# Shift limits of a non-autonomous system

Dawoud Ahmadi<EMAIL_ADDRESS><EMAIL_ADDRESS>Mahdi
Aghaee<EMAIL_ADDRESS>
###### Abstract

Let $t=t_{1}t_{2}\cdots$ be an element of the full shift with shift map $\tau$
on a finite set of characters $\mathcal{A}$ and let $\Sigma=\text{
closure}\\{\tau^{i}(t):\;i\in\mathbb{N}\cup\\{0\\}\\}$. Let
$f_{t}=f_{t_{1},\,\infty}=\cdots\circ f_{t_{2}}\circ f_{t_{1}}$ be a non-
autonomous system over a compact metric space $X$ where $t_{i}\in\mathcal{A}$.
The set ${\bf{F}}_{t}^{+}=\\{f_{\tau^{i}(t)}:\;i\in\mathbb{N}\\}$ is called
the shifted family of $f_{t}$. If $t$ is a transitive point of the full shift
on $\mathcal{A}$, then by introducing a natural topology,
$\overline{{\bf{F}}_{t}^{+}}$ is a classical IFS; otherwise,
$\overline{{\bf{F}}_{t}^{+}}=\\{f_{\sigma}=f_{\sigma_{1},\,\infty}:\;\sigma\in\Sigma\\}$
is a generalized IFS. We will show that if $f_{t}$ has some various shadowing
and specification properties, then this is true for
$f_{\sigma}\in\overline{{\bf{F}}^{+}_{t}}$; however, this claim is not true
for other properties such as transitivity, mixing and exactness. Also, if
$\Sigma$ is sofic and $x\in X$ is periodic point for some
$f_{\sigma}\in\overline{{\bf{F}}^{+}_{t}}$, then there is a periodic
$\sigma^{\prime}\in\Sigma$ such that $x$ is periodic for
$f_{\sigma^{\prime}}\in\overline{{\bf{F}}^{+}_{t}}$.

Keywords: iterated function systems (IFS), non-autonomous system,
specification, shadowing.
2010 Mathematics Subject Classification: 37B55, 37B10 .

##  1 Introduction

Assume that the functions defining a non-autonomous dynamical system are
coming from $\mathcal{F}=\\{f_{0},\ldots,f_{k-1}\\}$. Then we write such a
system as $f_{t_{1},\,\infty}=\cdots f_{t_{2}}\circ f_{t_{1}}$ where
$t_{i}\in\mathcal{A}=\\{0,\,1,\ldots,k-1\\}$ (our priority is to have
$\mathcal{A}$ finite, but if our results is valid for infinite case, we do
mention explicitly). On the other hand,
$t=t_{1}t_{2}\cdots\in\mathcal{A}^{\mathbb{N}}$ can be considered as an
element of the subshift
$\Sigma=\overline{\mathcal{O}^{+}(t)}=$closure$\\{\tau^{i}(t):\,i\in\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}\\}$
where $\tau$ is the shift map and $\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)$ is
a _general iterated function system_ (IFS). That means the combinations of
maps are not freely as in the _classical IFS_ and are done according to
restrictions imposed by $\Sigma$. By this settings a classical IFS is
$\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma_{|\mathcal{F}|})$ where
$\Sigma_{|\mathcal{F}|}$ is the full shift on $|\mathcal{F}|$ symbols. Here
for $\sigma=\sigma_{1}\sigma_{2}\cdots\in\Sigma$, we call the non-autonomous
dynamical system $f_{\sigma_{1},\,\infty}$ a _shift limit_ of
$f_{t_{1},\,\infty}$ and we are interested to see what dynamical properties of
$f_{t_{1},\,\infty}$ are shared with $f_{\sigma_{1},\,\infty}$. Some
properties that we have considered are specification, shadowing, chaos and in
particular different periodic behavior appearing in non-autonomous dynamical
systems. Our approach is to consider a typical transitive $t$ in a subshift
$\Sigma=\overline{\mathcal{O}^{+}(t)}$ and to see how the aforementioned
dynamical properties of $f_{t_{1},\,\infty}$ is carried to
$f_{\sigma_{1},\,\infty}$ for $\sigma=\sigma_{1}\sigma_{2}\cdots\in\Sigma$. If
that does so, then we say that specific property _along_ $t$ is preserved
along $\sigma$ as well.

Let us recall that many aspects of both dynamical systems, that is non-
autonomous dynamical systems and IFS’s have been worked out in other
investigations. Salman and Das [10] discussed specification property in non-
autonomous discrete systems and Vasisht and Das [14] studied specification and
shadowing properties for non-autonomous Systems. Pravec [7] considered some
different periodic points for non-autonomous dynamical systems. Specification
and shadowing for classical iterated function systems have been studied in [8,
15]. The general IFS was considered by the authors [1] and so fairly new
subject. There, a good deal of attention has been paid to transitivity. For
transitivity of non-autonomous systems, one can consult [13, 9, 11]. It is
worth to mention that we learned a lot from Kolyada and Snoha [3]. However,
their main interest is on entropy that we have not considered here.

The present paper is organized as follows. After introducing some required
basic notions in Preliminaries, in Section 3, we show that specification and
shadowing along a transitive orbit of a subshift, acting as our starting non-
autonomous system, will be preserved along all other shift limits. In Section
4, after recalling different types of periodicity in a non-autonomous system,
we see how these different periodic behaviors are projected to shift limits.

##  2 Preliminaries

Throughout the paper, $(X,\,d)$ will be a compact metric space and
$\mathcal{A}$ is a set of finite characters . Also $\mathbb{N}$ will be the
set of natural numbers and ${\bf N}$ the set of all non-autonomous systems
(NDS) on $X$. We are dealing with those NDS’s whose functions are in
$\\{f_{j}:X\rightarrow X:\;f_{j}$ continuous $j\in\mathcal{A}\\}$. So for
$t=t_{1}t_{2}\cdots$ where $t_{i}\in\mathcal{A}$, denote by

$f_{t_{1},\,\infty}=\cdots\circ f_{t_{2}}\circ f_{t_{1}}$ (1)

the NDS along $t$ and let $f_{t_{i},\,\infty}=\cdots\circ f_{t_{i+1}}\circ
f_{t_{i}}$ be the $i$ th shifted NDS of $f_{t_{1},\,\infty}$. Set

${\bf{F}}^{+}_{t}={\bf{F}}^{+}_{t}(f_{t_{1},\,\infty})=\\{f_{t_{i},\,\infty}:\;i\in\mathbb{N}\\}\subseteq{\bf
N}$ (2)

to be the _shifted family_ of the non-autonomous system $f_{t_{1},\infty}$.
Let $D(f,\,g)=\sup_{x\in X}d(f(x),\,g(x))$ be the metric on $C(X)$, the set of
continuous self maps on $X$. Then

$\rho(f_{t_{1},\,\infty},\,f_{t^{\prime}_{1},\,\infty})=\sum_{n\in\mathbb{N}}\frac{D(f_{t_{1}\cdots
t_{n}},\,f_{t^{\prime}_{1}\cdots t^{\prime}_{n}})}{2^{n-1}}$

is a metric on ${\bf N}$ where $f_{t_{1}\cdots
t_{n}}=f_{t_{n}}\circ\cdots\circ f_{t_{1}}$. An NDS is injective or surjective
if each function defining the NDS are injective or surjective respectively.

Now let $(\Sigma,\,\tau)$ be the one sided subshift on $\mathcal{A}$ whose
shift map is $\tau$ ( See §2.2 for a brief review of symbolic dynamics) and
equip $(\Sigma\times{\bf N})$ with the product topology. We are interested in
the dynamics of the space of skew product ${\bf S}\subset(\Sigma\times{\bf
N})$

${\bf F}:{\bf S}\to{\bf S}$ (3)

where ${\bf F}$ is defined as ${\bf
F}(\sigma,\,f_{\sigma_{1},\,\infty})=(\tau(\sigma),\,f_{\tau(\sigma)_{1},\,\infty})$.
Note that by the above settings, ${\bf S}$ is compact and ${\bf F}$ is
continuous. When $t\in\mathcal{A}^{\mathbb{N}}$, then
$\Sigma=\overline{\mathcal{O}^{+}(t)}$ is a subshift obtained by the orbit of
$t$ and ${\bf
F}^{+}_{t}=\text{pr}_{2}(\mathcal{O}^{+}(t,\,f_{t_{1},\,\infty}))$. This means
${\bf F}^{+}_{t}$ is the projection on the second component of the orbit of
the point $(t,\,f_{t_{1},\,\infty})$ under ${\bf F}$. The following is
immediate.

######  Proposition 2.1.

$\overline{{\bf{F}}^{+}_{t}}=\\{f_{\sigma_{1},\,\infty}:\;\sigma=\sigma_{1}\sigma_{2}\cdots\in\overline{\mathcal{O}^{+}(t)}\\}$.

The above proposition demonstrates a close relation between an NDS and the
theory of iterated function systems on a general subshift which we will give a
brief introduction in the next subsection.

###  2.1 “Generalized” iterated function systems.

The classical _iterated function system_ (IFS) consists of all possible
combinations of finitely many continuous self maps
$\mathcal{F}=\left\\{f_{0},\,\ldots,\,f_{k-1}\right\\}$ on $X$. In other
words, a classical IFS consists of $X$ and all maps such as

$f_{u}=f_{u_{n}}\circ\cdots\circ f_{u_{1}}:X\rightarrow X,\quad u=u_{1}\cdots
u_{n}\in\mathcal{L}(\Sigma_{k})$ (4)

where $\Sigma_{k}$ is the full shift on $\mathcal{A}=\\{0,\,1,\cdots,k-1\\}$
and $\mathcal{L}(\Sigma_{k})$ is the language of $\Sigma_{k}$. (See below for
a brief introduction of symbolic dynamics.) If one replaces $\Sigma_{k}$ with
a subshift $\Sigma\subseteq\Sigma_{k}$ in (4), then it will be a “generalized”
iterated function system which we refer to it again as IFS and it is denoted
by

$\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma),$ (5)

where $\mathcal{F}=\left\\{f_{0},\,\ldots,\,f_{k-1}\right\\}$.

When $\mathfrak{I}$ is the classical IFS; or equivalently, when $\Sigma$ is a
full shift, then still we use $\eqref{eq IFS}$ but the subshift will be shown
by $\Sigma_{|\mathcal{F}|}$. The classical IFS is usually denoted by
$(X,\,\mathcal{F})$ by other authors.

The _forward orbit_ of a point $x\in X$ in an IFS, is defined as
$\mathcal{O}^{+}(x)=\\{f_{u}(x):\;u\in\Sigma\\}$. Also to each
$\sigma=\sigma_{1}\sigma_{2}\cdots\in\Sigma$, a NDS
$f_{\sigma}=f_{\sigma_{1},\,\infty}$ will be associated that it is called the
_IFS along $\sigma$_. The trajectory (or orbit) of $x$ under $f_{\sigma}$ is
given by the sequence

$\mathcal{O}^{+}(x)=\\{x,\,f_{\sigma_{1}}(x),\,f_{\sigma_{1}\sigma_{2}}(x),\,\ldots,\,f_{\sigma_{1}\cdots\sigma_{n}}(x),\,\ldots\\}$

where $f_{\sigma_{1}\cdots\sigma_{n}}=f_{\sigma_{n}}\circ\cdots\circ
f_{\sigma_{1}}$, $n\geq 1$. In general,
$f_{\sigma_{m}\cdots\sigma_{n}}=f_{\sigma_{n}}\circ\cdots\circ
f_{\sigma_{m}}$, $n\geq m$. Notice that when $\mathcal{F}$ consists of just
one function, then $f_{\sigma}$ is a “conventional” discrete dynamical system.
For further properties of an IFS see [1].

###  2.2 Symbolic dynamics

Since we are dealing with a general subshift in an IFS such as (5), we require
a brief recall of the symbolic dynamics. A basic source for this theory is
[4]; here we just give a brief recall. Let
$\mathcal{A}=\\{0,\,1,\cdots,k-1\\}$ be the set of finite characters also
called alphabet. Let $\Sigma_{k}=\mathcal{A}^{\mathbb{Z}}$ (resp.
$\mathcal{A}^{\mathbb{N}}$) be the collection of all bi-infinite (resp. right-
infinite) sequences of symbols from $\mathcal{A}$. The map
$\tau:\Sigma_{k}\rightarrow\Sigma_{k}$ defined by
$\tau(\sigma)_{i}=\sigma_{i+1}$ is called the _shift map_ and the pair
$(\Sigma_{k},\,\tau)$ is the _full shift_ on $k$ symbols. Any invariant closed
subspace $\Sigma$ of $\Sigma_{k}$ is called a _subshift_.

A _word_ or a _block_ over $\mathcal{A}$ is a finite sequence of symbols from
$\mathcal{A}$. Denote by $\mathcal{L}_{n}(\Sigma)$ the set of all _admissible_
words of length $n$ and call
$\mathcal{L}(\Sigma):=\bigcup_{n=0}^{\infty}\mathcal{L}_{n}(\Sigma)$ the
_language_ of $\Sigma$. A subshift is characterized by its language or its
forbidden words. For $u\in\mathcal{L}_{k}(\Sigma)$, let the cylinder
${}_{\ell}[u]_{\ell+k-1}$$=_{\ell}[u_{\ell}\cdots u_{\ell+k-1}]_{\ell+k-1}$ be
the set
$\\{\sigma=\cdots\sigma_{-1}\sigma_{0}\sigma_{1}\cdots\in\Sigma:\sigma_{\ell}\cdots\sigma_{\ell+k-1}=u\\}$.
If $\ell=0$, we drop the subscripts and we just write $[u]$. A point
$\sigma\in\Sigma$ is _transitive_ if every word in $\Sigma$ appears in
$\sigma$ infinitely many often and a subshift $\Sigma$ is _irreducible_ iff
$\Sigma$ has a transitive point. Equivalently, $\Sigma$ is irreducible if for
every ordered pair of words $u,\,v\in\mathcal{L}(\Sigma)$ there is a word
$w\in\mathcal{L}(\Sigma)$ so that $uwv\in\mathcal{L}(\Sigma)$.

If forbidden set of $\Sigma$ is finite, then $\Sigma$ is called a _shift of
finite type_ (SFT) and any of its factor is called _sofic_. A word
$w\in\mathcal{L}(\Sigma)$ is called _synchronizing_ if
$uwv\in\mathcal{L}(\Sigma)$ whenever $uw,wv\in\mathcal{L}(\Sigma)$. A
_synchronized system_ is an irreducible shift which has a synchronizing word.
Any sofic is synchronized.

A synchronized system $\Sigma$ may be represented with a directed labeled
graph called _cover_. Points of $\Sigma$ is the recording of labels on
infinite walks and the closure of all such points. Covers are not unique and
there is a “minimal” representation which is called _Fischer cover_. Any
Fischer cover is right resolving: the edges starting at any vertex carry
different labels. For sofics, the Fischer cover is a right resolving labeled
graph with the fewest vertices among all covers. Also, a subshift is sofic iff
it has a finite cover.

##  3 Specification property and shadowing

On the basis of Proposition 2.1, we will concentrate on the study of a
“generalized” IFS given in (5). First some definitions for adjusting the
existing definitions to our notations.

######  Definition 3.1.

Let $\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)\;$be an IFS. A point $x\in X$ is
periodic of period $p$ along $\sigma=\sigma_{1}\sigma_{2}\cdots\in\Sigma$ if
for any $\ell\in\mathbb{N}$, $f_{\sigma_{1}\cdots\sigma_{\ell p}}(x)=x$. The
point $x$ is weakly periodic along $\sigma$, if there are infinitely many $i$
such that $f_{\sigma_{1}\cdots\sigma_{i}}(x)=x$.

Later in Section 4, we will consider another form of periodicity in NDS;
nonetheless, the above definition suffices our needs for time being.

######  Definition 3.2.

Let $\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)$ be an IFS. Then $\mathfrak{I}$
is said to have

  1. 1.

_Specification property (SP)_ along $\sigma\in\Sigma$, if for every
$\epsilon>0$ there is $N(\epsilon)$ such that for any
$x_{1},\,x_{2},\,\ldots,\,x_{s}$ with $s\geq 2$ and any sequence $0=j_{1}\leq
k_{1}<j_{2}\leq k_{2}<\cdots<j_{s}\leq k_{s}$ of integers with
$j_{n+1}-k_{n}\geq N(\epsilon)$, $n=1,\,2,\,\ldots,\,s-1$, there is a point
$x\in X$ such that for $1\leq m\leq s$ and $j_{m}\leq i\leq k_{m}$

$d(x,\,x_{1})<\epsilon,\quad
d(f_{\sigma_{1}\cdots\sigma_{i}}(x),\,f_{\sigma_{1}\cdots\sigma_{i}}(x_{m}))<\epsilon.$
(6)

  2. 2.

_Strong specification property (SSP)_ along $\sigma$, if $x\in X$ satisfying
(6) is periodic.

  3. 3.

_Local weak specification property (LWS)_ along $\sigma\in\Sigma$, if for
every $\epsilon>0$ there are $N\in\mathbb{N}$ and $\delta>0$ such that if
$x_{1},\,\ldots,\,x_{s}\in X$ satisfies
$d(f_{\sigma_{1}\cdots\sigma_{n}}(x_{k}),\,x_{k+1})$ $<\delta$ with $n\geq N$,
then there is $x\in X$ such that for $0\leq k\leq s-1$ and any $kn\leq
j<(k+1)n$

$d(x,\,x_{1})<\epsilon,\quad
d(f_{\sigma_{1}\cdots\sigma_{j}}(x),\,f_{\sigma_{1}\cdots\sigma_{j-kn}}(x_{k+1}))<\epsilon.$
(7)

An IFS has SP, SSP or LWS, if it has SP, SSP or LWS respectively along all
$\sigma\in\Sigma$.

Definitions of SP and LWS are adapted from [10, Definition 2.6] and [14,
Definition 3.2] respectively and surely (6) as well as (7) can be written in
one equation, but this is done for compatibility of our notations to the rest
of this note.

######  Definition 3.3.

Let $\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)\;$be an IFS and $U$, $V$
arbitrary non-empty open sets in $X$. Then $\mathfrak{I}$ is called _mixing_
(resp. _exact_) _along an orbit_ $\sigma\in\Sigma$, if there is
$N\in\mathbb{N}$ such that for $n\geq N$,
$f_{\sigma_{1}\cdots\sigma_{n}}(U)\cap V\neq\emptyset$ (resp.
$f_{\sigma_{1}\cdots\sigma_{n}}(U)=X$).

By [5, Theorem 2.1] and similar to conventional dynamical systems we have

######  Proposition 3.4.

For a surjective $\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)$, specification
property implies mixing along any orbit.

######  Remark 3.5.

  1. 1.

Any conventional mixing system which does not have specification property,
shows that the converse of the above proposition is not necessarily true.

  2. 2.

Let $X=\\{a,\,b\\}$ and $f:X\rightarrow X$ a continuous function defined as
$f(a)=f(b)=a$. Then this conventional system has specification property,
though it is not mixing. Thus this simple example shows that surjectivity in
Proposition 3.4 cannot be omitted.

Next we will show that when $\Sigma$ is irreducible, specification property
for $\mathfrak{I}$ is equivalent to having this property just along one
transitive point in $\Sigma$; in particular, the definition of specification
property can then be reduced to specification along a transitive
$t\in\Sigma_{|\mathcal{F}|}$ [2, Definition 4.2].

######  Proposition 3.6.

Assume $\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)$ is surjective and has
(strong) specification property along a transitive
$t=t_{1}t_{2}t_{3}\cdots\in\Sigma$. Then $\mathfrak{I}$ has (strong)
specification property along any orbit.

###### Proof.

We will prove that SP holds along any
$\sigma=\sigma_{1}\sigma_{2}\cdots\in\Sigma$. Similar proof works for SSP as
well.

For a given $\epsilon>0$, let $N=N(\epsilon)$ be the natural number provided
by the definition of specification property along $t$. We will show that this
$N$ works as well for the required definition of specification property along
$\sigma$. To this end fix $\sigma\in\Sigma$ and choose any sequence
$x_{1},\,x_{2},\ldots,x_{s}$ in $X$ for $s\geq 2$ and consider the arbitrary
sequence

$0=j_{1}\leq k_{1}<j_{2}\leq k_{2}<\cdots<j_{s}\leq k_{s}$ (8)

where $|j_{i+1}-k_{i}|\geq N$. Then choose $\ell\in\mathbb{N}$ so that
$k^{\prime}_{0}=\ell-N\geq 0$ and $t_{\ell+j}=\sigma_{j}$ for $0\leq j\leq
k_{s}$. This is possible, for $t$ is a transitive point in $\Sigma$. Replace
(8) with

$0=j^{\prime}_{0}\leq k^{\prime}_{0}<j^{\prime}_{1}\leq
k^{\prime}_{1}<j^{\prime}_{2}\leq k^{\prime}_{2}<\cdots<j^{\prime}_{s}\leq
k^{\prime}_{s}$ (9)

where $j^{\prime}_{i}=j_{i}+\ell$ and $k^{\prime}_{i}=k_{i}+\ell$ for $1\leq
i\leq s$. Set $u=t_{1}t_{2}\cdots t_{\ell-1}$ and pick $y_{i}\in X$ so that
$f_{u}(y_{i})=x_{i}$, $1\leq i\leq s$ and let $y_{0}$ be any point in $X$.

Now apply the specification property along $t$ for the above $\epsilon>0$, the
sequence $y_{0},\,y_{1},\,y_{2},\ldots,y_{s}$ and sequence 9 to have a point
$x^{\prime}\in X$ such that

$d(x^{\prime},\,y_{0})<\epsilon,\quad d(f_{t_{1}\cdots
t_{i}}(x^{\prime}),\,f_{t_{1}\cdots t_{i}}(y_{m}))<\epsilon;\quad 0\leq m\leq
s,\quad j^{\prime}_{m}\leq i\leq k^{\prime}_{m}.$

This in turn implies that

$d(f_{\sigma_{1}\cdots\sigma_{i}}(f_{u}(x^{\prime})),\,f_{\sigma_{1}\cdots\sigma_{i}}(f_{u}(y_{m}))<\epsilon;\quad
1\leq m\leq s,\quad j_{m}\leq i\leq k_{m}.$

Hence for $x=f_{u}(x^{\prime})$, we have

$d(x,\,x_{1})<\epsilon,\quad(f_{\sigma_{1}\cdots\sigma_{i}}(x),\,f_{\sigma_{1}\cdots\sigma_{i}}(x_{m}))<\epsilon$

for each $1\leq m\leq s$ and any $j_{m}\leq i\leq k_{m}$. Thus specification
property holds along $\sigma$ and by the fact that $\sigma$ was arbitrary,
$\mathfrak{I}$ has specification property as required. ∎

Thus if $\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)$ has SSP along a transitive
$t\in\Sigma$, then the periodic points are dense along any $\sigma\in\Sigma$
and in particular, the dynamics along all orbits have Devaney chaos [12].
Equivalently,

######  Corollary 3.7.

If a surjective NDS such as (1) has SSP, then $f_{\sigma_{1},\,\infty}$ has
Devaney chaos for any $\sigma$ in the limit set of ${\mathcal{O}^{+}(t)}$.

###### Proof.

This follows from Proposition 3.6 and the fact that periodic points are dense
in systems with SSP and also the fact that such systems are mixing and thus
transitive. ∎

So far we have shown that specification and strong specification along a
transitive orbit implies the same property along all other orbits. The rest of
this section is devoted to prove similar results for shadowing and LWS.

######  Definition 3.8.

The sequence $\\{x_{0},\,x_{1},\,\ldots\\}\subseteq X$ is a _ $\delta$-pseudo
orbit along $\sigma=\sigma_{1}\sigma_{2}\cdots\in\Sigma$_ for some $\delta>0$,
if $d(f_{\sigma_{i}}(x_{i-1}),\,x_{i})<\delta$ for all $i\geq 1$. We say that
$\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)\;$has the _shadowing property along
$\sigma$_ if for every $\epsilon>0$ there exists $\delta=\delta(\epsilon)$
such that for every $\delta$-pseudo orbit $\\{x_{0},\,x_{1},\,\ldots\\}$ along
$\sigma$ there is $y\in X$ such that

$d(y,\,x_{0})<\epsilon,\quad
d(f_{\sigma_{1}\cdots\sigma_{i}}(y),\,x_{i})<\epsilon,\quad\text{ for all
}i\geq 1.$

######  Proposition 3.9.

Assume $\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)$ is surjective and has
shadowing along a transitive $t\in\Sigma$. Then $\mathfrak{I}$ has shadowing
along any orbit.

###### Proof.

Let $\sigma=\sigma_{1}\sigma_{2}\cdots$ be arbitrary and for $n\in\mathbb{N}$,
let $\zeta_{n}=\sigma_{1}\sigma_{2}\cdots\sigma_{n}$. Since $t$ is transitive
and any $\zeta_{n}$ appears infinitely many times along $t$, one may write $t$
as

$t=t_{1}\cdots\zeta_{1}\cdots\zeta_{2}\cdots\zeta_{n}\cdots\zeta_{n+1}\cdots=t_{1}\cdots\zeta_{1}\cdots\zeta_{2}\cdots\sigma_{1}\sigma_{2}\cdots\sigma_{n}\cdots\zeta_{n+1}\cdots.$
(10)

Set $\underline{t}_{n}=t_{1}\cdots t_{|\underline{t}_{n}|}$ to be the initial
finite block of $t$ before $\zeta_{n}$ and $\bar{t}_{n}$ the infinite block
after $\zeta_{n}$. Thus

$t=t_{1}\cdots\zeta_{n}\cdots=\underline{t}_{n}\zeta_{n}\bar{t}_{n}.$

Let $\epsilon>0$ and let $\delta>0$ be provided by the definition of
$\frac{\epsilon}{2}$-shadowing for $t$ and let
$\alpha:=\\{x_{i}\\}_{i\in\mathbb{N}}$ be a $\delta$-pseudo orbit along
$\sigma$. Also for any $n$, let $\underline{x}_{n}$ be a point such that
$f^{-1}_{\underline{t}_{n}}(\underline{x}_{n})=x_{1}$.

Denote by $\mathcal{S}_{u}(x)$ the sequence representing the true orbit of $x$
along $u$ where $u$ is a finite or infinite block in $\mathcal{L}(\Sigma)$.
Then the sequence

$\gamma_{n}:=\mathcal{S}_{\underline{t}_{n}}(\underline{x}_{n}),\,x_{1},\,x_{2},\,\ldots,\,x_{|\zeta_{n}|},\,\mathcal{S}_{\bar{t}_{n}}(x_{|\zeta_{n}|})$

is a $\delta$-pseudo orbit along $t$. Hence there is a $z_{n}\in X$ tracing
$\gamma_{n}$ within an $\frac{\epsilon}{2}$ neighborhood of this trajectory
along $t$ and set $y_{n}=f_{\underline{t}_{n}}(z_{n})$. As a result we will
have

${d(y_{n},\,x_{1})<\frac{\epsilon}{2},\quad
d(f_{\sigma_{1}\cdots\sigma_{n}}(y_{n}),\,x_{n+1})<\frac{\epsilon}{2},\qquad
1<n<|\zeta_{n}|.}$ (11)

Let $y$ be a limit point of $\\{y_{n}\\}$ and without loss of generality
assume that $\\{y_{n}\\}$ converges to $y$. Pass to the limit in (11) to have

${d(y,\,x_{1})<\epsilon,\quad
d(f_{\sigma_{1}\cdots\sigma_{n}}(y),\,x_{n+1})<\epsilon,\qquad n>1.}$

Now this implies that $\alpha$ is $\epsilon$-shadowed by $y$ along $\sigma$. ∎

There are examples that shadowing occurs along orbits which are not
transitive.

######  Example 3.10.

Let $\mathfrak{I}=(X=[0,\,1],\,\\{f_{0},\,f_{1}\\},\,\Sigma_{F})$ where
$f_{0}(x)=2x\mod 1$ and $f_{1}$ any homeomorphism say $f_{1}(x)=x$. Then if
$\sigma$ terminates at $1^{\infty}$ i.e. if
$\sigma=\sigma_{1}\cdots\sigma_{n}1^{\infty}$ for some $n\in\mathbb{N}$, then
$\mathfrak{I}$ along $\sigma$ does not have shadowing property while along all
those $\sigma\in\Sigma$ terminating at $0^{\infty}$, $\mathfrak{I}$ has
shadowing property.

Authors in [14, Theorem 4.1] prove that LWS along $\sigma$ is equivalent to
shadowing property along $\sigma$. This implies the following.

######  Corollary 3.11.

Let $\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)\;$be a surjective IFS. Then LWS
along a transitive $t\in\Sigma$ implies LWS along any $\sigma\in\Sigma$.

Consider the set of properties

$\Gamma=\\{\text{SP },\text{SSP },\text{LWS},\text{ shadowing}\\}$ (12)

and let $t\in\Sigma$. Then proof of propositions 3.6 and 3.9 allows us to
restate those propositions as: If
$\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)\;$has property $\gamma\in\Gamma$
along $t$, then $\mathfrak{I}$ has property $\gamma$ along $\sigma$ in the
limit set of $\mathcal{O}^{+}(t)$. In particular, since we never used the
property that $\Sigma$ is generated by a finite set of characters, thus we
have

######  Corollary 3.12.

Let $\Gamma$ be as in (12), $f_{t_{1},\,\infty}$ as in (1) defined with
possibly infinitely many surjective functions $f_{t_{i}}:X\to X$ having
property $\gamma\in\Gamma$. Then, $f_{\sigma_{1},\,\infty}$ has property
$\gamma$ for any $\sigma$ in the limit set of $\mathcal{O}^{+}(t)$.

For $\gamma\in\Gamma$ in (12) define

$S_{\gamma}=S_{\gamma}(\mathfrak{I})=\\{\sigma\in\Sigma:\;\mathfrak{I}\text{
has }\gamma\text{ property along }\sigma\\}.$

######  Proposition 3.13.

Assume $\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)\;$is a surjective IFS and
$\Sigma$ an irreducible subshift. Then, either $S_{\gamma}=\Sigma$ or
$S_{\gamma}$ is a meager set.

###### Proof.

The proof is a result of propositions 3.6, 3.9 and the fact that the
transitive points in $\Sigma$ are generic and intersection of two generic sets
is again generic. ∎

In the following example, we show that the same conclusion in propositions 3.6
and 3.9 does not hold if one considers exact or mixing.

######  Example 3.14.

Let $\mathfrak{I}=([0,\,1],\,\\{f_{0},\,f_{1}\\},\,\Sigma_{2})$ where $f_{0}$
is identity and $f_{1}(x)=2x\mod 1$ on $[0,\,1]$. This is not hard to see that
$\mathfrak{I}$ is exact along any transitive point $t\in\Sigma_{2}$; however
since $f_{0}$ is not transitive, $\mathfrak{I}$ is not exact and so mixing
along $\sigma=0^{\infty}\in\Sigma_{2}$.

##  4 Periodicity in non-autonomous systems and IFS

The notion of a periodic point in the case of a conventional dynamical system
$(X,\,f)$ is very natural and intuitive. This is not the case for an IFS or a
non-autonomous system. One may check [7] where a survey of the periodic points
for a non-autonomous system is offered.

The most prevalence definition of a periodic point for an NADS is the one
given in Definition 3.1. Though this lacks the fact that if $x$ is periodic
for $f_{\sigma_{1},\,\infty}$, then $f_{\sigma_{1}}(x)$ is not necessarily
periodic which is fairly unintuitive with respect to a conventional dynamical
system.

######  Definition 4.1.

A point $x\in X$ is

  1. 1.

_regularly periodic_ for $f_{\sigma_{1},\,\infty}$ of period $p$, if for any
$i\geq 1$

$f_{\sigma_{1}\cdots\sigma_{i}}(x)=f_{\sigma_{1}\cdots\sigma_{i+p}}(x).$

  2. 2.

_orbitally periodic_ for $f_{\sigma_{1},\,\infty}$ of period $p$, if there is
a non-empty word $u$, $|u|=p$ such that $\sigma=u^{\infty}$ and $x$ is
regularly periodic of period $p$. Thus $x$ is orbitally periodic for
$f_{\sigma_{1},\,\infty}$ iff $(\sigma,\,x)\in{\bf S}$ is a periodic point of
${\bf F}$ in (3).

We have the following implications

orbitally periodic $\displaystyle\Rightarrow$ $\displaystyle\text{ regularly
periodic }\Rightarrow\text{ periodic (Definition \ref{def peridic}) }$
$\displaystyle\Rightarrow$ $\displaystyle\text{ weakly periodic (Definition
\ref{def peridic})}.$

It is not hard to give examples to show that none of the above implications
are reversible. We give an example for one of them and the others can be also
easily constructed.

######  Example 4.2.

Let $X$ be the unit circle and $\mathcal{F}=\\{f_{0},\,f_{1}\\}$ where
$f_{0}(x)$ is the rotation by an irrational $2\theta$ and $f_{1}(x)$ is the
rotation by $-\theta$ and

$\sigma=011\cdots\overbrace{00\ldots 0}^{\text{$n$
times}}\overbrace{11\ldots\ldots 1}^{\text{$2n$ times}}00\cdots.$

Then any point is weakly periodic but not periodic.

For all sorts of periodicity defined in definitions 3.1 and 4.1, the eventual
periodicity can be defined accordingly. For instance, a point $x\in X$ is
eventually weakly periodic if there is $i$ such that
$y=f_{\sigma_{1}\cdots\sigma_{i}}(x)$ is weakly periodic.

######  Proposition 4.3.

Suppose that $x$ is periodic for $f_{\sigma_{1},\,\infty}$ where
$\sigma_{i}\in\mathcal{A}=\\{0,\,1,$ $\,\ldots,k-1\\}$. Then for any $j$,
$y=f_{\sigma_{1}\cdots\sigma_{j}}(x)$ is eventually weakly periodic.

###### Proof.

Let $x$ be periodic of period $p$ and for some
$q\in\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$ write $j=qp+\ell$, $0\leq\ell<p$.
Then the set $\\{f_{\sigma_{ip+1}\cdots\sigma_{ip+\ell}}(x):\;i\geq q\\}$ is
finite; because, $\\{\sigma_{ip+1}\cdots\sigma_{ip+\ell}:\;i\geq
q\\}\subset\mathcal{L}_{\ell}(\Sigma_{k})$ is finite and thus for infinitely
many $i$, the values of $f_{\sigma_{ip+1}\cdots\sigma_{ip+\ell}}(x)$ are
identical. ∎

######  Lemma 4.4.

Let $\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)$ be an IFS where $\Sigma$ is an
$M$-step SFT. Assume that $x$ is periodic of period $p$, then there is
$\sigma^{\prime}={u^{\prime}}^{\infty}\in\Sigma$ where $x$ is orbitally
periodic for $f_{\sigma^{\prime}_{1},\,\infty}$ with $|u^{\prime}|=qp$ for
some $q\in\mathbb{N}$.

###### Proof.

Without loss of generality assume that $p\geq M$; otherwise, we may choose
$a\in\mathbb{N}$ so large with $ap\geq M$ and considering $x$ as a periodic
point of period $ap$.

Set $u_{i}:=\sigma_{(k_{i}p)+1}\sigma_{(k_{i}p)+2}\cdots\sigma_{k_{i+1}p}$ and
note that $f_{u_{i}}(x)=x$. But our alphabet is finite, thus for infinitely
many $i$’s, $u_{i}$’s are the same and call it $u$. This means that there is a
$w$ with $uwu\subset\sigma$ and $|w|$ an integral multiple of $p$. Now by the
fact that $u$’s length is larger than $M$, it is synchronizing and so
$\sigma^{\prime}=(uw)^{\mathbb{N}_{0}}\in\Sigma$ and that in turn implies that
$x$ is periodic of period $p^{\prime}=|uw|$ along $\sigma^{\prime}$. ∎

Let $\Sigma$ and $\Sigma^{\prime}$ be two subshifts on the character sets
$\mathcal{A}$ and $\mathcal{A}^{\prime}$ respectively. Then, $\Sigma^{\prime}$
is a factor of $\Sigma$ or $\Sigma$ is an extension of $\Sigma^{\prime}$ if
there is a map $\Psi:\mathcal{L}_{m+n+1}(\Sigma)\to\mathcal{A}^{\prime}$
called the _block map_ of memory $m$ and anticipation $n$ and a _code map_
$\psi=\Psi_{\infty}:\Sigma\to\Sigma^{\prime}$ such that
$\psi(\sigma)_{i}=\sigma^{\prime}_{i}=\Psi(\sigma_{i-m}\cdots\sigma_{i+n})$.

######  Definition 4.5.

[1] Let $\varphi$ be a continuous map from $X$ onto $Y$. Then
$\mathfrak{I}^{\prime}=(Y,\mathcal{G}=\\{g_{0},\,\ldots,\,g_{\ell^{\prime}}\\},\,\Sigma^{\prime})$
is a _factor_ of
$\mathfrak{I}=(X,\mathcal{F}=\\{f_{0},\,\ldots,\,f_{\ell}\\},\,\Sigma)$ if
$\Sigma^{\prime}$ is a factor of $\Sigma$ as above and for all $y\in Y$ and
$x\in\varphi^{-1}(y)$

$\varphi\circ
f_{\sigma_{i-m}\cdots\sigma_{i+n}}(x)=g_{\sigma_{i}^{\prime}}(y).$

######  Remark 4.6.

Let $({\bf S}_{\mathfrak{I}},\,{\bf F}_{\mathfrak{I}})$ be the conventional
dynamical system associated to $\mathfrak{I}$ as in §2. In that case, if
$\mathfrak{I}^{\prime}$ is a factor of $\mathfrak{I}$ given in Definition 4.5,
then $({\bf S}_{\mathfrak{I}^{\prime}},\,{\bf F}_{\mathfrak{I}^{\prime}})$ is
a factor of $({\bf S}_{\mathfrak{I}},\,{\bf F}_{\mathfrak{I}})$. Surely, there
are many other factors of $({\bf S}_{\mathfrak{I}},\,{\bf F}_{\mathfrak{I}})$
which do not look like this. Recall that we have this situation for subshifts:
many factors of a subshift are not necessarily subshifts.

######  Proposition 4.7.

Let $\mathfrak{I}$ and $x\in X$ be as in Lemma 4.4, but $\Sigma$ an
irreducible sofic. Then the conclusion of that lemma is valid.

###### Proof.

Let $G$ be the Fischer cover (minimal right resolving graph representing the
subshift) of $\Sigma$. Then, $G$ is a finite graph for our sofic shift and let
$\Sigma_{G}$ be the associated edge shift (the subshift arising from the
infinite walk on $G$ and recording labels after assigning different labels to
different edges). This $\Sigma_{G}$ is a 1-step SFT and an extension of
$\Sigma$. Thus by letting $\varphi:X\rightarrow X$ to be the identity map,
$\mathfrak{I}$ is a factor of
$\mathfrak{I}^{\prime}=(X,\,\mathcal{F},\,\Sigma_{G})$ and $x$ is a periodic
point of $\mathfrak{I}^{\prime}$ as well. Now apply Lemma 4.4 and recall that
periodic points are preserved by factor maps. ∎

Recall that SFT’s $\subsetneq$ sofics $\subsetneq$ synchronized systems.

######  Example 4.8.

The conclusion of Proposition 4.7 is not valid for the case when $\Sigma$ is a
non-sofic synchronized system.

###### Proof.

Let $\Sigma$ be the synchronized subshift whose cover is presented in Figure
1. In this cover, $m=m_{0}m_{1}\cdots$ is a fixed point of the Morse
substitution and thus $\Sigma_{m}=\overline{\mathcal{O}^{+}(m)}$ is a minimal
subshift [6]. Let us briefly remind the Morse substitution. Set

$\varrho(v)=\begin{cases}01,\quad\text{if }v=0,\\\ 10,\quad\text{if
}v=1,\end{cases}$

to be the substitution map which means for any word $u=u_{0}\cdots u_{k}$ in
$\\{0,\,1\\}^{\mathbb{N}}$, $\varrho(u)=\varrho(u_{0})\cdots\varrho(u_{k})$.
This gives a primitive substitution with two fixed points

$\displaystyle m$ $\displaystyle=0\mapsto 01\mapsto 0110\mapsto
01101001\mapsto\cdots$ $\displaystyle m^{\prime}$ $\displaystyle=1\mapsto
10\mapsto 1001\mapsto 10010110\mapsto\cdots$

Now, the closure of orbits of either of these fixed points under the shift map
gives a minimal subshift.

Let $\mathfrak{I}=(S^{1},\,\mathcal{F}=\\{f_{0},\,f_{1},\,f_{2}\\},\,\Sigma)$
where $f_{0}:S^{1}\rightarrow S^{1}$ is an irrational rotation,
$f_{1}=f_{0}^{-1}$ and $f_{2}$ another irrational rotation say
$f_{2}=f_{0}\circ f_{0}$. Since $\Sigma_{m}=\overline{\mathcal{O}^{+}(m)}$ is
minimal, the only periodic points of $\Sigma$ are the cycles presented in the
cover and in particular $m$ is not periodic; however $f_{m}(s)$, the orbit of
$s$ along $m$, is periodic of period two for all $s\in S^{1}$. In fact, if
$\sigma=\sigma_{0}\sigma_{1}\cdots\in\Sigma$ is any periodic point (or any
point with $\sigma_{i}=2$ for infinitely many $i$), then the orbit of any
$s\in S^{1}$ is transitive along $\sigma$.

$\ldots$$\ldots$$\ {}^{\ {}^{m_{0}}}$$\ {}^{\ {}^{m_{1}}}$$\ {}^{\
{}^{m_{2}}}$$\ {}^{\ {}^{2}}$$\ {}^{\ {}^{m_{3}}}$$\ {}^{\ {}^{m_{4}}}$$\
{}^{\ {}^{2}}$$\ {}^{\ {}^{m_{2n-1}}}$$\ {}^{\ {}^{m_{2n}}}$$\ {}^{\ {}^{2}}$
Figure 1: The cover for $\Sigma$. Here, $m=m_{0}m_{1}\cdots$ is a fixed point
of the Morse substitution.

∎

######  Proposition 4.9.

Let $\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)\;$be an IFS. If $x\in X$ is
periodic of period $p$ along a transitive $t=t_{1}t_{2}\cdots\in\Sigma$, then
for any $\sigma\in\Sigma$ there is a $1\leq k<p$ such that $x$ is periodic
along $\sigma^{\prime}=\tau^{k}(\sigma)$ where $\tau$ is our shift map.

###### Proof.

Let $\sigma=\sigma_{1}\sigma_{2}\cdots$ and for $n\in\mathbb{N}$, let
$\zeta_{n}=\sigma_{1}\sigma_{2}\cdots\sigma_{n}$. Write $t$ as in (10). If
$n>p$, then by the fact that for $\ell\in\mathbb{N}$, $f_{t_{1}\cdots t_{\ell
p}}(x)=x$, there is $1\leq i<p$ such that for some $\ell$

$\displaystyle t_{1}\cdots t_{\ell p}=$ $\displaystyle
t_{1}\cdots\zeta_{1}\cdots\zeta_{2}\cdots\sigma_{1}\sigma_{2}\cdots\sigma_{i},\quad\text{and}$
$\displaystyle f_{t_{1}\cdots t_{\ell p}}(x)=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!f_{t_{1}\cdots\zeta_{1}\cdots\zeta_{2}\cdots\sigma_{1}\sigma_{2}\cdots\sigma_{i}}(x)=x.$
(13)

The word $\sigma_{1}\sigma_{2}\cdots\sigma_{i}$ appearing in (4) is the
initial segment of $\zeta_{n}$ of length $i\;(<p<n)$. Then since we have
infinitely many $\zeta_{n}$, infinitely many of $i$’s in (4) must be identical
and denote that by $k$ and the corresponding $\zeta_{n}$ by $\zeta_{k_{i}}$.
Now along $\zeta_{k_{i}}$ as a subword of $t$ and for any $\ell\in\mathbb{N}$
where $k+\ell p\leq k_{i}$, we have $f_{\sigma_{k+1}\cdots\sigma_{k+\ell
p}}(x)=x$ and the conclusion follows by an induction argument and the fact
that $k_{i}\to\infty$. ∎

Example 4.8 is a case where the IFS has periodic points but not along any
transitive $t\in\Sigma$.

As an application of Proposition 4.9, consider
$\mathfrak{I}=(S^{1},\,\\{f_{0},\,f_{1}\\},\,\Sigma_{F})$ where $f_{0}$ is an
irrational rotation and $f_{1}(z)=z^{2}$ and $\Sigma_{F}$ the full shift and
hence sofic. Here, $\mathfrak{I}$ has a set of dense periodic points along
$\sigma=1^{\infty}\in\Sigma$ and no periodic points along
$0^{\infty}\in\Sigma$ and so no periodic points along any transitive
$t\in\Sigma$.

Let us end this note with a remark on considering the classical IFS as a
conventional dynamical system.

######  Remark 4.10.

As it was pointed out earlier in §2, one may consider an IFS as a conventional
dynamical system $({\bf S},\,{\bf F})$ where ${\bf F}$ is defined in (3). In
this situation, if our subshift is a full shift $\Sigma_{|\mathcal{F}|}$, then
dynamical properties such as transitivity, mixing, having shadowing for
$\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma_{|\mathcal{F}|})$ is characterized by
the respective properties for $({\bf S},\,{\bf F})$ and vice versa. In virtue
of Proposition 4.7, this claim is even true for a point $x$ being periodic.
However, from point of dynamical considerations, a general IFS
$\mathfrak{I}=(X,\,\mathcal{F},\,\Sigma)\;$and $({\bf S},\,{\bf F})$ are two
different entities.

## References

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# Multiple Photodetachment of Oxygen Anions via K-Shell Excitation and
Ionization:
Direct Double-Detachment Processes and Subsequent Deexcitation Cascades

S. Schippers<EMAIL_ADDRESS>I. Physikalisches
Institut, Justus-Liebig-Universität Gießen, Heinrich-Buff-Ring 16, 35392
Giessen, Germany  A. Hamann  I. Physikalisches Institut, Justus-Liebig-
Universität Gießen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany  A. Perry-
Sassmannshausen  I. Physikalisches Institut, Justus-Liebig-Universität Gießen,
Heinrich-Buff-Ring 16, 35392 Giessen, Germany  T. Buhr  I. Physikalisches
Institut, Justus-Liebig-Universität Gießen, Heinrich-Buff-Ring 16, 35392
Giessen, Germany  A. Müller  I. Physikalisches Institut, Justus-Liebig-
Universität Gießen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany  M. Martins
Institut für Experimentalphysik, Universität Hamburg, Luruper Chaussee 149,
22761 Hamburg, Germany  S. Reinwardt  Institut für Experimentalphysik,
Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany  F. Trinter
Institut für Kernphysik, Goethe-Universität Frankfurt am Main, Max-von-Laue-
Straße 1, 60438 Frankfurt am Main, Germany  Molecular Physics, Fritz-Haber-
Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
S. Fritzsche  Helmholtz-Institut Jena, Fröbelstieg 3, 07743 Jena, Germany
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-
Wien-Platz 1, 07743 Jena, Germany  GSI Helmholtzzentrum für
Schwerionenforschung, Planckstraße 1, 64291 Darmstadt, Germany

###### Abstract

Experimental cross sections for $m$-fold photodetachment ($m=2-5$) of oxygen
anions via $K$-shell excitation and ionization were measured in the photon-
energy range of 525–1500 eV using the photon-ion merged-beams technique at a
synchrotron light source. The measured cross sections exhibit clear signatures
of direct double detachment, including double $K$-hole creation. The shapes of
the double-detachment cross sections as a function of photon energy are in
accord with Pattard’s [J. Phys. B 35, L207 (2002)] empirical scaling law. We
have also followed the complex deexcitation cascades that evolve subsequently
to the initial double-detachment events by systematic large-scale cascade
calculations. The resulting theoretical product charge-state distributions are
in good agreement with the experimental findings.

##  I Introduction

Atomic anions are peculiar quantum systems since their extra electron is
(weakly) bound by short-range forces resulting from the induced polarization
of the atomic electron shells  [1, 2, 3]. In the
O-($1s^{2}\,2s^{2}\,2p^{5}\;{}^{2}P_{3/2}$) ground level, the binding energy
of the most loosely bound electron is only 1.461 eV [4], while the ionization
energy of neutral oxygen atoms amounts to 13.681 eV [5]. A fundamental process
that allows one to probe electron-electron correlation effects in atomic
anions is single photodetachment (SD), i.e., the removal of an electron by
absorption of a single photon. Recent studies on multiple photodetachment of
F- [6] and C- [7] have somewhat unexpectedly revealed that atomic anions are
also ideal systems for studying direct double detachment (DD), where one
photon simultaneously ejects two electrons (see also [8]). For F-, the cross
section for simultaneous ejection of a $1s$ and a $2p$ electron could be
extracted from the measured data for photon energies that range from below the
threshold for direct double detachment of a $1s$ and a $2p$ electron
($1s$+$2p$ DD) to well beyond the cross-section maximum. This result has
already stimulated a successful effort to theoretically describe the cross
section for this many-particle effect with two electrons ending up in the
continuum [9]. For C-, the cross section for double-core hole formation by a
single photon ($1s$+$1s$ DD) was measured. Surprisingly, this cross section
was about an order of magnitude larger than what was expected on the basis of
previous results on the double core-hole creation in carbon-containing
molecules [10].

In order to shed more light on the role of DD processes in the inner-shell
photodetachment of atomic anions, the present study extends the above
mentioned previous work to the O- anion. Double detachment, or double
ionization for neutral atoms and positive ions, has been repeatedly studied in
the literature, mostly for helium-like systems. Since a historical account has
already been given in Ref. [6] we shall mention here only the work that is
most closely related to the present investigation, i.e., studies on DD of H-
[11], He- [12], and K- [13] where the corresponding cross sections were
scanned over very narrow photon-energy ranges of a few meV in the vicinity of
the respective thresholds. Earlier studies on net double and triple inner-
shell detachment of O- ions also considered only quite limited photon-energy
ranges and focussed on the $1s\,2s^{2}\,2p^{6}$ resonance at 525.6 eV that is
associated with $1s\to 2p$ photoexcitation [14, 15].

When compared with the previous measurements, the present photon-energy range
of 525–1500 eV is much wider and comprises the thresholds for direct single
detachment of a $1s$ electron ($1s$ SD) at 529.6 eV [15] as well as the
thresholds for $1s$+$2p$ DD, $1s$+$2s$ DD, and $1s$+$1s$ DD. The threshold for
the latter process occurs at about 1100 eV (see below). Our experimental cross
sections $\sigma_{m}$ for $m$-fold photodetachment of O- ions with $m=2,3,4,5$
exhibit signatures of all these processes. $m$-fold photoionization results in
the production of multiply positively charged O(m-1)+ ions and can be
represented as

$h\nu+\textrm{O}^{-}\to\textrm{O}^{(m-1)+}+m\,e^{-}.$ (1)

In addition to experimental cross sections for the production of O(m-1)+ ions,
we present theoretical results for O- photoabsorption. Using a recently
developed theoretical toolbox [16, 17], we model the complex deexcitation
cascades that follow the creation of single or double inner-shell holes and
thus give rise to photon-energy-dependent distributions of product-ion charge
states.

##  II Experiment

The measurements were carried out using the photon-ion merged-beams technique
[18] at the PIPE facility [19, 20, 21], which is a permanently installed end-
station at the photon beamline P04 [22] of the synchrotron radiation source
PETRA III operated by DESY in Hamburg, Germany. Oxygen anions were produced by
a Cs-sputter ion source [23] with a sputter target made of solid aluminum-
oxide and a sputter potential of about 2 kV. After acceleration to a kinetic
energy of 6 keV, the ions were passed through an analyzing dipole magnet which
was adjusted such that 16O- ions were selected for further transport to the
photon-ion merged-beams interaction region. The ion current in the interaction
region was up to 350 nA (18 nA) with the ion beam being uncollimated
(collimated to about $2\times 2$ mm2). These values are an order of magnitude
larger than what could be achieved in our previous study where a different
type of ion source was used [15].

In the interaction region, where the residual-gas pressure was in the mid
$10^{-10}$ mbar range, the ion beam was coaxially merged with the counter-
propagating soft x-ray photon beam over a distance of about 1.7 m. O(m-1)+
ions as obtained from multiple photodetachment (Eq. 1) were separated from the
primary ion beam by a second dipole magnet. Inside this magnet, a Faraday cup
collected the primary ion beam, while the charge-selected product ions were
directed to the detector chamber. Along their flight path, they first passed
through a spherical 180-degree out-of-plane deflector to suppress background
from stray electrons, photons, and ions before the ions then entered a single-
particle detector with nearly 100% detection efficiency [24].

The dark-count rate of our detector amounted to $\sim$0.02 Hz (see also [25]).
In each product-ion channel, this contributed to a photon-energy-independent
background count rate. Other contributions to the background count rate arose
from collisions of the primary ions with residual-gas particles, where
electrons can be lost from the projectiles by stripping reactions. This
contribution to the energy-independent background was particularly noticeable
in the double-detachment channel, but practically negligible for triple,
fourfold, and fivefold detachment. For double detachment, the background was
determined by recording O+ product ions while alternatingly switching the
photon beam on and off by opening and closing a fast beam shutter in the
photon beamline.

Relative cross sections for $m$-fold photodetachment were obtained by
normalizing the background-subtracted count rates of O(m-1)+ product ions on
the primary O- ion current and on the photon flux measured with a calibrated
photodiode. The energy-dependent photon flux reached its peak of $3\times
10^{13}$ s-1 at photon energies around 850 eV and at a photon-energy spread of
$\sim$1 eV. This spread also depended on the photon energy ranging from
$\sim$0.4 eV at 500 eV up to $\sim$2.5 eV at 1500 eV for a nominal width of
the monochromator exit slit of 1000 $\mu$m. The relative cross sections were
put on an absolute scale by using our previously measured absolute cross
sections for single and double detachment of O- as reference values [15]. The
systematic uncertainty of the experimental absolute cross sections is
estimated to be $\pm$15% at 90% confidence level [19].

The photon-energy scale was calibrated by absorption measurements in nitrogen
gas using the lowest vibrational component of the $1s\to\pi^{*}$ resonance at
400.86(3) eV (see discussion by Müller et al. [26] and references therein) as
a calibration point. A second (intrinsic) calibration point was provided by
the O-($1s\to 2p$) resonance at 525.6(1) eV [15]. In addition, the Doppler
shift that is associated with the ion-beam velocity was taken into account.
It amounted to 0.467 eV at 520 eV and to 1.346 eV at 1500 eV. The remaining
uncertainty of the calibrated photon-energy scale is estimated to be $\pm$0.2
eV for photon energies of less than 600 eV. Since there are no calibration
points at higher energies, the uncertainty increases with increasing photon
energy. At a photon energy of 1500 eV, it is estimated to amount to about
$\pm$2 eV.

##  III Computations

Table 1: Computed threshold energies for single detachment (SD) and double detachment (DD) of O-. The energy ranges comprise all fine-structure levels that belong to the specified configurations.  Configuration | Process | Energy (eV)
---|---|---
$1s^{1}\,2s^{2}\,2p^{5}$ |  $1s$ SD | 529.8–532.5
$1s^{1}\,2s^{2}\,2p^{4}$ |  $1s$+$2p$ DD | 545.2–553.8
$1s^{1}\,2s^{1}\,2p^{5}$ |  $1s$+$2s$ DD | 561.9–574.0
$2s^{2}\,2p^{5}$ |  $1s+1s$ DD | 1160.2
Table 2: Simplified scheme of the cascade that sets in subsequent to $1s$ SD of O-($1s^{2}\,2s^{2}\,2p^{5}$). The scheme contains all energetically allowed single-step autoionizing transitions and dipole-allowed radiative transitions. Right arrows and down-right arrows denote Auger transitions from the configuration to the left and radiative transitions from a configuration above, respectively. The ground configurations of the different product ions, which are usually reached via multiple pathways, are printed in bold face. | O | O+ | O2+
---|---|---|---
| $1s^{1}\,2s^{2}\,2p^{5}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{3}}$ |  |  |
|  |  |  |  $\to$ $1s^{2}\,2s^{1}\,2p^{4}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$
|  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{3}}$ |  |  |
|  |  |  |  $\to$ $1s^{2}\,2s^{0}\,2p^{5}$ |  $\to$ $1s^{2}\,2s^{1}\,2p^{3}$
|  |  |  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$
|  |  |  |  |  $\Lsh$ $1s^{2}\,2s^{1}\,2p^{4}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$
|  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{3}}$ |  |  |
|  |  $\Lsh$ $1s^{2}\,2s^{2}\,2p^{4}$ |  |  |  |  |  |
|  |  $\Lsh$ $1s^{2}\,2s^{1}\,2p^{5}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{3}}$ |  |  |
|  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{4}}$ |  |  |  |  |  |
Table 3: Simplified scheme of the cascade that sets in subsequent to $1s$+$2p$ DD of O-($1s^{2}\,2s^{2}\,2p^{5}$). The scheme contains all energetically allowed single-step autoionizing transitions and dipole-allowed radiative transitions. See Tab. 2 for further details. | O+ | O2+ | O3+
---|---|---|---
| $1s^{1}\,2s^{2}\,2p^{4}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$ |  |  |
|  |  |  |  $\to$ $1s^{2}\,2s^{1}\,2p^{3}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{1}}$
|  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$ |  |  |
|  |  |  |  $\to$ $1s^{2}\,2s^{0}\,2p^{4}$ |  $\to$ $1s^{2}\,2s^{1}\,2p^{2}$
|  |  |  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{1}}$
|  |  |  |  |  $\Lsh$ $1s^{2}\,2s^{1}\,2p^{3}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{1}}$
|  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$ |  |  |
|  |  $\Lsh$ $1s^{2}\,2s^{2}\,2p^{3}$ |  |  |  |  |  |
|  |  $\Lsh$ $1s^{2}\,2s^{1}\,2p^{4}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$ |  |  |
|  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{3}}$ |  |  |  |  |  |
Table 4: Simplified scheme of the cascade that sets in subsequent to $1s$+$2s$ DD of O-($1s^{2}\,2s^{2}\,2p^{5}$). The scheme contains all energetically allowed single-step autoionizing transitions and dipole-allowed radiative transitions. See Tab. 2 for further details. In principle, the O3+($1s^{2}\,2s^{1}\,2p^{2}$) $\to$ O4+($1s^{2}\,2s^{2}$) and O3+($1s^{2}\,2s^{0}\,2p^{3}$) $\to$ O4+($1s^{2}\,2s\,2p$) transitions could be conceived, but these are energetically not possible. | O+ | O2+ | O3+ |
---|---|---|---|---
| $1s^{1}\,2s^{1}\,2p^{5}$ |  $\to$ $1s^{1}\,2s^{2}\,2p^{3}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{1}}$ |
|  |  |  |  |  |  |  |  $\to$ $1s^{2}\,2s^{1}\,2p^{2}$ |
|  |  |  |  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{1}}$ |
|  |  |  |  |  |  |  |  $\to$ $1s^{2}\,2s^{0}\,2p^{3}$ |
|  |  |  |  |  |  |  |  |  $\Lsh$ $1s^{2}\,2s^{1}\,2p^{2}$ |
|  |  |  |  |  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{1}}$ |
|  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$ |  |  |  |
|  |  |  |  |  $\to$ $1s^{2}\,2s^{1}\,2p^{3}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{1}}$ |
|  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$ |  |  |  |
|  |  |  |  |  $\to$ $1s^{2}\,2s^{0}\,2p^{4}$ |  $\to$ $1s^{2}\,2s^{1}\,2p^{2}$ |
|  |  |  |  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{1}}$ |
|  |  |  |  |  |  $\Lsh$ $1s^{2}\,2s^{1}\,2p^{3}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{1}}$ |
|  |  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$ |  |  |  |
|  |  $\Lsh$ $1s^{1}\,2s^{2}\,2p^{4}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$ |  |  |  |
|  |  |  |  |  $\to$ $1s^{2}\,2s^{1}\,2p^{3}$ |  |  |  |
|  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$ |  |  |  |
|  |  |  |  |  $\to$ $1s^{2}\,2s^{0}\,2p^{4}$ |  $\to$ $1s^{2}\,2s^{1}\,2p^{2}$ |
|  |  |  |  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{1}}$ |
|  |  |  |  |  |  $\Lsh$ $1s^{2}\,2s^{1}\,2p^{3}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{1}}$ |
|  |  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$ |  |  |  |
|  |  $\Lsh$ $1s^{2}\,2s^{1}\,2p^{4}$ |  $\to$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$ |  |  |  |
|  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{3}}$ |  |  |  |  |  |  |
|  |  $\Lsh$ $1s^{2}\,2s^{0}\,2p^{5}$ |  $\to$ $1s^{2}\,2s^{1}\,2p^{3}$ |  |  |  |
|  |  |  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{2}}$ |  |  |  |
|  |  |  $\Lsh$ $1s^{2}\,2s^{1}\,2p^{4}$ |  |  |  |  |  |  |
|  |  |  |  $\Lsh$ $\boldsymbol{1s^{2}\,2s^{2}\,2p^{3}}$ |  |  |  |  |  |  |

Extended energy-level and transition-rate calculations need to be performed to
model the successive (photo-) detachment from negative ions. For inner-shell-
excited atoms and ions, especially the multi-configuration Dirac-Hartree-Fock
(MCDHF) method [27, 28] has been found to be a versatile tool to model the –
radiative and non-radiative – decay and to describe the interplay of different
atomic processes in course of their relaxation. In this work, all calculations
have been performed by means of the Jac toolbox, the Jena Atomic Calculator
[16], which has been expanded recently to follow rather long ionization
pathways. Not much needs to be said about how such cascade computations are
implemented in practice [17, 29], and we just summarize a few major steps that
have been carried out. The numerical results of these simulations will then be
discussed together with the measurements below.

The photoexcitation and ionization of an electron requires first of all
insight into the $1s^{-1}$ and $(1s\,n\ell)^{-2}$ threshold energies and the
associated cross sections. This refers to both the excitation and decay of the
core-excited $1s\,2s^{2}\,2p^{6}\;{}^{2}S_{1/2}$ level of the O- ion and the
core-detached O-, that is $1s\,2s^{2}\,2p^{5}\;{}^{1,3}P_{J}$ core-excited
levels of the neutral atom, as well as of various doubly core-excited
configurations. In particular, the $1s\,2s^{2}\,2p^{4}\,3p$,
$1s\,2s^{2}\,2p^{4}$, and $1s\,2s\,2p^{5}$ configurations play a prominent
role  in the cascade decay of $1s$ core-detached neutral oxygen and have been
included in the computations. The probabilities for different shake
transitions have been determined from the orbital overlap of the
$1s^{2}\,2s^{2}\,2p^{5}$ ground configuration and the
$1s2s^{2}2p^{5}+1s\,2s^{2}\,2p^{4}\,(3p+4p)$ configurations, and show that up
to 10% of the $2p$ electrons are shaken to orbitals with higher $n$ or even
into the continuum. A similar shake probability is expected for the $2s$
electrons, although their contributions to the photodetachment cross sections
are more difficult to assess. Below, we make use of these shake probabilities
for estimating the ion distributions at photon energies
$\hbar\,\omega\>\gtrsim\>450$ eV.

Apart from the excitation energies and absorption strengths for creating a
$1s$ inner-shell hole, the main computational focus was placed upon the
stepwise relaxation and, hence, the (relative) ion distributions that can be
directly compared with the experiment. This stepwise relaxation can be
formally described by an atomic cascade, that connects ions of different
charge states to each other via different (decay) processes. This relaxation
proceeds until a given number of electrons is released and/or the ions cannot
further decay to any lower level. Such cascades therefore require to
automatically determine all single configurations that may energetically occur
due to various photoemission and autoionization processes from the initially
chosen hole configurations/levels, and to include a proper number of shake
configurations. To support a detailed analysis of different models, we
therefore distinguish in Jac between (so-called) cascade computations and
simulations, cf. Ref. [17].

In practice, any computation starts from setting up a cascade tree, i.e., the
list of configurations (Tabs. 2–4) and associated levels that likely
contribute to the relaxation of the ions. These levels are then divided into
(blocks of) multiplets from which the various decay pathways can be readily
derived. A pathway hereby refers to a sequence of two or more levels, which
can be subsequently occupied in the course of the relaxation and whose number
rapidly increases with (i) the number of open shells involved for an ion and
(ii) the depth of the cascade. The major computational effort refers, however,
to the representation of the fine-structure levels for each intermediate step
(block) of the cascade as well as to the computation of the transition
amplitudes.

To establish a hierarchy of useful cascade models, different approaches are
distinguished in Jac in order to support and analyze a systematically improved
representation of the fine structure and amplitudes. Unlike for Si- ions [30],
the rather simple shell structure of O- with just a $1s^{2}\,2s^{2}\,2p^{5}$
ground configuration enables us to apply a self-consistent treatment of all
blocks of the cascades (approach B), in addition to just an averaged single-
configuration approach (approach A). Indeed, approach A, as the simplest,
neglects all configuration mixing between the bound-state levels and also
restricts the computations to just a single set of continuum orbitals for each
step of the cascade [31]. In contrast, approach B applies an individual self-
consistent field and set of orbitals for all blocks of the cascade, while each
electron configuration in the decay tree still forms a single multiplet with a
well-defined fine structure (configuration mixing) by diagonalizing the
Hamiltonian matrix of this configuration. This single-configuration approach
is expected to already provide a quite reasonable description of the strongest
decay paths. For analyzing the ion distribution below, we calculated the
stepwise decay of the three initial configurations $1s\,2s^{2}\,2p^{5}$ (Tab.
2), $1s\,2s^{2}\,2p^{4}$ (Tab. 3), and $1s\,2s\,2p^{5}$ (Tab. 4) and for the
release of up to five electrons. These ion distributions are then combined
with the shake probabilities to estimate the (relative) $m$-fold
photodetachment cross sections and for comparison with the experiment. The
results from these simulations are shown and discussed below in Sec. IV.
Despite the improved modeling with approach B, the quite limited
representation of the fine-structure levels and the need for including
different shake transitions are likely the main reason for the remaining
deviations of the predicted ion distributions from the measurements. In
addition to the cascades that follow the creation of one $1s$ hole, we also
treated in a similar manner the cascades that result from double core-hole
creation. In these calculations, all together 23 configurations with 3–7
electrons and 83 cascade steps were considered, a tabulation of which similar
to Tabs. 2–4 would exceed one page.

##  IV Results and Discussion

Figure 1: Experimental cross sections for double (a), triple (b), and
quadruple (c) photodetachment of O- ions. The open symbols represent fine
scans over a narrower energy range and the more widely spaced full symbols
represent coarse scans that extend to higher energies of up to 1500 eV (see
Fig. 3). The vertical dash-dotted lines at 529.6, 543.9, and 572.5 eV mark the
thresholds for $1s$ SD, $1s$+$2p$ DD, and $1s$+$2s$ DD respectively. The full
lines result from a simultaneous fit of empirical cross section formulae for
these three ionization processes to the three experimental  fine-scan data
sets (see text) with the individual contributions being represented by the
dashed lines as explained by the legend in panel (a).

Figure 1 displays the measured cross sections $\sigma_{2}$, $\sigma_{3}$, and
$\sigma_{4}$ for double, triple, and fourfold detachment of O-. The
$1s\,2s^{2}\,2p^{6}\;{}^{2}S_{1/2}$ resonance at 525.6 eV can be discerned in
each of the three cross sections. Its resonance parameters were already
determined from our previously measured high-resolution data [15]. We used
this previous analysis for determining the experimental photon-energy spread
$\Delta E$ by fitting a Voigt profile to the resonance. In the fit, the
Gaussian full width at half maximum (FWHM), $\Delta E$, was treated as a free
parameter. The fit gave rise to a spread of $\Delta E\approx 0.4$ eV.

In our previous work on multiple photodetachment of C- ions [7], we discovered
a number of previously unknown resonances, in particular, for the triple-
detachment cross section. Within the limits of the present statistical
uncertainties, none of the cross sections displayed in Fig. 1 shows any signs
of further resonance features in addition to the
$1s\,2s^{2}\,2p^{6}\;{}^{2}S_{1/2}$ resonance at 525.6 eV. In the search for
additional resonances, we have also performed scans of $\sigma_{3}$ with lower
photon-energy spread and higher point density, but these did not result in any
further resonances. We therefore conclude that the O- anion does not feature
any strong excitation channels to more highly excited autoionizing levels
beyond the $1s\,2s^{2}\,2p^{6}\;{}^{2}S_{1/2}$ level.

In our earlier O- experiment [15], the photon-energy range extended only to a
few eV above the threshold for $1s$ SD at 529.6 eV. The new measurements
extend to much higher energies where the cross sections $\sigma_{3}$ [Fig.
1(b)] and $\sigma_{4}$ [Fig. 1(c)] exhibit additional thresholds at 543.9 and
572.5 eV. According to our atomic-structure calculations (Tab. 1), these
thresholds, which are not discernible in $\sigma_{2}$ [Fig. 1(a)], correspond
to the simultaneous removal of two electrons by one photon, i.e, to direct
$1s$+$2p$ DD and to $1s$+$2s$ DD, respectively.

The cross section $\sigma_{2}$ for net double detachment does not exhibit any
significant DD features, since the Auger processes, that follow the initial
double core-hole creation, further increase the charge state of the
intermediate O+ ion (created by DD of O-) with a probability of almost 100%.
Therefore, above the threshold for $1s$ SD at 529.6 eV, the cross section
$\sigma_{2}$ is practically exclusively due to this single-detachment process.
For the purpose of curve fitting, we have parameterized the cross section for
$1s$ SD following a prescription of Verner et al. [32]. We have determined the
according parameter values by a fit to $\sigma_{2}$ 111Parameters for
representing the O- $1s$ SD cross section according to Eqs. 1 and 2 of Ref.
[32]: $E_{\mathrm{th}}=529.6$ eV, $l=0$, $E_{0}=188.525$ eV,
$\sigma_{0}=16.05$ kb, $y_{a}=400$, $P=-0.71946$, $y_{w}=94.7123$. The
resulting curve $\sigma^{(SD)}_{1s}(E)$ is displayed as the short-dashed line
in Fig. 1(a). Above the threshold, the fit matches the experimental data. The
agreement is however less convincing at the threshold. This can be attributed
to the fact that the Verner formula, which was originally designed for
photoionization cross sections of positively charged ions [32], does not
account for the more gently rising photodetachment thresholds of negative
ions. Nevertheless, the fit provides a realistic extrapolation of the 1s SD
cross section towards higher energies as required below.

Table 5: Parameter values for expressing the apparent DD cross sections $\tilde{\sigma}_{m}^{(DD)}=F_{m}\sigma^{(DD)}(E)$ with $\tilde{\sigma}_{m}^{(\mathrm{max})}=F_{m}\sigma^{(\mathrm{max})}$ via Eq. 2 with $\alpha=1.1269$ as resulting from the fits discussed in the text. The specific contributions of the $1s$ SD cross section to $\sigma_{2}$, $\sigma_{3}$, and $\sigma_{4}$ are $F_{2}^{(1s)}=0.8223(47)$, $F_{3}^{(1s)}=0.1723(26)$, and $F_{4}^{(1s)}=0.00540(22)$. Numbers in parentheses denote one-sigma uncertainties obtained from the fit. The values for $1s$+$1s$ DD bear additional unknown uncertainties associated with “background” subtraction (see text). Process |  $E^{(\mathrm{th})}$ (eV) |  $E^{(\mathrm{max})}$ (eV) |  $\tilde{\sigma}_{3}^{(\mathrm{max})}$ (kb) |  $\tilde{\sigma}_{4}^{(\mathrm{max})}$ (kb) |  $\tilde{\sigma}_{5}^{(\mathrm{max})}$ (kb)
---|---|---|---|---|---
$1s$+$2p$ DD | 543.92(29) | 575.0(10) | 58.4(13) | 3.25(14) |
$1s$+$2s$ DD | 572.51(59) | 680.3(26) | 28.1(38) | 12.2(02) |
$1s$+$1s$ DD | 1010(46) | 1627(324) |  |  | 0.081(17)

The $1s$ SD process also contributes to the cross sections for net triple and
fourfold detachment displayed in Figs. 1(b) and 1(c), respectively. However,
above the thresholds for $1s$+$2p$ DD and $1s$+$2s$ DD, these latter processes
dominate the cross sections $\sigma_{3}$ and $\sigma_{4}$. The cross section
for net fourfold detachment rises from 5 kb at the $1s$+$2s$ DD threshold to a
maximum value of 14 kb at about 650 eV. The corresponding rise is barely
visible in the cross section for triple detachment, which nevertheless
exhibits a strong contribution by $1s$+$2p$ DD leading to a cross-section rise
from $\sim$70 kb at the $1s$+$2p$ DD threshold to $\sim$105 kb at the
$1s$+$2s$ DD threshold.

As demonstrated earlier [6, 7, 34], the DD contributions to the measured cross
sections can be represented as functions of the photon energy $E$ by a semi-
empirical formula, that has been devised by Pattard [35]:

$\sigma^{(DD)}(E)=\sigma^{\mathrm{(max)}}x^{\alpha}\left(\frac{\alpha+7/2}{x\alpha+7/2}\right)^{\alpha+7/2},$
(2)

where $x=({E-E^{(\mathrm{th})}})/({E^{(\mathrm{max})}-E^{(\mathrm{th})}})$,
$E^{(\mathrm{th})}$ is the threshold energy and  $\alpha=1.1269$ is the
Wannier exponent. Its numerical value [36] is the appropriate one for the
charge of the doubly detached intermediate O+ ion, which results from direct
double detachment of O-. Furthermore in Eq. 2, $\sigma^{(\mathrm{max})}$ is
the cross-section maximum, which occurs at the energy $E^{(\mathrm{max})}$.
Numerical values for these parameters were obtained by fitting

$\displaystyle\sigma_{m}^{(\mathrm{fit})}(E)$ $\displaystyle=$ $\displaystyle
F_{m}^{(1s)}\sigma^{(SD)}_{1s}(E)+$ $\displaystyle
F_{m}^{(1s+2p)}\sigma^{(DD)}_{1s+2p}(E)+F_{m}^{(1s+2s)}\sigma^{(DD)}_{1s+2s}(E)$

simultaneously to the three experimental cross sections $\sigma_{m}$ shown in
Fig. 1. The energy-independent factor $F_{m}^{(1s)}$ adjusts the relative
contribution of the $1s$ SD cross section $\sigma^{(SD)}_{1s}(E)$ as
appropriate for each $\sigma_{m}$. The factors $F_{m}^{(1s+2p)}$ and
$F_{m}^{(1s+2s)}$ have the same role for the $1s+2p$ and $1s+2s$ DD processes,
respectively. However, the fit does not allow one to disentangle these weight
factors from the corresponding DD cross sections. Therefore, Tab. 5, which
tabulates the fit results, provides parameters for the _apparent_ DD cross
sections $\tilde{\sigma}_{m}^{(DD)}(E)=F_{m}\sigma^{(DD)}(E)$ with their
cross-section maxima
$\tilde{\sigma}_{m}^{(\mathrm{max})}=F_{m}\sigma^{(\mathrm{max})}$ being
specific for each $m$-fold detachment channel (cf. Eq. 1).

The $1s$+$2p$ DD and $1s$+$2s$ DD threshold energies that result from the fit
(Tab. 5) have a combined fit and systematic uncertainty of less than 1 eV.
Within this uncertainty, they agree with the theoretical predictions (Tab. 1).
All cross-section maxima decrease when going from triple to fourfold
detachment. For $1s$ SD, $1s$+$2p$ DD, and $1s$+$2s$ DD, they are reduced by
factors of 0.031, 0.056, and 0.43, respectively, i.e., the relative importance
of $1s$+$2s$ DD is higher for fourfold than for triple detachment. For triple
detachment, the cross-section maximum of $1s$+$2p$ DD is twice as large as the
one of $1s$+$2s$ DD, while for fourfold detachment the latter is nearly a
factor of four larger than the former. The higher relative yield of O3+ ions
from the $1s$+$2s$ DD process as compared to $1s$+$2p$ DD can be understood
already qualitatively by comparing the respective cascade trees in Tabs. 4 and
3. In particular, the chain of the autoionizing transitions
O+($1s^{1}\,2s^{1}\,2p^{5}$) $\to$ O2+($1s^{1}\,2s^{2}\,2p^{3}$) $\to$
O3+($1s^{2}\,2s^{2}\,2p^{1}$), which starts with a fast Super-Coster-Kronig
process, is characteristic for $1s$+$2s$ DD (Tab. 4).

Figure 2: Experimental (full circles) and theoretical (open diamonds) product-ion charge fractions $f_{q}$ (Eq. 4). The photon-energy ($E_{\mathrm{ph}}$) axis is compressed towards higher energies using $x=\log(E_{\mathrm{ph}}/\mathrm{eV}-400)$ for the abscissa. Table 6: Theoretical product-ion charge-state distributions resulting from the present cascade calculations for $1s$ SD ($F_{m}^{(1s)}$), $1s$+$2p$ DD ($F_{m}^{(1s+2p)}$), $1s$+$2s$ DD ($F_{m}^{(1s+2s)}$), and $1s$+$1s$ DD ($F_{m}^{(1s+1s)}$). The quantity $q$ denotes the product-ion charge state which equals $m-1$ (Eq. 1). $q$ | $m$ | $F_{m}^{(1s)}$ | $F_{m}^{(1s+2p)}$ | $F_{m}^{(1s+2s)}$ | $F_{m}^{(1s+1s)}$
---|---|---|---|---|---
0 | 1 | 0.011 | 0.0 | 0.0 | 0.0
1 | 2 | 0.861 | 0.029 | 0.069 | 0.001
2 | 3 | 0.128 | 0.786 | 0.478 | 0.069
3 | 4 | 0.0 | 0.185 | 0.413 | 0.120
4 | 5 | 0.0 | 0.0 | 0.040 | 0.810
5 | 6 | 0.0 | 0.0 | 0.0 | 0.0

Our fine-structure-resolved cascade computations, which in addition to single-
step autoionizing transitions also consider competing radiative transitions,
can be compared quantitatively with the experimental findings. Figure 2
compares experimental and theoretical product-ion charge fractions

$f_{q}=\frac{\sigma_{q+1}}{\sigma_{\Sigma}},$ (4)

where $q=m-1$ is the product-ion charge state resulting from net $m$-fold
detachment and

$\sigma_{\Sigma}=\sum_{m=2}^{5}\sigma_{m}=\sum_{q=1}^{4}\sigma_{q+1}.$ (5)

Theoretical $f_{q}$ values have been computed for the photon energies 520,
560, 650, and 1300 eV (Fig. 2), which are above the thresholds for $1s$ SD,
$1s$+$2p$ DD, $1s$+$2s$ DD, and $1s$+$1s$ DD, respectively. In these
computations, the $F_{m}$ values which result from our cascade calculations
and which are provided in Tab. 6 have been weighted with the relative cross
sections for the individual SD and DD processes as discussed in the following.

At 520 eV, a $1s$ hole can be created only via $1s$ SD. Accordingly,
$f_{q}(520\mathrm{~{}eV})=F_{m}^{(1s)}$. The subsequent single-step cascade
tree produces O2+ as the highest-charged product ion (Tab. 2). The theoretical
O+ and O2+ fractions of 86% and 13% (Tab. 6) agree well with the corresponding
experimental $F_{m}^{(1s)}$ values (caption of Tab. 5). A fraction of 1% is
theoretically predicted for neutral oxygen, which cannot be observed in the
present experimental configuration. The small experimentally observed O3+
fraction can only be explained if higher-order processes are taken into
account such as multiple Auger processes (see, e.g., [37, 25]). Such processes
were, however, not considered in the present cascade calculations.

At 560 eV, the $1s$+$2p$ DD process is energetically allowed in addition to
$1s$ SD. Now the maximum calculated charge state, that can be reached by
single-step deexcitation cascades, is 3+ (Tab. 3). In the absence of a
rigorous theoretical treatment of direct double-detachment processes, we just
assumed in our calculations that the $1s$+$2p$ DD process contributes by 10%
to the total detachment cross section, which is of the same order of magnitude
as the shake probabilities mentioned in Sec. III. Accordingly,
$f_{q}(560\mathrm{~{}eV})=0.9F_{m}^{(1s)}+0.1F_{m}^{(1s+2p)}$. The resulting
fractions of net double, triple, and fourfold detachment agree surprisingly
well with the experimental findings considering the coarseness of this
approach. The same holds at 650 eV, which is above the threshold for $1s$+$2s$
DD and where our cascade calculations also predict a nonzero contribution by
net fivefold detachment. In analogy to the $1s$+$2p$ DD process, $1s$+$2s$ DD
also received a weight of 10% such that
$f_{q}(650\mathrm{~{}eV})=0.8F_{m}^{(1s)}+0.1F_{m}^{(1s+2p)}+0.1F_{m}^{(1s+2s)}$.
At 1300 eV, i.e., above the threshold for $1s$+1$s$ DD, a 1% contribution from
this latter process has been assumed in addition and, correspondingly,
$f_{q}(1300\mathrm{~{}eV})=0.79F_{m}^{(1s)}+0.1F_{m}^{(1s+2p)}+0.1F_{m}^{(1s+2s)}+0.01F_{m}^{(1s+1s)}$.

Figure 3: Results from coarse photon-energy scans over a wide range. The
differently colored data points represent the experimental cross sections
$\sigma_{m}$ for multiple ($m$-fold, Eq. 1) detachment of O- ions for $m$=2
(olive), $m$=3 (blue), $m$=4 (magenta), and $m$=5 (orange). The error bars
account only for the statistical experimental uncertainties. The full lines
represent a power-law with the exponent $-3.3\pm 0.1$ as obtained from a fit
to $\sigma_{3}$ (see text).

In our measurements, we scrutinized the weak O4+ product-ion channel in the
search for a signature of double $K$-hole formation via the $1s$+$1s$ DD
process. According to our calculations, the threshold for direct double
$K$-shell detachment occurs at 1160 eV (Tab. 1). Around this energy, the cross
section $\sigma_{5}$ exhibits a noticeable rise which is not present in
$\sigma_{3}$ and $\sigma_{4}$ (Fig. 3). These latter two cross sections show
identical high-energy behaviors, which can be described by a power law with an
exponent of $-3.3$. This number was obtained from a fit to $\sigma_{3}$, and
the resulting fit curve was then scaled to $\sigma_{4}$ and $\sigma_{5}$ by
multiplication with appropriate factors (full lines in Fig. 3).

Figure 4: Cross-section difference resulting from the subtraction of the
scaled power-law fit from the measured cross section $\sigma_{5}$ (see Fig.
3). The full line results from a fit of the Pattard formula (Eq. 2) to the
data points. The fit results are listed in Tab. 5.

The data points in Fig. 4 were obtained by subtracting the appropriately
scaled power law from the measured cross section $\sigma_{5}$. This cross-
section difference $\Delta\sigma_{5}$ is zero up to about 1000 eV from where
about it rises towards higher energies. A fit of Eq. 2 to the data points
yields a threshold value of $1010\pm 46$ eV (Tab. 5) which is somewhat lower
than our calculated value. Conceding an additional (unknown) uncertainty
related to the subtraction of the power-law curve, it seems nevertheless
plausible that the cross-section difference depicted in Fig. 4 is indeed due
to $1s$+$1s$ DD.

The fit of the Pattard formula (Eq. 2) yields a maximum value of the $1s$+$1s$
DD cross section of $81\pm 17$ b. This is significantly smaller than the
maximum cross section of 3 kb which we have found earlier for fivefold
detachment of C- [7]. This can be (at least partly) attributed to the general
strong decrease of double-photoionization cross sections with increasing
nuclear charge [38]. Moreover, it cannot be excluded that $1s$+$1s$ DD also
contributes to the production of final ion charge states other than O4+. Our
cascade calculations result in a product charge-state distribution that
consists of 7% O2+, 12% O3+, 81% O4+, and 0% O5+ ions ($F_{m}^{(1s+1s)}$ in
Tab. 6). At the present level of statistical uncertainty, a $\sim$10%
contribution of $1s$+$1s$ DD to $\sigma_{4}$ can remain unnoticed considering
the scatter of the experimental $\sigma_{4}$ data points above 1000 eV (Fig.
3). We also scrutinized the O5+ channel, but the count rate was too low for a
measurement of $\sigma_{6}$ with satisfying statistical uncertainty in a
reasonable amount of time.

It should be noted that the measurement of the small cross section
$\sigma_{5}$, which is in the sub-kilobarn range, is particularly challenging.
In addition, at photon energies above 1000 eV, the photon flux at the PETRA
III photon beamline P04 decreases rapidly as the photon energy increases, so
that it becomes difficult to measure small cross sections with sufficient
statistical accuracy when going to higher energies. In the present experiment,
the lowest O4+ count rate was only 0.2 Hz. Nevertheless, this is still an
order of magnitude larger than the dark-count rate ($\sim$0.02 Hz, see Sec.
II) of our single-particle detector. This rules out that the rise of
$\sigma_{5}$ beyond $\sim$1100 eV is an artifact of our product-ion detection
scheme. In this context we mention our recent measurement of the similarly
small cross section for sixfold photoionization of Ar+ ions [25].

##  V Summary and Conclusions

Employing the photon-ion merged-beams technique at a high-flux beamline of one
of the world’s brightest synchrotron-radiation facilities, we were able to
measure cross sections for the single-photon multiple inner-shell
photodetachment of oxygen anions over a wide energy range extending to well
beyond the threshold for double $K$-shell detachment. Owing to a high product-
ion selectivity and a near 100% product-ion detection efficiency, we were able
to measure cross sections for net double, triple, fourfold, and fivefold
ionization. These turned out to be dominated by direct double-detachment
processes where one photon simultaneously ejects two electrons. As in our
previous studies on F- [6] and C- [7], the experimental cross-section shapes
are very well described by the empirical scaling formula suggested by Pattard
[35].  In future experiments, more detailed information might be obtained from
electron spectroscopy. Although this is rather demanding, mainly because of
the low densities of ionic targets, first steps in this direction have already
been taken [39, 40, 41].

We identified the various single and double detachment thresholds by large-
scale atomic-structure calculations which also account for the complex
deexcitation cascades that set in after the initial core-hole creation. The
theoretical results for the product charge-state distributions agree well with
the experimental findings despite of the fact that only single Auger and
radiative dipole transitions were considered in the computation of the
deexcitation cascades. The calculations do not predict any yield of O4+
product ions below the threshold for $1s$+$1s$ double ionization although a
small experimental cross for fivefold detachment could be measured. This
suggests that many-electron processes such as detachment accompanied by shake-
up or double-Auger processes are decisive for the production of O4+ ions. In
the future, we will incorporate such processes into our systematic approach
for the computation of deexcitation cascades [17]. Even more theoretical
development work is required for a more quantitative understanding of the
various double-ionization processes at play.

###### Acknowledgements.

We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association
HGF, for the provision of experimental facilities. Parts of this research were
carried out at PETRA III and we would like to thank Kai Bagschik, Frank
Scholz, Jörn Seltmann, and Moritz Hoesch for assistance in using beamline P04.
We are grateful for support from Bundesministerium für Bildung und Forschung
within the “Verbundforschung” funding scheme (grant nos. 05K19GU3 and
05K19RG3) and from Deutsche Forschungsgemeinschaft (DFG, project nos.
389115454 and SFB925/A3).

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