H\"older continuity of the parametrization $\psi_j$ obtained by the above theorem.
\begin{corollary}\label{intro-cor}
 Let $\lambda_M$ be the maximal eigenvalue of $M$. Let $\lambda_{max}$ and $\lambda_{min}$ be the maximal modulus and minimal modulus of the eigenvalues of $A$, respectively.
For any $0<\epsilon<\lambda_{min}-1$,

We will close this section by three examples, where the substitution rules induced linear GIFS are given directly. We shall discuss
in subsequent paper on how to obtain substitution rules, and once we have a substitution rule, the GIFS is constructed automatically. In the following examples, we calculate the value of H\"older exponent and it will be interesting to compare this with the value obtained from Corollary \ref{intro-cor}.

\begin{itemize}
\item  $\psi_j$ is $\frac{\ln (\lambda_{\max}+\epsilon)}{\ln \lambda_M}$-H\"older continuous if $\|\psi_j(x)-\psi_j(y)\|\geq 1$;

\item $\psi_j$ is $\frac{\ln (\lambda_{\min}-\epsilon)}{\ln \lambda_M}$-H\"older continuous if $\|\psi_j(x)-\psi_j(y)\|\leq 1$.
\end{itemize}
 The matrices $M$ and $A$ have the same meaning as in the above theorem.
\end{corollary}
\begin{figure}[htbp]
\subfigure[\text{Unit square $Q=\cup_{i=1}^6S_i(Q)$.}]{\includegraphics[width=0.33 \textwidth]{Squarefilling010}}
\subfigure[\text{Initial cycle.}]{\includegraphics[width=0.33  \textwidth]{SquarefillingThree01}}\subfigure[\text{First generation.}]{\includegraphics[width=0.33  \textwidth]{SquarefillingThree02}}\\
\subfigure[\text{Second generation.}]{\includegraphics[width=0.33 \textwidth]{SquarefillingThree03}}\subfigure[\text{Third generation.}]{\includegraphics[width=0.33 \textwidth]{SquarefillingThree04}}\subfigure[\text{ Parametrized square.}]{\includegraphics[width=0.33  \textwidth]{SquarefillingThree05}}
\caption{The space-filling curve of unit square given by IFS $\{S_i\}_{i=1}^6$.}\label{Ex3_fig1}
\end{figure}

In the following examples, we always use ${\bf e}_1, {\bf e}_2$ to denote the standard basis of $\mathbb{R}^2$.  Denote the maximal eigenvalue of $M$ by $\lambda_M$. And denote by the H\"older exponent with respect to Euclidean norm H\"older$_E$.

\begin{example}\label{Ex3}
\begin{figure}[htbp]
\includegraphics[height=7.5 cm]{GIFS_graph.eps}
\caption{The directed graph $G$ with vertex set $\{v_1, v_2, v_3\}$ and labelled edges obtained by the substitution rule \eqref{Ex3_subrule}. The label $(S,i)$ with $S\in \{S_i\}_{i=1}^6$ and $i\in \mathbb{N}$ means that $S$ is the contraction for the $i$-th edge starting from this vertex. The graph here determins a GIFS.}\label{GIFS_graph}
\end{figure}
{\bf A unit square.}  Let $Q$ be the unit square generated by the IFS $S_i(x)=A^{-1}(x+d_i),~~d_i\in \mathcal{D}$, where $\mathcal{D}=\{0,{\bf e}_2, {\bf e}_1+{\bf e}_2, {\bf e}_1, 2{\bf e}_1,2{\bf e}_1+{\bf e}_2\}$, and the expanding matrix is $A=\left(\begin{matrix}
3&0\\
0&2\\
\end{matrix}\right)$.
Let $a_1,a_2,a_3$ be the three vertice of the unit square, see Figure \ref{Ex3_fig1}(b). Denote by $v_i$ the edge from $a_i$ to $a_{i+1}, i=1, 2, 3$ (assume $a_4=a_1$). By the similar idea as Dai, Rao, and Zhang \cite{DaiRaoZhang15}, we can construct the following substitution rule which means to replace an edge $v_i$ by a trail which shares the same starting point and ending point with $v_i$.
\begin{equation}\label{Ex3_subrule}
\begin{split}
v_1 \longrightarrow\vphantom{a}&S_1(v_1)+ S_1(v_2)+S_3(v_1)+S_4(v_3)+S_4(v_1)+S_5(v_1),
\\
v_2 \longrightarrow\vphantom{a}&S_5(v_2)+ S_5(v_3)+S_4(v_2)+S_6(v_1)+S_6(v_2),
\\
v_3 \longrightarrow\vphantom{a}&S_6(v_3)+ S_3(v_2)+S_3(v_3)+S_2(v_2)+S_2(v_3)+S_2(v_1)+S_1(v_3),
\end{split}
\end{equation}
where we use the symbol $'+'$ to connect the consecutive edges or sub-trails.
It is shown in \cite{DaiRaoZhang15} that the above substitution rule induces a GIFS (see Figure \ref{GIFS_graph}) which is linear. We will show the set equation form of the GIFS in Section \ref{Sec_Example}.
The associated matrix of the substitution which is defined as the associated matrix of the directed graph $G$ obtained by the substitution is $$M=\left(\begin{matrix}
4&1&1\\
1&3&2\\
1&1&4\\
\end{matrix}\right),\quad\lambda_M=6,\quad\text{H\"older}_E=\log_6 2.$$
Compared with the unit square parametrized using the method as Hilbert or Peano which have the H\"older exponent $\frac{1}{2}$, the parametrization obtained here doesn't have a better smoothness.
\end{example}

\begin{figure}[htbp]
\includegraphics[width=0.38 \textwidth]{plane-filing_Deking0.eps}
\includegraphics[width=0.3 \textwidth]{plane-filing_Deking02.eps}\includegraphics[width=0.3 \textwidth]{plane-filing_Deking03.eps}\\
\includegraphics[width=0.4 \textwidth]{plane-filing_Deking04.eps}
\includegraphics[width=0.4 \textwidth]{plane-filing_Deking055.eps}
\caption{The figures show the first four approximation of $E_1$.}\label{Ex1_Fig2}
\end{figure}

\begin{example}\label{Ex1}
{\bf Dekking's plane filling curve \cite{Dekking82b}}. It is induced by the following substitution:
$$\begin{matrix}
\sigma: & a\mapsto abadadab; \quad b\mapsto cbcbadab; \quad c\mapsto cbcbcdadcbcd;\quad d\mapsto adcd.
\end{matrix}
$$
The set equation form of this substitution can be found in Section \ref{Sec_Example} and we obtain a linear GIFS as well. In this example, the expanding matrix is $A=\left(\begin{matrix}
4&0\\
0&2\\
\end{matrix}\right)$ and associated matrix of the substitution is
$$M=\left(\begin{matrix}
4&2&1&1\\
2&3&3&0\\
0&2&5&1\\
2&1&3&2\\
\end{matrix}\right),  \quad\lambda_M=8,\quad\text{H\"older}_E=\frac{1}{3}.$$
Then we can check that H\"older$_E$ is between the two H\"older exponents obtained by Corollary \ref{intro-cor}.
\end{example}

\begin{example}\label{Ex2}
\begin{figure}[h]
\includegraphics[width=0.35 \textwidth]{McmullensetA}\quad\quad \includegraphics[width=0.35 \textwidth]{Mcmullenset_005}
\caption{The left is Mcmullen set $T$ and the right is $\cup_{i=1}^5 E_i$.}\label{Ex2_fig1}
\end{figure}
\begin{figure}[h]
\includegraphics[width=0.35 \textwidth]{Mcmullenset_ske}\includegraphics[width=0.35 \textwidth]{Mcmullenset_substi}
\caption{Substitution rule of the Mcmullen set }\label{Ex2_fig2}
\end{figure}
{\bf A McMullen set}.  We consider the McMullen set depicted in Figure~ \ref{Ex2_fig1}, left. Denote the four vertices of the unit square by $a_1, a_2, a_3, a_4$.
Denote
$v_i=\overrightarrow{a_ia_{i+1}}, i=1,2,3,4~ (\text{ assume } a_5=a_1),$
and we get the following substitution rule (which is obtained from replacing the edge from $a_i$ to $a_{i+1}$ (left, Figure \ref{Ex2_fig2}) by the trail from $a_i$ to $a_{i+1}$ (right, Figure \ref{Ex2_fig2})).
\begin{flalign*}
 v_1 \longrightarrow\vphantom{a}&S_1(v_1)+ S_1(v_2)+S_4(v_4)+S_4(v_1),
\\
v_2 \longrightarrow\vphantom{a}&S_4(v_2)+ S_4(v_3)+S_2(v_2)+S_5(v_1)+S_5(v_2),
\\
v_3 \longrightarrow\vphantom{a}&S_5(v_3)+ S_5(v_4)+S_3(v_2)+S_3(v_3),
\\
v_4\longrightarrow\vphantom{a}&S_3(v_4)+S_3(v_1)+S_2(v_3)+S_2(v_4)+S_2(v_1)+S_1(v_3)+S_1(v_4).
\end{flalign*}
In this example, the expanding matrix is $A=\left(\begin{matrix}
3 &0\\
0 &2\\
\end{matrix}\right)$ and the associated matrix of the substitution is $$M=\left(\begin{matrix}
2&1&0&2\\
1&3&1&0\\
0&1&2&2\\
1&0&1&3\\
\end{matrix}\right), \quad\lambda_M=5,  \quad \text{H\"older}_E=\log_5 2.$$
Figure
\ref{Ex2_fig3} shows the visualization of the filling curve of the Mcmullen set $T$. To give a self-avoiding visualization, we round off the corners of the approximating curves.

\begin{figure}[h]
\includegraphics[width=0.3 \textwidth]{Mcmullenset_1}
\includegraphics[width=0.3 \textwidth]{Mcmullenset_2}\includegraphics[width=0.3 \textwidth]{Mcmullenset_3}
\caption{The first three approximations to the filling curve of Mcmullen set of Example \ref{Ex2}.}
\label{Ex2_fig3}
\end{figure}

\end{example}

Inspired by \cite{DaiRaoZhang15}, we shall do some further study on space-filling curve of self-affine fractals, such as space-filling curves of McMullen sets and of Rauzy fractals. To this matter, we try to find a similar method as we did for self-similar sets to construct a linear GIFS from a self-affine set or from a Rauzy fractal (See Figure \ref{Rauzy_fig} where we shows the approximations of the filling curve of a Rauzy fractal). Here we want to emphasis that to do the parametrization of a Rauzy fractal is based on invariant sets of a graph-directed iterated system, that is to say, we construct a linear GIFS from a given GIFS. (\cite{DaiRaoZhang15} focused on constructing a linear GIFS from a given IFS.)
 However, to construct a linear GIFS from the original iterated function system or from graph-directed iterated function system is a complicated work, so we will not do the part of constructions of linear GIFS and only give the set equations of linear GIFS in present paper. Hopefully, it will be shown in our following work. Figure \ref{Rauzy_fig} shows an interesting example of Rauzy fractal.

This paper is organised in the following way. In Section \ref{Preli}, we recall some definitions and notations, and then we give some additional explanations of the Examples~ \ref{Ex3}, \ref{Ex1}, and \ref{Ex2} in Section \ref{Sec_Example}. The last section  is devoted to the verification of our main Theorem \ref{Main1}.

\section{Preliminaries}\label{Preli}

\subsection{The symbolic space related to a graph $G$} Let $G=(\mathcal A,\Gamma)$ be a directed graph.
 A sequence of edges in $G$, denoted by ${\boldsymbol {\omega}}=\omega_1+\omega_2+\dots+\omega_n$, is called a \emph{walk}, if the
  terminal state of $\omega_i$ coincides with the initial state of  $\omega_{i+1}$ for $1\leq i \leq n-1$. A walk is called a \emph{trail}, if all the edges appearing in the walk are distinct. A trail is
called a \emph{path} if all the vertices are distinct.
 For $i\in \mathcal{A}$, let
 $$
 \Gamma_i^k,\ \Gamma_i^\ast, \text{ and } \Gamma_{i}^{\infty}
 $$
  be the set of
 all walks of length $k$, the set of all walks of finite length, and the set of all infinite walks,
  starting at state $i$, respectively.
Note that $\Gamma_i^\ast=\bigcup_{k\geq 1}\Gamma_i^k$.

 For ${\boldsymbol {\omega}}=(\omega_k)_{k=1}^\infty$, define by ${\boldsymbol {\omega}}|_n=\omega_1+\omega_2+\dots+\omega_n$  the prefix of ${\boldsymbol {\omega}}$ of length $n$.
Moreover, call
$[\omega_1\dots\omega_n]:=\{{\boldsymbol \gamma}\in \Gamma_i^\infty;~~{\boldsymbol \gamma}|_n=\omega_1+\dots+\omega_n\}$
the \emph{cylinder} associated with a walk $\omega_1+\dots +\omega_n$.

For a walk  ${\boldsymbol \gamma}=\gamma_1+\dots +\gamma_n\in\Gamma_i^n$, set $g_{{\boldsymbol {\gamma}}}:=g_{\gamma_1}\circ g_{\gamma_2}\dots\circ g_{\gamma_n}$, then we denote
$$
E_{\boldsymbol \gamma}:=g_{{\boldsymbol {\gamma}}}(E_{t({\boldsymbol {\gamma}})}),
$$
where $t({\boldsymbol {\gamma}})$ denotes the terminal state of the path $\boldsymbol \gamma$ (which equals$\gamma_n$ here). Iterating \eqref{GIFS1} $k$-times, we obtain
\begin{equation}\label{uni-set3}
E_i=\bigcup_{{\boldsymbol \gamma}\in \Gamma_i^k}E_{\boldsymbol \gamma}.
\end{equation}

We define the projection $\pi: \Gamma_1^{\infty}\times\dots\times\Gamma_N^{\infty} \rightarrow \mathbb{R}^d\times\dots\times\mathbb{R}^d$, where $\pi_i: \Gamma_i^{\infty} \rightarrow \mathbb{R}^d$ is given by
\begin{equation}\label{eq-projection}
\{\pi_i({\boldsymbol \omega})\}:=\bigcap_{n\geq 1}E_{\boldsymbol {\omega|_{n}}}.
\end{equation}
 For $x\in E_i$, we call ${\boldsymbol \omega}$ a \emph{coding} of $x$ if $\pi_i({\boldsymbol \omega})=x$. It is easy to see that $\pi_i(\Gamma_{i}^{\infty})=E_i$.

\subsection{Pseudo-norm and Hausdorff measure in pseudo-norm}

Denote by $B(x,r)$ the open ball with center $x$ and radius $r$. Recall that $A$ is the expanding matrix with $|\det A|=q$, then $V=A(B(0,1))\setminus B(0,1)$ is homeomorphic to an annulus.  For $\delta\in(0,\frac{1}{2})$,  choose a positive $\mathbb{C}^\infty$-function $\phi_{\delta}(x)$  with support in $B(0,\delta)$ such that $\phi_{\delta}(x)=\phi_{\delta}(-x)$ and $\int\phi_{\delta}(x)dx=1$, and then define a pseudo-norm $\|\cdot\|_{\omega}$ in $\mathbb R^d$ by
$$\|x\|_{\omega}=\sum_{n\in \mathbb Z}q^{-n/d}h(A^{n}x),$$
where $h(x)=\chi_V*\phi_{\delta}(x)=\int_{\mathbb R^d}\chi_V(x-y)\cdot\phi_{\delta}(y)dy$.

We list some basic properties of $\|\cdot\|_{\omega}$.

\begin{proposition}(See \cite[Proposition 2.1]{HeLau08}) \label{Pro-Weak}
The function $\|\cdot\|_{\omega}$ satisfies the following conditions.

(i) $\|x\|_{\omega}\geq 0$; $\|x\|_{\omega}=0$ if and only if $x=0$.

(ii) $\|x\|_{\omega}=\|-x\|_{\omega}$;

(iii) $\|Ax\|_{\omega}=q^{1/d}\|x\|_{\omega}\geq\|x\|_{\omega}$ for all $x\in \mathbb R^d$.

(iv) There exists  a constant $\beta>0$ such that $\|x+y\|_{\omega}\leq \beta \max \{\|x\|_{\omega}, \|y\|_{\omega}\}$ for any $x,y \in \mathbb R^d$.

\end{proposition}

The pseudo-norm $\|\cdot\|_{\omega}$ is comparable with the Euclidean norm $\|x\|$ .

\begin{proposition}\label{weaknorm}
(See \cite[Proposition 2.4]{HeLau08})  Let  $\lambda_{\max}$ and  $\lambda_{\min}$  be  the maximal modulus and minimal modulus of the eigenvalues of $A$, respectively. For any $0<\varepsilon<\lambda_{\min}-1$, there exists $C>0$(depends on $\varepsilon$) such that
$$ C^{-1}\|x\|^{\ln q/d \ln(\lambda_{\max}+\varepsilon)}\leq \|x\|_{\omega}\leq C\|x\|^{\ln q/d \ln(\lambda_{\min}-\varepsilon)},\quad\text{ if } \|x\|>1;$$
$$C^{-1}\|x\|^{\ln q/d \ln(\lambda_{\min}-\varepsilon)}\leq \|x\|_{\omega}\leq C\|x\|^{\ln q/d \ln(\lambda_{\max}+\varepsilon)},\quad\text{ if } \|x\|\leq 1.$$
\end{proposition}

The Hausdorff measure  with respect to the pseudo-norm $\|\cdot\|_{\omega}$  was given by He and Lau \cite{HeLau08} as follows.
For $E\subset\mathbb R^d$, set $\text{diam}_{\omega}E=\text{sup}\{\|x-y\|_{\omega};~~x,y\in E\}$ to be the $\omega$-diameter of $E$. For $s\geq 0, \delta>0$, set
$$\mathcal{H}_{\omega,\delta}^s(E)=\inf\big\{\sum_{i=1}^\infty (\text{diam}_{\omega}E_i)^s;~~E\subset \bigcup_i E_i,~~ \text{diam}_{\omega}E_i\leq\delta\big\},$$
$$\mathcal{H}_{\omega}^s(E)=\lim_{\delta\rightarrow 0}\mathcal{H}_{\omega,\delta}^s(E).$$
$\mathcal H_{\omega}^s$ has the translation invariance property and the scaling property \cite{HeLau08}, that is,
$$\mathcal H_\omega^s(E+x)=\mathcal H_\omega^s(E) \text{ and } \mathcal H_{\omega}^s(A^{-1}E)=q^{-s/d}\mathcal H_{\omega}^{s}(E).$$
Thus the Hausdorff dimension with respect to $\|\cdot\|_{\omega}$ can be defined by
$$\dim_{\omega} E=\inf\{s;~~\mathcal H_{\omega}^s(E)=0\}=\sup\{s;~~\mathcal H_{\omega}^s(E)=\infty\}.$$

\subsection{Linear GIFS}

Let $({\mathcal{A}},\Gamma,\mathcal{G})$ be a GIFS with vertex set $\mathcal{A}$, edge set $\Gamma$ and mapping set $\mathcal{G}$.
Let $\Gamma_i=\Gamma_i^1$ be the set of outgoing edges from the state $i$.
We call $({\mathcal{A}},\Gamma, \mathcal{G}, \prec)$ an \emph{ordered GIFS},
if  $\prec$ is a partial order on $\Gamma$ such that
\begin{enumerate}
  \item[($i$)] $\prec$ is a linear order when restricted on $\Gamma_j$ for every $j\in {\mathcal A}$;
  \item[($ii$)]  elements in $\Gamma_i \text{ and } \Gamma_j$ are not comparable if  $i \neq j$.
\end{enumerate}
(See \cite{RaoZhangS16} for detail.)

 The  order $\prec$  induces a lexicographical order on each
 $\Gamma_i^k$.

  Observe that $(\Gamma_i^k, \prec)$ is a linear order;
  two paths  ${\boldsymbol \gamma},{\boldsymbol \omega}\in \Gamma_{i}^k$ are said to be \emph{adjacent}
  if there is no walk between them  with respect to the order $\prec$.

\begin{definition}(see \cite{RaoZhangS16})\label{Linear} {\rm An ordered GIFS $({\mathcal{A}},\Gamma,\mathcal{G},\prec)$   with invariant sets $\{E_i\}_{i=1}^N$
 is called a \emph{linear} GIFS,  if for all $i\in {\mathcal A}$ and $k\geq 1$,
 $$
 E_{\boldsymbol \gamma}\cap E_{\boldsymbol \omega}\neq \emptyset
 $$
  for adjacent walks ${\boldsymbol \gamma},{\boldsymbol \omega}$  in $\Gamma_{i}^k$ .}
\end{definition}
For $i\in \mathcal{A}$, a walk ${\boldsymbol {\omega}}\in\Gamma_i^{\infty}$ is called the \emph{lowest} walk, if ${\boldsymbol {\omega}}|_n$ is the lowest walk in $\Gamma_i^n$ for all $n$; in this case, we call $a=\pi_i({\boldsymbol {\omega}})$ the \emph{head} of $E_i$.
Similarly, we define the highest walk ${\boldsymbol {\omega}}'$ of $\Gamma_i^\infty$, and we call $b=\pi_i({\boldsymbol {\omega}}')$ the
  the \emph{tail} of $E_i$.

\begin{definition}(see \cite[Definition 4.1]{RaoZhangS16} )\label{Chain}{\rm
An ordered GIFS is said to satisfy the \emph{chain condition}, if for any $i\in {\mathcal A}$, and any two adjacent edges
$\omega, \gamma\in \Gamma_i$ with $\omega \prec \gamma$,
$$g_\omega(\text{tail of } E_{t(\omega)})=g_\gamma(\text{ head of }E_{t(\gamma)}).$$
}
\end{definition}
\begin{lemma}\label{Text_linear}
An ordered GIFS is a linear GIFS if and only if it satisfies the chain condition.
\end{lemma}
{\bf Remark.} Definition \ref{Linear}, Definition \ref{Chain} and Lemma \ref{Text_linear} still make sense when  $({\mathcal{A}},\Gamma,\mathcal{G})$ is a single matrix GIFS.

\section{Examples}\label{Sec_Example}
In this section, we will give the set equation form of substitution rules which we showed in the previous examples and explain simply that these induced GIFS are linear.
\begin{example}\label{Ex31}
{\bf A unit square.} Through the edge-to-trail substitution rule, we get the following GIFS in the set equation form.
\begin{flalign*}
  E_1\circeq\vphantom{a}&S_1(E_1)\cup S_1(E_2)\cup S_3(E_1)\cup S_4(E_3)\cup S_4(E_1)\cup S_5(E_1),
\\
  E_2\circeq\vphantom{a}& S_5(E_2)\cup S_5(E_3)\cup S_4(E_2)\cup S_6(E_1)\cup S_6(E_2),\\
  E_3\circeq\vphantom{a}& S_6(E_3)\cup S_3(E_2)\cup S_3(E_3)\cup S_2(E_2)\cup S_2(E_3)\cup S_2(E_1)\cup S_1(E_3).
\end{flalign*}
Here we use $``\circeq"$ to emphasize that it is an ordered GIFS.
By the result of \cite{DaiRaoZhang15}, the above induced GIFS is a linear GIFS, or we can check it by Lemma \ref{Chain} directly.

\end{example}

\begin{example}\label{Ex11}
\begin{figure}[h]
\includegraphics[width=0.4 \textwidth]{plane-filing_Deking01.eps}\includegraphics[width=0.4 \textwidth]{plane-filing_Deking06.eps}
\caption{The left figure is $E=\cup_{i=1}^4E_i$, and the right one is $E_4$. Example \ref{Ex1}.}
\end{figure}

 {\bf Dekking's plane filling curve}   \cite{Dekking82b}.
Here we continue Example \ref{Ex1}. We translate the substitution of Dekking's example to the following set equation
{\small
\begin{flalign*}
AE_1\circeq\vphantom{a}& E_1\cup(E_2+{\mathbf e}_1)\cup(E_1+{\mathbf e}_1+{\mathbf e}_2)\cup(E_4+2{\mathbf e}_1+{\mathbf e}_2)
\cup(E_1+2{\mathbf e}_1)\cup\\ &(E_4+3{\mathbf e}_1)
\cup(E_1+3{\mathbf e}_1-{\mathbf e}_2)\cup(E_2+4{\mathbf e}_1-{\mathbf e}_2),\\
AE_2\circeq\vphantom{a}& E_3\cup(E_2-{\mathbf e}_1)\cup(E_3-{\mathbf e}_1+{\mathbf e}_2)\cup(E_2-2{\mathbf e}_1+{\mathbf e}_2)
\cup(E_1-2{\mathbf e}_1+2{\mathbf e}_2)\cup\\
&(E_4-{\mathbf e}_1+2{\mathbf e}_2)\cup(E_1-{\mathbf e}_1+{\mathbf e}_2) \cup(E_2+{\mathbf e}_2),\\
AE_3\circeq\vphantom{a}& E_3\cup(E_2-{\mathbf e}_1)\cup(E_3-{\mathbf e}_1+{\mathbf e}_2)\cup(E_2-2{\mathbf e}_1+{\mathbf e}_2)\cup(E_3-2{\mathbf e}_1+2{\mathbf e}_2)\cup
\\ &(E_4-3{\mathbf e}_1+2{\mathbf e}_2)\cup(E_1-3{\mathbf e}_1+{\mathbf e}_2)\cup(E_4-2{\mathbf e}_1+{\mathbf e}_2)\cup(E_3-2{\mathbf e}_1)\cup
\\ &(E_2-3{\mathbf e}_1)\cup(E_3-3{\mathbf e}_1+{\mathbf e}_2)\cup(E_4-4{\mathbf e}_1+{\mathbf e}_2),
\\
AE_4\circeq\vphantom{a}& E_1\cup(E_4+{\mathbf e}_1)\cup(E_3+{\mathbf e}_1-{\mathbf e}_2)\cup(E_4-{\mathbf e}_2).
\end{flalign*}
}
To check the chain condition, we need to calculate the heads and tails of $E_1, E_2, E_3, E_4$. We denote the head of a set $K$ by $h(E)$ and tail of a set $t(E)$. Then we have
 \begin{align*}
 h(E_1)=0,~t(E_1)={\mathbf e}_1,& ~h(E_2)=0, ~t(E_2)={\mathbf e}_2,\\
h(E_3)=0,~ t(E_3)=-{\mathbf e}_1,& ~h(E_4)=0,~t(E_4)=-{\mathbf e}_2.
\end{align*}
Thus it is easy to check that it satisfies the chain condition. Figure \ref{Ex1_Fig2} shows the proceeding of filling curve of $E_1$.
\end{example}
\begin{example}\label{Ex2plus}

{\bf A McMullen set}.
The McMullen set $T$ (see Figure \ref{Ex2_fig1}, left) in Example \ref{Ex2} is given by $T=\bigcup_{i=1}^5 S_i(T)$ with $$\Big\{S_i(x)=A^{-1}(x+d_{i})\Big\}_{i=1}^5,\quad d_1=0, d_2={\mathbf e}_1, d_3=2{\mathbf e}_1, d_4={\mathbf e}_2, d_5={\mathbf e}_1+{\mathbf e}_2.$$

Through the substitution rule in Example \ref{Ex2}, we obtain the following set equation form of a GIFS.
\begin{flalign*}
  E_1\circeq\vphantom{a}&S_1(E_1)\cup S_1(E_2)\cup S_4(E_4)\cup S_4(E_1),
\\
  E_2\circeq\vphantom{a}& S_4(E_2)\cup S_4(E_3)\cup S_2(E_2)\cup S_5(E_1)\cup S_5(E_2),\\
  E_3\circeq\vphantom{a}& S_5(E_3)\cup S_5(E_4)\cup S_3(E_2)\cup S_3(E_3),\\
 E_4\circeq\vphantom{a}&S_3(E_4)\cup S_3(E_1)\cup S_2(E_3)\cup S_2(E_4)\cup S_2(E_1)\cup S_1(E_3)\cup S_1(E_4).
\end{flalign*}
In the same way as Example \ref{Ex11}, we check that the above GIFS satisfied the chain condition. Then it is a linear GIFS and clearly the open set condition is satisfied. Actually the union of the invariant sets $\bigcup_{i=1}^4 E_i$ is the McMullen set $T$.
\end{example}

\section{Proof of Theorem \ref{Main1}}\label{Sec_Main1}
In this section,

 we prove Theorem \ref{Main1} by constructing an auxiliary GIFS (which we call measuring-recording GIFS), which is very similar to
 the proof in \cite{RaoZhangS16}.
However, the theorem related to the open set condition of Mauldin and Williams \cite{MauldinWilliams88} does  not hold when $A$ is not a similitude.
So we need to use the result of Luo and Yang \cite{LuoYang10} to modify the proof.

\subsection{Markov measure induced by GIFS}
Let $(\mathcal{A},\Gamma,\mathcal{G})$ be single-matrix GIFS with expanding matrix $A$ and  $\{E_i\}_{i=1}^N$ be the invariant sets. Denote $q=|\text{det} (A)|$.
And $M=(m_{ij})_{1\leq i,~j\leq N}$ is the associated matrix of the directed graph $(\mathcal{A},\Gamma)$. Due to the following lemma from \cite{LuoYang10}, we can construct the Markov measure.

  \begin{lemma}\label{YL} (\cite[Theorem 1.2]{LuoYang10})
  For  a single matrix GIFS $(\mathcal{A},\Gamma,\mathcal{G})$,  let $\lambda$ be the maximal eigenvalue of $M$.  If $M$ is primitive and the OSC holds, then for any $1\leq i \leq N$,

  (i) $\alpha=\dim_\omega E_i=d\frac{\log\lambda}{\log q}$;

  (ii) $0<\mathcal H_\omega^{\alpha}(E_i)<\infty$.

  (iii)  The right hand side of \eqref{GIFS1} is a disjoint union in sense of the measure of $\mathcal H_\omega^{\alpha}$.
\end{lemma}
{\bf Remark $(1)$.}  By item $(iii)$ of the above lemma, we immediately have
 $${\mathcal H}_\omega^{\alpha}(E_{\boldsymbol \omega}\cap E_{\boldsymbol \gamma})=0$$
  for any incomparable ${\boldsymbol \omega},{\boldsymbol \gamma} \in \Gamma_i^\ast$. (Two walks are said to be \emph{comparable} if one of them is a prefix of the other.)

{\bf Remark $(2)$.} Since $E_i=\bigcup_{j=1}^N\bigcup_{e\in \Gamma_{ij}}g_e(E_j)$, using Remark (1),  we get
  $$\mathcal H_\omega^{\alpha}(E_i)=\sum_{j=1}^{N}\sum_{e\in \Gamma_{ij}}\mathcal H_\omega^{\alpha}(g_e(E_j))=\lambda^{-1}\sum_{j=1}^{N}\sharp\Gamma_{ij}\mathcal H^{\alpha}_\omega(E_j).$$
This shows that $(\mathcal{H}_{\omega}^{\alpha}(E_1),\dots,\mathcal{H}_{\omega}^{\alpha}(E_N))$ is an eigenvector with respect to $\lambda$ of $M$.

In the rest of the section, we will always assume that ${\mathcal  G}$ satisfies the conditions of Lemma \ref{YL}. Then $0<\mathcal H_\omega^{\alpha}(E_i)<\infty$ for all $1\leq i \leq N$.

Now, we define Markov measures on the symbolic spaces $\Gamma_i^\infty$, $i\in\mathcal{A}$.
For arbitrary edge $e\in \Gamma$ such that $e\in \Gamma_{ij}$, set
\begin{equation} \label{our-weight}
p_e=\frac{\mathcal H_{\omega}^{\alpha}(E_j)}{\mathcal H_{\omega}^{\alpha}(E_i)}\lambda^{-1}.
\end{equation}
Using Remark $(2)$ of Lemma \ref{YL},
it is easy to verify that $(p_e)_ {e\in \Gamma}$ satisfies
\begin{equation}\label{weights}
\sum_{j\in {\mathcal A}}\sum_{e\in \Gamma_{ij}} p_e=1, \text{ for all } i\in \mathcal{A}.
\end{equation}
We  call $(p_e)_ {e\in \Gamma}$ a \emph{probability weight vector}.
 Let $\mathbb{P}_i$ be a Borel measure on $\Gamma_i^\infty$ satisfying the relations
\begin{equation}\label{measure}
\mathbb{P}_i([\omega_1\dots\omega_n])=\mathcal H_{\omega}^{\alpha}(E_i) p_{\omega_1}\dots p_{\omega_n}
\end{equation}
for all cylinder $[\omega_1\dots\omega_n]$. The existence of such measures is guaranteed by \eqref{weights}. We call $\{{\mathbb P_i}\}_{i=1}^N$ the \emph{Markov measures} induced by the GIFS ${\mathcal G}$.

Denote the restriction of ${\mathcal H}^{\alpha}_{\omega}$ on $E_i$ by $\mu_i={\mathcal H}^{\alpha}_{\omega}|_{E_i} , \text{ for }i=1,\dots, N.$ The following Lemma gives the relation between the Markov measure and the restricted Hausdorff measure.

 \begin{lemma}(see \cite{MauldinWilliams88, LuoYang10})\label{lem-Markov}
Suppose the single-matrix graph IFS $(\mathcal A, \Gamma, \mathcal G )$ satisfies the OSC and
  the associated matrix $M$ is primitive.
 Let $\pi_i: \Gamma_i^\infty\to E_i$ be the projections defined by \eqref{eq-projection}. Then
$
\mu_i=\mathbb{P}_i\circ \pi_i^{-1}.
$
 \end{lemma}

\subsection{The construction of measure-recording GIFS}

Let $(\mathcal{A},\Gamma,\mathcal{G},\prec)$ be a linear GIFS such that the open set condition is fulfilled and the associated matrix is primitive, then
$0<{\mathcal  H}_\omega^{\alpha}(E_i)<\infty$ for all $i$, where $\alpha=d\frac{\log\lambda}{\log q}$ by Lemma \ref{YL}.

For $i\in \mathcal{A}$, we list the edges in $\Gamma_i$ in the ascendent order with respect to $\prec$, i.e.,
$$
\gamma_1\prec \gamma_2\prec\dots\prec \gamma_{\ell_i}.
$$
 Recall that $t(\gamma)$ denotes the terminate vertex of an edge $\gamma$. Then by \eqref{GIFS1}, we can rewrite $E_i$ as
$$
E_i\circeq g_{\gamma_1}(E_{t(\gamma_1)})\cup \cdots \cup g_{\gamma_{\ell_i}}(E_{t(\gamma_{\ell_i})}).
$$
Here we use `$\circeq$' to emphasize the order of the union of the right side.

Denote by
$F_i=[0, {\mathcal  H}_\omega^{\alpha}(E_i)]$ an interval on $\mathbb{R}$, then by equation \eqref{weights}, we have
\begin{equation}\label{F_i}
F_i=\Big[0,~~\mathcal H_{\omega}^{\alpha}(g_{\gamma_1}(E_{t(\gamma_1)}))\Big]\cup\dots\cup\Big[\sum_{j=1}^{{\ell}_i-1}\mathcal H_{\omega}^{\alpha}(g_{\gamma_j}(E_{t(\gamma_j)})), ~~\sum_{j=1}^{{\ell}_i}\mathcal H_{\omega}^{\alpha}(g_{\gamma_j}(E_{t(\gamma_j)})) \Big].
\end{equation}
We define the mappings,
$$
f_{\gamma_k}(x)=q^{-\alpha/d}x+b_k: ~~\mathbb{R}\longrightarrow\mathbb{R}, \quad 1\leq k \leq \ell_i,
$$
where $b_k=\sum_{j=1}^{k-1}\mathcal H_{\omega}^{\alpha}(E_{t(\gamma_j)})q^{-\alpha/d}$. Then $F_i$ satisfies the following equation by \eqref{F_i}
\begin{equation}\label{partition-F}
F_i\circeq f_{\gamma_1}(F_{t(\gamma_1)})\cup \cdots \cup f_{\gamma_{\ell_i}}(F_{t(\gamma_{\ell_i})}).
\end{equation}
Repeating these procedures for all $i\in {\mathcal A}$, equation \eqref{partition-F} gives us an ordered GIFS on $\mathbb{R}$. Set $\mathcal{F}=\{f_{\gamma}: \mathbb{R}\longrightarrow\mathbb{R};~~ \gamma\in\Gamma\},$
and denote this GIFS by
$$
(\mathcal{A}, \Gamma, {\mathcal  F}, \prec),
$$
and call it the \emph{measure-recording GIFS} of $({\mathcal A}, \Gamma, {\mathcal G}, \prec)$. And the invariant sets of the measure-recording  GIFS are $\{F_i\}_{i=1}^N$. (See \cite{RaoZhangS16}.)

Obviously, the measure-recording GIFS has the same graph and the same order as the original GIFS; also keeps the Hausdorff measure information of the original GIFS.  And it is easy to check ${\mathcal F}$ satisfies the open set condition. In fact, the open intervals $\{U_i=(0,\mathcal H_{\omega}^{\alpha}(E_i))\}_{i=1}^N$ are the according open sets.

For an edge $e\in \Gamma$, the contraction ratio of $f_e$ is $q^{-\alpha/d}=\lambda^{-1}$, then it is easy to check $(\mathcal{L}(F_1),\dots,\mathcal{L}(F_N))$ is an eigenvector of $M$ with respect the eigenvalue $\lambda^{-1}$. Thus the Markov measure induced by the measure-recording GIFS coincides with $\{\mathbb{P}_i\}_{i=1}^N$ induced by the original GIFS.

Let
$$
\pi_i: \Gamma_i^\infty \rightarrow E_i \text{ and }
\rho_i: \Gamma_i^\infty \rightarrow F_i,\quad i=1,\dots, N,
$$
 be  projections w.r.t. the GIFS $({\mathcal  G})$ and $({\mathcal  F})$, respectively, (see \eqref{eq-projection}). Define
\begin{equation}\label{psi-map}
\psi_i:=\pi_i\circ \rho_i^{-1}.
\end{equation}
In \cite{RaoZhangS16}, it is shown that $\psi_i$ is a well-defined mapping from $F_i$ to $E_i$ since we consider a linear GIFS.

Now, we prove Theorem \ref{Main1} by showing that the mapping $\psi_i$ is an optimal parametrization of $E_i$.
\medskip

\noindent {\bf Proof of Theorem \ref{Main1}.}
We use the same notations as before,
let ${\mathcal  F}$ be the measure-recording GIFS of ${\mathcal  G}$. Through the discussion before, we denote the common Markov measure
induced by ${\mathcal G}$ and ${\mathcal F}$ by ${\mathbb P}_i$. $\psi_i=\pi_i\circ\rho_i^{-1}$ is the well-defined mapping from $F_i$ to $E_i$. Let
 $\nu_i={\mathcal L}|_{F_i}$ be the restriction of the Lebesgue measure on $F_i$ and $\mu_i={\mathcal H}_{\omega}^{\alpha}|_{E_i}$ be the restriction of the weak Hausdorff measure on $E_i$, then  $\nu_i=\mathbb{P}_i\circ\rho_i^{-1}$, $\mu_i=\mathbb{P}_i\circ\pi_i^{-1}$ by Lemma \ref{lem-Markov}.

The fact that $\psi_i$ is almost one to one and measure preserving follows by the same arguments as in the self-similar case and we refer to the proof of \cite[Theorem 1.1]{RaoZhangS16}.

We have to prove the $1/\alpha$-H\"older continuity of $\psi_i$.  From the previous construction, we know that $F_i=[0,\mathcal H_{\omega}^{\alpha}(E_i)]$. Now we choose two different points $x_1, x_2$ from $F_i$ which are determined by ${\boldsymbol {\omega}}=(\omega_i)_{i=1}^\infty$ and ${\boldsymbol {\gamma}}=(\gamma_i)_{i=1}^\infty$, respectively, that is,
$x_1=\rho_i({\boldsymbol {\omega}}), x_2=\rho_i({\boldsymbol {\gamma}})$. Let $k$ be the smallest integer such that $x_1, x_2$ belongs to two different cylinders. Set ${\boldsymbol {\omega}}|_k=\omega_1+\dots+\omega_k$, we know that ${\boldsymbol {\gamma}}|_k$ is only different from ${\boldsymbol {\omega}}|_k$ at last edge, i.e., ${\boldsymbol {\gamma}}|_k=\omega_1+\dots+\omega_{k-1}+\gamma_k$.
We consider two cases according to ${{\boldsymbol {\omega}}}|_k$ and ${{\boldsymbol {\gamma}}}|_k$ are adjacent or not.
First, we consider that ${\omega}_k$ and ${\gamma}_k$ are not adjacent. (See Figure \ref{unadjacent}.)
\begin{figure}[h]
  \centering

  \includegraphics[width=0.55 \textwidth]{adjacent_009.eps}
 \vspace{-1.0 cm}  \caption{ ${{\boldsymbol {\omega}}}$ and ${{\boldsymbol {\gamma}}}$ are not adjacent.} \label{unadjacent}
\end{figure}
 Then there is a cylinder
${\boldsymbol {\eta}}=\omega_1+\dots +\omega_{k-1}+\eta_k$ between ${\boldsymbol {\omega}}|_k$ and ${\boldsymbol {\gamma}}|_k$, so
$$
\|x_1-x_2\|\geq \text{diam}~F_{\boldsymbol \eta}\geq
 h  \cdot (q^{-\alpha/d})^k,
$$
where $h=\min\{{\mathcal  H}_\omega^{\alpha}(E_i);~i=1,\dots, N\}$.
Since $x_1$ and $x_2$ belong to $\rho([\omega_1\omega_2\dots\omega_{k-1}])$ and denote ${\boldsymbol {\omega}}_*=\omega_1+\dots +\omega_{k-1}$, the images of $x_1$ and $x_2$ under $\pi_i\circ\rho_i^{-1}$, which denote by $y_1$ and $y_2$, respectively,
belong to $\pi_i([{{\boldsymbol {\omega}}}_*])=E_{{{\boldsymbol {\omega}}}_*}$.
Then we have
\begin{equation}\label{Holder-1}
\begin{split}
\|y_1-y_2\|_{\omega}
&\leq \text{diam}_{\omega} E_{{\boldsymbol \omega}^*}
\leq\big(\text{max}_{1\leq m \leq N} \text{ diam}_{\omega}E_i\big)\cdot q^{-\frac{k-1}{d}}\\
&=D\cdot q^{1/d}\cdot(q^{-1/d})^k
\leq
D\cdot q^{1/d}\cdot (1/h)^{\frac{d}{\alpha}}\|x_1-x_2\|^{\frac{1}{\alpha}},
\end{split}
\end{equation}
where $D=\max_{1\leq i\leq N} \text{ diam}_{\omega} E_i.

Now, we consider the case that ${{\boldsymbol {\omega}}}_k$ and ${{\boldsymbol {\gamma}}}_k$ are adjacent. (See Figure \ref{adjacent} (left).)
\begin{figure}[h]

  \includegraphics[width=0.47 \textwidth]{adjacent_008}
  \includegraphics[width=0.47 \textwidth]{adjacent_007}
  \vspace{-1.0 cm}
 \caption{${{\boldsymbol {\omega}}}$ and ${{\boldsymbol {\gamma}}}$ are adjacent.}\label{adjacent}
\end{figure}
Let $x_3$ be the intersection of $F_{{\boldsymbol {\omega}}|_k}$ and
 $F_{{\boldsymbol {\gamma}}|_k}$. Let $k'$ be the smallest integer such that $x_1$ and $x_3$ belong to different cylinders of rank $k'$, say, $x_1\in \rho_i([{\boldsymbol {\omega}}'])$ and $x_3\in \rho_i([{\boldsymbol {\omega}}''])$ (see Figure \ref{adjacent} (right)), then
 $\|x_1-x_3\|\geq \text{diam}~F_{{\boldsymbol {\omega}}''}$
 since $x_3$ is an endpoint.
 Let $y_3=\psi_i(x_3)$. Similar to Case 1,  we have
 $$
 \|y_1-y_3\|_{\omega}\leq
D\cdot q^{1/d}\cdot (1/h)^{\frac{d}{\alpha}}\|x_1-x_3\|^{\frac{1}{\alpha}}.
$$
By the same argument, we have
 $$
\|y_2-y_3\|_{\omega}\leq
D\cdot q^{1/d}\cdot (1/h)^{\frac{d}{\alpha}}\|x_2-x_3\|^{\frac{1}{\alpha}}.
$$
Hence, by the fact $x_3$ is located between $x_1$ and $x_2$,
\begin{equation} \label{Holder-2}
\begin{split}
 \|y_1-y_2\|_{\omega}&\leq \beta \cdot\text{ max}\{\|y_1-y_3\|_{\omega},\|y_3-y_2\|_{\omega}\} \\
&\leq \beta\cdot D\cdot q^{1/d}\cdot (1/h)^{\frac{d}{\alpha}}\cdot\text{max }\{\|x_1-x_3\|^{\frac{1}{\alpha}},\|x_2-x_3\|^{\frac{1}{s}}\}\\
&\leq \beta\cdot D\cdot q^{1/d}\cdot (1/h)^{\frac{d}{\alpha}}\cdot \|x_1-x_2\|^{\frac{1}{\alpha}},
\end{split}
\end{equation}
where the first inequality is from Proposition \ref{Pro-Weak} (iv).

Therefore, \eqref{Holder-1} and \eqref{Holder-2} verify  the $1/\alpha$- H\"older continuity of $\psi_i$.
$\Box$

\begin{figure}[h]
\includegraphics[width=0.35 \textwidth]{affine_Rauzy_frac}\includegraphics[width=0.35 \textwidth]{affine_Rauzy_01}\includegraphics[width=0.35 \textwidth]{affine_Rauzy_02}\\
\includegraphics[width=0.4 \textwidth]{affine_Rauzy_03}
\includegraphics[width=0.4 \textwidth]{affine_Rauzy_04}
\caption{The first figure is the Rauzy fractal given by the substition $
\sigma: 1\longrightarrow 12321,2 \longrightarrow 321,3 \longrightarrow 2$. The following three Figures show the first three approximations of the filling curve of this Rauzy fractal.}
\label{Rauzy_fig}
\end{figure}

\bibliographystyle{siam}

redpajama_set_name: RedPajamaArXiv


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text:
Will Gorbachev Find World War II's Long-Lost POWs?
19-07-1989 , by Bill Paul , ed. The Wall Street Journal
Mikhail Gorbachev just took an extraordinary first step in stripping away the lies of the Stalinist regime concerning the fate of Swedish diplomat Raoul Wallenberg, the hero who saved thousands of Hungarian Jews from the Nazis.Last month, Yuri Kashlev, the chief Soviet negotiator at a Paris human rights conference, said the Soviet Union has no evidence that Mr. Wallenberg is dead. No witnesses, no documentation. Thus did the Gorbachev government cast doubt on the official Soviet explanation, devised by Stalin's lieutenants, that Mr. Wallenberg died in 1947 while in a Soviet prison. (Many former Soviet prisoners have reported seeing Mr. Wallenberg as recently as 1987.)Mr. Kashlev went even further. He indicated that a new investigation is under way into Mr. Wallenberg's disappearance while in Soviet custody at the end of World War II. And he said that if new information is discovered, the Soviet government will share it with the world.Mr. Kashlev's comments raise the possibility not only of Mr. Wallenberg's being found alive but also of the return of hundreds, maybe even thousands, of German, Japanese and even American prisoners of war who have been interned in the Soviet Union for the past 44 years.In 1950, the United Nations established the Ad Hoc Commission on Prisoners of War to account for and, wherever possible, repatriate all remaining POWs from World War II. Seven years later, the commission admitted defeat. It blamed the Soviet Union which, contrary to its pledges under the 1949 Geneva Convention, refused to participate in this humanitarian endeavor. Albania, Bulgaria, Czechoslovakia, Hungary, Poland and Romania also refused to cooperate with the UN commission.Thus, 12 years after the war, the German people still didn't know what had happened to 87,353 German soldiers « whose presence in captivity in the Soviet Union at one time or another had been established, but whose subsequent fate was not known, » according to the commission's 1957 report. An 11,330 confirmed German prisoners in Poland also were still unaccounted for, as are others in other East-bloc countries.The commission further reported that, as of Oct. 1, 1955, « There were about 11,177 Japanese nationals believed to be in the Soviet Union whose fate was unknown, » plus 35,565 Japanese believed to be in China — another country that refused to help the commission.Italy, according to the UN report, still had nearly 1,000 confirmed prisoners of the Soviets unaccounted for. Belgium, Denmark, France, Luxembourg and the Netherlands all still had nationals under detention in the Soviet Union; some are believed to have since been repatriated.Separately, in February 1946, the U.S. War Department notified the American Red Cross that 5,414 American soldiers known by the U.S. to have been in German POW camps still had not been accounted for. While the War Department offered no explanation for why such a large number of GIs had vanished without a trace, the fact is that the Soviet Army « liberated » nearly 29,000 U.S. soldiers from German POW camps during the waning days of the war — when Moscow and Washington were acting more like belligerents than allies. By comparison, as of February 1946, there were only 86 U.S. ex-prisoners in Asia who hadn't yet been accounted for.While in Germany the other week, Mr. Gorbachev seems to have promised to account for all these long-lost POWs. Indeed, upon Mr. Gorbachev's arrival in Bonn, the Soviet Union released the names of 1,500 German soldiers who died while in Soviet POW camps. This marked the first time a Soviet government had accounted for missing German soldiers. On the same trip, Mr. Gorbachev's wife, Raisa, spotlighted the POW issue by laying a wreath at a cemetery for Russian POWs who died while in German hands.It's not just soldiers from World War II who need to be accounted for. The U.S. government must also be concerned about American soldiers in the Korean War who were taken to the Soviet Union and never seen again. On May 5, 1954, the U.S. Embassy in Moscow sent a note to the Soviet minister of foreign affairs saying that the U.S. has « recently received reports which support earlier indications that American prisoners of war who had seen action in Korea have been transported to the Union of Soviet Socialist Republics and that they are now in Soviet custody. « A week later, the Soviet foreign ministry replied that the U.S. contention « is devoid of any foundation whatsoever and is clearly farfetched. »More recently, Vietnam and Laos have been the scene of numerous live-sighting reports of American POWs. The most recent of these reported sightings was by a Japanese monk, released by Hanoi in January, who last month said he had been imprisoned with U.S. soldiers. Hanoi, of course, denied it, but as usual the Vietnamese denial was careful to say that no « Americans » are being held.That raises a critical point. While the 1949 Geneva Convention protects the rights of prisoners who have been accused of war crimes, Hanoi's intepretation of the convention allows it to strip a POW of his rights — including even his nationality — the moment he is accused of being a war criminal. Indeed, Vietnam maintains that a POW accused of war crimes is an ordinary Vietnamese criminal who must serve a life sentence with no hope for release. Moreover, Vietnam maintains it has no obligation under the convention to inform the world that it is holding such « criminals. »During the war, North Vietnam routinely accused the Americans it captured of being war criminals. Thus, Hanoi could still be holding hundreds of U.S. soldiers and yet publicly deny holding any American prisoners.If he expands the Wallenberg investigation, Mr. Gorbachev could make an emphatic statement about human rights by reaching out around the world and accounting for every missing soldier.—
(Copyright (c) 1989, Dow Jones & Co., Inc.)
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Mickey's Not-So-Scary Halloween Party Tickets Now On Sale!
May 12, 2017 By Chelsea T.
May is already flying by, which means Halloween is still a couple months away, but it is in fact creeping closer and closer! Walt Disney World released their dates in April for their annual Halloween event, Mickey's Not-So-Scary Halloween Party – and the tickets are now on sale.
The party has grown to be the most popular Halloween event in the country, bringing in both locals and tourists from all over the world. Beginning on August 25th, party-goers will be able to attend the after-hours event on 32 evenings, instead of the usual 29.
Returning for 2017 is the "Happy HalloWishes" fireworks display, the "Hocus Pocus Villain Spelltacular" stage show that takes place in front of Cinderella Castle and Mickey's Boo-to-You Halloween Parade.


(NaturalNews) Oxygen has been a critical component in the evolution of life on earth, and humans are no exception as the gas has enabled nutrients to be used more efficiently, in turn allowing for the essential generation of energy required for development and growth. While oxygen is a crucial part of life, it is also implicated in the generation of free radicals that promote abnormal cellular destruction, cancer proliferation and advanced aging in humans.
Many environmental factors such as pollution, cigarette smoke, and radiation turn the oxygen molecules found in mitochondria into free radicals. These unstable molecules destroy the normal molecules that form our cells, such as lipids, proteins, and even DNA, by turning them into free radicals.
The destructive nature of free radicals is behind the genesis of many chronic diseases including hypertension, diabetes, cardiovascular disease, dementia, and cancer. This knowledge has prompted researchers to investigate the potent antioxidant properties of fruits and vegetables in the fight against these often fatal illnesses.
Many prior studies designed to investigate the impact of food based antioxidants on human health have been disappointing because most phytonutrients are unable to penetrate the mitochondrial power factories housed in each cell. Free radicals continue to damage the delicate mitochondria, causing energy production to stop and the cell to collapse and die.
Researchers from the Federation of American Societies for Experimental Biology (FASEB) have released the results of a study conducted with avocado oil to determine absorption by the mitochondria and subsequent neutralization of deadly free radicals. Avocado oil is highly resistant to oxidation, and has been shown to help neutralize the metabolic effects of iron, a primary cause of cellular oxidation.
Lead study design author, Dr. Christian Cortez-Rojo noted that "avocado oil causes accelerated respiration in mitochondria, which indicates that the use of nutrients for producing energy for cell functions remains effective even in cells attacked by free radicals and that mitochondria can produce little amounts of damaging free radicals." Avocados have been maligned for decades as poor dietary advice about fat consumption has prevailed in modern medical culture.
The study concluded that avocado lowers the blood concentration of cholesterol and certain fats that are increased in diabetic patients and that may lead to stroke or heart attack. Adding small amounts of avocado to your daily diet can provide excellent antioxidant support to protect mitochondria from early decline and protect against chronic diseases of aging.


Airport Transportation Services in Bloomfield Hills, MI Area Contact us today!
We would like the opportunity to earn your business by providing next level service with a smile. We offer a multitude of services ranging from Airport Shuttle Service to Corporate Special Events. We specialize in Concert Venues, Dinner/Date Night out, Weddings, Parties and more. We offer both flat rates as well as hourly, depending on your needs. We are a small, local Michigan business with six high-end Vehicles. We offer Sedans, SUV's, and will soon be acquiring a 15 passenger van for larger group events. We also offer the latest in technology and conveniences for our clientele. This makes booking appointments a breeze. When you call us you'll speak with either the Owner, Robert, or his Partner, Scott. You're not treated like just another number with us. When calling Robert or Scott as a returning Client, you will be greeted by name. Each one of our clients is treated with a personal touch and the utmost respect. Our Clients have 3 different options for booking reservations with us. You're welcome to Call, SMS Text Message, or use our Automated System here on our site. Our automated system will not only send you a confirmation email, it will also send you an email 2 hours ahead of your pickup time. This email will contain information letting you know who your driver is, as well as a photo, what vehicle they're driving, and a direct contact phone number for the driver. All of these features are designed to ensure accountability as well as reliability. The system also alerts you by SMS Text Messaging when your Driver is both on the way, and has arrived at your desired pickup location. Please expect your driver to arrive approx. 15 minutes earlier than your arranged time. We offer unparalleled service to accommodate your every need.
All Detroit Metro Airport Pick Ups are done INSIDE OF THE TERMINAL per the Detroit Airport Authority. As a commercial transportation company we are NOT ALLOWED to do ANY Pickup's from Curbside PickUp area. As a result, your Driver will be waiting for you inside the Baggage Claim Area holding either a Tablet or Phone with your Name welcoming you to Detroit. Your Driver will be waiting in the Preassigned Transportation Area.
Located in Baggage Claim between the Hanging Flight Monitors and The Coffee Bean on the 1st Floor.
Located on the Lower Level and Driver waiting area right outside of the Security Doors.
In front of Legends Bar between carousels 2 and 3.
Between carousels 4 and 5.
Last is International Arrivals ONLY Driver will be waiting just outside of the Security Doors as you enter the Baggage Claim Department.
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Home › Home and Kitchen › Dc Comics Robin Finger Puppet And Refrigerator Magnet - By The Unemployed Phi..
Dc Comics Robin Finger Puppet And Refrigerator Magnet - By The Unemployed Phi..
HISTORICAL AND ADORABLE - Each of our finger puppets bears the likeness of one historical, literary, political, or pop-culture figure. See the product description below for more info.
A PUPPET AND A MAGNET - Our 4" tall polyester puppets double as delightful fridge magnets thanks to the magnet in their heads. We thought so already, but each one truly is a magnetic personality.
EDUCATIONAL AND CHILD SAFE - Every puppet comes with a removable fold out tag that displays a portrait, a quote, and a brief summary of that character or historical figure's life and major contributions. They are child safe, and recommended for ages 4 and up.
A PERFECT GIFT - For someone who has everything, why not a cute puppet of their most beloved personality. We have over 100 different puppets. Whether it's the Batman fan, or diehard Hannah Arendt enthusiast in your life, we have something for everyone.
FROM THE UNEMPLOYED PHILOSOPHERS GUILD - Don't worry. We are employed, just not as philosophers. We're a small, Brooklyn based company specializing in gifts for the sophisticated gift giver.
Robin can team up with Batman to fight crime as the "Dynamic Duo" or patrolyour fridge and locker. This puppet Boy Wonder is 4" tall and highly skilledin keeping your finger safe. Brought to You by The Unemployed Philosophers Guild The origins of the Unemployed Philosophers Guild are shrouded in mystery. Someaccounts trace the Guild's birth to Athens in the latter half of the 4thcentury BCE. Allegedly, several lesser philosophers grew weary of the endlessSocratic dialogue endemic in their trade and turned to crafting householdimplements and playthings. (Hence the assertions that Socrates quaffed hishemlock poison from a Guild-designed chalice, though vigorous debate surroundsthe question of whether it was a "disappearing" chalice.) Others argue that the UPG dates from the High Middle Ages, when thePhilosophers Guild entered the world of commerce by selling bawdy pamphlets topilgrims facing long lines for the restroom. Business boomed until 1211 whenPope Innocent III condemned the publications. Not surprisingly, this led toincreased sales, even as half our membership was burned at the stake. More recently, revisionist historians have pinpointed the birth of the Guildto the time it was still cool to live in New York City's Lower East Side. Twobrothers turned their inner creativity and love of paying rent towardsfulfilling the people's needs for finger puppets, warm slippers, coffee cups,and cracking up at stuff.


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MS in Data Science
USF's one-year Master of Science in Analytics (MSAN) program delivers a rigorous curriculum focused on mathematical and computational techniques in the emerging field of data science. Read more
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