text:
\section{Introduction}
\label{sec-intro}

Uniformly expanding Markov maps of the interval represent a paradigm
for chaotic \dsy s. They make up a fairly large class of non-trivial
maps, and possess the standard ingredient for chaos, namely, \hyp
ity---insofar as expansivity can be understood as the one-dimensional
version of \hyp ity. On the other hand, they are simple enough to be
more or less fully understood via the techniques of the modern theory
of \dsy s; cf. the excellent textbook by Boyarsky and G\'ora
\cite{bg}.

In infinite \erg\ theory, the analogues of such maps are the uniformly
expanding Markov maps of $\R$ which preserve an infinite \me.  Perhaps
not surprisingly, very little is known about them, at least to this
author. (The \sy s studied by Bugiel \cite{b1, b2} are very similar,
but they are designed to preserve a finite \me.)

In this note we consider a rather large class of uniformly expanding
Markov maps of the real line; an example is illustrated in
Fig.~\ref{pwe} below. We assume that they preserve the Lebesgue \me,
which is no big loss of generality for maps that preserve an
absolutely continuous \me\ with bounded density. We also assume a
condition, cf. (A2) below, that rules out periodicity.  The remaining
assumptions are rather standard and not unduly restrictive.

We are interested in the mixing properties of such \dsy s. To start
with, we consider exactness, which is generally regarded as the
strongest form of mixing, and has the advantage of being defined in
the same way in both finite and infinite \erg\ theory. We prove a
theorem (Theorem \ref{thm-exa}) that essentially says that every
irreducible component of the map is exact, and gives easy sufficient
conditions for exactness.

We then apply the notions of mixing for infinite-\me-preserving \dsy s
recently introduced by the author in \cite{limix}. We give these
notions, within the present scope, in Section \ref{sec-im} below and
refer the reader to \cite{limix, liutam, lpmu} for a more thorough
discussion. (The last reference, in particular, uses a more intuitive
notation and contains several results that are used in this article.)

For this part, we specialize to a much narrower but still nontrivial
class of maps. We consider both \emph{quasi-lifts} of expanding circle
maps, i.e., piecewise smooth, translation invariant maps $\R \into
\R$, whose quotient on a fundamental domain is an expanding map of the
circle (see Fig.~\ref{ql}), and \emph{finite modifications} thereof,
namely, maps that differ form a quasi-lift of an expanding circle map
only on a bounded domain (see Fig.~\ref{fm}). In both cases, we prove
versions of \emph{global-local \m} and \emph{global-global \m}.  Very
loosely, global-local \m\ means that any \emph{global \ob} (roughly, a
bounded \fn) and any \emph{local \ob} (an integrable \fn) decorrelate
in time. Global-global \m\ means that the same happens for any two
global \ob s.

Of course, we have stronger results for the more specialized class of
\sy s, that is, the quasi-lifts. In particular, we prove a property,
called \textbf{(GLM2)} in what follows, that can be recast like this:
For any global \ob\ $F: \R \into \C$ and any Lebesgue-absolutely
continuous \pr\ \me\ $\mu$, if $T_*^n \mu$ denotes the push-forward of
$\mu$ via the map $T^n$, then
\begin{equation}
  \label{glm2-intro}
  \lim_{n \to \infty} T_*^n \mu(F) = \avg(F),
\end{equation}
where $\avg(F)$ represents, in a sense that is specified below, the
average of $F$ over $\R$. Thus, (\ref{glm2-intro}) can be regarded as
a sort of weak convergence of $T_*^n \mu$, the statistical state of
the \sy\ at time $n$, to the ``equilibrium state'' $\avg$, which is
independent of the initial condition $\mu$, with the global \ob s
playing the role of test \fn s.

We are unable to prove this strong property for all finite
modifications of quasi-lifts of circle maps, but we certainly believe
it to be true for a large class of such \sy s. For this reason, we
give an example for which a very strong version of \textbf{(GLM2)} can
be indeed be shown. As described in Section \ref{sec-fmrw}, this
example represents a random walk in $\Z$, and in this sense it
illustrates a whole family of maps.

\skippar

This is how the paper is organized. In Section \ref{sec-setup} we
introduce our maps and state their exactness properties.  In Section
\ref{sec-im} we give our definitions of infinite \m\ and apply them to
certain subclasses of maps. In Section \ref{sec-fmrw} we give an
example of a finite modification of a quasi-lift that verifies a
strong form of global-local \m. Since some proofs are rather
cumbersome, we have gathered all the proofs in Section
\ref{sec-proofs}, further moving the more standard lemmas to the
Appendix.

\bigskip\noindent
\textbf{Acknowledgments.}\ I thank Stefano Isola for a few
illuminating discussions and Sara Munday for her help in the preparation
of the paper. I also acknowledge the hospitality of the
Courant Institute of Mathematical Sciences at New York University,
where part of this work was done.

\section{Setup and exactness}
\label{sec-setup}

We start with a collection of real numbers $( a_j )_{j \in \Z}$, such
that, for some $c_1, c_2 > 0$ and all $j$,
\begin{itemize}
\item[(A1)] $c_1 \le a_{j+1}-a_j \le c_2$.
\end{itemize}
Let $I_j := [a_j, a_{j+1}]$. We call $\{ I_j \}_{j \in \Z}$ a
partition of $\R$ even though formally it is not---the substance of
what we discuss in this paper would not change if we made the cleaner
yet more cumbersome choice $I_j := [a_j, a_{j+1})$.

Let us consider $T: \R \longrightarrow \R$, a Markov map relative to
$\{ I_j \}$. More precisely, we assume that there exists $\bigj \in
\Z^+$ such that
\begin{itemize}
\item[(A2)] $T |_{(a_j, a_{j+1})}$ has a unique extension $\br_j: I_j
  \longrightarrow B_j$, which is twice differentiable and bijective
  onto $B_j := \bigcup_{k=k_{1j}}^{k_{2j}} I_k$, with $k_{1j} \le j
  \le k_{2j}$ and $2 \le k_{2j} - k_{1j} +1 \le \bigj$.
\end{itemize}
The significance of (A2) is that (safe for the endpoints $a_j$ and
$a_{j+1}$) $T$ maps $I_j$ onto an interval that is made up of at least
2 and at most $\bigj$ intervals of the Markov partition, including
$I_j$.

Let $\br_j'$ and $\br_j''$ denote, respectively, the first and second
derivatives of $\br_j$, and $\| \cdot \|_\infty$ the sup norm in
$\R$. There exist constants $\lambda > 1$ and $\eta > 0$ such that
\begin{itemize}
\item[(A3)] $\left\| \br_j' \right\|_\infty \ge \lambda$;
\item[(A4)] $\ds \left\| \frac{\br_j''}{\br_j'} \right\|_\infty \le
  \eta$.
\end{itemize}
Finally, if $\leb$ is the Lebesgue \me\ on $\R$, we require that
\begin{itemize}
\item[(A5)] $T$ preserves $\leb$, i.e., $\leb(T^{-1}A) = \leb(A)$, for
  any Borel set $A \subset \R$.
\end{itemize}
Observe that (A5) is equivalent to the condition that, for all $x \in
\R \setminus \bigcup_j \{ a_j \}$,
\begin{equation} \label{leb-pres}
  \sum_{y \in T^{-1} \{x\}} \frac1 {| T'(y) |} = 1.
\end{equation}
An example of a map satisfying (A1)-(A5) is depicted in
Fig.~\ref{pwe}.

\fig{pwe}{6cm}{A uniformly expanding Markov map $\R \into \R$.}

We now give a couple of definitions that will help us better
understand the features of the maps we have just defined.

\begin{definition} \label{def-irr-ap}
  The \emph{transition matrix} associated to the Markov map $T$ is the
  stochastic matrix $\mathcal{P} = (p_{jk})_{j,k \in \Z}$, with
  $p_{jk} := \leb( T^{-1} I_k \,|\, I_j)$. We say that $T$ possesses a
  certain property pertaining to homogeneous Markov chains if
  $\mathcal{P}$ does. In particular,
  \begin{itemize}
  \item[(a)] $T$ is \emph{irreducible} if the graph whose vertices are
    the elements of $\Z$ and whose bonds are the pairs $(j,k)$ with
    $p_{jk} > 0$ is connected. It is easy to see that this occurs \iff
    the only $T$-invariant set that can be written as a union of
    intervals $I_j$ is the whole of $\R$.

  \item[(b)] $T$ is \emph{aperiodic} if, $\forall j \in \Z$, the
    greatest common divisor of the numbers $n \in \Z^+$ such that
    $p_{jj}^{(n)} > 0$ is 1 \cite{s}. (Here $p_{jk}^{(n)}$ denote the
    entries of $\mathcal{P}^n$.)
  \end{itemize}
\end{definition}

Notice that (A2) implies $p_{jj} > 0$, which gives $p_{jj}^{(n)} > 0$,
for all $n \ge 1$, and thus aperiodicity. It is important to require
that ours maps be aperiodic, because periodicity is a clear obstacle
to exactness (see Remark \ref{rk-ssrw} below for a detailed example).

This is the main result of this section:

\begin{theorem} \label{thm-exa}
  If $\bor$ is the Borel $\sigma$-algebra of $\R$, the following hold
  for the \dsy\ $(\R, \bor, \leb, T)$ defined above:
  \begin{itemize}
  \item[(a)] The conservative part $\mathcal{C}$ is made up of a
    countable number of \erg\ components $X_\alpha$, each of which is
    a union of adjacent intervals from $\{ I_j \}_{j \in \Z}$ and is
    exact.
  \item[(b)] The dissipative part $\mathcal{D}$ is the disjoint union
    of at most two \erg\ components, $X_+$ and $X_-$. Each $X_\pm$, if
    not trivial, has infinite \me, is exact, and is such that, for
    a.a.\ $x \in X_\pm$, $\ds \lim_{n \to \infty} T^n(x) = \pm\infty$.
  \item[(c)] If the \dsy\ is not totally dissipative (i.e.,
    $\leb(\mathcal{C}) > 0$) and $T$ is irreducible, then the \sy\ is
    recurrent and exact.
  \end{itemize}
\end{theorem}

\proof See Section \ref{sec-proofs}.
\bigskip

We can use Theorem \ref{thm-exa} to find many examples of exact maps
of our type. Let us focus on the simplest cases that are not piecewise
linear. In what follows, the reference $\sigma$-algebra will always be
the Borel $\sigma$-algebra.

\subsection{Quasi-lifts of expanding circle maps}
\label{subs-ql}

Let us consider the case where the elements of the Markov partition
have the same size, e.g., $I_j := [aj, a(j+1)]$, for some $a>0$, and
$T$ acts in the same way on each of them, that is, for $x \in I_j$,
$\br_j(x) = \br_0(x-aj) + aj$. This is equivalent to
\begin{equation} \label{trn-comm}
  T \circ \trn = \trn \circ T,
\end{equation}
where $\trn(x) := x+a$. In other words, $T$ is translation invariant
(by the quantity $a$). See Fig.~\ref{ql}.

\fig{ql}{6cm}{An example of a quasi-lift of an expanding circle map.}

If $\mathcal{S}_a$ is the circle constructed by identifying the
endpoints of $[0,a]$, and $T_a: \mathcal{S}_a \into \mathcal{S}_a$ is
defined by $T_a(x) := T(x)$ mod $a$, it easy to see that $T_a$ is a
Lebesgue-preserving, uniformly expanding map of the circle, with
bounded distortion. It thus possesses all the good mixing properties
one expects \cite{bg}. $T$ is a sort of lift of $\mathcal{S}_a$ to
$\R$, which we might call \emph{quasi-lift}. Not that we are
interested in the actual lift, which fails to preserve the Lebesgue
\me. (Quasi-lifts and similar maps have often been used in nonlinear
physics as toy models for normal and anomalous diffusion; see, e.g.,
\cite{sj, k, ac, khk} and references therein.)

\begin{proposition} \label{prop-exa-ql}
  A quasi-lift of an expanding circle map, as defined above, is
  irreducible and exact.
\end{proposition}

\proof See Section \ref{sec-proofs}.

\subsection{Finite modifications of quasi-lifts}
\label{subs-fm}

Slightly more complex examples than the quasi-lifts of circle maps are
the so-called finite modifications of quasi-lifts of circle maps.  If
$T_0$ is a quasi-lift as defined earlier, $T: \R \into \R$ is a
\emph{finite modification} of $T_0$ if there exists $k' \in \Z^+$ such
that $T(x) = T_0 (x)$, for all $x \not\in \bigcup_{j=-k'}^{k'}
I_j$. We assume that $T$ verifies (A1)-(A5) as well.

Constructing maps of this kind is not hard; here's a possibility.
Denote $\br_{0j} := T_0 |_{I_j}$, $B_{0j} := \br_{0j}(I_j)$ (compare
with the notation at the beginning of Section \ref{sec-setup}) and set
$\mathcal{J} := \rset{j\in\Z} {B_{0j} \supset I_0}$; notice that, by
(A2), either $B_{0j} \supset I_0$ or $B_{0j} \cap I_0$ consists of at
most a point.  Now, pick a $C^2$ \fn\ $\psi: \R \into \R_0^+$ that is
compactly supported in $\mathrm{int}(I_0)$, the interior of $I_0$. For
$j \in \mathcal{J}$, denote by $\ph_{0j}: B_{0j} \into I_j$ the
inverse \fn\ of $\br_{0j}$, and set $\ph_j := \ph_{0j} + \delta_j
\psi$, which defines a \fn\ on $B_{0j}$. Here $(\delta_j)_{j \in
\mathcal{J}}$ is a collection of numbers so small, in absolute
value, that $\ph_j$ is a monotonic, hence bijective, \fn\ $B_{0j}
\into I_j$. Also, they satisfy
\begin{equation} \label{def-fm10}
  \sum_{j \in \mathcal{J}} \mathrm{sign} (\ph_{0j}' ) \, \delta_j = 0.
\end{equation}
(Notice that, by (A2), $\ph_{0j}$ is monotonic in $B_{0j} \supset
I_0$.)  Finally, define $\br_j := \ph_j^{-1}$, for $j \in
\mathcal{J}$, and $\br_j := \br_{0j}$, otherwise. This determines the
map $T$. An example of this construction is shown in Fig.~\ref{fm}.

\fig{fm}{6cm}{A finite modification of a quasi-lift of a circle map,
  constructed with the procedure given in Section \ref{subs-fm}.}

$T$ verifies (A1)-(A2) by construction. If the $\delta_j$ are
sufficiently small, (A3)-(A4) are verified as well. As for (A5),
for all $y \in \mathrm{int}(I_0)$,
\begin{equation} \label{def-fm20}
\begin{split}
  \sum_{{j \in \Z} \atop {\ph_j (y) \, \mathrm{defined}}}
  \hspace{-4pt}| \ph_j' (y) | &= \sum_{j \in \mathcal{J}}
  \mathrm{sign} (\ph_j') \, \ph_j'  \\
  &= \sum_{j \in \mathcal{J}} \mathrm{sign} (\ph_j' ) \,  (\ph_{0j}'
  (y) + \delta_j \psi(y)) \\
  &= \sum_{j \in \mathcal{J}} \mathrm{sign} (\ph_{0j}') \, \ph_{0j}'
  (y) + \psi(y) \sum_{j \in \mathcal{J}} \mathrm{sign} (\ph_{0j}') \,
  \delta_j \\
  &= \sum_{j \in \mathcal{J}} | \ph_{0j}' (y) | ,
\end{split}
\end{equation}
via (\ref{def-fm10}) and the fact that $\ph_j'$ and $\ph_{0j}'$ have
the same sign. If $y \in \mathrm{int}(I_k)$, with $k \ne 0$, the same
conclusion as (\ref{def-fm20}) holds by definition of $\br_k$, namely,
$\ph_k$. In other words, using the notation of (\ref{leb-pres}), we
have shown that, $\forall x \not\in \bigcup_j \{ a_j \}$,
\begin{equation}
  \sum_{y \in T^{-1} \{x\}} \frac1 {| T'(y) |} = \sum_{y \in T_0^{-1} \{x\}}
  \frac1 {| T_0'(y) |},
\end{equation}
namely, $T$ preserves the Lebesgue \me\ \iff $T_0$ does, which we have
assumed.

Coming back to the general case, a useful feature of these maps is
that, if $T$ is a finite modification of $T_0$, then $T^n$ is a finite
modification of $T_0^n$ (which is also a quasi-lift of a circle map,
by (\ref{trn-comm})).  In fact, (A2) shows that, if $x \in I_j$,
$T(x)$ can land at most $\bigj-1$ intervals away form $I_j$, hence,
for all $x \ne \bigcup_{j=-k'-n(\bigj-1)}^{k'+n(\bigj-1)} I_j$,
$T^n(x) = T_0^n(x)$.

Finally, a finite modification of a quasi-lift of an expanding circle
map need not be irreducible (it is rather easy to think of
counterexamples), but, if it is, it is also exact.

\begin{proposition} \label{prop-exa-fm}
  If $T$ is a map of the type defined in Section \ref{subs-fm}, and is
  irreducible, then it is exact.
\end{proposition}

\proof See Section \ref{sec-proofs}.

\section{Infinite mixing}
\label{sec-im}

In this section we specify, for the \sy s at hand, some of the notions
of infinite \m\ introduced in \cite{limix} (though we use the more
recent notation of \cite{lpmu}). Then we prove some of those
properties.

Let us call \emph{global \ob} any \fn\ $F \in L^\infty(\R; \C)$ such
that
\begin{equation} \label{im-10}
  \avg(F) := \lim_{r \to \infty} \, \frac1{2r} \int_{x_0 - r}^{x_0 + r}
  F(x) \, dx
\end{equation}
exists uniformly in $x_0$ (and independently of it, as the notation
suggests). Clearly, the class of all global \ob s forms a linear
space, containing, for example, all the \fn s
\begin{equation} \label{def-e-gamma}
  E_\gamma(x) := e^{\i \gamma x},
\end{equation}
for $\gamma \in \R$.

Another way to introduce this class, in the general spirit of
\cite{limix, liutam, lpmu}, would be to define the collection of sets
\begin{equation} \label{im-20}
  \scv := \lset{ \bigcup_{j=k}^\ell I_j } {k \le \ell},
\end{equation}
which we call the \emph{exhaustive family} relative to the partition
$\{ I_j \}$. The elements of $\scv$, denoted $V$, play the role of
``large boxes in phase space''.  Since global \ob s are bounded, it is
easy to see that $F$ verifies (\ref{im-10}) \iff
\begin{equation} \label{im-30}
  \lim_{M \to \infty} \sup_{v \in \scv \atop \leb(V) \ge M} \left|
  \frac1{\leb(V)} \int_V F(x) \, dx - \avg(F) \right| = 0.
\end{equation}
We describe this situation by saying that the average of $F$ over $V
\in \scv$, also denoted $\leb_V (F) := \leb(V)^{-1} \int_V F\, d\leb$,
converges in the \emph{infinite-volume limit} to $\avg(F)$. The
notation
\begin{equation} \label{im-40}
  \ivlim \leb_V (F) = \avg(F)
\end{equation}
is short for (\ref{im-30}). $\avg(F)$ is called the
\emph{infinite-volume average} of $F$.

We also call \emph{local \ob} any \fn\ $g \in L^1(\R; \C)$. For any
such $g$ we use the customary notation $\leb(g) := \int_\R g\, d\leb$.

Let us consider two (sub)classes $\go$ and $\lo$ of global and local
\ob s, respectively. Relative to $\go$ and $\lo$, one says that the
\dsy\ $(\R, \leb, T)$ is \m\ of type \textbf{(GLM1)} if, for all $F
\in \go$ and $g \in \lo$, with $\leb(g) = 0$,
\begin{displaymath}
  \lim_{n \to \infty} \leb ((F \circ T^n) g) = 0.
  \eqno{\mathrm{(GLM1)}}
\end{displaymath}
It is \m\ of type \textbf{(GLM2)} if, for all $F \in \go$ and $g \in
\lo$,
\begin{displaymath}
  \lim_{n \to \infty} \leb ((F \circ T^n) g) = \avg(F) \, \leb(g).
  \eqno{\mathrm{(GLM2)}}
\end{displaymath}
It is easy to see that \textbf{(GLM2)} is equivalent to
(\ref{glm2-intro}), and, manifestly, implies \textbf{(GLM1)}.  As they
involve the pairing of a global and a local \ob, we say that these are
two definitions of \emph{global-local \m}. (There exists another
definition of global-local \m, which is a uniform version of
\textbf{(GLM2)} and is denoted \textbf{(GLM3)} in \cite{lpmu}. We do
not consider it here.)

The following is a trivial consequence of a well-known theorem of Lin
\cite{li}.

\begin{proposition} \label{prop-glm1}
  An exact \dsy\ is {\rm\bfseries (GLM1)}-mixing for any choice of
  $\go \subseteq L^\infty$ and $\lo \subseteq L^1$.
\end{proposition}

\proof See \cite[Thm.~3.5\emph{(a)}]{lpmu}.

\begin{remark}
  One may rephrase the above by saying that an exact \dsy\ is
  \textbf{(GLM1)}-mixing w.r.t.\ $L^\infty$ and $L^1$. This is not
  formally correct, because $L^\infty$ is not a family of global \ob s
  (as it contains \fn s that do not possess an infinite-volume
  average), but the meaning is clear so we will use such an
  expression.
\end{remark}

When we consider the ``correlation'' between two global \ob s, we
study the so-called \emph{global-global \m}. We have two definitions
for it.  The \sy\ is called \m\ of type \textbf{(GGM1)} if, for all
$F,G \in \go$, $\avg ( (F \circ T^n) G )$ exists for all sufficiently
large $n$, and
\begin{displaymath}
  \lim_{n \to \infty} \, \avg ( (F \circ T^n) G ) = \avg(F) \, \avg(G).
  \eqno{\mathrm{(GGM1)}}
\end{displaymath}
It is called \m\ of type \textbf{(GGM2)} if, for all $F,G \in \go$,
\begin{displaymath}
  \ivlimtwo \, \leb_V ( (F \circ T^n) G ) = \avg(F) \, \avg(G).
  \eqno{\mathrm{(GGM2)}}
\end{displaymath}
The above limit, which we call \emph{joint infinite-volume and time
  limit}, is defined as
\begin{equation} \label{im-60}
  \lim_{M \to \infty} \sup_{{V \in \scv} \atop {{\leb(V) \ge M} \atop
  {n \ge M}}} \left| \frac1{\leb(V)} \int_V (F \circ T^n)(x) G(x) \,
  dx - \avg(F) \, \avg(G) \right| = 0.
\end{equation}
The second definition is in essence stronger than the first, as the
following proposition shows.

\begin{proposition} \label{prop-mgg2-1}
  If $F,G \in \go$ are such that $\avg( (F \circ T^n) G )$ exists for
  all $n$ large enough (depending on $F,G$), then
  \begin{equation} \label{mgg2-1-impl}
    \ivlimtwo \leb_V ((F \circ T^n) G) = b \quad \Longrightarrow
    \quad \lim_{n \to \infty} \avg ((F \circ T^n) G) = b.
  \end{equation}
  In particular, if the above hypothesis holds $\forall F,G \in \go$,
  then {\rm\bfseries (GGM2)} implies {\rm\bfseries (GGM1)}.
\end{proposition}

\proof See \cite[Prop.~2.3]{lpmu}.
\bigskip

Some of the very maps considered in this paper give a good sense of
the relative strength of \textbf{(GGM2)} and \textbf{(GGM1)}, as the
latter property will be trivially verified while the former will
remain an open question; cf.~Corollary \ref{cor-mix-fm}.

For an in-depth discussion on the meaning and relevance of the above
definitions we refer the reader to \cite{limix}. Here we just point
out that, in order for them to make sense as indicators of
``decorrelation'', it must be that, for all $F \in \go$ and $n \in
\N$, $\avg(F \circ T^n) = \avg(F)$. As shown in \cite{limix}, this is
guaranteed by the following hypothesis: for all $n \in \N$,
\begin{equation} \label{im-70}
  \ivlim \frac{ \leb( T^{-n} V \symmdiff V ) } { \leb(V) } = 0,
\end{equation}
in the sense of the infinite-volume limit, as in (\ref{im-30}).  The
above is easily verified for the \dsy s introduced in Section
\ref{sec-setup}: If $V = \bigcup_{j=k}^\ell I_j$, with $\ell-k$
sufficiently large, it is easy to see that
\begin{equation} \label{im-80}
  \bigcup_{j=k + n (\bigj-1)}^{\ell - n(\bigj-1)} \hspace{-12pt} I_j \
  \subset \ T^{-n} V \ \subset \hspace{-4pt} \bigcup_{j=k - n
  (\bigj-1)}^{\ell +  n(\bigj-1)}
  \hspace{-12pt} I_j ,
\end{equation}
whence
\begin{equation} \label{im-90}
  T^{-n} V \symmdiff V \ \subset \hspace{-4pt}
  \bigcup_{j=k - n (\bigj-1)}^{k +  n(\bigj-1)} \hspace{-12pt} I_j
  \hspace{6pt} \cup  \hspace{-4pt} \bigcup_{j=\ell - n
  (\bigj-1)}^{\ell +  n(\bigj-1)} \hspace{-12pt} I_j.
\end{equation}
Therefore, the numerator of (\ref{im-70}) is bounded above by $(4n
(\bigj-1) + 2) c_2$, while the denominator is bounded below by
$(\ell-k+1)c_1$; cf.\ (A1)-(A2). But the infinite-volume limit here
corresponds precisely to the limit $\ell-k \to +\infty$ (uniformly in
$k, \ell$), whence the assertion.

\skippar

The quasi-lifts of expanding circle maps introduced earlier verify all
definitions of infinite \m, both global-local and global-global, with
suitable choices of the global \ob s. Let us see which.

If $\psi$ is either a global or a local \ob, and $k \in \Z^+$, set
\begin{equation} \label{def-ak}
  \mathcal{A}_k \psi := \frac1k \sum_{j=0}^{k-1} \psi \circ \trn^j.
\end{equation}
By (\ref{trn-comm}), this operator commutes with the dynamics, namely
$\mathcal{A}_k (\psi \circ T) = (\mathcal{A}_k \psi) \circ T$. Now
define
\begin{align}
  \label{def-go1}
  \go_1 &:= \rset{F \in L^\infty \! } { \exists F_p = F_p \circ \trn
  : \lim_{k\to\infty} \| \mathcal{A}_k F - F_p \|_\infty = 0 }; \\
  \label{def-go2}
  \go_2 &:= \overline{ \mathrm{span}_\C \rset{F \in L^\infty \! }
  { \exists \beta \in \R : F \circ \trn = e^{\i a \beta}F } },
\end{align}
where the bar denotes closure in the $L^\infty$-norm.  In other words,
$\go_1$ is the space of all essentially bounded \fn s whose
$a\Z$-average converges uniformly to an $a$-periodic \fn; $\go_2$ is
the space generated by the $a$-quasiperiodic \fn s. Clearly, $\go_2
\subset \go_1$. To see that all these \fn s are global \ob s, we need
to verify that every $F \in \go_1$ possesses an infinite-volume
average $\avg(F)$, in the sense of (\ref{im-30}).  Since, in our case,
$V$ is of the form $[ak, a(\ell+1)]$, we see that
\begin{equation} \label{im-100}
  \frac1{\leb(V)} \int_V F\, d\leb = \frac1{a(\ell-k+1)} \int_{ak}^{a\ell}
  F \, d\leb = \frac1a \int_{I_0} \mathcal{A}_{\ell-k+1} F \circ \trn^k \,
  d\leb,
\end{equation}
which, by the hypotheses on $F$, converges to
\begin{equation} \label{im-110}
  \frac1a \int_{I_0} F_p \, d\leb =: \avg(F),
\end{equation}
as $\ell-k \to \infty$ (that is, as $\leb(V) \to \infty$, uniformly in
$V \in \scv$).

Examples of elements of $\go_2$ are the \fn s $E_\gamma$ seen in
(\ref{def-e-gamma}). An example of $F \in \go_1 \setminus \go_2$ is
given by $F := \sum_{j\in\Z} b_j 1_{I_j}$, with $(b_j)$ a non-periodic
sequence such that $\{ b_{2k}, b_{2k+1}\} = \{ 0, 1 \}$, for all $k
\in \Z$.

We have:

\begin{theorem} \label{thm-mix-ql}
  A quasi-lift of an expanding circle map, as defined in Section
  \ref{subs-ql}, is:
  \begin{itemize}
  \item[(a)] \m\ of type {\rm\bfseries (GLM1)} for $\go = L^\infty$
    and $\lo = L^1$;
  \item[(b)] \m\ of type {\rm\bfseries (GLM2)} w.r.t.\ $\go_1$ and
    $L^1$;
  \item[(c)] \m\ of type {\rm\bfseries (GGM1)} and {\rm\bfseries
    (GGM2)} w.r.t.\ $\go_2$.
  \end{itemize}
\end{theorem}

\proof See Section \ref{sec-proofs}.
\bigskip

The situation is not as good for finite modifications of quasi-lifts.

\begin{corollary} \label{cor-mix-fm}
  If a finite modification of a quasi-lift, cf.\ Section \ref{subs-fm},
  is irreducible, then it is:
  \begin{itemize}
  \item[(a)] \m\ of type {\rm\bfseries (GLM1)} for $\go = L^\infty$ and
    $\lo = L^1$;
  \item[(b)] \m\ of type {\rm\bfseries (GGM1)} w.r.t.\ $\go_2$.
  \end{itemize}
\end{corollary}

\proof See Section \ref{sec-proofs}.

\section{Example: finite modification of a random walk}
\label{sec-fmrw}

As seen earlier, the mixing properties of finite modifications of
quasi-lifts are harder to prove, in general, than those of
quasi-lifts.  In particular this holds for the interesting property
that we have called \textbf{(GLM2)}.  However, one expects
\textbf{(GLM2)} to hold true for a large class of maps. In this
section we present one such case. Even though we pick a specific
example, the technique generalizes easily to other maps of the same
kind.

We construct a piecewise expanding $T: \R \into \R$ which, in a very
precise sense that we describe momentarily, represents the (generally
inhomogeneous) random walk in $\Z$ defined by a transition matrix
$\mathcal{P} = (p_{jk})_{j,k \in \Z}$.  (Here $p_{jk} \in [0,1]$ is
the \pr\ that the walker jumps from the site $j$ to the site $k$; thus
$\sum_k p_{jk} = 1$, $\forall j\in\Z$.)  We assume that the walk only
admits finite jumps, i.e., there exists $\bigjj \in \Z^+$ such that
$p_{jk} = 0$, for all $|k-j|>\bigjj$.

For $j\in\Z$ and $k \in \{ j-\bigjj, j-\bigjj+1, \ldots, j+\bigjj \}$,
set $I_j := [j, j+1]$ and
\begin{equation} \label{fmrw-10}
  I_{jk}' := \left[ j + \sum_{i = j-\bigjj}^{k-1} p_{ji} \,,\, j +
  \sum_{i = j-\bigjj}^k p_{ji} \right],
\end{equation}
with the understanding that, when $k = j-\bigjj$, the first of the
above sums is zero. So $\{ I_j \}_{j\in\Z}$ is a partition of $\R$ and
$\{ I_{jk}' \}_{k=j-\bigjj}^{j+\bigjj}$ is a partition of $I_j$ into
intervals of length, respectively, $p_{j, j-\bigjj}, \, p_{j,
j-\bigjj+1}, \ldots, \, p_{j, j+\bigjj}$.  Notice that, if
$p_{jk}=0$, $I_{jk}'$ reduces to a point. We do not consider such
degenerate intervals.

Any $x\in\R$ that is not an endpoint of one of the intervals
(\ref{fmrw-10}), belongs to one, and only one, $I_{jk}'$, with
$p_{jk}>0$. For such $x$, define
\begin{equation} \label{fmrw-20}
  T (x) := \frac1 {p_{jk}} \left( x - j - \sum_{i = j-\bigjj}^{k-1} p_{ji}
  \right) + k
\end{equation}
(for all other $x$, the definition of $T(x)$ is irrelevant). In other
words, $T$ maps $\mathrm{int}(I_{jk}')$ affinely onto
$\mathrm{int}(I_k)$. Observe that $\leb( T^{-1} I_k \,|\, I_j) = \leb(
I_{jk}' ) = p_{jk}$, which shows that $\mathcal{P}$ is also the
transition matrix of $T$, in the sense of Definition \ref{def-irr-ap}.

This explains in what sense $T$ represents the random walk defined by
$\mathcal{P}$. In fact, fix $j_0 \in \Z$ and chose a random $x_0 \in
I_{j_0}$, with respect to $\leb_0$, the Lebesgue \me\ on
$I_{j_0}$. Call $j_1$ the integer such that $x_0 \in I_{j_0,j_1}'$
(this is unique, save for a negligible set of $x_0$). Then $x_1 :=
T(x_0) \in I_{j_1}$. Calling $j_2$ the (almost surely unique) integer
such that $x_1 \in I_{j_1,j_2}'$ ensures that $x_2 := T(x_1) \in
I_{j_2}$; and so on. Let us call $(J_n)_{n\in\N}$ the stochastic
process defined by $J_n(x_0) := j_n$. This is the sought random walk
because of the Markov property
\begin{equation} \label{fmrw-30}
\begin{split}
  & \leb_0 ( J_{n+1} = j_{n+1}  \,|\,  J_n = j_n, \ldots,  J_1 = j_1 ) \\
  =\ & \leb_0 ( x_{n+1} \in I_{j_{n+1}} \,|\, x_n \in I_{j_n},
  \ldots,  x_1 \in I_{j_1} ) \\
  =\ & \frac{ \leb_0 \left( \bigcap_{i=1}^{n+1} T^{-i} I_{j_i} \right) }
  { \leb_0 \left( \bigcap_{i=1}^n T^{-i} I_{j_i} \right) } \\
  =\ & \frac{ p_{j_0, j_1} \, p_{j_1, j_2} \cdots \, p_{j_{n-1}, j_n}
  \, p_{j_n, j_{n+1}} } { p_{j_0, j_1} \, p_{j_1, j_2} \cdots
  \, p_{j_{n-1}, j_n} } \\
  =\ &  p_{j_n, j_{n+1}}.
\end{split}
\end{equation}

Now, clearly $\{ I_j \}$ is a Markov partition for $T$ and verifies
(A1). On the other hand, (A2) is generally not true, because $\br_j$
is likely to have corners at the endpoints of the intervals
$I_{jk}$. This, however, makes little difference in the proofs, as the
lack of regularity is restricted to a countable set of points;
cf.\ Sections \ref{sec-im} and \ref{sec-proofs}. What is more
important is the irreducibility of the map, as per Definition
\ref{def-irr-ap}\emph{(a)}. This will be easily checked in the example
given below, and is however guaranteed if we make the following
assumption: for every $j$, there exist $k_{1j}, k_{2j} \in \Z$, with
$k_{1j} < j < k_{2j}$, such that $p_{j, k_{1j}} \, p_{j, k_{1j}+1}
\cdots \, p_{j, k_{2j}} > 0$. Notice that, due to the strict
inequalities on $k_{1j}, k_{2j}$, this is stronger than (A2).

(A3) holds provided $p_{jk} \le \eta^{-1}$, for some $\eta>1$ and all
$j,k$ (the lack of regularity of $\br_j$ at finitely many points is
irrelevant to the $L^\infty$-norm). (A4) is trivially true.

Before discussing the fundamental assumption (A5), we introduce the
Perron-Frobenius operator for $T$, which is defined by the identity
\begin{equation} \label{fmrw-40}
  \int_\R (F \circ T) g \, d\leb =  \int_\R F (Pg) \, d\leb,
\end{equation}
valid for all $F \in L^\infty$ and $g \in L^1$. It is well
known that, for a.e.\ $x \in \R$,
\begin{equation} \label{fmrw-50}
  (Pg) (x) =  \sum_{y \in T^{-1} \{x\}} \frac{g(y)} {| T'(y) |}.
\end{equation}
In our specific case, for all $k \in \Z$ and $x \in
\mathrm{int}(I_k)$,
\begin{equation} \label{fmrw-60}
  (Pg) (x) = \sum_{{j\in\Z} \atop {p_{jk}>0}} p_{jk} \, g (\br_j^{-1}(x)),
\end{equation}
where $\br_j^{-1}$ is the inverse of the \fn\ defined in (A2), which
is well defined on $x$.

In view of (\ref{leb-pres}) and (\ref{fmrw-50}), (A5) is verified \iff
$P1 = 1$, with $1(x) := 1$, i.e., by (\ref{fmrw-60}), \iff $\sum_j
p_{jk} = 1$ for all $k$, namely, when the random walk is doubly
stochastic; cf.~\cite{lrwyep, klo, lrwwe} and references therein.

\begin{remark} \label{rk-ssrw}
  This family of maps provides easy examples for the importance of the
  aperiodicity assumption for the exactness of a Markov map.  For
  instance, take $\mathcal{P}$ to be the transition matrix of the
  simple symmetric random walk in $\Z$. Denoting $X_\mathrm{even} :=
  \bigcup_{j\in\Z} I_{2j}$ and $X_\mathrm{odd} := \bigcup_{j\in\Z}
  I_{2j+1}$, it is clear that $T^{-1} X_\mathrm{even} =
  X_\mathrm{odd}$ and $T^{-1} X_\mathrm{odd} =
  X_\mathrm{even}$. Therefore the tail $\sigma$-algebra $\mathscr{T}$
  contains both $X_\mathrm{even}$ and $X_\mathrm{odd}$, and is not
  trivial. (It is indeed not hard to show that $\mathscr{T} = \{
  \emptyset, X_\mathrm{even}, X_\mathrm{odd}, \R \}$.)
\end{remark}

The specific map we consider in this section is the one defined by the
transition matrix
\begin{equation} \label{fmrw-70}
  \mathcal{P}  :=  \frac19 \left(
  \begin{array}{ccccccccccccc}
    \ddots\!\! & \!\!\ddots\!\! & \!\!\ddots\!\! & \!\!\ddots\!\!
    & \!\!\ddots\!\!
    & & & & & & & & \\
    & 1 & 2 & 3 & 2 & 1 & & & & & & & \\
    & & 1 & 2 & 3 & 2 & 1 & & & & & & \\\hline
    & & & 1 & 2 & 5 & 1 & 0 & & & & & \\
    & & & & 1 & 1 & 5 & 1 & 1 & & & & \\
    & & & & & 0 & 1 & 5 & 2 & 1 & & & \\\hline
    & & & & & & 1 & 2 & 3 & 2 & 1 & & \\
    & & & & & & & 1 & 2 & 3 & 2 & 1 & \\[-3pt]
    & & & & & & & &
    \!\!\ddots\!\! & \!\!\ddots\!\! & \!\!\ddots\!\! & \!\!\ddots\!\!
    & \!\!\ddots
  \end{array}
  \right),
\end{equation}
whose entries are null outside of the shown diagonal strip. Evidently,
$\mathcal{P}$ is stochastic and symmetric, hence doubly
stochastic. Also, its rows $(p_{jk})_k$ fail to be translations of
each other only for $j \in \{ -1, 0, 1 \}$ (as indicated in
(\ref{fmrw-70})), which makes $T$ a finite modification of a
quasi-lift.  More examples of this kind can be constructed using the
ideas of \cite[App.~A]{lrwyep}.

Given a stochastic vector $\pi := (\pi_j)_{j \in \Z}$ (this means,
$\pi_j \in [0,1]$ and $\sum_j \pi_j = 1$), let us define the local \ob
\begin{equation} \label{fmrw-80}
  g_\pi := \sum_{j \in \Z} \pi_j 1_{I_j}.
\end{equation}
Clearly $g_\pi$ is a density w.r.t.\ the Lebesgue \me, namely, $g_\pi
\ge 0$ and $\leb(g_\pi)=1$.

\begin{proposition} \label{prop-fmrw-decr}
  If $\pi$ is stochastic vector that is symmetric (i.e., $\pi_j =
  \pi_{-j}$, $\forall j \in \Z$) and decreasing on $\N$, and $g_\pi$
  is its corresponding density via (\ref{fmrw-80}), then the evolution
  of $g_\pi$ via the Perron-Frobenius operator is also of the type
  (\ref{fmrw-80}). More precisely, $P^n g_\pi = g_{\pi^{(n)}}$, where
  $\pi^{(n)} := \pi \mathcal{P}^n$ is the evolution at time $n$ of the
  initial state $\pi$ for the random walk described above.  Moreover,
  $\pi^{(n)}$ is symmetric and decreasing on $\N$.
\end{proposition}

\proof See Section \ref{sec-proofs}.
\bigskip

With this proposition we can prove a very strong instance of
\textbf{(GLM2)} for our \sy. Define $\scv' := \lset{[-k, k] \subset
\R} {k \in \Z^+}$. This exhaustive family is smaller than the one we
have introduced in (\ref{im-20}), which, in our specific case, reads
$\scv = \lset{[k, \ell] \subset \R} {k < \ell \in \Z}$. In a sense, up
to inessential variations, $\scv'$ is the smallest collection of sets
that make sense as an exhaustive family, because it contains only one
increasing sequence of sets that covers the phase space
$\R$. Therefore, the class of \fn s
\begin{equation} \label{def-go0}
  \go' := \rset{F \in L^\infty(\R; \C)} {\exists \avg'(F)
  := \lim_{k \to \infty} \, \frac1{2k+1} \int_{-k}^k F\, d\leb}
\end{equation}
is essentially the largest class of global \ob s one can imagine for
the \dsy\ at hand, because $\avg'$ is the infinite-volume average
w.r.t.\ $\scv'$; cf.\ (\ref{im-30})-(\ref{im-40}).

\begin{remark}
  It is worthwhile to point out that $\go'$ is not simply a larger
  class of global \ob s than $\go_1$ and $\go_2$ of Section
  \ref{sec-im}. By using $\scv'$ in lieu of $\scv$, here, we have
  changed the notion of infinite-volume average from $\avg$, cf.\
  (\ref{im-10})-(\ref{im-40}), to $\avg'$, cf.\ (\ref{def-go0}), and
  therefore extended the very concept of global \ob. (Notice that, if
  $\avg(F)$ exists, $\avg'(F) = \avg(F)$.)
\end{remark}

\begin{proposition} \label{prop-fmrw-mix}
  The map $T$ defined above is irreducible and exact.  Also, in
  addition to the statements of Corollary \ref{cor-mix-fm}, it is
  {\rm\bfseries (GLM2)} relative to $\go'$ and $L^1$.
\end{proposition}

\proof See Section \ref{sec-proofs}.

\begin{remark}
  No form of global-global \m\ can hold for $T$ w.r.t. $\go'$.  This
  is an instance of a general phenomenon that we have called
  \emph{surface effect} in \cite[Sect.~3]{limix}. Very briefly, a
  counterexample is constructed by choosing, e.g., $F = G = \Theta$,
  the Heaviside \fn, which belongs in $\go'$, with $\avg(\Theta) =
  1/2$. Since $T$ has bounded dynamics, meaning $| T(x)-x | \le K$,
  $\forall x$ (in our specific case $K<3$), it is clear that $\Theta
  \circ T^n(x) = \Theta(x)$, for all $|x| > Kn$. So, for all $n\in
  \N$,
  \begin{displaymath}
    \ivlim \leb_V ((\Theta \circ T^n) \Theta) = \avg ( \Theta^2 ) =
    \frac12 \ne \avg(\Theta)^2,
  \end{displaymath}
  contradicting both \textbf{(GGM1)} and \textbf{(GGM2)} (recall that,
  in the latter, the convergence in $n$ is uniform w.r.t.\ the one in
  $V$, and viceversa).
\end{remark}

The above says no more and no less than, $\scv'$ is the wrong
exhaustive family for the global-global \m\ of quasi-lifts, or finite
modifications thereof; cf.\ \cite[Sect.~3]{limix}.  We still expect
\textbf{(GGM1-2)} to hold for general classes of global \ob s in the
sense of Section \ref{sec-im}, i.e., relative to the exhaustive family
$\scv$.

\section{Proofs}
\label{sec-proofs}

In this section we give the proofs of all the results stated in the
previous sections. Some of more standard lemmas that we use are in
turn moved to the Appendix.

\bigskip

\proofof{Theorem \ref{thm-exa}} For $\bj = (j_0, j_1, \ldots, j_{n-1})
\in \Z^n$, set
\begin{equation} \label{exa-5}
  I^{(n)}_\bj := I_{j_0} \cap T^{-1} I_{j_1} \cdots \cap
  T^{-n+1} I_{j_{n-1}}.
\end{equation}
By (A1)-(A3), $\{ I^{(n)}_\bj \}_{\bj \in \Z^n}$ is a Markov partition
for $T^n$ (as always, modulo the endpoints of the intervals) and
$\leb(I^{(n)}_\bj) \le c_2 \lambda^n$.

Let $\mathscr{H} := \{ \emptyset, \mathcal{C}, \mathcal{D}, \R \}$ be
the $\sigma$-algebra of the Hopf decomposition; $\mathscr{I} :=
\rset{A \in \bor} {T^{-1}A = A \mbox{ mod } \mu}$ the $\sigma$-algebra
of the invariant sets; and $\mathscr{T} := \bigcap_{n=0}^\infty T^{-n}
\bor$ the tail $\sigma$-algebra. Clearly, $\mathscr{H} \subseteq
\mathscr{I} \subseteq \mathscr{T}$.

Given a positive-\me\ set $A$ and any of its density points $x$, for
$n \ge 1$ let $\itwo{n}{x}$ denote the only element of $\{ I^{(n)}_\bj
\}$ such that $x \in \itwo{n}{x}$ (in case $x$ belongs to two such
elements, being an endpoint of both, we make the convention that
$\itwo{n}{x}$ is the interval on the right). We have
\begin{equation} \label{exa-10}
  \lim_{n \to \infty} \leb(A \,|\, \itwo{n}{x} ) = 1.
\end{equation}
Setting $B_{n,x} := \itwo{n+1}{x} \setminus A$, (\ref{exa-10}) is
equivalent to $\lim_{n \to \infty} \leb(B_{n,x} \,|\, \itwo{n+1}{x}) =
0$, whence, by Corollary \ref{cor-dist} of the Appendix, $\lim_{n \to
  \infty} \leb(T^n B_{n,x} \,|\, T^n \itwo{n+1}{x}) = 0$.  Now,
$\forall y \in \R$, denote $\ione{y} := \itwo{1}{y}$. Notice that
$T^n$ is a bijection $\itwo{n+1}{x} \into \ione{T^n(x)}$, therefore
$T^n B_{n,x} = \ione{T^n(x)} \setminus T^n A$. Thus,
\begin{equation} \label{exa-20}
  \lim_{n \to \infty} \leb(T^n A \,|\, \ione{T^n(x)} ) = 1.
\end{equation}

\skippar

For the proof of \emph{(a)} we posit $\leb(\mathcal{C}) > 0$
(otherwise there is nothing to prove) and consider any set $A \in
\mathscr{I}$ with $\leb(A) > 0$. The $m$-typical point $x \in A$ will
be both a density point of $A$ and a recurring point to $\ione{x}$.
This means that there exists a sequence $(n_k)$ such that $T^{n_k}(x)
\in \ione{x}$, equivalently, $\ione{T^{n_k}(x)} = \ione{x}$. Since $A$
is invariant (\ref{exa-20}) gives that $\leb(A | \ione{x}) = 1$, which
proves that the \erg\ decomposition of $\mathcal{C}$ is coarser than
$\{ I_j \}$.

To prove that each \erg\ component $X_\alpha \subseteq \mathcal{C}$ is
a union of adjacent intervals $I_j$, suppose the contrary: there exist
$j_1< j_2 < j_3 \in \Z$ such that $I_{j_1}, I_{j_3} \subseteq
X_\alpha$ and $I_{j_2} \cap X_\alpha$ contains at most the endpoints
of $I_{j_2}$. Now, consider any $j < j_2$ with $I_j \subseteq
X_\alpha$. By (A2), $T(I_j)$ contains $I_j$. If it contained $I_k$,
for some $k > j_2$, then (since $T(I_j)$ is connected) it would
contain $I_{j_2}$ as well, contradicting the definition of
$j_2$. Thus, $X_\alpha \cap (-\infty, a_{j_2}]$ is $T$-invariant, and
$X_\alpha$ can be further split into \erg\ components, contradicting
our assumption.

Now for the exactness.  The hypotheses on $T$ imply that there exists
a $\delta \in (0, 1/2)$ such that, $\forall j \in \Z$, if $B \subseteq
I_j$ with $\leb(B | I_j) > 1 - \delta$, then $\leb(TB | I_j) > 1/2$,
entailing that $\leb(TB \cap B) > 0$. (With the help of Section
\ref{subs-dist} of the Appendix, we see that a good estimate for
$\delta$ is $c_1 / (2D ((\bigj - 1) c_2 + c_1))$.  In fact, in the
worst-case scenario, $I_j$ is an interval of length $c_1$ and is
mapped onto the union of itself and $\bigj - 1$ intervals of length
$c_2$, cf.\ (A1)-(A2); $T$ maps $I_j \setminus B$ entirely into $I_j$,
expanding it at the maximum rate, which, by Lemma \ref{lem-dist}, is
smaller than $D$ times the average expansion rate, which is $((\bigj -
1) c_2 + c_1) / c_1$. This yields the estimate.)

Take any $A \in \mathscr{T}$ with $\leb(A)>0$ and choose a typical
point $x$ as before. By (\ref{exa-20}), there exists $n$ such that
$\leb(T^n A | \ione{T^n(x)}) > 1-\delta$. Hence $\leb(T^{n+1} A \cap
T^n A) > 0$. Thus, Proposition \ref{prop-mn} of the Appendix shows
that all the \erg\ components are exact.

As for \emph{(b)}, let us assume that $\leb(\mathcal{D}) > 0$ (again,
otherwise there would be nothing to prove). Define
\begin{align}
  X_+ &:= \rset{x \in \mathcal{D}} {\limsup_{n \to \infty} \, T^n(x)
  = +\infty}, \\
  X_- &:= \rset{x \in \mathcal{D}} {\liminf_{n \to \infty} \, T^n(x) =
  -\infty}.
\end{align}
It must be that $\leb(\mathcal{D} \setminus (X_+ \cup X_-)) = 0$, or
there would exist $R > 0$ such that $\leb(\mathcal{D} \cap [-R,R]) >
0$ and $\limsup_{n \to \infty} |T^n(x)| < R$ for a positive \me\ of $x
\in \mathcal{D}$. The forward \o\ of any such $x$ visits $\mathcal{D}
\cap [-R,R]$ an infinite number of times, contradicting Lemma
\ref{lem-tech1} of the Appendix. Similarly, $\leb(X_+ \cap X_-) = 0$,
because the contrary implies that, for $R > \bigj c_2/2$, a positive
\me\ of points $x \in \mathcal{D}$ visit $\mathcal{D} \cap [-R,R]$
infinitely often in the future (observe that (A1)-(A2) imply that,
$\forall x \in \R$, $|T(x) - x| \le \bigj c_2$). Once again, this is
forbidden by Lemma \ref{lem-tech1}, assuming w.l.g.\ that
$\leb(\mathcal{D} \cap [-R,R]) > 0$. This argument also implies that,
for $x \in X_+$, $\liminf_{n \to \infty} T^n(x) \ge R$, and, for $x
\in X_-$, $\limsup_{n \to \infty} T^n(x) \le -R$. Since $R$ can be
arbitrarily large, $X_\pm = \rset{x \in \mathcal{D}} {\lim_{n \to
\infty} \, T^n(x) = \pm \infty}$.

It remains to prove that $X_+$ cannot be split in more than one \erg\
component, the argument for $X_-$ being analogous.  Given $A \subseteq
\mathcal{D}$, we say that a set of the type $C := \bigcup_{j=i}^{\ell}
I_j$ is \emph{$A$-prevalent} if $\ell-i+1 \ge \bigj$, cf.\ (A2), and,
$\forall j = i, \ldots, \ell$, $\leb(A | I_j) > 1/2$.

Assume that $A, B \subset X_+$ are disjoint, $T$-invariant and of
positive \me. We prove that, for a typical $x \in A$, $x_n := T^n(x)$
belongs to an $A$-prevalent set, for all $n$ large enough. In fact,
(\ref{exa-20}) and the invariance of $A$ imply that, for $n$ large,
$\leb(A | \ione{x_n}) > 1 - \delta'$, where $\delta'$ is chosen so
small that $T^{\bigj-1} (A \cap \ione{x_n})$ has density larger than
$1/2$ in all the intervals $I_j$ that make up the set $C = T^{\bigj-1}
\ione{x_n}$. (Let us explain this point. By (A2), $C =
\bigcup_{j=i}^\ell I_j$, for some $i < \ell$; we are saying that
$\leb( T^{\bigj-1} (A \cap \ione{x_n}) | I_j) > 1/2$, $\forall j = i,
\ldots, \ell$.) Notice that $\delta'$ can be chosen to depend only on
$\bigj$ and the other parameters of $T$, not on $x$ and $n$. If we
show that
\begin{equation} \label{exa-30}
  \ell-i+1 \ge \bigj,
\end{equation}
we have proved that $C$ is an $A$-prevalent set that contains
$x_{n+\bigj-1}$ (observe that $x_{n+\bigj-1} \in T^{\bigj-1} (A \cap
\ione{x_n}) \subset C$).  This would give the assertion made in the
previous paragraph, because the above argument holds for \emph{all}
$n$ bigger than a certain value that depends on $x$. But
(\ref{exa-30}) is easily verified: By (A2), $( T^k \ione{x_n} )_{k \ge
  0}$ is an increasing sequence of sets that are unions of adjacent
intervals from $I_j$. The sequence must be strictly increasing,
otherwise, for some $k$, $T^k \ione{x_n}$ would be an invariant set,
contradicting that $x_{n+k} \to +\infty$, as $k \to \infty$. This
implies that $C = T^{\bigj-1} \ione{x_n}$ is made up of at least
$\bigj$ intervals, which is precisely (\ref{exa-30}).

So there are infinitely many $A$-prevalent sets in any right half-line
of $\R$.  On the other hand, applying the above to a typical $y \in
B$, we have that $y_n := T^n(y)$ belongs to a $B$-prevalent set, for
all large $n$. But $y_n \to +\infty$, and the distance between $y_n$
and $y_{n-1}$, in terms of intervals, is at most $\bigj-1$. This means
that, for $n$ big enough, a point $y_n$ must fall in an $A$-prevalent
set. But this is a contradiction, since $A$-prevalent sets and
$B$-prevalent sets cannot overlap.  Hence, $A$ and $B$ cannot be
disjoint and there is only one \erg\ component.

The proof of \emph{(c)} follows immediately from \emph{(a)}. If
$\leb(\mathcal{C}) > 0$, there is at least one non-trivial \erg\
component $X_\alpha$, which is a union of intervals $I_j$. By the
irreducibility of $T$, $X_\alpha = \R$.
\qed

\bigskip

\proofof{Proposition \ref{prop-exa-ql}} (\ref{trn-comm}) and (A2)
imply immediately that $T$ is irreducible.

As to the exactness, if the \sy\ has a non-trivial conservative part,
Theorem \ref{thm-exa}\emph{(c)} implies that it is conservative and
exact. If the system is dissipative, Theorem \ref{thm-exa}\emph{(b)}
states that there are at most two \erg\ components, $X^+$ and $X^-$,
and each of them is exact. Suppose that neither has \me\ zero. By
(\ref{trn-comm}), both components are invariant for the action of
$\trn$, thus $\leb (X^\pm | I_j)$ is constant in $j$.  But the proof
of Theorem \ref{thm-exa} shows that there exists a (sufficiently
large) $j$ such that $\leb (X^+ | I_j) > 1/2$. This, then, holds for
all $j \in \Z$. The same can be proved for $\leb (X^- | I_j)$. It
follows that our assumption was wrong and $T$ is exact.
\qed

\bigskip

\proofof{Proposition \ref{prop-exa-fm}} We use the notation of Section
\ref{subs-fm}. If $T$ has a non-trivial conservative part, then it is
conservative and exact by Theorem \ref{thm-exa}\emph{(c)}. So, let us
assume that $T$ is dissipative and see which consequences this has on
the dynamics of $T_0$.

We denote by $\mathcal{C}_0$ the conservative part of $T_0$, and by
$X_{0\pm}$ the sets defined in Theorem \ref{thm-exa}\emph{(b)},
relative to it. The proof of Proposition \ref{prop-exa-ql} shows that
$T_0$ is exact and $\R$ coincides, mod $\leb$, with either
$\mathcal{C}_0$, or $X_{0+}$, or $X_{0-}$. If first were the case, by
\erg ity, the forward $T_0$-\tr y of a.e.\ $x \in \R$ would intersect
$B := \bigcup_{j=-k'}^{k'} I_j$. This occurs in particular for $x \in
\R \setminus B$, where $T_0$ and $T$ coincide. So the forward $T$-\tr
y of any such $x$ also intersects $B$, which contradicts the
dissipativity of $T$ (because, by Theorem \ref{thm-exa}\emph{(b)}, if
$T$ is dissipative, for any finite-\me\ $B$ there is a
positive-\me\ set of points whose forward \tr ies never see $B$).

So either $X_{0+}$ or $X_{0-}$ has full \me. Suppose, w.l.g., that it
is $X_{0+}$. We want to prove that the same occurs for $X_+$,
equivalently, $\leb(X_-)=0$. Assume the contrary and define, for
$k \in \N$,
\begin{equation}
  X_-^{(k)} := \rset{x \in X_-} {T^n(x) \in \bigcup_{j < -k'} I_j, \
  \forall n \ge k }.
\end{equation}
Clearly, $X_-^{(k)} \subseteq X_-^{(k+1)}$ and $\bigcup_{k \in \N}
X_-^{(k)} = X_-$. Therefore, there exists a $k$ such that $\leb(
X_-^{(k)} ) > 0$.  On the other hand, $X_-^{(k)} \subseteq X_{0-}$,
because, $\forall x \in X_-^{(k)}$, $\lim_{n \to \infty} T^n(x) =
-\infty$ and $T^n(x) = T_0^n(x)$, $\forall n \ge k$ (as all such
points lie outside of $B$).

This yields $\leb(X_{0-}) > 0$, which is a contradiction. Thus our
assumption was wrong and $X_+ = \R$ mod $\leb$.  Hence, $T$
is exact by Theorem \ref{thm-exa}\emph{(b)}.
\qed

\bigskip

\proofof{Theorem \ref{thm-mix-ql}} Assertion \emph{(a)} comes from
Proposition \ref{prop-glm1}, because $T$ is exact by Proposition
\ref{prop-exa-ql}.

As for \emph{(b)}, fix $F \in \go_1$ and $k \in \Z^+$. Recalling
definition (\ref{def-ak}), it is easy to check that
\begin{equation} \label{ql-10}
  \frac{ \leb((F \circ T^n) 1_{[0,ak]}) } {ak} = \frac1a \int_{I_0}
  (\mathcal{A}_k F) \circ T^n \, d\leb
\end{equation}
(use that $1_{[0,ak]} = \sum_{j=0}^{k-1} 1_{I_0} \circ \trn^{-j}$, and
that $T$ and $\trn$ commute and are both $\leb$-invariant).  By
definition of $\go_1$, for every $\eps>0$, there exists a large enough
$k$ such that, uniformly in $n$,
\begin{equation} \label{ql-20}
  \left| \frac1a \int_{I_0} (\mathcal{A}_k F) \circ T^n \, d\leb
  - \frac1a \int_{I_0} F_p \circ T^n \, d\leb \right| \le \eps.
\end{equation}
Recalling from Section \ref{subs-ql} the definition of the
\me-preserving \dsy\ $(\mathcal{S}_a, \leb, T_a)$, we see that, since
$F_p$ is $a$-periodic,
\begin{equation} \label{ql-30}
  \frac1a \int_{I_0} F_p \circ T^n \, d\leb = \frac1a
  \int_{\mathcal{S}_a} F_p \circ T_a^n \, d\leb = \frac1a
  \int_{\mathcal{S}_a} F_p \, d\leb = \avg(F),
\end{equation}
as per definition (\ref{im-110}) (with the slight abuse of notation
whereby the projection of $F_p$ to $\mathcal{S}_a \cong I_0$ is still
called $F_p$).

The following lemma is an easy consequence of the exactness of $T$.

\begin{lemma} \label{lem-pmu36}
  If, for some $b \in \C$ and $\eps \ge 0$, the limit
  \begin{displaymath}
    \limsup_{n \to \infty} \left| \frac{ \leb((F \circ T^n) g) } {\leb(g)}
    - b \right| \le \eps
  \end{displaymath}
  holds for some $g \in L^1$ (with $\leb(g) \ne 0$), then it holds for
  all $g \in L^1$ (with $\leb(g) \ne 0$).
\end{lemma}

\proof See \cite[Lem.~3.6]{lpmu}.
\bigskip

Equations (\ref{ql-10})-(\ref{ql-30}) show that the lemma can be
applied, for any $\eps>0$, with $b = \avg(F)$ and $g = 1_{[0,ak]}$,
with $k$ depending on $\eps$.  Therefore, for all $g$ with $\leb(g)
\ne 0$,
\begin{equation}
  \lim_{n \to \infty} \left| \leb((F \circ T^n) g) - \avg(F) \leb(g)
  \right| = 0.
\end{equation}
The case $\leb(g) = 0$ was already covered when we proved
\textbf{(GLM1)}. This ends the proof of \emph{(b)}.

By simple density arguments, \textbf{(GGM2)} will be verified once the
limit (GGM2) is proved for any pair of quasiperiodic \ob s $F,G$. More
precisely, assume $F \circ \trn = e^{\i a \beta}F$ and $G \circ \trn =
e^{\i a \gamma}G$. Notice that, if $\beta \ne 0$ mod $2\pi / a$, then
$\avg(F) = 0$. The analogous implication holds for $G$.

Set $g = a^{-1} G 1_{[0,a]}$.  Since $V$ is of the form $V = [ak,
a(\ell+1)]$, let us write
\begin{equation} \label{ql-40}
  G 1_V = \sum_{j=k}^\ell G 1_{I_j} = \sum_{j=k}^\ell ((G \circ \trn^j)
  1_{I_0}) \circ \trn^{-j} = a \sum_{j=k}^\ell e^{\i a \gamma j} g
  \circ \trn^{-j},
\end{equation}
whence, using (\ref{trn-comm}) and the translation-invariance of the
Lebesgue \me,
\begin{align}   \label{ql-50}
  \leb_V ((F \circ T^n)G) &= \frac1 {a(\ell-k+1)} \int_\R (F \circ T^n)
  G 1_V \, d\leb \nonumber \\
  &= \frac1 {\ell-k+1} \sum_{j=k}^\ell \int_\R (F \circ T^n)
  e^{\i a \gamma j} (g \circ \trn^{-j}) \, d\leb \nonumber \\
  &= \frac{ \sum_{j=k}^\ell e^{\i a(\beta + \gamma)j} } {\ell-k+1}
  \int_\R (F \circ T^n) g \, d\leb .
\end{align}
In the last term above, the factor in front of the integral is bounded
by 1 uniformly in $k, \ell$, namely, in $V$, while the integral does
not depend on it.  In fact, the latter term is $\leb((F \circ T^n) g)$
and, by \textbf{(GLM2)}, converges to $\avg(F) \leb(g)$, as $n \to
\infty$ (since $\go_2 \subset \go_1$).

We now have three cases:
\begin{enumerate}
\item \underline{$\beta \ne 0$ mod $2\pi / a$}. In this case $\avg(F)
  = 0$, therefore (\ref{ql-50}) converges to 0, as $n \to \infty$,
  uniformly in $V$. In particular, it converges to 0 in the joint
  infinite-volume and time limit; cf.\ (\ref{im-60}).

\item \underline{$\beta = 0$ mod $2\pi / a$ and $\gamma \ne 0$ mod
  $2\pi / a$}.  In this case $\avg(G) = 0$ and the factor in front
  of the integral in (\ref{ql-50}) vanishes when $\ell-k \to \infty$;
  that is, uniformly in $V$ as $\leb(V) \to \infty$, i.e., in the
  infinite-volume limit. On the other hand, the integral is bounded by
  $\| F \|_\infty \| g \|_1$, uniformly in $n$. This implies that,
  again, in the joint infinite-volume and time limit, (\ref{ql-50})
  converges to 0.

\item \underline{Both $\beta$ and $\gamma$ are 0 mod $2\pi / a$}. In
  this case the factor in front of the integral is identically 1,
  (\ref{ql-50}) no longer depends on $V$ and, for $n \to \infty$,
  tends to $\avg(F) \leb(g) = \avg(F) \avg(G)$, which is the same as
  the joint infinite-volume and time limit, here.
\end{enumerate}
In all these cases, the limit (GGM2) is verified.

In view of Proposition \ref{prop-mgg2-1}, \textbf{(GGM1)} will be
shown once we have proved that $\avg((F \circ T^n)G)$ exists for all
$F,G \in \go_2$. Once again, since $\avg$ is a continuous \fn al in
the $L^\infty$-norm, it is enough to prove the assertion for $F,G$
quasiperiodic. But, in that case, $F \circ T^n$ is quasiperiodic by
(\ref{trn-comm}), which implies the same for $(F \circ T^n)G$, which
thus has an infinite-volume average.  This completes the proof of
assertion \emph{(c)} of Theorem \ref{thm-mix-ql}.
\qed

\bigskip

\proofof{Corollary \ref{cor-mix-fm}} As in the previous proof,
assertion \emph{(a)} comes from Propositions \ref{prop-exa-fm} and
\ref{prop-glm1}.

Now for \emph{(b)}. Consider $F, G \in \go_2$ and fix $n \in \Z^+$.
Since $T^n$ is a finite modification of $T_0^n$, as emphasized in
Section \ref{subs-fm}, the \fn\ $(F \circ T^n) G$ differs from $(F
\circ T_0^n) G$ by a compactly supported and bounded \fn\ of
$\R$. This shows that $\avg( (F \circ T^n) G )$ exists \iff $\avg( (F
\circ T_0^n) G )$ does, and they are equal. Therefore, Theorem
\ref{thm-mix-ql}\emph{(c)} proves \textbf{(GGM1)} in this case as
well.
\qed

\bigskip

\proofof{Proposition \ref{prop-fmrw-decr}} We prove all the assertions
for $n=1$ and the proposition will follow by induction. For the sake
of the notation, let us denote $\pi' := \pi^{(1)} = \pi \mathcal{P}$.

By (\ref{fmrw-60}) and (\ref{fmrw-80}), for all $x \in
\mathrm{int}(I_k)$, $(P g_\pi)(x) = \sum_j \pi_j p_{jk} =:
\pi'_k$. This means that $P g_\pi = g_{\pi'}$.  Also, by the symmetry
properties of $\pi$ and $\mathcal{P}$,
\begin{equation} \label{fmrw-p10}
  \pi'_{-k} = \sum_j \pi_j p_{j,-k} = \sum_j \pi_{-j} p_{j,-k} =
  \sum_j \pi_j p_{-j,-k} = \sum_j \pi_j p_{jk} = \pi'_k.
\end{equation}
Finally, using both the symmetry and the half-monotonicity of $\pi$,
\begin{align}
  \pi'_0 - \pi'_1 &= [( \pi_{-2} + \pi_{-1} + 5\pi_0 + \pi_1 + \pi_2) -
  (\pi_0 + 5\pi_1 + 2\pi_2 + \pi_3)]/9 \nonumber \\
  &= ( 4\pi_0 - 3\pi_1 - \pi_3 ) /9 > 0; \\[6pt]
  \pi'_1 - \pi'_2 &=  [( \pi_0 + 5\pi_1 + 2\pi_2 + \pi_3) -
  (\pi_0 + 2\pi_1 + 3\pi_2 + 2\pi_3 +\pi_4)]/9 \nonumber \\
  &= (3\pi_1 - \pi_2 - \pi_3 - \pi_4)/9 > 0;
\end{align}
and, for $k \ge 2$,
\begin{align}
  \pi'_k - \pi'_{k+1} &= [( \pi_{k-2} + 2\pi_{k-1} + 3\pi_k +
  2\pi_{k+1} + \pi_{k+2}) \nonumber \\
  &\phantom{= [} - ( \pi_{k-1} + 2\pi_k + 3\pi_{k+1} + 2\pi_{k+2} +
  \pi_{k+3})]/9 \nonumber \\
  &= ( \pi_{k-2} + \pi_{k-1} + \pi_k - \pi_{k+1} - \pi_{k+2} -
  \pi_{k+3} )/9 >0.
\end{align}
Therefore $\pi'$ is decreasing on $\N$ and the proposition is
proved.
\qed

\bigskip

\proofof{Proposition \ref{prop-fmrw-mix}} The irreducibility of $T$ is
apparent from the expression of the transition matrix $\mathcal{P}$ of
its associated random walk, see (\ref{fmrw-70}). The exactness comes
from Proposition \ref{prop-exa-fm}. (It is easy to see that $T$ is
also conservative, because it represents a finite modification of a
recurrent random walk.)

Thus, Proposition \ref{prop-fmrw-mix} will be proved once the limit
(GLM2) is proved for any $F \in \go'$ and some $g \in L^1$, with
$\leb(g) \ne 0$; see Lemma \ref{lem-pmu36}. We choose $g = g_\pi$, as
defined in (\ref{fmrw-80}), with $\pi$ symmetric and decreasing on
$\N$ (e.g., $\pi_j := \delta_{0j}$). Without loss of generality (for
(GLM2) is trivial for $F =$ constant), we further assume that
$\avg'(F) = 0$.

Set $f_j := \int_{I_j} F \, d\leb$. We observe that, $\forall \eps>0$,
$\exists \ell \in \N$ such that, $\forall k \ge \ell$,
\begin{equation} \label{fmrw-p50}
  \frac1{2k+1} \left| \sum_{j=-k}^k f_j \right| = \left|
  \avg_{[-k,k]}(F) \right| \le \frac{\eps}2.
\end{equation}
By (\ref{fmrw-40}) and Proposition \ref{prop-fmrw-decr} we have
\begin{equation} \label{fmrw-p60}
\begin{split}
  \int_\R (F \circ T^n) g \, d\leb &=  \int_\R F g_{\pi^{(n)}} \, d\leb \\
  &= \sum_{j \in \Z} f_j \, \pi_j^{(n)} \\
  &= \sum_{k=0}^\infty \left(\pi_k^{(n)} - \pi_{k+1}^{(n)} \right)
  \sum_{j=-k}^k f_j \\
  &=: S_\ell + S_\ell' ,
\end{split}
\end{equation}
where $S_\ell$ and $S_\ell'$ correspond to restricting the outer
summation to $\sum_{k=0}^{\ell-1}$ and $\sum_{k=\ell}^{\infty}$,
respectively. Observe that the third equality of (\ref{fmrw-p60})
comes from disintegrating the density $( \pi_j^{(n)} )$ in
``horizontal slices'' of width $k$ and height $\pi_k^{(n)} -
\pi_{k+1}^{(n)}$.

By (\ref{fmrw-p50}) we obtain
\begin{equation} \label{fmrw-p70}
\begin{split}
  \left| S_\ell' \right| &\le \sum_{k=\ell}^\infty \left(\pi_k^{(n)} -
  \pi_{k+1}^{(n)} \right) \left| \sum_{j=-k}^k f_j \right| \\
  &\le \frac{\eps}2 \sum_{k=\ell}^\infty \left(\pi_k^{(n)} -
  \pi_{k+1}^{(n)} \right) (2k+1) \le \frac{\eps}2,
\end{split}
\end{equation}
because $\sum_{k \in \N} (\pi_k^{(n)} - \pi_{k+1}^{(n)}) (2k+1) =
\sum_{j \in \Z} \pi_j^{(n)} = 1$, as in (\ref{fmrw-p60}).

To estimate $S_\ell$ we need a property of the \dsy\ which is an easy
consequence of its exactness.

\begin{lemma} \label{lem-llm}
  For all $f \in L^\infty \cap L^1$ and $g \in L^1$,
  \begin{displaymath}
    \lim_{n \to \infty} \leb( (f \circ T^n)g ) = \lim_{n \to \infty}
    \leb( f (P^n g) ) = 0.
  \end{displaymath}
\end{lemma}

\proof See \cite[Thm.~3.5\emph{(b)}]{lpmu}. (In our terminology, see
\cite{lpmu}, the above notion is called \emph{local-local \m}, or
\textbf{(LLM)}, and is easily seen to be equivalent to the zero-type
property of \cite{hk}.)

\bigskip

Let us apply Lemma \ref{lem-llm} with $f = 1_{[-\ell+1, \ell]}$ (and
$g = g_\pi$, as already chosen). There exists $N \in \N$ such that,
$\forall n \ge N$,
\begin{equation} \label{fmrw-p80}
  \int_{-\ell+1}^\ell \! P^n g \, d\leb = \sum_{j=-\ell+1}^{\ell-1} \pi_j^{(n)}
  \le \frac{\eps} {2 \| F \|_\infty}.
\end{equation}
Therefore
\begin{equation} \label{fmrw-p90}
\begin{split}
  \left| S_\ell \right| &\le \sum_{k=0}^{\ell-1} \left(\pi_k^{(n)} -
  \pi_{k+1}^{(n)} \right) \left| \sum_{j=-k}^k f_j \right| \\
  &\le \| F \|_\infty \sum_{k=0}^{\ell-1} \left(\pi_k^{(n)} -
  \pi_{k+1}^{(n)} \right) (2k+1) \\
  &\le \| F \|_\infty \! \sum_{j=-\ell+1}^{\ell-1} \pi_j^{(n)} \le
  \frac{\eps}2 .
\end{split}
\end{equation}
Since $N$ is chosen depending on $\ell$, which in turns depends on
$\eps$, (\ref{fmrw-p60}), (\ref{fmrw-p70}) and (\ref{fmrw-p90}) prove
the assertion.
\qed

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
The Dirac equation unified special relativity with quantum mechanics for fermions.
It was a step further beyond the Schr\"odinger equation due to its invariance with respect to Lorentz boosts.
Several early experimental successes of it were the accurate prediction of the intrinsic magnetic moment of the electron,
 the prediction of antiparticles and the correct description for the fine-structure of the spectrum of the hydrogen atom.
These were based on a form of the Dirac equation which describes relativistic electrons.
There exists a similar form of it, which offers a phenomenological description of some compound particles like the neutron and the proton in the relativistic domain.

Despite its successes regarding the relativistic behavior of the electron, certain solutions of the Dirac equation predict some unusual effects, which have not been experimentally measured so far for neither the electron, the neutron or any other particle~\cite{Thaller}.
The best known of them are (i) Klein's paradox~\cite{Thaller} --- the tunneling of a relativistic electron into an infinite potential barrier, and (ii) Zitterbewegung (ZB) \cite{Thaller} --- helicoidal motion of the expectation value of the position $\hat{\vec{r}}$ and oscillatory behavior of the expectation values of the momentum $\hat{\vec{p}}$, the spin $\hat{\vec{S}}$ and the orbital angular momentum of a free electron.
Recently these two effects have gathered renewed attention by the scientific community, thanks to several proposals for their experimental emulation in physical systems like solid-state quantum wells~\cite{Schliemann05}, trapped ions~\cite{Lamata,Casanova,Bermudez,Gerritsma}, graphene~\cite{ZawadskiGraphene} and optical waveguides~\cite{Longhi}.

Most of the recent work on ZB~is focused on electrons.
In contrast here we consider the Dirac equation for neutral particles propagating relativistically in external magnetic and/or electric fields~\cite{Thaller} along the direction of the applied field.
The Dirac equation~\cite{Thaller} for these particles deviates from the Dirac equation for the electron in three aspects: (i) the term coupling the electrostatic potential to the charge of the particle dissapears; (ii) the terms coupling the vector potential to the charge and momentum are not present;  (iii) a term coupling the anomalous magnetic dipole moment (MDM) to the magnetic field and a term coupling the eventual electric dipole moment (EDM) to the electrostatic field appear.

\section{ZB~of free particles}

The Dirac Hamiltonian for a neutral particle in the absence of external fields reads
\begin{equation}
\hat{H} = c\hat{\alpha}_x\!\cdot\!\hat{p}_x + \hat{\beta} m_0c^2\ ,\label{EQ:FreeHam}
\end{equation}
where $c$ is the speed of light, $m_0$ is the rest mass of the particle, $\hat{\alpha}_x$ and $\hat{\beta}$ are the Dirac matrices.
The spectrum of the Hamiltonian is doubly degenerate and its eigenvalues are $E_{\pm}=\pm E$, with $E=\sqrt{(cp_x)^2+(m_0c^2)^2}$.
It is well known~\cite{Thaller,AdvancedSakurai} that if the initial state is a superposition of both positive and negative energy eigenstates, in the process of evolution the expectation values of velocity,  position and spin will oscillate with the same frequency
\begin{equation}\label{free-ZB}
\omega^{\text{zb}}=\frac{2E}{\hbar}=\frac{2}{\hbar}\sqrt{(cp_x)^2+(m_0c^2)^2}.
\end{equation}
This is the free-particle ZB.

There exists a lower bound  $\omega=2mc^2/\hbar$ for the free-particle ZB~of the three quantities $\hat{\vec{S}}$, $\hat{\vec{\alpha}}$ and $\hat{\vec{r}}$.
In the literature~\cite{Zawadski} it is usually accepted that the frequency of the free-particle ZB~is approximately $\omega=2mc^2/\hbar$.
However, for relativistic speed of propagation, $cp_x$ becomes comparable to $mc^2$.
Then there will be considerable contribution of $cp_x$ to the ZB~frequency $\omega^{\text{zb}}$, Eq.~\eqref{free-ZB}.
The difference
\begin{equation}
\Delta\omega^{\text{zb}}=\frac{2}{\hbar}\sqrt{(cp_x)^2+(m_0c^2)^2}-\frac{2mc^2}{\hbar}\; ,
\end{equation}
between the rest frame ($p_x=0$) and the observer frame of reference amounts to a \emph{blue shift} of the ZB~frequency: it is demonstrated in Fig.~\ref{FIG:WZB}.
Therefore, the free-particle ZB~frequency has the opposite behavior of the earlier reported~\cite{TenevB} relativistic red shift effect for the frequency of Larmor precession.
It is a consequence of the relativistic energy-momentum-mass relationship $E^2=(pc)^2 + (mc^2)^2$, and the fact that ZB~is observed due to an interference between positive and negative energy solutions.

\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{WZB1}
\end{center}
\caption{Demonstration of blue shift effect of free particle ZB~frequency $\omega^{\text{zb}}$, Eq.~\eqref{free-ZB}.
The ZB~frequency $\omega^{\text{zb}}$ increases with the particle velocity $v$, and respectively, its momentum $p=\gamma mv$, $\gamma=1/\sqrt{1-(v/c)^2}$.}\label{FIG:WZB}
\end{figure}

\section{Modification of ZB~due to lifted spin degeneracy}

Both Larmor precession of spin along an external field and spin-ZB~manifest themselves as spin precession.
However, despite their similarity there are important differencies between them.
Spin-ZB~is a consequence of interference between positive and negative energy components of the Dirac equation, while Larmor precession is a consequence of lifting of spin-degeneracy by external fields.
It is natural then to expect some modifications of at least spin-ZB, and eventually position-ZB, when spin degeneracy is lifted by external fields in the Dirac equation.

We address these problems by examining a relativistic model of a neutral particle possessing MDM, and eventually EDM, placed in an external field, in which the static magnetic and/or electric fields are applied along the propagation direction.
Working with the $x$-direction in SI units the Hamiltonian takes the form
\begin{equation}
\hat{H} = c\hat{\alpha}_x \hat{p}_x + \hat{\beta}mc^2 + 2\hat{\beta}\hat{S}_x\left(d E - \mu B\right)  \; , \label{EQ:1DRelHam}
\end{equation}
where $\hat{S}_x$ is the $x$-component of the spin vector operator in relativistic theory, $E$ is the electric field, $B$ is the magnetic field, $d$ is the size of the EDM and $\mu$ is the size of the MDM of the relativistic particle.
The four distinct eigenvalues of the Hamiltonian are
\begin{subequations}\label{EQ:EigenE}
\begin{eqnarray}
E_{\pm}^{\uparrow}   = \pm \sqrt{(cp)^2  + (mc^2 + \Delta)^2}\; ,  \\
E_{\pm}^{\downarrow} = \pm \sqrt{(cp)^2  + (mc^2 - \Delta)^2}\; ,
\end{eqnarray}
\end{subequations}
where \mbox{$\Delta = d E -\mu B$}. The four distinct eigenvalues of the Hamiltonian correspond to four different frequencies in the matrix elements between the eigenstates,
\begin{subequations}
\begin{eqnarray}
 \omega_{\text{L}}&=& \frac{1}{\hbar}(E_{+}^{\uparrow}-E_{+}^{\downarrow})= \frac{1}{\hbar}(E_{-}^{\downarrow}-E_{-}^{\uparrow})\; , \\
 \omega^{\text{zb}}_1 &=&  \frac{1}{\hbar} (E_{+}^{\uparrow}-E_{-}^{\uparrow})\; ,  \\
 \omega^{\text{zb}}_2 &=&  \frac{1}{\hbar} (E_{+}^{\uparrow}-E_{-}^{\downarrow}) =  \frac{1}{\hbar} (E_{+}^{\downarrow}-E_{-}^{\uparrow})\; , \\
 \omega^{\text{zb}}_3 &=&  \frac{1}{\hbar} (E_{+}^{\downarrow}-E_{-}^{\downarrow})\; .
\end{eqnarray}
\end{subequations}
Using the plane wave eigenstates $|l,s\rangle$  of the Hamilatonian (\ref{EQ:1DRelHam}), an arbitrary wavepacket can be represented as the superposition
\begin{equation}\label{EQ:WavePacket}
|\Psi\rangle = \;\;\mathclap{\displaystyle\int_p}\mathclap{\textstyle\sum}\;\; \sum_{s=\uparrow,\downarrow}\sum_{l=\pm}c_{l,p,s}|l,s\rangle e^{\frac{\text{i}}{\hbar}(p\cdot x-E_{l}^{s})},
\end{equation}
which includes both positive and negative energy eigenvectors~\cite{AdvancedSakurai}.

\subsection{Modification to spin-ZB}

The expectation value $\langle\Psi|\hat{S}_x|\Psi\rangle$ of the component of the spin vector operator along the propagation direction does not contain an oscillating component.
Its value $\langle\Psi|\hat{S}_x|\Psi\rangle=\;\mathclap{\displaystyle\int_p}\mathclap{\textstyle\sum}\;\left( \sum_{l=\pm}\sum_{s=\uparrow,\downarrow}|c_{p,l,s}|^2\langle s,l|\hat{S}x|s,l\rangle \right)$ is a constant of motion, determined by the initial conditions.
This is natural since in this geometry $\hat{S}_x$ is the helicity of the particle.
The expectation value of the $y$ and $z$ components of the spin operator read ($j=y,z$)
\begin{align}
\langle\hat{S}_j\rangle & = \;\;\mathclap{\displaystyle\int_p}\mathclap{\textstyle\sum}\;\; \sum_{l=\pm} 2\text{Re}(c^*_{p,l,\uparrow}c_{p,l,\downarrow}\langle\uparrow,l| \hat{S}_j|l,\downarrow\rangle e^{l i\omega_{L}t}) + \nonumber \\
&+ \;\;\mathclap{\displaystyle\int_p}\mathclap{\textstyle\sum}\;\; \sum_{s\neq s'} 2\text{Re} ( c^*_{p,+,s'}c_{p,-,s}\langle s',+|\hat{S}_j|-,s\rangle e^{i\omega_{\text{zb}}^2t} ).
\end{align}
They contain two oscillating terms with different frequencies: a relativistic Larmor precession frequency,
\begin{align}
\omega_L &=\frac{1}{\hbar}\sqrt{(cp)^2  + (mc^2 + \Delta)^2}\nonumber\\
&-\frac{1}{\hbar}\sqrt{(cp)^2  + (mc^2 - \Delta)^2},
\end{align}
 and a modified spin-ZB~frequency,
\begin{align}
 \omega_2^{\text{zb}} &= \frac{1}{\hbar}\sqrt{(cp)^2  + (mc^2 + \Delta)^2} \nonumber\\
  &+\frac{1}{\hbar}\sqrt{(cp)^2  + (mc^2 - \Delta)^2}.
\end{align}
The presence of two different frequencies for every plane wave component leads to a beating pattern with an angular frequency
\begin{equation}
\omega^{\text{sb}}=\omega_2^{\text{zb}}-\omega_L=\frac{2}{\hbar}\sqrt{(cp)^2+(mc^2-\Delta)^2}.
\end{equation}

Earlier~\cite{TenevB} we have reported the existence of an upper bound $2mc^2/\hbar$ for the Larmor precession frequency of a relativistic neutral particle.
On the other hand, the lower bound for the free-particle spin-Zitterbwegung is not modified by the coupling of EDM and MDM to external fields and therefore it is still $2mc^2/\hbar$.
However, in a realistic situation the two limits can not be achieved by $\omega_L$ and $\omega_2^{\text{zb}}$ due to the Heisenberg uncertainty principle. Thus the frequency
\begin{equation}
\omega_{\text{forbidden}}=\frac{2mc^2}{\hbar}
\end{equation}
is a forbidden frequency for spin-precession within the considered model.
Below it, spin precession is due to Larmor precession, whereas above it, it is because of ZB.
For the Hamiltonian~\eqref{EQ:1DRelHam} the transverse spin components precess with the two frequencies at the same time leading to the beating pattern with beating frequency $\omega^{\text{sb}}$.

\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{SB2}
\end{center}
\caption{Demonstration of blue shift effect for the spin-ZB~frequency $\omega^{\text{zb}}_2$, red shift for Larmor precession frequency $\omega_L$ and blue shift effect for their difference - the spin beating frequency $\omega^{\text{sb}}$. The figure is plotted assuming $\Delta=0.4\; mc^2$}\label{FIG:SB}
\end{figure}

The modified ZB~frequency $\omega_2^{\text{zb}}$ exhibits a relativistic blue-shift effect similar to the one discussed for free-ZB~frequency and contrary to the Larmor precession frequency\cite{TenevB}, which is subjected to red-shift effect.
The beating frequency  $\omega^{\text{sb}}$ also exhibits blue shift effect --- it is larger for particles with larger velocity and respectively momentum.
The behavior of the three frequencies versus the velocity is illustrated in Fig.~\ref{FIG:SB}.

\subsection{Modifications to velocity and position ZB}

While modification of spin-Zitterbwegung by MDM-B or EDM-E coupling is expected, the modification of orbital dynamics by the coupling of an external magnetostatic/electrostatic field to the spin-degree of freedom is more surprising.
We investigate these by computing the expectation values of $\hat{\alpha}_j$ and $\hat{r}_j$ ($j=x,y,z$) with respect to the superposition~\eqref{EQ:WavePacket}.
We first calculate the modified velocity-ZB~in the Schr\"odinger representation and then the position-ZB~from the expression
\begin{equation}
\frac{d}{dt}\int\Psi^+(x,t)\hat{x}\Psi(x,t)d^3x = c\int\Psi^+(x,t)\hat{\alpha}_x\Psi(x,t)d^3x
\end{equation}
 and integration~\cite{AdvancedSakurai} of the expectation value of velocity with respect to time and multiplication by $c$.

It turns out that Eq.~\eqref{EQ:1DRelHam} leads to two different types of velocity and position ZB:
 (a) \emph{longitudinal} for the $x$ component of the velocity and position operators here,
 and (b) \emph{transverse} for the $y$- and $z$- components of the velocity and position operators.
They are characterized by different precession frequencies.

The expectation value $\langle\alpha_x\rangle$ for the longitudinal velocity-ZB~reads
\begin{eqnarray}\label{EQ:VZSDL}
\langle\hat{\alpha}_x\rangle &=&\;\;\mathclap{\displaystyle\int_p}\mathclap{\textstyle\sum}\;\; \sum_{l,s} \frac{cp_x}{E^s_{p,l}} +
 \;\mathclap{\displaystyle\int_p}\mathclap{\textstyle\sum}\;\; N_1(p_x) 2\text{Re}\left[c^*_{+,p,\uparrow}c_{-,p,\uparrow}e^{i\omega_1^{\text{zb}}t}\right] + \nonumber \\
 &+&\;\;\mathclap{\displaystyle\int_p}\mathclap{\textstyle\sum}\;\; N_2(p_x) 2\text{Re}\left[c^*_{+,p,\downarrow}c_{-,p,\downarrow}e^{i\omega_3^{\text{zb}}t}\right]\; ,
\end{eqnarray}
where
$N_1(p_x)=\frac{E_0^{\uparrow}cp_x}{\sqrt{E_+^{\uparrow}E_-^{\uparrow}(E_+^{\uparrow}+E_0^{\uparrow})(E_-^{\uparrow}+E_{0}^{\uparrow})}}$, $E_0^{\uparrow}=mc^2+\Delta$ and $N_2(p_x)=\frac{E_0^{\downarrow}cp_x}{\sqrt{E_+^{\downarrow}E_-^{\downarrow}(E_+^{\downarrow}+E_0^{\downarrow})(E_-^{\downarrow}+E_{0}^{\downarrow})}}$,
$E_0^{\downarrow}=mc^2-\Delta$.
The first term of Eq.~\eqref{EQ:VZSDL} describes the group velocity of the wavepacket, obtained as the weighted sum of the averaged velocities of the components of the wavepacket~\eqref{EQ:WavePacket}.
The other two terms describe the velocity-ZB~for the Hamiltonian~\eqref{EQ:1DRelHam}.
Excluding the initial position, the expectation value of the transverse velocity-ZB~reads
\begin{eqnarray}\label{EQ:PZSDL}
\langle\hat{r}_x\rangle &=&\;\;\mathclap{\displaystyle\int_p}\mathclap{\textstyle\sum}\;\;\sum_{l,s} \frac{cp_x}{E^s_{p,l}}c t +
 \;\;\mathclap{\displaystyle\int_p}\mathclap{\textstyle\sum}\;\; N_1(p_x) 2\text{Re}\left[\frac{c^*_{+,p,\uparrow}c_{-,p,\uparrow}}{i\omega_1^{\text{zb}}}e^{i\omega_1^{\text{zb}}t}\right] + \nonumber \\
 &+& \;\;\mathclap{\displaystyle\int_p}\mathclap{\textstyle\sum}\;\; N_2(p_x) 2\text{Re}\left[\frac{c^*_{+,p,\downarrow}c_{-,p,\downarrow}}{i\omega_3^{\text{zb}}}e^{i\omega_3^{\text{zb}}t}\right]\; ,
\end{eqnarray}
where again the first term describes classical motion while the other two describe position-ZB.

\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{LZ1}
\end{center}
\caption{Demonstration of blue shift effects fir  longitudinal velocity and position ZB~frequencies $\omega^{\text{zb}}_{1}$ and $\omega^{\text{zb}}_3$, and red shift effect for their difference - the orbital beating frequency $\omega^{\text{ob}}_1$. Plotted assuming $\Delta=0.4\; mc^2$. Clearly the longitudinal ZB~frequency $\omega^{\text{zb}}_1$ can take values below $2mc^2/\hbar$, as well as the ZB~beating frequency $\omega^{\text{ob}}_1$.}\label{FIG:LZ}
\end{figure}

In contrast to the free particle case~\cite{Thaller,AdvancedSakurai} where there is just one frequency for velocity- and position-ZB, in Eq.~(\ref{EQ:VZSDL}) and Eq.~(\ref{EQ:PZSDL}) there are two different frequencies of longitudinal velocity-ZB~$\omega^{\text{zb}}_1=2\sqrt{(cp)^2+ (mc^2+\Delta)^2}$ and $\omega^{\text{zb}}_3=2\sqrt{(cp)^2+(mc^2-\Delta)^2}$. Both of them are different from the free particle ZB~frequency $\omega^{\text{zb}}=2\sqrt{(cp)^2+ (mc^2)^2}$, by a term determined by the size of the spin-splitting $\Delta$.
In the rest frame of the particle they become
\begin{subequations}
\begin{eqnarray}\label{EQ:LongiFreq}
\omega^{\text{zb}}_1(0)=\frac{2mc^2}{\hbar} + \frac{2\Delta}{\hbar}\; , \\
\omega^{\text{zb}}_3(0)=\frac{2mc^2}{\hbar} - \frac{2\Delta}{\hbar}\; . \label{EQ:LongiFreqB}
\end{eqnarray}
\end{subequations}
Obviously,  $\omega^{\text{zb}}_1$ is always greater than the lower limiting frequency of free-particle ZB~${2mc^2}/{\hbar}$, while $\omega^{\text{zb}}_2$ can take values which are smaller;
 this is illustrated in Fig.~\ref{FIG:LZ}.
In principle, by application of strong enough magnetic field the longitudinal velocity/position-ZB~frequency can be lowered to the observable range for current technology.
Thus when the spin degeneracy of the particle is lifted by an external field,  ${2mc^2}/{\hbar}$ is not a lower limit for velocity/position-ZB, while it is still a lower limit for spin-ZB, as shown above.

Furthermore, the presence of two different frequencies in the expressions for longitudinal velocity/position-ZB~leads to the appearance of a beating pattern with an orbital beating frequency
\begin{align}
\omega^{\text{ob}}_1 = \omega^{\text{zb}}_1 - \omega^{\text{zb}}_3 &= 2\sqrt{(cp)^2+ (mc^2+\Delta)^2} \nonumber\\
 &- 2\sqrt{(cp)^2+(mc^2-\Delta)^2} \; .
\end{align}
The beating frequency is just twice the Larmor precession frequency of spin-splitting $\omega^{\text{ob}}_1=2\omega_{\text{L}}$, which makes the predicted effect readily observable with current technology.
With respect to Lorentz transformations of the coordinate systems the two longitudinal velocity/position ZB~frequencies exhibit \emph{blue shift effect} similar to the one for the spin-ZB~frequency.
In contrast, the beating frequency, as their difference, manifests relativistic \emph{red shift effect}.
This is the opposite behavior of the beating frequency $\omega^{\text{sb}}$ of spin-ZB.
These dependences of the ZB~frequencies on velocity are plotted on Fig.~\ref{FIG:LZ}.

The expectation values of the $y$ and $z$ components of velocity/position have similar structures as the expectation values of spin-ZB.
They have terms with two frequencies $\omega_L$ and $\omega^{\text{zb}}_2$, which are different from the free-particle ZB~and from the longitudinal velocity/position-ZB~frequencies $\omega^{\text{zb}}_1$ and $\omega^{\text{zb}}_3$.
The position expectation values reads
\begin{align}
\langle\hat{r}_j\rangle & =\;\;\mathclap{\displaystyle\int_p}\mathclap{\textstyle\sum}\ \ \sum_{l=\pm} 2\text{Re}\left[\frac{c^*_{p,l,\uparrow}c_{p,l,\downarrow}}{l\text{i}\omega_L}\langle\uparrow,l| \hat{\alpha}_j|l,\downarrow\rangle e^{l i\omega_{L}t} \right] + \nonumber\\
&+\;\;\mathclap{\displaystyle\int_p}\mathclap{\textstyle\sum}\ \ \sum_{s\neq s'} 2\text{Re}\left[\frac{c^*_{p,+,s'}c_{p,-,s}}{\text{i}\omega_2^{\text{zb}}}\langle s',+|\hat{\alpha}_j|-,s\rangle e^{i\omega^{\text{zb}}_2t}  \right]\label{EQ:TrExp}
\end{align}
The second term in Eq.~\eqref{EQ:TrExp}, with the angular frequency $\omega_2^{\text{zb}}$, is easily identified as position-ZB.
It is the result of an overlap between the components of positive and negative energies and of opposite spin/helicity clearly visible in the expectation value of $\hat{\alpha}_j$, $j=y,z$.
The transverse position-ZB~frequency $\omega^{zb}_2$ has the same properties, as the one discussed in the context of spin-ZB~and, in particular, it is subjected to the blue-shift effect.

The first term in Eq.~\eqref{EQ:TrExp} with the angular frequency equal to the Larmor frequency $\omega_L$ is harder to be identified as traditional ZB.
While it describes a phenomenon similar to traditional ZB~--- oscillation in the expectation values of velocity and position, it does not stem from interference between solutions with positive and negative energy.
As it is clear from the structure of the matrix element $\langle\uparrow,l| \hat{\alpha}_j|l,\downarrow\rangle$ ($l=\pm$) of the velocity operator $\hat{\alpha}_j$ ($j=y,z$), it stems from matrix elements with the same sign of energy but with opposite spin orientations.
Explicitly, the matrix elements $\langle\uparrow,l| \hat{\alpha}_j|l,\downarrow\rangle$ are given by
\begin{subequations}
\begin{align}
\langle-,\downarrow|\alpha_y|-,\uparrow\rangle &+ \langle+,\uparrow|\alpha_y|+,\downarrow\rangle = \frac{cp_x(\hbar\omega_L-2\Delta)}{i\zeta} \nonumber \\
& + \frac{cp_x(\hbar\omega_L+2\Delta)}{i\eta},\\
\langle-,\downarrow|\alpha_z|-,\uparrow\rangle &+ \langle+,\uparrow|\alpha_z|+,\downarrow\rangle = \frac{cp_x(\hbar\omega_L-2\Delta)}{\zeta} \nonumber \\
& - \frac{cp_x(\hbar\omega_L+2\Delta)}{\eta},
\end{align}
\end{subequations}
where $\zeta = 2\sqrt{E_-^{\uparrow}E_-^{\downarrow}(E_-^{\uparrow}+E_0^{\uparrow})(E_-^{\downarrow}+E_0^{\downarrow})}$
 and $\eta =2\sqrt{E_+^{\uparrow}E_+^{\downarrow}(E_+^{\uparrow}+E_0^{\uparrow})(E_+^{\downarrow}+E_0^{\downarrow})} $.

Since $\zeta \gg \eta$ the amplitude of the oscillations will be dominated by the matrix elements $\langle-,\downarrow|\alpha_j|-,\uparrow\rangle$ ($j=y,z$) between the solutions with negative energy.
Furthermore, the matrix elements depend on the size of the relativistic Larmor frequency $\omega_L$, the interaction energy $\Delta$ and the momentum of the particle.
In the rest frame of the particle ($p_x=0$) the oscillations disappear.
They also disappear when the interaction energy $\Delta$ between the particle MDM/EDM and external static field is zero.
This is a feature of relativistic spin splitting.
The lifting of spin degeneracy leads to precessionary motion in the plane transverse to the direction of spin splitting.
This shows that by lifting the spin degeneracy and by the orbital effect of spin splitting it might be possible to observe ZB~with a frequency smaller than the limiting frequency ${2mc^2}/{\hbar}$ of a free particle.

The combination of two frequencies in the expectation values of $\langle\hat{r}_j\rangle$ leads to the appearance of a beating pattern with a beating frequency $\omega^{ob}_2=\omega^{zb}_2-\omega_L$.
It has the same properties as the beating frequency of the modified spin-ZB.

The lowering of the ZB~frequency $\omega^{\text{zb}}_2$ below the free particle limit ${2mc^2}/{\hbar}$, as well as the beating frequencies, lay down a route for eventual detection of ZB~phenomena in conventional experiments.
This is further helped by the red-shift effect for the longitudinal beating frequency $\omega^{\text{ob}}$.
An alternative route for experimental studies of the above effects is quantum simulations.
In a previous work \cite{TenevA}, we have proposed the emulation of the Dirac equation with EDM in trapped ions.
The same scheme of simulation can be applied to the emulation of the effects considered here.
After the preparation of an appropriate initial state the dynamics can be simulated in the proposed way~\cite{TenevA}.
The different ZB~frequencies can then be measured using the technique of mapping spatial variables onto the internal states and subsequent measurements of them \cite{Gerritsma}.

\section{Conclusions}

In this work we pointed out the existence of a \emph{motional blue-shift effect} for the ZB~frequency of a free particle.
We predicted the existence of a ``forbidden frequency'' of spin precession in 1D and showed that it is not modified by the presence of external fields.
We also predicted a modification of the frequency of spin-ZB.
We showed that the beating frequency stemming from the combination of spin-ZB~and Larmor precession exhibits a \emph{motional blue-shift effect}.
Furthermore, we predicted the existence of two different position ZB~phenomena: transverse and longitudinal.
The longitudinal position-ZB~is characterized by two frequencies different from the free particle one ${2mc^2}/{\hbar}$ and leading to a beating pattern.
We showed that the two ZB~frequencies exhibit a \emph{motional blue-shift effect}, while the beating frequency exhibits a \emph{motional red-shift effect}.
The transverse ZB~is also characterized by two frequencies and a beating pattern similar to the spin-ZB.
However, while the higher frequency can be interpreted as ZB~frequency, the lower can not be identified as conventional ZB, because it stems from matrix elements with the same sign of energy but opposite spin orientations.
We referred to it as orbital effect of spin splitting.

This work has been supported by the EU COST project IOTA and the Bulgarian NSF grants D002-90/08 and DMU03/107.