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substitution sequences, nonprimitivity, singular continuous spectrum, Schroedinger operators
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\newcommand{\Q}{\ensuremath{{\textup{Q\hspace{-2.1mm}\rule{0.3mm}{2.8mm}\;}}}}
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\begin{document}

\title[Singular Continuous Nonprimitive Operators]{Singular Continuous
Spectrum for a Class of Nonprimitive Substitution Schr\"odinger Operators}

\author{C\'{e}sar R. de Oliveira}
\thanks{CRO was partially supported by CNPq (Brazil).} 
\address{Departamento de Matem\'{a}tica -- UFSCar, S\~{a}o Carlos, SP,
13560-970 Brazil}
\email{oliveira@dm.ufscar.br}


\author{Marcus V. Lima}
\address{Departamento de Matem\'{a}tica -- UFSCar, S\~{a}o Carlos, SP,
13560-970 Brazil\\  AFA, Pirassununga, SP, 13630--000  Brazil }
\email{lima@dm.ufscar.br}

\subjclass{81Q10(11B,47B39)}

\maketitle



\begin{abstract} We present a class of discrete Schr\"odinger operators, with
potentials derived from nonprimitive substitutions, that has purely singular
continuous spectrum. We give sufficient conditions on the substitution rule
assuring singular continuous spectrum, either for a generic set in the hull
of the potential or for a set of total invariant measure. 
\end{abstract}





\section{Introduction and Results} From the topological point of view the
``building blocks'' of dynamical systems are the minimal ones, i.e., those
for which all orbits are dense; from the measure theoretical point of view
such building blocks are the uniquely ergodic systems, i.e., those with just
one probability invariant measure. In some cases these two properties are
simultaneously verified and such dynamical systems are called {\em strictly
ergodic}, a property that may present deep consequences. An important recent
application of this concept is on the spectral study of a class of (discrete)
Schr\"odinger operators, whose potentials take finite many values, generated
by primitive substitution rules, with the predominance of pure singular
continuous spectrum~[1--16]. In this work we give new examples of
Schr\"odinger operators with substitution potentials with purely singular
continuous spectrum, but here we show that many techniques used to
investigate the primitive case (including strict ergodicity) can be adapted
to a class of nonprimitive substitutions; it expands and generalizes results 
for a particular substitution presented in our previous paper~\cite{deOL}. We
also mention that, from the physical point of view, the substitution
Schr\"odinger operators are quantum one-dimensional models of quasicrystals,
whose potentials are neither random nor periodic, but exhibit
self-similarities.

Let's clarify what is called a substitution potential and primitivity. Since
we consider only potentials taking two values we shall restrict our
discussion to substitutions $\xi:{\mathcal A}\mapsto{\mathcal A}^*$ on an
alphabet ${\mathcal A}=\{ a,b\}$; ${\mathcal A^*}$ is the set of all words of
finite length on the alphabet. If we denote by ${\mathcal A^{\N}}$ the set of
all sequences of letters in ${\mathcal A}$, the substitution
$\xi$ have natural extensions, by concatenation, to
${\mathcal A}^*$ and ${\mathcal A^\N}$; for example,
$\xi(ab)=\xi(a)\xi(b)$. A {\sl substitution sequence} is a fixed point
${\bar \eta}$ of
$\xi$ in ${\mathcal A}^\N$, i.e., $\xi({\bar \eta})={\bar \eta}$. We consider
on ${\mathcal A}^\N$ (${\mathcal A}^\Z$) the point convergence topology
generated by the metric
\[ d(u,v)=\sum_{n} \frac{|u_n-v_n|}{2^{|n|}},\;\;u=(u_n), v=(v_n),
\] with $n\in\N$ ($n\in\Z$). Then, we take~$a$ as a seed, suppose that
$\xi(a)$ begins with
$a$ and that the length of $\xi^k(a)$ goes to infinity as $k\mapsto \infty$
($\xi^k$ denotes the
$k$-th iterate of
$\xi$), assuring the existence of a fixed point for~$\xi$ (see~\cite{Q},
Proposition~V.1).

Given a substitution sequence~${\bar \eta}$, consider the periodic sequences
in~${\mathcal A}^\Z$
\[
\eta_n=\cdots\xi^n(a)\xi^n(a).\xi^n(a)\xi^n(a)\cdots,
\] with the dot indicating the position of the term of index zero.
If~$\xi(a)$ begins and ends with~$a$, then~$(\eta_n)$ is a Cauchy sequence
and one gets a well-defined limit sequence
\[ {\eta}=\lim_{n\to\infty} \eta_n
\]   in~${\mathcal A}^\Z$, called the bilateral substitution sequence
generated by~$\xi$. Such conditions will be satisfied by the substitution
class~$\zeta_N$ that will be considered in this work. 

Let 
$T:{\mathcal A}^{{\Z}}\mapsto{\mathcal A}^{{\Z}}$ be the left shift
$(Tu)_n=u_{n+1}$; the {\sl hull} of
$\eta$ in~${\mathcal A}^{\Z}$ is defined as
$$
\Omega=\Omega(\eta)={\textup{closure}}\{T^n \eta:\;n\in{{\N}}\}.   
$$ We notice that for a recurrent sequence~$\eta$ its hull $\Omega$ is a
compact and $T$-invariant subset of ${\mathcal A}^{{\Z}}$ with
$T\Omega(\eta)=\Omega(\eta)$ (\cite{Q}, Chapter~V), so that the dynamical
system~$(\Omega(\eta),T)$ is well-defined.

Given a bijective real function $f:{\mathcal A}\mapsto\R$, we associate a
sequence in
$l^\infty(\Z)$ to each $\omega=(\omega_n)_{n\in\Z}\in\Omega$ by
$(f(\omega_n))_{n\in\Z}$, i.e., we just replace letters by numbers, which we
again denote by $\omega$ and call it a {\em substitution potential}.
Therefore, every point $\omega\in\Omega$ defines a bounded self-adjoint
operator
${H_{\omega}}$ on $l^2(\Z)$~by
\begin{equation}
\label{hamiltonian} ({H_{\omega}}u)_n=u_{n+1}+u_{n-1}+{\omega}_{n}u_n.
\end{equation}
 Our goal is the investigation of the spectral properties of the Schr\"odinger
operator~(\ref{hamiltonian}) for a special class of substitutions defined
below.

Let $M$ be the matrix of the substitution $\xi$ whose entries $M_{ab}$ are
given by the number of occurrences of the symbol~$a$ in
$\xi(b)$, for
$a,b\in{\mathcal A}$. Recall that a matrix $M$ is primitive if all entries in
a power of $M$ are strictly positive. A substitution rule $\xi$ is called
primitive if its corresponding substitution matrix is primitive. It is known
that for primitive substitution the subshift dynamical system
$(\Omega,T)$ is strictly ergodic (see~\cite{Q}, Chapter~V).

 Here we consider a subclass of the following collection of substitutions
$\zeta=\zeta_N$, for each $N\geq1$, on ${\mathcal A}=\{a,b\}$, given by
\begin{equation}
\label{nonprimitive}
\zeta(a)= \underbrace {a...a}_{A_1}\underbrace {b...b}_{B_1}\underbrace
{a...a}_{A_2}\cdots
\underbrace {b...b}_{B_N}\underbrace {a...a}_{A_{N+1}},\hspace{1cm}\zeta(b)=b,
\end{equation} with $A_j,B_j\geq 1,\;j=1,\cdots,N$ and $A_{N+1}\geq1$, being
the number of factors in each block. Such substitution is clearly
nonprimitive, for any positive power of its matrix of substitution has the
form
$$
\left[ \begin{array}{cc}
 A&0  \\ B & 1   \end{array}\right]
$$ for some positive integers $A,B$.

It is possible to generate periodic sequences with appropriate choices
of~$\zeta$ in~(\ref{nonprimitive}); for instance, the
substitution~$\zeta(a)=abba$,
$\zeta(b)=b$. The aperiodicity of the substitution sequences is significant
for our spectral results; the following lemma presents a very simple and
rather general condition on~$\zeta$ assuring that~$\eta$ is not periodic.

\begin{Lemma}\label{nonperiodic} If $A_j\ge 2$ for some $1\le j\le (N+1)$,
then the resulting substitution sequence~$\eta$ is not ultimately periodic.
\end{Lemma}

\begin{proof} It is possible to restrict the argument for strictly positive
index of~$\eta$, i.e., to~$\bar
\eta$. For notational convenience we suppose that~$A_1=2$; the other cases
can be handled in a similar way. Notice first that all (constant) blocks
of~$a's$ (and~$b's$) in~$\eta$ are finite. Suppose that~$\eta$ is periodic
with minimal period~$\tau\ge2$ ($\eta$ is clearly nonconstant) and denote the
first minimal period block, represented by the first~$\tau$ letters
of~$\bar\eta$, by~$B$. Let
$n_c$ be the unique integer such that ($|Q|$ denotes the length
of~$Q\in\mathcal{A}^*$; e.g., $|B|=\tau$. It is also supposed
that~$\zeta^0(a)=a$.)
\[ |\zeta^{n_c-1}(a)|<\tau \le|\zeta^{n_c}(a)|.
\]
 The first entries of~$\bar\eta$ are
\[
\zeta^{n_c+1}(a)=\zeta^{n_c}(a)\zeta^{n_c}(a)b\dots
b\zeta^{n_c}(a)\dots\zeta^{n_c}(a)\cdots b\dots
b\zeta^{n_c}(a)\dots\zeta^{n_c}(a).
\]

The choice of~$n_c$ entails that, besides at position~$1$, a new period
block~$B$ starts at position~($|\zeta^{n_c}(a)|+1$). Since the period
block~$B$ can not appear shifted by any amount different from an integer
multiple of~$\tau$ (this would force a constant sequence~$\bar\eta$), one
sees that~$|\zeta^{n_c}(a)|$ is an integer multiple of~$\tau$. Therefore, a
new period block~$B$ should start at position~($2|\zeta^{n_c}(a)|+1$), but
this can not occur since~$B$ starts with the letter~$a$ and at
position~($2|\zeta^{n_c}(a)|+1$) of~$\bar\eta$ there is a~$b$.
Therefore~$\bar\eta$ (and consequently~$\eta$) is not periodic.

Suppose now that~$\eta$ is ultimately periodic, so that there is an
integer~$k\ge1$ such that~$\eta$ is periodic, with period~$\tau$, after
its~$k$th position ($k\ge 2$). Choose an integer~$m$ such
that~$|\zeta^{m}(a)|>k$ and~$|\zeta^{m}(a)|>\tau$. 
 The first entries of~$\bar\eta$ are
\[
\zeta^{m+1}(a)[\zeta^{m+1}(a)]b\dots b\zeta^{m+1}(a)\dots\zeta^{m+1}(a)\cdots
b\dots b\zeta^{m+1}(a)\dots\zeta^{m+1}(a),
\] and writing out the above second block~$\zeta^{m+1}(a)$ one gets, for the
beginning of~$\bar\eta$,
\[
\zeta^{m+1}(a)[\zeta^{m}(a)\zeta^{m}(a)b\dots
b\zeta^{m}(a)\dots\zeta^{m}(a)\cdots b\dots
b\zeta^{m}(a)\dots\zeta^{m}(a)]b\dots b.
\]

The choice of~$m$ implies that a minimal period block~$B$ starts at
position~$(\zeta^{m+1}(a)+1)$ and also
at~$(|\zeta^{m+1}(a)|+|\zeta^{m}(a)|+1)$, and with the letter~$a$. The same
argument used above implies that another period block~$B$ should start at
position~$(|\zeta^{m+1}(a)|+2|\zeta^{m}(a)|+1)$, i.e., $B$ should start with
a~$b$. This contradiction shows that~$\bar\eta$ and~$\eta$ are not ultimately
periodic.
\end{proof}


Although the substitution $\zeta$ is not primitive, it will be possible to
adapt some techniques used for the primitive case in order to prove spectral
results for~(\ref{hamiltonian}), with potentials along the hull of a
subclass~$\zeta$ of~(\ref{nonprimitive}) (to be specified later), always
assuming from now on that
\begin{enumerate}
\item[\it (i)]$\Omega_{\zeta}$  denotes the hull of the substitution~$\zeta$
defined by~(\ref{nonprimitive}), so that
$(\Omega_{\zeta},T)$ indicates the subshift dynamical system associated 
to~$\zeta$.
\item[\it (ii)]the function $f:{\mathcal A}\mapsto\R$ defining the
substitution potential is bijective (in order to avoid constant potentials).
\end{enumerate}


Some studies of the spectral properties of~(\ref{hamiltonian}), with
potentials~$(\omega_n)$ generated by~$\zeta_N$ satisfying~$(i)$ and~$(ii)$,
will rely upon dynamical features of~$(\Omega_{\zeta},T)$ and an adaptation
of the so-called Gordon argument~(see, for instance,
\cite{DP1} and  Section~10.2 of~\cite{CFKS}); these are the contents of
Propositions~\ref{strictly},
\ref{absolutely} and~\ref{GordonProp}, whose proofs are postponed to other
Sections of this work. However, they will be used below to present our
spectral results, which are summarized in two theorems.


\begin{Propositon}
\label{strictly}
 The subshift dynamical system~$(\Omega_{\zeta},T)$ is strictly ergodic. We
shall denote its unique probability ergodic measure by~$\mu$.
\end{Propositon}

From general arguments presented, for example, in Queff\'elec's
book~\cite{Q}, Chapter~V, it follows from the strict ergodicity
of~$(\Omega_{\zeta},T)$ that every word from ${\mathcal A}^*$ occurring in
some
$\omega\in\Omega_{\zeta}$  also occurs in every
$\omega'\in\Omega_{\zeta}$, and with a well-defined and strictly positive
frequency. 



As usual, we shall indicate the spectrum of a self-adjoint operator~$H$ by
$\sigma(H)$, and by
$\sigma_{ac}(H)$ and $\sigma_p(H)$ the absolutely continuous and point
spectrum of
$H$, respectively. Recall that a {\sl generic set} is a dense $G_\delta$
(countable intersections of open sets).

The above dynamical properties will imply the following spectral result

\begin{Propositon}
\label{absolutely}
 If~$A_j\ge2$ for some~$1\le j\le (N+1)$, then
$\sigma_{ac}({H_{\omega}})=\emptyset$ for any
$\omega\in\Omega_{\zeta}$.
\end{Propositon}


A curious property of some sequences is palindromicity, which plays an
important role in the exclusion of eigenvalues for discrete Schr\"odinger
operators~(\ref{hamiltonian}). A word in~${\mathcal A}^*$ is called a
palindrome~\cite{W,HKS} if it is the same when read backwards, and an element
of~${\mathcal A}^\Z$ is palindromic if it contains arbitrary long
palindromes. A compact and
$T$-invariant minimal subset of~${\mathcal A}^\Z$ is called palindromic if it
contains one palindromic sequence.
\begin{Propositon}
\label{palindromicProp}  If~$\Omega_{\zeta}$ is palindromic, or if~$\zeta(a)$
is a palindrome,  then there exists a generic set
$\Upsilon\subset\Omega_{\zeta}$ such that
$\sigma_p({H_{\omega}})=\emptyset$ for any $\omega\in\Upsilon$.
\end{Propositon}

\begin{proof} Since the proof is essentially contained in~\cite{HKS} we just
sketch it. If~$\zeta(a)$ is a palindrome then~$\zeta(w)$ is also a palindrome
for any palindrome~$w$, so that a
$\zeta$-fixed point~$\eta$ has palindromes of arbitrary long length
and~$\Omega_{\zeta}$ is palindromic. Therefore the second alternative
hypothesis in the proposition implies the first one which, by its turn,
guarantees the presence of a dense set~${\mathcal S}\subset\Omega_{\zeta}$ of
{\em strong palindromes} with~$\sigma_p(H_\omega)=\emptyset$ for
any~$\omega\in{\mathcal S}$ (Proposition~2.1 and Theorem 5.1 of~\cite{HKS}).
The generic property in the proposition follows by the set of results known
as Wonderland Theorem~\cite{Si1} (see also Theorem 7.1 in~\cite{HKS}). 
\end{proof}

Our motivating case was the particular substitution $\zeta^{(p)}:\mathcal
A\to \mathcal A^*$,
$\zeta^{(p)}(a)=aabaa$, $\zeta^{(p)}(b)=b$, discussed in~\cite{deOL}.
According to~Lemma~\ref{nonperiodic} and Proposition~\ref{strictly}, this is
an example of a nonperiodic substitution sequence of the
type~(\ref{nonprimitive}) for which~$\Omega_{\zeta^{(p)}}$ is palindromic,
since~$\zeta^{(p)}(a)$ is a palindrome.


\begin{Theorem}\label{palindromicTeor}  Suppose $A_j\ge2$ for some $1\le j\le
(N+1)$. If~$\Omega_{\zeta}$ is palindromic, or if~$\zeta(a)$ is a palindrome,
then there exists a generic set
$\Upsilon\subset\Omega_{\zeta}$ such that for any $\omega\in\Upsilon$ the
operator
${H_{\omega}}$ has purely singular continuous spectrum.
\end{Theorem}
\begin{proof} Combine Propositions~\ref{absolutely}
and~\ref{palindromicProp}. 
\end{proof}

Besides palindromicity, another way to exclude point spectrum is via the
following adaptation of Gordon's argument~\cite{DP1,Su1,D1,BG1,BG2,CFKS}.

\begin{Propositon}\label{GordonProp} 
 If either $(A_1+A_{N+1})\geq4$ or  $A_j\geq4$ for some~$1\leq j\leq (N+1)$
(or both), 
 then there exists a measurable set~$\Xi\subset\Omega_{\zeta}$, with
$\mu(\Xi)=1$, such that
$\sigma_{p}({H_{\omega}})=\emptyset$ for any $\omega\in\Xi$.
\end{Propositon}

\begin{Theorem}
\label{GordonTeor}  If either $(A_1+A_{N+1})\geq4$ or  $A_j\geq4$ for
some~$1\leq j\leq (N+1)$, then there exists a measurable
set~$\Xi\subset\Omega_{\zeta}$, with $\mu(\Xi)=1$, such that for any
$\omega\in\Xi$ the operator ${H_{\omega}}$ has purely singular continuous
spectrum.
\end{Theorem}
\begin{proof} Combine Propositions~\ref{absolutely} and~\ref{GordonProp}. 
\end{proof}

The above theorems present (very easy to verify) sufficient conditions for
the occurrence of singular continuous spectra. For example, the substitutions
$a\to abaaa$, $a\to abbaaaaba$, $a\to aaabababba$ ($b\to b$ in all cases)
result in pure singular continuous spectrum of~$H_\omega$ for~$\omega$ in
sets of total invariant measure in the respective hulls. The  palindromic
nonprimitive substitution
 $\zeta(a)=abaaba$ in~(\ref{nonprimitive}) gives rise to singular continuous
spectrum for the operator~(\ref{hamiltonian}) in a generic set of potentials
in its hull; we do not have any results on the measure theoretical side in
this case. The operator~$H_\omega$, with potentials in the hull of the
substitution
$\zeta^{(p)}(a)=aabaa$, has pure singular continuous spectrum for both in a
generic set and in a set of full invariant measure of its hull (notice that
this improves results previously obtained in~\cite{deOL}).


The reminder of this work is organized as follows. In Section~\ref{SESection}
we prove Propositions~\ref{strictly} and~\ref{absolutely}, and in
Section~\ref{GordonSection} we give the proof of
Proposition~\ref{GordonProp}. In the final Section we present additional
remarks.


\section{Strict Ergodicity}

\label{SESection} In this section we prove Propositions~\ref{strictly}
and~\ref{absolutely}. Before proceeding to the proof of
Proposition~\ref{strictly}, we recall that~$|Q|$ is the length of $Q \in
\mathcal A^*$ and say that such
$Q$ is called a $k$-block in~$\omega \in \Omega_{\zeta}$ if~$Q$ is a word
contained in~$\omega$ and
$|Q|=k.$ If $P$ and $Q$ belong to~${\mathcal A}^*$, $\#_{P}Q$ denotes the
number of occurrences of $P$ in
$Q.$ An important point in Proposition~\ref{strictly} is to show that
\[
\lim_{j \to \infty}  {\frac {{\#}_{{\zeta}^n(a)} ( |{\zeta}^{n+j}(a) |
{\textup{-block}} )}{|\zeta^{n+j}(a)|}}
\] exists, is nonzero and independent of the  $|{\zeta}^{n+j}(a) |
{\textup{-block}}$ considered; this  implies that~($\Omega_{\zeta},T$) is
uniquely ergodic~(\cite{Pe}, Section~4.5). To perform the required estimates,
we define
$f_{N+1} ={\sum_{i=1}^{N+1}A_i}$ and $h_{N}={\sum_{i=1}^{N}B_i}$; the $j$th
power of~$f_{N+1}$ will be denoted by~$f_{N+1}^j$, and the ratio 
\[R_j= {{(f_{N+1}^j-1})/( {f_{N+1}-1}});\] notice that~$f_{N+1}\ge 2$.


\begin{Lemma}\label{lema1}  The iterates of the  substitution~$\zeta$ satisfy
\begin{enumerate}
\item [(a)]
$\quad |\zeta^j(a)|=f_{N+1}^j + h_{N} R_j$ 
\item [(b)] $\quad {\#}_a({\zeta}^j(a))=f_{N+1}^j$.
\end{enumerate} 
\end{Lemma}

\begin{proof}  {\it (a)} It is clear that $|{\zeta(a)}|={\displaystyle {
f_{N+1} + h_{N} }}$ and
\begin{eqnarray*} |{\zeta^2(a)}|&=&{\displaystyle { f_{N+1}(f_{N+1} + h_{N})
+ h_{N}}} \\  &=& {\displaystyle { f_{N+1}^2 + h_{N} (f_{N+1} +1)}};
\end{eqnarray*} an induction argument gives
\begin{eqnarray*}|{\zeta^j(a)}|&=&{\displaystyle { f_{N+1}^j + h_{N}
[f_{N+1}^{j-1}+ \cdots  +1]}} \\  &=&{\displaystyle { f_{N+1}^j + h_{N} R_j}}.
\end{eqnarray*}  

{\it (b)} It is a simple induction.  
\end{proof}


Define $C_A$ and $C_{B_k}$ by
\[\#_{\zeta^n(a)}(\zeta^n(a) \zeta^n(a)) = 2 + C_A
\]
\[
\#_{\zeta^n(a)}(\zeta^n(a) \underbrace {b...b}_{B_k}
\zeta^n(a))=2+C_{B_k},\;\;\;1\le k\le N.
\]

\begin{Lemma}\label{CACB} If $C_A>0$, then $C_{B_q}=0\;\;\forall 1\le q\le
N.$ Conversely, if 
$C_{B_q}>0$ for some~$q$, then~$C_A=0$.
\end{Lemma}

\begin{proof}
  If $C_A>0$ there exists $2 \leq j \leq N$ such that
\[
\underbrace {a...a}_{A_j-A_{N+1}}\underbrace {b...b}_{B_j}\underbrace
{a...a}_{A_{j+1}}\cdots
\underbrace {a...a}_{A_{N+1}+A_1}\cdots \underbrace
{b...b}_{B_{j-1}}\underbrace {a...a}_{A_j-A_1} =
\underbrace {a...a}_{A_1}\underbrace {b...b}_{B_1}\underbrace
{a...a}_{A_2}\cdots
\underbrace {b...b}_{B_N}\underbrace {a...a}_{A_{N+1}};
\]  then
\begin{equation}\label{aa1} A_k= \left\{ \begin{array}{lll} A_{k+j-1} &k+j-1
< N+1 \\ A_1+A_{N+1} &k+j-1=N+1 \\ A_{k+j-1-N} &k+j-1>N+1 \end{array}
\;\;\;\; 2 \leq k \leq N, \right.
\end{equation} and
\begin{equation}\label{aa2} B_m= \left\{ \begin{array}{ll}B_{j+m-1} &k+j-1
\leq N \\  B_{j+m-1-N} &k+j-1>N \end{array} \;\;\;\; 1
\leq m \leq N. \right.
\end{equation} If $C_{B_q}>0$ for some~$q$ there exists $2 \leq i \leq N$
such that
\[
\underbrace {a...a}_{A_i}\underbrace {b...b}_{B_i}\underbrace
{a...a}_{A_{i+1}}\cdots
\underbrace {a...a}_{A_{N+1}}\underbrace{b...b}_{B_q}\underbrace
{a...a}_{A_1} \ldots \underbrace {b...b}_{B_{i-2}}\underbrace
{a...a}_{A_{i-1}} = \underbrace {a...a}_{A_1}\underbrace
{b...b}_{B_1}\underbrace {a...a}_{A_2}\cdots
\underbrace {b...b}_{B_N}\underbrace {a...a}_{A_{N+1}}
\] then
\begin{equation}\label{aba1} A_l= \left\{ \begin{array}{ll} A_{l+i-1} &l+i-1
\leq N+1 \\  A_{l+i-1-(N+1)} &l+i-1>N+1
\end{array}\;\;\;\; 1 \leq l \leq N, \right.
\end{equation} and
\begin{equation}\label{aba2} B_m= \left\{ \begin{array}{lll} B_{m+i-1} &m+i-1
< N+1 \\ B_q &m+i-1=N+1 \\ B_{m+i-1-(N+1)} &m+i-1>N+1 \end{array} \;\;\;\; 1
\leq m \leq N. \right.
\end{equation}

If the conditions~(\ref{aa1}) and~(\ref{aba1}) are simultaneously satisfied
we have (the number over the equal sign indicates the equation used in each
step)
\[
(A_1+A_{N+1})\stackrel{(\ref{aa1})}{=}A_{N-j+2}\stackrel{(\ref{aba1})}{=}A_{N-j+i+1}\stackrel{(\ref{aa1})}{=}
A_i\stackrel{(\ref{aba1})}{=}A_1
\] if~$j\geq i$, and
\[
(A_1+A_{N+1})\stackrel{(\ref{aa1})}{=}A_{N-j+2}\stackrel{(\ref{aba1})}{=}A_{i-j}\stackrel{(\ref{aa1})}{=}
A_{i-1}\stackrel{(\ref{aba1})}{=}A_{N+1}
\] if~$j<i.$ Since $A_{N+1}>0$ and $A_1>0$, we have got contradictions and
the Lemma is proven. 

We remark that the maximum possible values of $C_A$ and $C_{B_q}$ are $(N-1)$
and $N,$ respectively (and each of them occurs for suitable choices
of~$\zeta_N$).
\end{proof}



\begin{proof} ({\it Proposition \ref{strictly}}) To begin with we show
that~$(\Omega_{\zeta},T)$ is uniquely ergodic. Due to Lemma~\ref{CACB} it is
enough to consider two cases: 
\begin{enumerate}
\item[(i)]$
\#_{\zeta^n(a)}(\zeta^n(a) \zeta^n(a)) = 2+C_A, \;\;\;
\#_{\zeta^n(a)}(\zeta^n(a) B_k\zeta^n(a))=2;
$ 
\item[(ii)]$
\#_{\zeta^n(a)}(\zeta^n(a) \zeta^n(a))=2, \;\;\;
\#_{\zeta^n(a)}(\zeta^n(a) {B_k} \zeta^n(a)) = 2+C_{B_k}.
$
\end{enumerate}  Notice that both cases include $C_A=C_{B_k}=0$, for any~$k$,
as particular instances.


In the case (i), by Lemma~\ref{lema1}, 
\[
\#_{\zeta^n(a)}(\zeta^{n+j}(a)) = f_{N+1}^j + (f_{N+1}-(N+1))C_AR_{j}, \;\; 
\] and by counting the occurrences of $\zeta^n(a)$ in 
$|\zeta^n(a)|-{\textup{blocks}}$ we find
\begin{eqnarray*} f_{N+1}^j +(f_{N+1}-(N+1))C_AR_{j}-C_A-1   \leq
{\#}_{{\zeta}^n(a)}(|{\zeta}^{n+j}(a)|{\textup{-block}})  \\ \le f_{N+1}^j +
(f_{N+1}-(N+1))C_AR_{j}+C_A-1,
\end{eqnarray*} so that
\begin{eqnarray*}
\lim_{j \to \infty}  {\frac {{\#}_{{\zeta}^n(a)}
 ( |{\zeta}^{n+j}(a) | {\textup{-block}} ) }{ |\zeta^{n+j}(a)|}} &=& \lim_{j
\to \infty} {\frac {f_{N+1}^j + (f_{N+1}-(N+1))C_AR_{j}}{ f_{N+1}^{j+n}+
h_{N} R_{j+n}}}  \\ &=&  {\frac {f_{N+1} +
(f_{N+1}-(N+1))C_A-1}{f_{N+1}^{n}[f_{N+1}+h_{N}-1]} }>0.
\end{eqnarray*} 

In the case (ii) there is an integer~$M_k\ge0$, which depends on~$N$, such
that ($M_k$ takes into account the existence of eventual $B_l=B_k$ for~$k\ne
l$)
\[
\#_{\zeta^n(a)}(\zeta^{n+j}(a)) = f_{N+1}^j + M_kR_j.
\]
 Again by counting
\begin{eqnarray*} f_{N+1}^j + M_kR_j -C_{B_k}-1 &\le&
{\#}_{{\zeta}^n(a)}(|{\zeta}^{n+j}(a)|{\textup{-block}})\\ &\le& f_{N+1}^j +
M_kR_j+C_{B_k}-1,
\end{eqnarray*}  so that 
\begin{eqnarray*}
\lim_{j \to \infty}  {\frac {{\#}_{{\zeta}^n(a)}
 ( |{\zeta}^{n+j}(a) | {\textup{-block}} ) }{ |\zeta^{n+j}(a)|}} &=& \lim_{j
\to \infty} {\frac {f_{N+1}^j + M_kR_j}{ f_{N+1}^{j+n}+ h_{N} R_{j+n}}} 
\\ &=&  {\frac {f_{N+1} + M_k-1}{f_{N+1}^{n}[f_{N+1}+h_{N}-1]} }>0.
\end{eqnarray*}   (If there are other~$C_{B_q}\ne 0,\: B_q\ne B_l$ for~$k\ne
l$, just replace~$M_k$ by~$\sum_q M_q$   in the above average, with the sum
over~$q$ such that~$B_q\ne B_l$.) Therefore~$(\Omega_{\zeta},T)$ is uniquely
ergodic.


Now we show that~$(\Omega_{\zeta},T)$ is minimal. Given a word $B \in
{\mathcal A^*}$ occurring in~$\eta$, there is an
$m \in {\N}$ such that $B \subset \zeta^{m}(a)$. Now, $\zeta^{m}(a)$ occurs
in~$\eta$ with  bounded gaps  (with bound~$\le$~$\max\{B_1,B_2,\cdots, B_{N}
\}+|\zeta^{m}(a)|$); this implies that $\eta$ is almost periodic, and since
from the very definition of~$\zeta$ it follows that~$\eta$ is also recurrent,
one concludes that~$(\Omega_{\zeta},T)$ is minimal~(\cite{Q}, Section~IV.2).
Therefore
$(\Omega_{\zeta},T)$ is strictly ergodic. 
\end{proof}

\begin{proof} {\it (Proposition~\ref{absolutely})} The conclusions of
Proposition~\ref{absolutely} will follow  from Proposition~\ref{strictly} and
two additional results. The first one is due to Last and Simon~\cite{LS}
which, in this setting, states that for strictly ergodic potentials the
absolutely continuous spectrum is independent of~$\omega\in\Omega_{\zeta}$.
The second result we use is due to Kotani~\cite{K}: for nonperiodic ergodic
potentials taking only a finite number of values the set of potentials with
no absolutely continuous spectrum has full ergodic measure. By
Proposition~\ref{strictly} our (nonperiodic) substitution potentials are
strictly ergodic, so by Last-Simon
$\sigma_{ac}({H_{\omega}})$ is constant on~$\Omega_{\zeta}$; since by Kotani
and Lemma~\ref{nonperiodic} there are potentials with
$\sigma_{ac}({H_{\omega}})=\emptyset$, it follows that
$\sigma_{ac}({H_{\omega}})=\emptyset$ for any~$\omega\in\Omega_{\zeta}$. 
\end{proof}

\section{A Gordon Type Argument}
\label{GordonSection}

 Now we will show that if $(A_1+A_{N+1})\geq4$ or  $A_j\geq4$ for some~$1\leq
j\leq N+1$, then~$\sigma_{p} (H_{\omega}) =
\emptyset$ for~$\omega$ in a set of full invariant measure~$\mu$ in the
hull~$\Omega_{\zeta}$, i.e., we prove Proposition~\ref{GordonProp}. We shall
follow a Gordon type argument similar to the one used
in~\cite{DP1,Su1,D1,BG1,BG2,CFKS}.

Let's first sketch the idea of the proof. For a fixed~$\omega\in {\mathcal
A^{\Z}}$, if  there is a sequence~$m\to\infty$ such that, for each such~$m$,
\[ {\omega}_{k-m}={\omega}_{k}={\omega}_{k+m},\;\;\;\;1\le k\le m,
\] then the Schr\"odinger operator~(\ref{hamiltonian}) has no eigenvalues;
this is the Gordon argument (see the above references). It will be shown
that, under the assumptions of Proposition~\ref{GordonProp}, the measurable
sets 
\[ B(n) \equiv \{\omega\in\Omega_{\zeta} : 
 {\omega}_{(k-|\zeta^n(a)|)}={\omega}_{k}={\omega}_{(k+|\zeta^n(a)|)}, 1
\leq k \leq |\zeta^n(a)| \},   
\] satisfy 
\[
 \mu(\limsup_{n\to\infty} B(n))>0,
\]
 and since $\limsup_{n\to\infty} B(n)$ is contained in the shift invariant
set $\{
\omega\in\Omega_{\zeta} : \sigma_{p} (H_{\omega}) = \emptyset \}$, it will
follow, by an argument in ergodic theory, that the latter set has full
invariant measure.
 

If~$\eta$ is the bilateral substitution sequence generated by~$\zeta$, it is
then natural to investigate the sets 
\[ J(n) \equiv \{ {\textup{distinct\;words\;of\;}} \eta {\textup{\;of\;
the\;form\;}} vvv  {\textup{\;with\;}}  |v|=|\zeta^{n}(a)|\}
\] and, in particular, the values of their measures~$\mu(J(n))$, which can be
recovered by ergodic averages. The following Lemmas will be used to estimate
such averages.

\begin{Lemma}\label{G} 
 If~$(A_1+A_{N+1}) \ge 4$ or~$A_j\ge 4$ for some~$1\le j\leq N+1$, then for
all $n\ge1$
$\#J(n)$ (the cardinality of~$J(n)$) satisfies
$$B_1+A_2|\zeta^{n-1}(a)| \le {\#}J(n) \le |\zeta^n(a)|.$$ 
\end{Lemma}

\begin{proof} The occurrence of $aaaa$ in $\eta$ yields existence of blocks  
$$\zeta^n(a)\zeta^n(a)\zeta^n(a)\zeta^n(a)$$ that contain all different
blocks of the form $vvv,$ with
$|v|=|\zeta^n(a)|$. Thus we consider cyclic permutations of $\zeta^n(a)$ to
determine the number of different blocks $vvv$. For each block $u_1u_2 \cdots
u_l$ let
$$ P(u_1u_2 \cdots u_l) = u_2u_3 \cdots u_lu_1.
$$ Thus
$$
\#J(n)= \min \{i \ge 1: \zeta^n(a)=P^i(\zeta^n(a)) \}\le |\zeta^n(a)|.
$$ 

For a general substitution of the type~(\ref{nonprimitive}), by successively
applying $P$ to
$\zeta(a)$, the first opportunity to recover~$\zeta(a)$ is (see the examples
at the end of this proof)
$$
\underbrace {a...a}_{A_2-A_{N+1}}\underbrace {b...b}_{B_2}\cdots
\underbrace{a...a}_{A_1+A_{N+1}}
\underbrace {b...b}_{B_1} \underbrace {a...a}_{A_2-A_1}.
$$ Hence we have $B_1+A_2 \le \#J(1) \le |\zeta(a)|$. By the very definition
of the substitution, this property generalizes for any  positive integer~$n$
$$ B_1+A_2|\zeta^{n-1}(a)| \le \#J(n) \le |\zeta^n(a)|. 
$$  Both extreme conditions can occur. For example, if all~$B_j$ are pairwise
different or $A_j\ne A_1+A_{N+1},\forall 2\le j\le N$, one gets 
$\#J(n)=|\zeta^n(a)|$; while the case in which all $B_j$ are equal and
$A_j=A_1+A_{N+1},\forall 2\le j\le N,$ one gets
$\#J(n)=B_1+A_2|\zeta^{n-1}(a)|$. 
\end{proof}


\begin{Lemma}\label{aaaa}  If $vvv$ occurs in~$\eta$ with $|v|=|\zeta^n(a)|$,
then
$$\begin{array}{c}
\#_{vvv}\zeta^{n+j}(a) \ge \#_{aaaa}\zeta^j(a)=\hspace{8cm}  \\ 
=f_{N+1}^{j-1}\sum_{\{i:A_i \geq 4 \}
}(A_i-3)+R_{j-1}E_{N}\sum_{i=1}^{N+1}(A_i-1),   
\end{array}
$$ where
$$ E_{N}= \left \{ \begin{array}{llll} \hspace{1cm} 0 &\;\;{\textup{if}}\;\;
A_1+A_{N+1}<4 \\
\hspace{1cm}  1 &\;\;{\textup{if}}\;\; A_1=A_{N+1}= 2 \\
\hspace{1cm} 3 &\;\;{\textup{if}}\;\; \min \{A_1,A_{N+1}\} \ge 3 \\  \min
\{A_1,A_{N+1}\} &\;\;{\textup{otherwise}}\;\; 
 \end{array} \right. 
$$
\end{Lemma}

\begin{proof} By the construction of the blocks $vvv$ with
$|v|=|\zeta^n(a)|$, in the proof of Lemma~\ref{G}, we have
$$
\#_{vvv}\zeta^{n+j}(a)\ge \#_{\zeta^n(aaaa)}\zeta^{n+j}(a) =
\#_{aaaa}\zeta^j(a).
$$ Direct computation gives  {$\#_{aaaa}\zeta(a)=\sum_{\{i:A_i \geq 4 \}
}(A_i-3)$}, and
$$
\#_{aaaa}\zeta^2(a)=f_{N+1}\sum_{\{i:A_i \geq 4 \}
}(A_i-3)+E_{N}\sum_{i=1}^{N+1}(A_i-1).
$$ Then, by induction one concludes that 
\begin{eqnarray*}
\#_{aaaa}\zeta^j(a)&=&\\ &=&f_{N+1}^{j-1}\sum_{\{i:A_i \geq 4 \}
}(A_i-3)+[f_{N+1}^{j-2}+
\cdots +1]E_{N}\sum_{i=1}^{N+1}(A_i-1) \\ &=& f_{N+1}^{j-1}\sum_{\{i:A_i
\geq 4 \} }(A_i-3)+R_{j-1}E_{N}\sum_{i=1}^{N+1}(A_i-1).  
\end{eqnarray*} 
\end{proof}

The Borel $\sigma$-algebra in $(\Omega_{\zeta},T)$ is generated by the
cylinders
$$ [u_0 \cdots u_{l-1}]_{[m, \cdots , m+l-1]} \equiv \{ \omega \in
\Omega_{\zeta} : {\omega}_{m+i} = u_i, 0
\leq i \leq l-1
\}.$$ Since~${\Omega_{\zeta}}$ is strictly ergodic, it is know
that~(\cite{Q}, Section~V.4) the unique invariant probability measure
$\mu$ satisfies 
$$ d(u_0
\cdots u_{l-1})\equiv\mu([u_0 \cdots u_{l-1}]_{[m, \cdots, m+l-1]})=\lim_{n
\to \infty}\frac{\#_{u_0 \cdots u_{l-1}}\zeta^n(a)}{|\zeta^n(a)|}.
$$

\begin{proof} {\it (Proposition~\ref{GordonProp})} By Lemma~\ref{aaaa}, for
$vvv$ occurring in $\eta$ with
$|v|=|\zeta^n(a)|$ 
\begin{eqnarray*} d(vvv)&=&\lim_{j \to \infty} { {\#_{vvv}\zeta^{n+j}(a)}
\over {|\zeta^{n+j}(a)|} } \\  & \ge &
\lim_{j
\to \infty} { {\#_{aaaa}\zeta^{j}(a)} \over {|\zeta^{n+j}(a)|} } 
    \\ &=& 
\lim_{j \to \infty} { {f_{N+1}^{j-1}\sum_{\{i:A_i \geq 4\}}(A_i-3)+
R_{j-1}E_{N}\sum_{i=1}^{N+1}(A_i-1)  } \over {f_{N+1}^{n+j}+ R_{n+j}h_{N}  }
} \\ &=&
\frac{d_N}{{f_{N+1}^{n+1}(f_{N+1}+ h_{N} - 1)} },
 \end{eqnarray*} with $d_N= [{f_{N+1}-1] \sum_{\{i:A_i \geq 4\}}(A_i-3) +
E_{N}\sum_{i=1}^{N+1}(A_i-1)}$. Since the set $B(n)$ is the finite union of
disjoint cylinder sets $[vvv]_{I_n}$, with  
$vvv\in J(n)$, where $I_n=\left[1-|\zeta^n(a)|,2|\zeta^n(a)|\right]$,
\begin{eqnarray*}\mu(B(n))&=& \sum_{vvv \in J(n)}{\mu}([vvv]_{I_n}) \\ &=&
\sum_{vvv \in J(n)}d(vvv) \\ &\ge& [B_1+A_2 (f_{N+1}^{n-1} + h_{N}  R_{n-1}
)]d(vvv)
\end{eqnarray*} By writing out $d(vvv)$ one gets
$$
\lim_{n\to\infty}\mu(B(n))\ge  \frac{A_2{(f_{N+1} + h_{N} -1)d_N}}{{f_{N+1}^4
-2f_{N+1}^3 + f_{N+1}^2 + (f_{N+1}^2 h_{N})(f_{N+1}-1)}}>0.   
$$ Noticing that~\cite{D1,DP1}
$$\limsup B(n) \subseteq C_p \equiv \{ \omega\in\Omega_{\zeta} : \sigma_{p}
(H_{\omega}) = \emptyset \},
$$
 $C_p$ is subshift invariant, $\mu$ is ergodic, and
$$
\mu(C_p)\ge \mu(\limsup B(n))\ge \lim \mu(B(n))>0,
$$ one concludes that
$\sigma_{p}(H_{\omega})= \emptyset$  $\mu$-a.e. $\omega \in \Omega_{\zeta}$,
since subshift invariants sets in
$(\Omega_{\zeta},T)$  have either null or full ergodic measure. 
\end{proof}



\section{Final Remarks}
\label{RemarksSection}
\begin{enumerate}
\item Notice that the integer 1 is always an eigenvalue of the  matrix of
substitution~$M(\zeta)$, and the other eigenvalue is the
 positive integer $\sum_{k=1}^{N+1} A_k>1$, clearly a non-Pisot number.
Notice also that besides nonprimitive~$\zeta$ is not irreducible.

\item We remark that Propositions~\ref{palindromicProp} and~\ref{GordonProp},
used to exclude point spectrum, can be adapted for ultimately periodic
sequences. The nonperiodicity were important here for the exclusion of the
absolutely continuous spectrum in Proposition~\ref{absolutely}.

\item The generalization of the above results for the case~$\zeta(b)=bb...b$
in~(\ref{nonprimitive}), with more than one factor,  is not direct, since in
this case there are words composed only of $b$'s of arbitrary long length and
$(\Omega_{\zeta},T)$ is not strictly ergodic.

\item As a direct corollary of our results it follows that for palindromic
substitutions~$\zeta$, with
$A_1+A_{N+1}\geq4$ or  $A_j\geq4$ for some~$1\leq j\leq (N+1)$, the
spectrum~$\sigma(H_\omega)$ is pure singular continuous for~$\omega$ in a
generic set as well as in a set of full invariant measure.

\item Another possibility to exclude point spectrum directly for the
fixed-point sequence~$\eta$ of the substitution~$\zeta$ is to apply the
techniques presented in references~\cite{Su1,BG1}.

\item Given a particular substitution~$\zeta_N$ (under the hypotheses of
Proposition~\ref{GordonProp}) it is possible to find the value of
$\lim_{n\to\infty}\mu(B(n))$. E.g., if all~$B_j$ are pairwise different  or
$A_j\ne A_1+A_{N+1},\forall j$, then this limit is given by
$$
\frac{{(f_{N+1} + h_{N} -1)d_N}}{{f_{N+1}^3 -2f_{N+1}^2 + f_{N+1} +
(f_{N+1}h_{N})(f_{N+1}-1)}}.  
$$
\item An open problem is to characterize the spectral type~\cite{Q} of the
autocorrelation measure for the nonperiodic substitution sequences generated
by~$\zeta_N$. How does it depend on~$A_1,\cdots,A_{N+1},B_1,\cdots,B_N$?


\end{enumerate}





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