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\title{Singular continuous spectrum in ergodic theory}
\author{Oliver Knill\thanks{Division of Physics, Mathematics and Astronomy,
Caltech, 91125 Pasadena, CA, USA}
}
\date{}
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\maketitle
\begin{abstract}
We prove that in the weak topology of measure preserving transformations,
a dense $G_{\delta}$
has purely singular continuous spectrum in the orthocomplement of the
constant functions. In the uniform topology, a dense $G_{\delta}$
of aperiodic transformations has singular continuous spectrum.
We show that a dense $G_{\delta}$ of shift-invariant
measures has purely singular continuous spectrum. These results stay
true for $\ZZ^d$ actions of measure preserving transformations.
There exist smooth unitary cocycles over an
irrational rotation which have purely singular continuous spectrum.
\end{abstract}
\pagestyle{myheadings}
\thispagestyle{plain}
\section{Introduction}
The spectral theory for the unitary operators belonging to
measure-preserving transformations started with the works of
Koopman and von Neumann and turned out to be very fruitful in
ergodic theory (see for example \cite{CFS,Queffelec,Sinai}).
A general problem in spectral theory for Schr\"odinger
operators is to distinguish pure point (=discrete) spectrum ($\sigma_{pp}$),
singular continuous spectrum ($\sigma_{sc}$) and absolutely continuous
spectrum ($\sigma_{ac}$).
As in the theory of Schr\"odinger operators,
the spectral analysis of a transformation
is in general not easy and evenso we will see that
genericity results are easy to find, it is harder to find
{\it concrete} examples of a measure preserving
transformation with specific spectral properties.
Such situations occur at other places in ergodic theory and mathematical physics:
in the spectral theory for Schr\"odinger operators, singular continuous
spectrum is often generic \cite{Sim95,Sim95a,De+94,De+94a}
for concrete examples of operators $L$
with $\sigma(L)=\sigma_{sc}(L)$ see \cite{AvSi82,JiSi94,Hof+94}.
Polygonal billiards are examples of difficult spectral problem
there is no ergodic polygonal billiard known,
evenso ergodicity is generic for such systems (see \cite{Gut86}). \\
{\bf Discrete spectrum}: \\
Transformations with discrete spectrum are known to be
isomorphic to group translations
on the dual group of the spectrum and the spectrum is always simple.
On the other hand, the set of dynamical systems which have no
eigenvalues (except the trivial
eigenvalue $1$ belonging to the constant eigenfunctions)
coincides with the class of weakly mixing transformations. Therefore,
pure point or absence of pure point spectrum is dynamically characterized. \\
{\bf Absolutely continuous spectrum}: \\
Also absolutely continuous spectrum has relations
with dynamical properties evenso no characterization is known.
K-systems and ergodic automorphisms of
commutative compact groups have absolutely continuous spectrum of countable
multiplicity. Sufficient conditions for absence of $\sigma_{ac}$
can be given in terms of periodic approximations \cite{Sinai}.
If a transformation has positive metric entropy, a theorem of
Sinai \cite{Sinai} assures that it contains a
Bernoulli shift as a factor. This implies a result of Rohlin saying that such
a transformation has some
absolutely continuous spectrum of infinite multiplicity. Much research
has been devoted to the multiplicity of the spectrum (see
for example \cite{Rob83,Rob85,CFS,Iwa92}).
Rohlin mentions (\cite{Roh49}, p. 218)
that the set of transformations $T$ with $\sigma(T)=\sigma_{ac}(T)$
are meager both in the uniform and the
weak topology of measure preserving transformations, a result which can
also be proven by periodic approximation. There seems not to exist
a necessary and sufficient condition for purely absolutely continuous spectrum.
This can be illustrated by
the still open question of finding substitution systems with purely absolutely
continuous spectrum (see \cite{ReRa93}). \\
{\bf Singular continuous spectrum}. \\
While automorphisms with
pure point spectrum or purely absolutely continuous spectrum are easy to
give, examples of dynamical systems with purely singular continuous spectrum
are harder to find. Robinson (\cite{Rob83} Theorem 7.1 and the
remark in the intruduction)
constructed interval exchange transformations with purely singular
continuous spectrum.
Katok-Stepin have shown (\cite{KaSt70} p. 201) that a generic
volume preserving homeomorphism of a manifold
has purely singular continuous spectrum,
strengthening so Oxtoby-Ulams's theorem about generic ergodicity.
With their methods \cite{KaSt67}, it can be deduced
(see \cite{Ka+75} remark 4.1.1),
that a generic measure preserving transformation has purely
singular continuous spectrum. We will prove
this here also with other methods. \\
{\bf Mixed spectrum}: \\
Many systems have mixed spectrum. For example,
for certain skew products, there is both countable absolutely continuous and
discrete spectrum \cite{Iw+93}.
For substitution systems, there are conditions for distinguishing
discrete and mixed discrete-continuous spectrum. For example,
for the Thue-Morse system $0 \mapsto 01, 1 \mapsto 10$, there is
mixed spectrum \cite{Queffelec}. Substitutions systems are not
strongly mixing and have therefore not purely absolutely continuous
spectrum (\cite{Parry} p. 50). Continuous spectrum and so some
singular continuous spectrum are known for the
substitution systems $0 \mapsto 01010, 1 \mapsto 011$
(see \cite{Queffelec} p. 131) and for $0 \mapsto 001$,
$1 \mapsto 11100$ \cite{DeKe78}.
The Rudin-Shapiro system is an example of a substitution containing
some absolutely continuous spectrum
of multiplicity $2$ \cite{Queffelec}.
Toeplitz flows can have discrete spectrum \cite{JaKe69},
there are examples with fast periodic approximation and so no
absolutely continuous spectrum \cite{IwLa94} and examples with some absolutely
continuous spectrum of finite multiplicity \cite{Lem88}. \\
Given a transformation with mixed spectrum, the conditional
expectation projection map leads to factors, which
have pure spectrum of each occurring spectral component.
Concrete examples with
purely singular continuous spectrum can therefore be obtained by
taking suitable factors of some substitution systems.
There are many other candidates (see also the discussion). \\
For other generic spectral properties of measure preserving transformations,
see \cite{DeLe92}.
%There, it is shown there (roughly speaking)
%that in the weak
%topology, a generic transformation $T$ has the property that most convolutions
%of spectral measures of iterates of $T$ are mutually singular. \\
In this note, spectral questions are discussed in the context of recent
developments of operator theory \cite{Sim95,Sim95a}.
\section{Measure preserving transformations}
A measure preserving automorphism $T$ of a probability space $(X,\Acal,m)$
defines the unitary Koopman operator $U_T: f \mapsto f(T)$ on $L^2(X)$.
Since $1$ is always an eigenvalue, it is convenient to restrict $U_T$ to
the orthogonal complement of the constant functions and to speak of
the spectrum of this operator.
Let $\Gcal$ be the complete topological
group of automorphisms of
$(X,\Acal,m)$ with the weak topology: $T_j$ converges to $T$ weakly, if
$m(T_j(A) \Delta T(A)) \rightarrow 0$ for all $A \in \Acal$
(see \cite{Halmos}).
\begin{thm}
\label{generic}
In the weak topology,
a dense $G_{\delta}$ of transformations $T$ in $\Xcal$ satisfies
$\sigma(U_T)=\sigma_{sc}(U_T)$.
\end{thm}
\begin{proof}
By Halmos mixing theorem \cite{Halmos},
a transformation $T \in \Gcal$ is weekly mixing ($\Leftrightarrow
T \times T$ is ergodic)
if and only if it has continuous spectrum.
By Halmos second category theorem \cite{Halmos},
the set of weakly mixing transformations (and so with continuous spectrum)
is a dense $G_{\delta}$ in $\Gcal$ in the weak topology.
Rohlin's lemma (see \cite{Halmos}) shows that
the set of periodic transformations (and so with pure point spectrum)
is dense in the uniform topology and so also dense in the weak topology.
If $T_n \rightarrow T$ weakly in $\Gcal$,
then the corresponding Koopman operators $U_{n}$ converge to $U$ in the
strong operator topology and so in the strong resolvent topology.
(We know that $U_{n} 1_A \rightarrow U 1_A$ for every $A \in \Acal$.
This stays true for step functions $\sum_n a_n 1_{A_n}$ which are dense
in $L^2(X,m)$. Therefore $U_n \rightarrow U$ in the strong operator topology
and by general principles in the strong resolvent sense.)
We can now apply
Simon's wonderland theorem \cite{Sim95,Sim95a}) in the following form:
let $\Xcal$ be a complete metric space of unitary operators, where the metric
is stronger than the strong resolvent topology. Assume
there exists a dense set with continuous spectrum and a dense set with
pure point spectrum, then there exists a dense $G_{\delta}$ of
purely singular continuous spectrum.
(The wonderland theorem is
formulated for selfadjoint operators. However, the proofs
that the two sets $\{ A \in \Xcal \; | \; , A$ has no eigenvalues $\}$,
$\{ A \in \Xcal \; | \; , A$ has no absolutely continuous spectrum $\}$
are both $G_{\delta}$', hold also of unitary operators.
Alternatively, the result for unitary operators can be obtained from
the result for selfadjoint operators by Cayley transforms.)
\end{proof}
Remarks. \\
1) As mentioned in the introduction, Theorem~\ref{generic} is also reachable with
Katok-Stepin's approximation
methods and is deducable (while not explicitely stated)
from the results in \cite{KaSt67}. \\
2) This theorem or the theorem of Katok-Stepin \cite{KaSt70} saying that
a generic volume preserving homeomorphism has purely singular continuous
spectrum can be used to get generic purely singular continuous spectrum
for other topological spaces of transformations:
take two topological spaces $X \subset Y$ of transformations which are
both strong enough to use the wonderland theorem.
Assume $X$ is dense in $Y$ in the topology
of $Y$ and a generic set in $X$ has purely singular continuous spectrum,
then a generic set in $Y$ has purely singular continuous spectrum. \\
2) Theorem~\ref{generic} illustrates the proof of Halmos genericity result:
it is enough to show that weakly mixing is dense in order to conclude
that it is a dense $G_{\delta}$. The density of weakly mixing follows
directly from the conjugacy lemma of Halmos
(see \cite{Alp81} for more details about
conjugates of antiperiodic transformations). In order to get generically
weakly mixing, one needs a dense set with no $\sigma_{ac}$ and a dense
set with no $\sigma_{pp}$. \\
Theorem~\ref{generic} can not true in the uniform topology
$d(T,S)=m( S(x) \neq T(x) )$ because ergodic transformations
are then nowhere dense \cite{Halmos}. We can however restrict ourself to
$\Kcal$, the closure of the aperiodic transformations in the
uniform topology.
\begin{thm}
In the uniform topology, $\Kcal$ contains
a dense $G_{\delta}$ of transformations $T$ with $\sigma(U_T)=\sigma_{sc}(U_T)$.
\end{thm}
\begin{proof}
$\Gcal_a$ is a complete metric
space since it is a closed set of a complete metric space.
By Rohlin's lemma, it contains a dense set of
periodic transformations.
The conjugacy class $[T]$ of
one aperiodic transformation $T$ is dense in $\Gcal_a$ in the uniform
topology (see \cite{Halmos,Friedman}).
The existence of weakly mixing transformations,
gives therefore a dense set of weakly mixing transformations.
The wonderland theorem implies again the claim.
\end{proof}
Remarks. \\
1) It follows that $\Kcal$ is
nowhere dense in $\Gcal$ in the uniform topology. \\
2) It follows that
all pure spectral types $\sigma_{sc}$, $\sigma_{ac}$, $\sigma_{pp}$
are dense in the weak topology in $\Xcal$ and dense in the uniform topology
in $\Kcal$. \\
3) Singular continuous spectrum can say something about the
dynamics. Theorem~(\ref{generic}) implies a
result of Rohlin \cite{Roh59} saying that
generically, a transformation has zero metric entropy.
To obtain this corollary, one uses
Rohlin's theorem \cite{Roh59} stating that
a transformation with positive metric entropy has some
countable Lebesgue spectrum. \\
4) Let $U$ be the unitary Koopman operator of a transformation $T$.
Consider the free Schr\"odinger operator $L=U+U^*$ on $L^2(X)$. Since
$U$ can be written as a function of $L$ as
$L/2+ i \sqrt{1-L^2/4}$, the operator $L$ has the same spectral
type as $U$ (every eigenvalue of $U$ corresponds to an eigenvalue of $L$
and converse) and $L$ has some absolutely continuous component if and only
if $U$ has. Theorem~\ref{generic} shows
that we can find such free Schr\"odinger operators on $L^2(X)$
with purely singular continuous spectrum. \\
5) With the induced product topology, the set $\Gcal^d$ of $\ZZ^d$-
actions is a complete metric space.
We say, $T \in \Gcal^d$ has purely singular continuous
spectrum, if all the transformations $T^n, n \neq 0$
have purely singular continuous spectrum in the orthocomplement of the
eigenspace of the eigenvalue $1$. Since the projection from $\Gcal^d$
to one coordinate is continuous and maps open sets into open sets, the
inverse image of a generic set is generic.
It follows that a dense $G_{\delta}$
of $\ZZ^d$-actions has purely singular continuous spectrum.
%The Rohlin lemma for $\ZZ^d$-actions \cite{FeLi76} is not needed.
%Natarajan who has shown
%that weakly mixing is generic for $\ZZ^2$
%actions (see \cite{Ka+75} 4.1.1). \\
Also in the multidimensional case, a $K$-system has countable Lebesgue spectrum
and systems with positive metric entropy have some Lebesgue
spectrum (see \cite{KaLi94}). It follows
that a generic $\ZZ^d$-action has zero entropy.
\section{Shift invariant measures}
Given a topological shift $(X=A^{\ZZ},T)$ over a finite alphabet
$A$. The space of shift-invariant measures is a complete metric space
with respect to the weak-* topology.
\begin{thm}
\label{genericmeasure}
There exists a generic set of $T$-invariant measures $m$ for which
the abstract dynamical system $(X,T,m)$
has purely singular continuous spectrum.
\end{thm}
\begin{proof}
The claim follows again with wonderland:
according to Parthasarathy \cite{Par61}, there exists a dense set
of periodic invariant measures.
Sigmund \cite{Sig72}
showed that there exists a generic set of invariant measures for which
the system is weakly mixing.
If $\mu_n \rightarrow \mu$,
then $U_n f \rightarrow U f$ for every continuous function $f$ and
so $U_n \rightarrow U$ strongly and so in the strong resolvent sense.
\end{proof}
Remarks.
1) As a corollary, we obtain an other result of Sigmund \cite{Sig71}
saying that
the set of shift-invariant measures with
zero metric entropy is a dense $G_{\delta}$. \\
2) Summarizing the other known results in literature (see \cite{DGS}):
A generic shift-invariant measure
on $X$ is ergodic, nonatomic, with support $X$, weakly mixing
and not strongly mixing, has zero metric entropy and singular continuous
spectrum. \\
3) An other corollary is that
if $f$ is an invertible Axiom-A diffeomorphism restricted
to a basic set $\Omega_i$,
there is a generic set of $f$ invariant measures on $\Omega_i$ which
have purely singular continuous spectrum (see \cite{Sig70,Sig72}). \\
4) Theorem~(\ref{genericmeasure}) stays true if the finite alphabet $A$ is
replaced by a compact metric space $Y$. The reason is that
there is a dense set of measures
in $Y^{\ZZ}$ which have support on some $A^{\ZZ}$, where $A$ is a finite
subset of $Y$. We can especially take $Y=Z^{(\ZZ^{d-1})}$, where $Z$ is
a compact space, to obtain the corollary that
a generic $\ZZ^d$-shift invariant measure in $Z^{(\ZZ^d)}$
has purely singular continuous spectrum and again with \cite{KaLi94}
zero metric entropy.
\section{Circle-valued cocycles}
Let $T_{\alpha}$ be the irrational rotation $x \mapsto x+\alpha \;
{\rm mod} \; 1$
on the circle $X=\TT^1$. Define the maps $a_{\beta,s}:
X \rightarrow \{|z|=1\}$
by $a_{\beta,s}(x)=e^{2\pi i s 1_{[0,\beta)}}$ and the unitary operators
$U_{\beta,s,\alpha} f(x) = a_{\beta,s} f(x+\alpha)$.
\begin{thm}
For a generic set of pairs $(\alpha,\beta)$, the operator
$U_{\beta,s,\alpha}$ has purely singular continuous spectrum.
\end{thm}
\begin{proof}
If $\alpha$ has bounded partial quotients and $s \neq 0$ and $\beta \notin
\alpha \RR/\ZZ$, then $U_{\beta,s,\alpha}$ has purely continuous spectrum
(\cite{Mer85} Theorem 2.4), (use also \cite{Bag88} Proposition 2.1).
If $\alpha$ is rational, then $U_{\beta,s,\alpha}$ has pure point
spectrum.
$U_{\beta,s,\alpha}$ depends continuously
on $(\alpha,\beta)$ in the strong operator topology.
The claim follows again from the wonderland theorem.
\end{proof}
Let $V$ be a measurable map from $\TT^1=\RR/\ZZ$ to $\RR$. Consider the
evolution of a function $u \in L^2(\TT^1)$ with time-dependent vector field
$$ H_t = i (\frac{d}{d\theta} + V(\theta) \delta(t+2 \pi n \alpha)) \; . $$
Mapping over one period gives
the Floquet map $Su(\theta) = u(\theta+1)$ of a solution $\dot{u}=Hu$.
This map is a circle-valued cocycle and we get
\begin{coro}
There are potentials $V=s \cdot 1_{[0,\beta)}$
for which the unitary Floquet operator of $H_t$ has purely singular continuous
spectrum.
\end{coro}
Given now $m \in \ZZ \setminus \{0\}$ and
$a_{\phi,m}(x)=e^{2\pi i \phi(x) + m x}$, where $\phi$ is in the topological
space $C(\TT)$ of continuous periodic functions. Denote with
$U_{\phi,m,\alpha}$ the associated unitary weighted composition
operator on $L^2(X)$: $(U_{\phi,m,\alpha} f)(x) = a_{\phi,m}(x) f(x+\alpha)$.
\begin{thm}
\label{continuouscocycle}
For $m \neq 0$, there exists a generic set of
$(\phi,\alpha) \in C(\TT) \times \RR$ such that the
operator $U_{\phi,m,\alpha}$ has purely singular continuous spectrum.
\end{thm}
\begin{proof}
The complete metric space $C(\TT)$ contains the dense set of functions $\phi$
which have a Fourier series $\hat{\phi}_n=o(1/n)$. For such $\phi$ and irrational
$\alpha$, the operator $U_{\phi,m\alpha}$ has no eigenvalues if $m \neq 0$
(\cite{LeMa94} Corollary 2).
If $\alpha$ is rational, then the operator $U$ has pure point spectrum.
Apply wonderland.
\end{proof}
Remark. In Theorem~\ref{continuouscocycle},
the space $C(\TT)$ can be replaced with any subset of $L^2(\TT)$
which is equiped with a topology in which functions with Fourier series
of order $o(1/n)$ are dense. Especially, this gives smooth cocycles with
purely singular continuous spectrum.
\section{Discussion and questions}
We have shown here how putting together known results in ergodic
theory and operator theory leads to new results in ergodic theory.
However, further investigations of
the nature of transformations with purely singular continuous spectrum is
necessary and many questions are open. \\
%Since, generically, the multiplicity of the spectrum
%is simple, one could guess that singular continuous spectrum
%is simple in general. A similar question is in general also open
%in the theory of random (multidimensional) Schr\"odinger operators. \\
For every $f \in L^2(X)$ is defined the correlation sequence
$c_n=(U^n f,f)$. The Fourier transform of
this sequence is the spectral measure $\mu_f$ called the correlation
measure. It has purely singular
continuous spectrum if $T$ has. How can one characterize sequences which are
the Fourier transform of singular continuous probability
measures on the circle? This question (see \cite{Mak94} and references therin
for selfaffine measures in $\RR^n$)
would shed light onto the problem to characterize
transformations $T$ with $\sigma(T)=\sigma_{ac}(T)$ or
$\sigma(T)=\sigma_{sc}(T)$. \\
If $T$ is the irrational rotation on the circle, it is known that
the set of $A \in \Acal$ for which
the induced transformation $T_A$ is weakly
mixing is generic with respect to the metric $d(A,B)=m(A \Delta B)$
\cite{Friedman,Pet73}.
Veech asked (see \cite{Pet73} p. 229),
whether the set of $A$ for which $\sigma_{pp}(T_A)=\sigma(T_A)$ is dense
(see also \cite{Con72}).
If the answer to this question were
yes, then the wonderland theorem would give
a generic set of $A$'s with $\sigma_{sc}(T_A)=\sigma(T_A)$. \\
It would be interesting to get smooth differential
equations $\dot{u}=A(u,v), \dot{v}=B(u,v)$ on the torus with
$A^2+B^2>0$ and preserving a smooth measure $\mu$ with flows having
purely singular continuous spectrum.
If the rotation number
$\lambda=\int A \; d\mu/\int B \; d\mu$ of the flow
satisfies a Diophantine condition, Kolmogorov showed
that the flow has purely discrete spectrum.
Also Katok-Stepin (see \cite{Sinai})
show with the theory of periodic approximations
that there exists a dense set of $\lambda$ such
that the flow has no absolutely continuous spectrum.
According to \cite{Sinai}, Kolmogorov (worked out by Sklover)
constructed examples which are weakly mixing. Krygin \cite{Kry74}
has given examples with mixed spectrum. \\
Consider a family of
Sinai Billiards in a square table with a one convex scatterer
(of fixed diameter and center) as
a parameter. If we take for the scatterer the Hausdorff topology
on the compact set of closed convex sets (compare \cite{Gruber}),
we expect to have a dense set of
mixing systems, when the scatterer is smooth and with positive curvature
everywhere and a dense set
with no absolutely continuous spectrum when the scatterer is a polygon.
This would lead for a generic set of convex scattereres purely singular
continuous spectrum and especially ergodicity. \\
The spectral question can also be asked in noncommutative ergodic theory:
an automorphism $S$ on the hyperfinite factor $(\Xcal,{\rm tr})$
preserving the trace defines a unitary operator $U_S$ on the completion
of the pre-Hilbert space
$(\Xcal, {\rm tr}( \cdot, \cdot ))$. An obvious problem is to
ask for the existence of a noncommutative example $S$ with
$\sigma(U_S)=\sigma_{sc}(U_S)$. \\
{\bf Acknowledgements}: I thank B. Simon for valuable discussions
and A. Hof for some important remarks.
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