# Pentagonal Domain Exchange

Shigeki Akiyama  Department of Mathematics, Faculty of Science, Niigata
University, Ikarashi-2 8050 Niigata, 950-2181 Japan
<EMAIL_ADDRESS>and  Edmund Harriss  Department of
Mathematical Sciences, 1 University of Arkansas, Fayetteville, AR 72701, USA
<EMAIL_ADDRESS>
###### Abstract.

Self-inducing structure of pentagonal piecewise isometry is applied to show
detailed description of periodic and aperiodic orbits, and further dynamical
properties. A Pisot number appears as a scaling constant and plays a crucial
role in the proof. Further generalization is discussed in the last section.

The first author is supported by the Japanese Society for the Promotion of
Science (JSPS), grant in aid 21540010.

Adler-Kitchens-Tresser [1] and Goetz [10] initiated the study of piecewise
isometries. This class of maps shows the way to possible generalizations of
results on interval exchanges to higher dimensions [16, 30]. In this paper we
examine the detailed properties of the map shown in Figure 1 from an algebraic
point of view.

Figure 1. A piecewise rotation $T$ on two pieces. The triangle is rotated
$2\pi/5$ around $a$ and the trapezium is rotated $2\pi/5$ around $b$. Periodic
points with short periods are shown below, in two colours to illustrate that
they cluster into groups, each forming a pentagon.

The goal of this paper is to see how this map is applied to show number
theoretical results. First we reprove that almost all orbits in the sense of
Lebesgue measure are periodic, and in addition, there are explicit aperiodic
points. Second we show that aperiodic points forms a proper dense subset of an
attractor of some iterated function system and are recognized by a Büchi
automaton (c.f. Figure 14). The dynamics acting on this set of aperiodic
points are conjugate to the $2$-adic odometer (addition of one) whose explicit
construction is given (Theorem 3). As a result, we easily see that all
aperiodic orbits are dense and uniformly distributed in the attractor. We
finally give a characterization of points which have purely periodic
multiplicative coding by constructing its natural extension (Theorem 6). In
doing so we obtain an intriguing picture Figure 16 that emerges naturally from
taking algebraic conjugates, whose structure is worthy of further study. We
discuss possible generalizations for 7-fold and 9-fold piecewise rotations in
Section 3.

A dynamical system is self-inducing if the first return map to some subset has
the same dynamics as the full map. The most important example is the
irrational rotation, presented as exchange of two intervals. An elementary
example begins with $\Phi$ shown in Figure 2.

Figure 2. An interval exchange map $\Phi$, where
$\lambda=\frac{1+\sqrt{5}}{2}$

For this interval exchange, now consider the second interval $B$. As shown in
Figure 3 this interval is translated to the left once, and to the right. Thus
$\Phi^{2}(B_{1})$ is back in $B$, the interval $B_{2}$ requires one more step,
but $\Phi^{3}(B_{2})$ also lies within $B$. This first return dynamics on $B$
is therefore conjugate to the dynamics on $A\cup B$.

Figure 3. The interval exchange $\Phi$ is self-inducing. The intervals $B_{1}$
and $B_{2}$ are swapped by the first return map of $\Phi$ on the interval $B$.

Self-inducing subsystem of two interval exchange corresponds to purely
periodic orbits of continued fraction expansion and they are efficiently
captured by the continued fraction algorithm.

This gives a motivation to study the interval exchange transform (IET) of
three or more pieces, trying to find higher dimensional continued fraction
with good Diophantine approximation properties. The study of self-inducing
structure of IET’s was started by a pioneer work of Rauzy [26], now called
Rauzy induction, and got extended in a great deal by many authors including
Veech [31] and Zorich [33], see [32] for historical developments.

Self-inducing piecewise isometries emerged from dynamical systems as a natural
generalization of IET [21, 1, 5, 11, 12, 8, 22] and the first return dynamics
appears in outer billiards [28, 6]. Like IET they provide a simple setting to
study many of the deep and perplexing behaviors that can emerge from a
dynamical system.

The self-inducing structure links such dynamical systems to number theoretical
algorithms, such as, digital expansions and Diophantine approximation
algorithms, and allows us to study their periodic orbits by constructing their
natural extensions. This idea leads to complex and beautiful fractal behavior.

Our target is the piecewise isometry in Figure 1, but to illustrate the bridge
formed between the two fields let us begin with a simple conjecture from
number theory:

######  Conjecture 1.

For any $-2<\lambda<2$, each integer sequence defined by $0\leq
a_{n+1}+\lambda a_{n}+a_{n-1}<1$ is periodic.

Since $a_{n+2}\in\mathbb{Z}$ is uniquely determined by
$(a_{n},a_{n+1})\in\mathbb{Z}^{2}$, we treat this recurrence as a map
$(a_{n},a_{n+1})\mapsto(a_{n+1},a_{n+2})$ acting on $\mathbb{Z}^{2}$. It is
natural to set $\lambda=-2\cos(\theta)$ to view this map as a ‘discretized
rotation’:

$\begin{pmatrix}a_{n+1}\\\ a_{n+2}\end{pmatrix}\sim\begin{pmatrix}0&1\\\
-1&-\lambda\end{pmatrix}\begin{pmatrix}a_{n}\\\ a_{n+1}\end{pmatrix}$

with eigenvalues $\exp(\pm\sqrt{-1}\theta)$. As the matrix is conjugate to the
planar rotation matrix of angle $\theta$, putting
$P=\begin{pmatrix}1&0\cr\cos\theta&-\sin\theta\end{pmatrix}$, we have

$P\begin{pmatrix}a_{n+1}\cr
a_{n+2}\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\cr\sin\theta&\cos\theta\end{pmatrix}P\begin{pmatrix}a_{n}\cr
a_{n+1}\end{pmatrix}+P\begin{pmatrix}0\cr\langle\lambda
a_{n+1}\rangle\end{pmatrix}$

where $\langle x\rangle$ is the fractional part of $x$. Therefore this gives a
rotation map of angle $\theta$ acting on a lattice $P\mathbb{Z}^{2}$ but the
image requires a bounded perturbation of modulus less than two to fit into
lattice points of $P\mathbb{Z}^{2}$. For conjecture 1 we expect that such
perturbations do not cumulate and the orbits stay bounded, equivalently, all
orbits become periodic.

A nice feature of the map $(a_{n},a_{n+1})\mapsto(a_{n+1},a_{n+2})$ is that it
is clearly bijective on $\mathbb{Z}^{2}$ by symmetry, while under the usual
round off scheme, the digital information should be more or less lost by the
irrational rotation. This motivates dynamical study of global stability of
this algorithm.

The conjecture is trivial when $\lambda=0,\pm 1$. Among non-trivial cases, the
second tractable case is when $\theta$ is rational and $\lambda$ is quadratic
over $\mathbb{Q}$. Akiyama, Brunotte, Pethő and Steiner [3] proved:

######  Theorem 1.

The conjecture is valid for $\lambda=\frac{\pm
1\pm\sqrt{5}}{2},\pm\sqrt{2},\pm\sqrt{3}.$

It seems hard to prove Conjecture 1 for other values. The case
$\lambda=\frac{1-\sqrt{5}}{2}$ was firstly shown by Lowenstein, Hatjispyros
and Vivaldi [21] with heavy computer assistance. A number theoretical proof
for $\frac{1+\sqrt{5}}{2}$ appeared in [2], whose proof is short but not so
easy to generalize. We try to give an accessible account using self-inducing
piecewise isometry in the case $\lambda=\omega=\frac{1+\sqrt{5}}{2}$, together
with its further dynamical behavior. The proof in Section 1 is basically in
[3]. However this version may elucidate the background idea and is directly
connected to the scaling constant of self-inducing structure of piecewise
isometry acting on a lozenge.

A Pisot number is an algebraic integer $>1$ whose conjugates have modulus less
than $1$. Throughout the paper, we will see the importance of the fact that
the scaling constant of self-inducing system is a Pisot number. Our all
discussions heavily depend on this fact. Indeed, Pisot scaling constants often
appear in self-inducing structures of several important dynamical systems, for
e.g., IET and substitutive dynamical systems. We discuss this point in Section
3. It is pretty surprising that we see this phenomenon in cubic piecewise
rotations as well. We hope this paper gives an easy way to access this
interesting area of mathematics.

We wish to show our gratitude to P.Hubert, W.Steiner and F.Vivaldi for helpful
comments and relevant literatures in the development of this manuscript.

##  1\. Proof of the periodicity for golden mean

Setting $\zeta=\exp(2\pi i/5)$, we have $\omega=-\zeta^{2}-\zeta^{-2}$ and
$1/\omega=\zeta+\zeta^{-1}$. The integer ring of $\mathbb{Q}(\zeta)$ coincides
with the ring $\mathbb{Z}[\zeta]$ generated by $\zeta$ in $\mathbb{Z}$,
$\mathbb{Z}[\zeta]$ is a free $\mathbb{Z}$-module generated by
$1,\zeta,\zeta^{2},\zeta^{3}$. Hereafter we use a different base as a
$\mathbb{Z}$-module:

######  Lemma 2.

$\mathbb{Z}[\zeta]$ is a free $\mathbb{Z}$-module of rank $4$ generated by
$1,\omega,\zeta,\omega\zeta$.

###### Proof.

From $\omega=-\zeta^{2}-\zeta^{-2}$, we have

$x_{1}+x_{2}\omega+(y_{1}+y_{2}\omega)\zeta=(x_{1}+y_{2})+(y_{1}+y_{2})\zeta+(y_{2}-x_{2})\zeta^{2}-x_{2}\zeta^{3}.$

On the other hand

$a_{0}+a_{1}\zeta+a_{2}\zeta^{2}+a_{3}\zeta^{3}=(a_{0}-a_{2}+a_{3})-a_{3}\omega+((a_{1}-a_{2}+a_{3})+(a_{2}-a_{3})\omega)\zeta.$

∎

Taking the complex conjugate, the same statement is valid with another basis
$1,\omega,\zeta^{-1},\omega\zeta^{-1}$. Thus each element in
$\mathbb{Z}[\zeta]$ has a unique expression:

$x-\zeta^{-1}y\qquad(x,y\in\mathbb{Z}[\omega]).$

Denote by $\langle x\rangle$ the fractional part of $x\in\mathbb{R}$. Then a
small computation gives

$\displaystyle 0\leq a_{n}+\omega a_{n+1}+a_{n+2}$ $\displaystyle<1$
$\displaystyle a_{n}+\omega a_{n+1}+a_{n+2}$ $\displaystyle=\langle\omega
a_{n+1}\rangle$ $\displaystyle\langle\omega
a_{n}\rangle-\frac{1}{\omega}\langle\omega a_{n+1}\rangle+\langle\omega
a_{n+2}\rangle$ $\displaystyle\equiv 0\pmod{\mathbb{Z}}$ $\displaystyle
x_{n}-(\zeta+\zeta^{-1})x_{n+1}+x_{n+2}$ $\displaystyle\equiv
0\pmod{\mathbb{Z}}$ $\displaystyle(x_{n+1}-\zeta^{-1}x_{n+2})$
$\displaystyle\equiv\zeta^{-1}(x_{n}-\zeta^{-1}x_{n+1})\pmod{\zeta^{-1}\mathbb{Z}}$

and $x_{n}=\langle\omega a_{n}\rangle$. Our problem is therefore embedded into
a piecewise isometry $T$ acting on a lozenge $[0,1)+(-\zeta^{-1})[0,1)$:

$T(x)=\begin{cases}x/\zeta&\text{Im}(x/\zeta)\geq 0\\\
(x-1)/\zeta&\text{Im}(x/\zeta)<0\end{cases}.$

The action of $T$ is geometrically described in Figure 1. The lozenge
$L=[0,1)+(-\zeta^{-1})[0,1)$ is rotated by the multiplication of $-\zeta^{-1}$
and then the trapezoid $\mathcal{Z}$ which falls outside $L$ is pulled back in
by adding $-\zeta^{-1}$. In total, the isosceles triangle $\Delta$ is rotated
clockwise by the angle $3\pi/5$ around the origin and the trapezoid
$\mathcal{Z}$ is rotated by the same angle but around the point
$\frac{1}{2}+i\frac{\sqrt{5(5+2\sqrt{5})}}{10}\simeq 0.5+0.6882i$ indicated by
a black spot, that is the intersection of two diagonals. Our aim is to show
that each point $x\in\mathbb{Z}[\zeta]\cap L$ gives a periodic $T$-orbit.

A. Goetz [10] gave a slightly different map. Ours is an ‘inclined’
modification of [16] and [3].

Clearly the map $T$ is bijective and preserves 2-dimensional Lebesgue measure
$\mu$. However the measure dynamical system $(L,\nu,\mathbb{B},T)$ (with the
$\sigma$-algebra $\mathbb{B}$ of Lebesgue measurable sets) is far from
ergodic. It turned out that orbits of $T$ is periodic for almost all points
but for an exceptional set of Lebesgue measure zero. Our goal is to prove that
the set $\mathbb{Z}[\zeta]$ has no intersection with this exceptional set.
This is not so obvious since $\mathbb{Z}[\zeta]$ is dense in $L$ because
$\mathbb{Z}[\omega]$ is dense in $\mathbb{R}$.

To illustrate the situation, it is instructive to describe an orbit of $1/3$.
See Figure 4.

Figure 4. The orbit of $1/3$

Later we will show that the orbit of $1/3$ is aperiodic and forms a dense
subset of the exceptional set of aperiodic points. Roughly speaking, our task
is to show that $\mathbb{Z}[\zeta]\cap L$ has no intersection with the fractal
set appeared Figure 4.

The key to the proof is a self-inducing structure with a scaling constant
$\omega^{2}$. We consider a region $L^{\prime}=\omega^{-2}L$ and consider the
first return map

$\hat{T}(x)=T^{m(x)}(x)$

for $x\in L^{\prime}$ where $m(x)$ is the minimum positive integer such that
$T^{m(x)}(x)\in L^{\prime}$. For any $x\in L^{\prime}$, the value $m(x)=1,3$
or $6$. We can show that

(1) $\omega^{2}\hat{T}(\omega^{-2}x)=T(x)$

for $x\in L$. The proof is geometric, shown in Figure 5. The return time
$m(x)=3$ in the open pentagon $\Delta^{\prime}=\omega^{-2}\Delta$ [this is
marked $\Delta$ in the figure] and $m(x)=6$ in the shaded pentagonal region
$D$ with three closed and two open edges. In the remaining isosceles triangle
in $\omega^{-2}L$ (whose two equal edges are closed and the other open), the
return time $m(x)$ is $1$.

Figure 5. Self Inducing structure

Note that the equation is valid for all $x\in L^{\prime}$. This makes the
later discussion very simple. Unfortunately this is not the case for other
quadratic values of $\gamma$ and we have to study the behavior of the boundary
independently, see [3].

Let $U$ be the 1-st hitting map to $L^{\prime}$ for $x\in L$, i.e.,
$U(x)=T^{m(x)}(x)$ for the minimum non-negative integer $m(x)$ such that
$T^{m(x)}(x)\in L^{\prime}$. Note that $U$ is a partial function, i.e., $U(x)$
is not defined when there is no positive integer $m$ such that $T^{m}(x)\in
L^{\prime}$. Since

$T(x)=\begin{cases}x/\zeta&x\in\Delta\\\ (x-1)/\zeta&x\in
T(\mathcal{Z})\setminus\Delta\end{cases},$

it is easy to make the map $U$ explicit:

$U(x)=\begin{cases}x&x\in L^{\prime}\\\ \left(x-1\right)/\zeta&x\in
T^{5}(D)\\\ \left(x-\zeta\right)/\zeta^{2}&x\in T^{4}(D)\\\
\left(x-\frac{\zeta}{\omega}\right)/\zeta^{3}&x\in T^{3}(D)\\\
\left(x+\frac{1}{\omega\zeta^{2}}\right)/\zeta^{4}&x\in T^{2}(D)\\\
x+\frac{1}{\omega\zeta}&x\in T(D)\\\ \text{Not defined}&x\in P_{0}\cup
P_{1}\cup P_{2}\end{cases}$

where $P_{0}$ is the largest open pentagon and $P_{1}$ and $P_{2}=P_{1}/\zeta$
are two second largest closed pentagons in Figure 6.

Figure 6. Period Pentagons

Set

$Q=\left\\{0,1,\zeta,\frac{\zeta}{\omega},-\frac{1}{\omega\zeta^{2}},-\frac{1}{\omega\zeta}\right\\}=\\{d_{0},d_{1},d_{2},d_{3},d_{4},d_{5}\\}\subset\mathbb{Z}[\zeta]$

to use later.

We introduce a crucial map $S$ which is the composition of the 1-st hitting
map $U$ and expansion by $\omega^{2}$, i.e. $S(x)=\omega^{2}U(x)$. Denote by
$\pi(x)$ the period of $T$-orbits of $x\in L$ and put $\pi(x)=\infty$ if $x$
is not periodic by $T$. (We easily see $\pi(x)=5$ in $P_{0}$ and $\pi(x)=10$
in $P_{1}\cup P_{2}$ unless $x$ is the centroid of the pentagon.) Then if
$\pi(x)$ and $\pi(S(x))$ are defined and finite, then we see that
$\pi(S(x))<\pi(x)$ which is a consequence of Equation (1). Therefore if
$\pi(x)$ is finite then we have a decreasing sequence

$\pi(x)>\pi(S(x))>\pi(S^{2}(x))>\dots$

of positive integers. This shows that there exists a positive integer $k$ such
that $S^{k}(x)$ is not defined. In this case we say that $S$-orbit of $x$ in
finite. We easily see that if $S$-orbit of $x\in L$ is finite, then clearly
$\pi(x)$ is finite by Equation (1). Thus we have a clear distinction: $x\in L$
is $T$-periodic if and only if its $S$-orbit is finite. Assume that $x\in
L\cap\mathbb{Z}[\zeta]$ gives an infinite $S$-orbit. When $U(x)$ is defined,
we have $U(x)=(x-d_{m(x)})/\zeta^{m(x)}$ with $m(x)=\\{0,1,2,3,4,5\\}$ and
$d_{i}\in Q$ for all $x\in L$. Thus we have

$S^{k}(x)=\omega^{2k}\frac{x}{\zeta^{\sum_{j=1}^{k}m_{j}}}-\sum_{i=1}^{k}\omega^{2(k-i+1)}\frac{d_{m_{i}}}{\zeta^{\sum_{j=i}^{k}m_{j}}}.$

By the assumption $S^{k}(x)$ is defined for $k=1,2,\dots$ and stays in $L$.
Consider the conjugate map $\phi$ which sends $\zeta\rightarrow\zeta^{2}$. As
$\phi(\omega)=-1/\omega$, we have

$\phi(S^{k}(x))=\frac{\phi(x)}{\omega^{2k}\zeta^{2\sum_{j=1}^{k}m_{j}}}-\sum_{i=1}^{k}\frac{d^{\prime}_{m_{i}}}{\omega^{2(k-i+1)}\zeta^{2\sum_{j=i}^{k}m_{j}}}$

with $d^{\prime}_{i}=\phi(d_{i})\in\phi(Q)$. Put $A=\max\\{|d^{\prime}_{i}|\
:\ d_{i}\in Q\\}$. Then we have

$|\phi(S^{k}(x))|\leq|\phi(x)|+\frac{A}{\omega^{2}-1}$

Thus we have $S^{k}(x),\phi(S^{k}(x))$ and their complex conjugates are
bounded by a constant which does not depend on $k$. This implies that the
sequence $(S^{k}(x))_{k}$ must be eventually periodic.

Summing up, for a point $x$ in $\mathbb{Z}[\zeta]$, its $S$-orbit is finite or
eventually periodic. When it is finite then its $T$-orbit is periodic and when
its $S$-orbit is eventually periodic then $T$-orbit is aperiodic. Thus we have
an algorithm for $x\in\mathbb{Z}[\beta]\cap L$ to tell whether its $T$-orbit
is periodic or not. Since

$|\phi(S^{k}(x))|\leq\frac{|\phi(x)|}{\omega^{2k}}+\frac{A}{1-\omega^{-2}},$

for any positive $\varepsilon$, the right hand side is bounded by

$\varepsilon+\frac{A}{\omega^{2}-1}$

for a sufficiently large $k$. This means that under the assumption that there
is an infinite $S$-orbit, the set

$\left\\{x\in\mathbb{Z}[\zeta]\cap L\ :\
|\phi(x)|\leq\varepsilon+\frac{A}{\omega^{2}-1}\right\\}$

contains $x$ with $\pi(x)=\infty$. Since this set is finite, it is equal to

$B=\left\\{x\in\mathbb{Z}[\zeta]\cap L\ :\
|\phi(x)|\leq\frac{A}{\omega^{2}-1}\right\\}$

for a sufficiently small $\varepsilon$. Since there are only finitely many
candidates in $B$, we obtain an algorithm to check whether an element
$x\in\mathbb{Z}[\zeta]\cap L$ with $\pi(x)=\infty$ exists. In fact, all
elements in $B$ gives a finite $S$-expansion, we are done.

The same algorithm applies to $\frac{1}{M}\mathbb{Z}[\zeta]$ with a fixed
positive integer $M$. In this way, we can also show that points in
$\frac{1}{2}\mathbb{Z}[\zeta]$ are periodic. We can find aperiodic orbits in
$\frac{1}{3}\mathbb{Z}[\zeta]$. For example, one can see that $1/3$ has an
aperiodic $T$-orbit because its $S$-orbit:

$\frac{1}{3},\frac{w^{2}}{3},-\frac{\zeta^{-1}}{3},-\frac{\omega^{2}\zeta^{-1}}{3}-\frac{2\zeta^{-1}}{3},-\frac{\omega^{-2}\zeta^{-1}}{3},-\frac{\zeta^{-1}}{3},\dots$

satisfies $S^{2}(1/3)=S^{6}(1/3)$.

It is crucial in the above proof that the scaling constant of the self-
inducing structure is a Pisot number. Scaling constants of piecewise
isometries often become Pisot numbers, moreover algebraic units. We discuss
these phenomena in Section 3.

##  2\. Coding of aperiodic $T$-orbits

Denote by $\mathbf{A}$ the set of all $T$-aperiodic points in $L$. By the
proof of the previous section, we have

$\mathbf{A}=\\{x\in L\ |\ S^{k}(x)\text{ is defined for all }k=1,2,\dots\\}.$

We also have $S(\mathbf{A})\subset\mathbf{A}$. This means that for
$x_{1}\in\mathbf{A}$, there is a $m_{i}\in\\{0,1,2,3,4,5\\}$ and
$x_{i}\in\mathbf{A}$ such that
$\omega^{2}\zeta^{-m_{i}}(x_{i}-d_{m_{i}})=x_{i+1}\in\mathbf{A}$ for
$i=1,2,\dots$. We therefore have an expansion

(2)
$x_{1}=d_{m_{1}}+\frac{\zeta^{m_{1}}}{\omega^{2}}\left(d_{m_{2}}+\frac{\zeta^{m_{2}}}{\omega^{2}}\left(d_{m_{3}}+\frac{\zeta^{m_{3}}}{\omega^{2}}\left(d_{m_{4}}+\frac{\zeta^{m_{4}}}{\omega^{2}}\dots\right.\right.\right.$

Conversely a sequence $\\{m_{i}\\}_{i=1,2,\dots}$ defines a single point of
$Y$. Therefore $\mathbf{A}$ must be a subset of the attractor $Y$ of the
iterated function system (IFS):

$Y=\bigcup_{i=0}^{5}\left(\frac{\zeta^{i}}{\omega^{2}}Y+d_{i}\right),$

an approximation of which is depicted in Figure 7(a).

(a) All digits (b) $d_{0},d_{2},d_{3},d_{5}$

Figure 7. Attractors containing $\mathbf{A}$

At this point we can assert that $2$-dimensional Lebesgue measure of aperiodic
points in $L$ must be zero, because $\omega^{4}\simeq 6.854\dots>6$.

We notice that the digits in $Q$ are not arbitrarily chosen because the image
of $S$ must be in $T(\mathcal{Z})$. Thus the digits $d_{1}$ and $d_{4}$
appears only at the beginning in the expression of Equation (2). Therefore it
is more suitable to study $\mathbf{A}\cap T(\mathcal{Z})$. The attractor

(3)
$Y^{\prime}=\left(\frac{1}{\omega^{2}}Y^{\prime}+d_{0}\right)\cup\left(\frac{\zeta^{2}}{\omega^{2}}Y^{\prime}+d_{2}\right)\cup\left(\frac{\zeta^{3}}{\omega^{2}}Y^{\prime}+d_{3}\right)\cup\left(\frac{\zeta^{5}}{\omega^{2}}Y^{\prime}+d_{5}\right)$

is depicted in Figure 7(b).

This iterated function system satisfies OSC by a pentagonal shape $K$ with
whose vertices are

$0,-\zeta^{-1},\zeta,-\zeta\omega^{-1}-\zeta^{-1},-\zeta^{2}\omega^{-1}$

as in Figure 8. We confirm that the pieces
$K_{m}=\frac{\zeta^{m}}{\omega^{2}}K+d_{m}$ do not overlap.

Figure 8. Open set condition

We consider the induced system of $(L,\mathbb{B},\nu,T)$ to $T(\mathcal{Z})$.
Denote by $\widetilde{T}$ the first return map on $T(\mathcal{Z})$. Then the
induced system $(T(\mathcal{Z}),\widetilde{T})$ is the domain exchange of two
isosceles triangle $A$ and $B$ depicted in Figure 9. The triangle $A$ has two
closed edges of equal length and one open edge, while $B$ has one closed edge
and two open edges of the same length. The open regular pentagon $P_{0}$ and
the triangle $B$ move together by $\widetilde{T}$ and can be merged into a
single shape.

Figure 9. Induced Rotation $\widetilde{T}$ on $T(\mathcal{Z})$

We see

(4) $\widetilde{T}(x)=\begin{cases}T^{2}(x)&x\in\Delta\\\ T(x)&x\in
T(\mathcal{Z})\setminus\Delta.\end{cases}$

Again we find self-inducing structure with the scaling constant $\omega^{2}$:

(5) $\omega^{2}\widetilde{T}(\omega^{-2}x)=\widetilde{T}(x)$

for all $x\in T(\mathcal{Z})$. This can be seen in Figure 10 with
$\alpha=\omega^{-2}A$, $\beta=\omega^{-2}B$ and $R=\omega^{-2}P_{0}$.

Figure 10. Self Inducing Structure of $(T(\mathcal{Z}),\widetilde{T})$

This induced dynamics $(T(\mathcal{Z}),\widetilde{T})$ is essential in
describing the set $\mathbf{A}$.

Readers may notice that we can find a self-inducing structure by smaller
scaling constant $\omega$ in Figure 7(b) by taking two connected pieces.
However this choice of inducing region is not suitable because the self-
inducing relation (with flipping) is measure theoretically valid, but has
different behavior on the boundary.

Let us introduce two codings. First is the coding of $T$-orbits of a point $x$
in $L$ in two symbols $\\{0,1\\}$:
$\mathbf{d}(x)=(\psi(T^{n}(x))_{n}\in\\{0,1\\}^{\mathbb{N}}$ where

$\psi(x)=\begin{cases}0&x\in\Delta\\\ 1&x\in\mathcal{Z}\end{cases}.$

For e.g., the $\mathbf{d}(1/3)=10110101011010101101101101\dots$ The second
coding is defined by
$\widetilde{\mathbf{d}}(x)=(\widetilde{\psi}(\widetilde{T}^{n}(x)))_{n}\in\\{a,b\\}^{\mathbb{N}}$
for $x\in T(\mathcal{Z})$ where

$\widetilde{\psi}(x)=\begin{cases}a&x\in\Delta\\\ b&x\in
T(\mathcal{Z})\setminus\Delta.\end{cases}$

For a point $x$ in $T(\mathcal{Z})$ we have two codings by $\\{a,b\\}$ and by
$\\{0,1\\}$. From Equation (4), the two codings are equivalent through the
substitution $a\rightarrow 01$, $b\rightarrow 1$. For a given coding of
$T$-orbit by $\\{0,1\\}$, there is a unique way to retrieve the coding of
$\widetilde{T}$-orbit by $\\{a,b\\}$, because the symbol $0$ must be followed
by $1$. For e.g., $T(1/3)=-2\zeta^{-1}/3\in T(\mathcal{Z})$ is coded in two
ways as:

$\widetilde{\mathbf{d}}(-2\zeta^{-1}/3)=a\>b\>a\>a\>a\>b\>a\>a\>a\>b\>a\>b\>a\>b\>a\>b\dots$

and

$\mathbf{d}(-2\zeta^{-1}/3)=01\>1\>01\>01\>01\>1\>01\>01\>01\>1\>01\>1\>01\>1\>01\>1\dots$

Hereafter we discuss the coding $\widetilde{\mathbf{d}}$. Observing the
trajectory of the region $\omega^{-2}(\Delta)$ and
$\omega^{-2}(T(\mathcal{Z})\setminus\Delta)$ by the first return map by the
iteration of $\widetilde{T}$ to the region $\omega^{-2}T(\mathcal{Z})$, it is
natural to introduce a substitution $\sigma_{0}$:

$a\rightarrow aaba,\quad b\rightarrow baba.$

on $\\{a,b\\}^{*}$ and we have

$\widetilde{\mathbf{d}}(\omega^{-2}x)=\sigma_{0}(\widetilde{\mathbf{d}}(x))$

for $x\in T(\mathcal{Z})$. More generally, following the analogy of the
previous section, the first hitting map to the region
$\omega^{-2}(T(\mathcal{Z}))$ provide us an expansion of a point $x\in
T(\mathcal{Z})$ exactly in the same form as (2) with restricted digits
$\\{d_{0},d_{2},d_{3},d_{5}\\}$. One can confirm that

(6)
$\widetilde{\mathbf{d}}\left(\frac{\zeta^{m}}{\omega^{2}}x+d_{m}\right)=\begin{cases}\sigma_{0}(\widetilde{\mathbf{d}}(x))&m=0\\\
a\oplus\sigma_{0}(\widetilde{\mathbf{d}}(\widetilde{T}(x)))&m=2\\\
ba\oplus\sigma_{0}(\widetilde{\mathbf{d}}(\widetilde{T}^{2}(x)))&m=3\\\
aba\oplus\sigma_{0}(\widetilde{\mathbf{d}}(\widetilde{T}^{3}(x)))&m=5\end{cases}$

where $\oplus$ is the concatenation of letters. Defining conjugate
substitutions by $\sigma_{1}=a\sigma_{0}a^{-1}$,
$\sigma_{2}=ba\sigma_{1}a^{-1}b^{-1}$ and
$\sigma_{3}=aba\sigma_{0}a^{-1}b^{-1}a^{-1}$, i.e.,

$\displaystyle\sigma_{0}(a)=aaba,$ $\displaystyle\quad\sigma_{0}(b)=baba$
$\displaystyle\sigma_{1}(a)=aaab,$ $\displaystyle\quad\sigma_{1}(b)=abab$
$\displaystyle\sigma_{2}(a)=baaa,$ $\displaystyle\quad\sigma_{2}(b)=baba$
$\displaystyle\sigma_{3}(a)=abaa,$ $\displaystyle\quad\sigma_{3}(b)=abab$

one may rewrite

$\widetilde{\mathbf{d}}\left(\frac{\zeta^{m}}{\omega^{2}}x+d_{m}\right)=\begin{cases}\sigma_{0}(\widetilde{\mathbf{d}}(x))&m=0\\\
\sigma_{1}(\widetilde{\mathbf{d}}(\widetilde{T}(x)))&m=2\\\
\sigma_{2}(\widetilde{\mathbf{d}}(\widetilde{T}^{2}(x)))&m=3\\\
\sigma_{3}(\widetilde{\mathbf{d}}(\widetilde{T}^{3}(x)))&m=5.\end{cases}$

We say that an infinite word $y$ in $\\{a,b\\}^{\mathbb{N}}$ is an $S$-adic
limit of $\sigma_{i}\ (i=0,1,2,3)$ if there exist
$y_{i}\in\\{a,b\\}^{\mathbb{N}}$ for $i=1,2,\dots$ such that

$y=\lim_{\ell\rightarrow\infty}\sigma_{m_{1}}\circ\sigma_{m_{2}}\circ\sigma_{m_{3}}\circ\cdots\circ\sigma_{m_{\ell}}(y_{\ell}).$

with $m_{i}\in\\{0,1,2,3\\}$. Since each element $x\in
T(\mathcal{Z})\cap\mathbf{A}$ has an infinite expansion (2) with digits
$\\{d_{0},d_{2},d_{3},d_{5}\\}$, we find $x_{i}\in\omega^{-2}T(\mathcal{Z})$
such that

$\widetilde{\mathbf{d}}(x)=\lim_{\ell\rightarrow\infty}\sigma_{m_{1}}\circ\sigma_{m_{2}}\circ\sigma_{m_{3}}\circ\cdots\circ\sigma_{m_{\ell}}(\widetilde{\mathbf{d}}(x_{\ell})).$

This shows that $\widetilde{\mathbf{d}}(x)$ is an $S$-adic limit of
$\sigma_{i}\ (i=0,1,2,3)$.

Note that from the definition (2) of $\sigma_{i}$, for a given $S$-adic limit
$y$ there is an algorithm to retrieve uniquely the sequence
$(\sigma_{m_{i}})_{i}$. Checking first four letters of $y$, we know the first
letter of $y_{1}$ and to determine $m_{1}$ we need first 6 letters. We can
iterate this process easily.

Summing up, we embedded the set $\mathbf{A}\cap T(\mathcal{Z})$ into the
attractor $Y^{\prime}$ of an IFS (3) and succeeded in characterizing the
coding of $\widetilde{T}$-orbits of points in this attractor as a set of
$S$-adic limits on $\\{\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{3}\\}$.
However recalling that points in closed pentagons $P_{1}$ and $P_{2}$ are
$T$-periodic and $Y^{\prime}$ is a non-empty compact set, we see from Figure
7(b) that $\mathbf{A}$ is a proper subset of $Y^{\prime}$.

We wish to characterize the set of aperiodic points in $Y^{\prime}$ and its
coding through $\widetilde{\mathbf{d}}$. Recalling the discussion in the
previous section, if $x\in T(\mathcal{Z})$ has periodic $T$-orbits if and only
if there exists a positive integer $k$ such that $S^{k}(x)\in P_{0}\cup
P_{1}\cup P_{2}$. The equivalent statement in the induced system
$(T(\mathcal{Z}),\widetilde{T})$ is that $x\in T(\mathcal{Z})$ is
$\widetilde{T}$-periodic if and only if there exists a positive integer $k$
such that $S^{k}(x)\in P_{0}\cup P_{1}$. Note that we have:

$\widetilde{T}(x)=\begin{cases}\zeta^{-1}(x-p)+p&x\in P_{0}\\\
\zeta^{-2}(x-q)+q&x\in P_{1}\\\ \end{cases}$

where $p=\frac{1}{2}+i\frac{\sqrt{5(5+2\sqrt{5})}}{10}$ (resp.
$q=i\sqrt{\frac{5+\sqrt{5}}{10}}$) is the center of $P_{0}$ (resp. $P_{1}$)
and consequently $\widetilde{T}^{5}(x)=x$ holds for $x\in P_{0}\cup P_{1}$. If
$x\in T(\mathcal{Z})$ and $x$ is $\widetilde{T}$-periodic, then there exist
$x_{i}\in T(\mathcal{Z})$ such that $x_{\ell}\in P_{0}\cup P_{1}$ and

$x=d_{m_{1}}+\frac{\zeta^{m_{1}}}{\omega^{2}}\left(d_{m_{2}}+\frac{\zeta^{m_{2}}}{\omega^{2}}\left(d_{m_{3}}+\frac{\zeta^{m_{3}}}{\omega^{2}}\dots\left(d_{m_{\ell}}+\frac{\zeta^{m_{\ell}}}{\omega^{2\ell}}x_{\ell}\right)\dots\right)\right),$

with $m_{i}\in\\{0,2,3,5\\}$. Thus the set of $\widetilde{T}$-periodic points
in $T(\mathcal{Z})$ consists of all the pentagons of the form

(7)
$d_{m_{1}}+\frac{\zeta^{m_{1}}}{\omega^{2}}\left(d_{m_{2}}+\frac{\zeta^{m_{2}}}{\omega^{2}}\left(d_{m_{3}}+\frac{\zeta^{m_{3}}}{\omega^{2}}\dots\left(d_{m_{\ell}}+\frac{\zeta^{m_{\ell}}}{\omega^{2\ell}}P_{j}\right)\dots\right)\right)$

with $j=0,1$ and $m_{j}\in\\{0,2,3,5\\}$. From the self-inducing structure
(5), it is easy to see that if two points $x,x^{\prime}$ are in the same
pentagon of above shape and none of them is the center, then they have exactly
the same periods. Moreover two $\widetilde{T}$-orbits keeps constant distance,
i.e.,
$\widetilde{T}^{n}(x)-\widetilde{T}^{n}(x^{\prime})=\zeta^{s}(x-x^{\prime})$
for some integer $s$. The period is completely determined in [3]. We have all
the periodic orbits in $T(\mathcal{Z})$ and therefore have a geometric
description of aperiodic points:

$\mathbf{A}\cap T(\mathcal{Z})=T(\mathcal{Z})\setminus\\{\text{All pentagons
of the form (\ref{Pent})}\\}.$

Subtraction of these pentagons from $T(\mathcal{Z})$ is described by an
algorithm. The initial set is $D_{0}=T(\mathcal{Z})\setminus P_{0}$ with two
open and three closed edges as in the left Figure 11. The interior ${\rm
Inn}(D_{0})$ gives another feasible open set to assure the open set condition
of the IFS of (3). Inductively we define the decreasing sequence of sets

$D_{i+1}=\bigcup_{m\in\\{0,2,3,5\\}}\left(\frac{\zeta^{m}}{\omega^{2}}D_{i}+d_{m}\right)$

for $i=0,2,\dots$. Then $D_{i}$ consists of $4^{i}$ pieces congruent to
$\omega^{-2i}D_{0}$ without overlapping. Note that since

$D_{1}=D_{0}\setminus(P_{1}\cup\omega^{-2}P_{0}\cup\frac{\omega^{-2}P_{0}-1}{\zeta}),$

$D_{1}$ is obtained by subtracting from $D_{0}$ one closed and two open
regular pentagons as in Figure 11.

Figure 11. Pentagon Removal Algorithm

To generate $D_{i+1}$, each $4^{i}$ pieces in $D_{i}$ are subdivided into $4$
sub-pieces by subtracting three small regular pentagons. Clearly all regular
pentagons of the shape (7) are subtracted by this iteration and we obtain

$\mathbf{A}\cap T(\mathcal{Z})=\bigcap_{i=0}^{\infty}D_{i}.$

This observation allows us to symbolically characterize aperiodic points in
$Y^{\prime}$. First, every point $x$ of $Y^{\prime}$ has an address
$d_{m_{1}}d_{m_{2}}\dots\in\\{d_{0},d_{2},d_{3},d_{5}\\}^{\mathbb{N}}$ by the
expansion (2). The address is unique but for countable exceptions. The
exceptional points forms the set of cut points of $Y^{\prime}$ having the
eventually periodic expansion:

$\displaystyle d_{0}d_{2}(d_{0})^{\infty}$ $\displaystyle\simeq$
$\displaystyle d_{3}d_{3}(d_{5})^{\infty}$ $\displaystyle
d_{3}(d_{0})^{\infty}$ $\displaystyle\simeq$ $\displaystyle
d_{2}(d_{5})^{\infty}$ $\displaystyle d_{2}d_{2}(d_{0})^{\infty}$
$\displaystyle\simeq$ $\displaystyle d_{5}d_{3}(d_{5})^{\infty}$

in the suffix of its address, which is understood by Figure 12 where
$K_{mn}=\frac{\zeta^{m}}{\omega^{2}}(\frac{\zeta^{n}}{\omega^{2}}K+d_{n})+d_{m}$.

Figure 12. Subdivision procedure

Note that if a point $x$ in $T(\mathcal{Z})$ is periodic, then there exists a
non-negative integer $k$ such that $S^{k}(x)\in P_{0}\cup P_{1}$. Moreover, if
$x\in Y^{\prime}\cap T(\mathcal{Z})$, then there exists a non-negative integer
$k$ such that $S^{k}(x)\in\partial(P_{1})$, because it can not be an inner
point of $P_{0}$ or $P_{1}$. In other words, such $x$ must be located in the
open edge of one of $4^{k}$ pieces of $D_{k}$. From Figure 11, one can
construct the following Figure 13 which recognize points of two open edges in
$\partial(D_{0})$. For construction, we introduce a new symbol set $\\{R,L\\}$
(right and left) to distinguish which open edge of $D_{k}$ is into focus.

To read the graph and obtain the previous sequences, ignore $\\{R,L\\}$ and
substitute $\\{0,2,3,5\\}$ with $\\{d_{0},d_{2},d_{3},d_{5}\\}$. A point $x\in
Y^{\prime}$ is periodic (or in the open edge of $D_{0}$) if and only if a
suffix of the address
$d_{m_{1}}d_{m_{2}}\dots\in\\{d_{0},d_{2},d_{3},d_{5}\\}^{\mathbb{N}}$ is in
Figure 13. Note that the points with double addresses are on the open edge of
some $D_{i}$ and consequently their suffixes are read in Figure 13. Figure 12
helps this construction. For e.g., the right open edge of $K_{5}$ consists of
the left open edge of $K_{53}$ and the right open edge of $K_{50}$, therefore
we draw outgoing edges from $5R$ to $3L$ and $0R$.

$\textstyle{2R}$$\textstyle{0R}$$\textstyle{0L,2L}$$\textstyle{3R,5R}$$\textstyle{5L}$$\textstyle{3L}$
Figure 13. $\widetilde{T}$-periodic expansions

As a result, the set of addresses of the points in $\mathbf{A}\cap
T(\mathcal{Z})$ are recognized by a Büchi automaton which is the complement of
the Büchi automaton of Figure 14. Here the double bordered states in Figure 14
are final states. Each infinite word produced by the edge labels
$\\{d_{0},d_{2},d_{3},d_{5}\\}$ on this directed graph is accepted, because it
visits infinitely many times the final states. We do not give here the exact
shape of its complement. It is known that complementation of a Büchi automaton
is much harder than the one of a finite automaton, because the subset
construction does not work (c.f. [29, 23]).

$\scriptstyle{3,5}$$\scriptstyle{0,2}$$\scriptstyle{0,2,3,5}$$\scriptstyle{3}$$\scriptstyle{0}$$\scriptstyle{0}$$\scriptstyle{3}$$\scriptstyle{5}$$\scriptstyle{2}$$\scriptstyle{3}$$\scriptstyle{0}$$\scriptstyle{5}$$\scriptstyle{2}$$\scriptstyle{5}$$\scriptstyle{2}$
Figure 14. Büchi automaton for periodic points in $Y^{\prime}$

Now consider the topology of $\\{a,b\\}^{\mathbb{N}}$ induced from the metric
defined by $2^{-\max_{x_{i}\neq y_{i}}i}$ for
$x=x_{1}x_{2}\dots,y=y_{1}y_{2}\dots\in\\{a,b\\}^{\mathbb{N}}$. Take a fixed
point $w=(w_{i})_{i=0,1,2,\dots}\in\\{a,b\\}^{\mathbb{N}}$ with
$\sigma_{0}(w)=w$. This is computed for e.g., by $\lim_{n}\sigma_{0}^{n}(a)$.
The shift map $V$ is a continuous map from $\\{a,b\\}^{\mathbb{N}}$ to itself
defined by $V((w_{i}))=(w_{i+1})$. Letting $X_{\sigma_{0}}$ be the closure of
the set $\\{V^{n}(w)\ |\ n=0,1,\dots\\}$, we can define the substitutive
dynamical system $(X_{\sigma_{0}},V)$ associated with $\sigma_{0}$. Since
$\sigma_{0}$ is primitive the set $X_{\sigma_{0}}$ does not depend on the
choice of the fixed point and $(X_{\sigma_{0}},V)$ is minimal and uniquely
ergodic (see [9]). Let $\tau$ be the invariant measure of
$(X_{\sigma_{0}},V)$. On the other hand, for the attractor $Y^{\prime}$ there
is the self-similar measure $\nu$, i.e., a unique probability measure (c.f.
Hutchinson [13]) satisfying

$\nu(X)=\frac{1}{4}\sum_{m\in\\{0,2,3,5\\}}\nu\left(\frac{\omega^{2}}{\zeta^{m}}(X-d_{m})\right)$

for $\nu$-measurable sets $\mathbb{B}_{Y^{\prime}}$ in $Y^{\prime}$.

######  Theorem 3.

The restriction of $\widetilde{T}$ to $Y^{\prime}$ is measure preserving and
$(Y^{\prime},\mathbb{B}_{Y^{\prime}},\nu,\widetilde{T})$ is isomorphic to the
$2$-adic odometer $(\mathbb{Z}_{2},x\mapsto x+1)$ as measure dynamical
systems:

(8) $\begin{CD}\mathbb{Z}_{2}@>{+1}>{}>\mathbb{Z}_{2}\\\
@V{\phi}V{}V@V{\phi}V{}V\\\ Y^{\prime}@>{\widetilde{T}}>{}>Y^{\prime}\end{CD}$

where $\phi:\mathbb{Z}_{2}\rightarrow Y^{\prime}$ is almost one to one and
measure preserving, which will be made explicit in the proof. Moreover the map

$\rho:x\mapsto\frac{x-(x\bmod{4})}{4}$

from $\mathbb{Z}_{2}$ to itself gives a commutative diagram:

(9) $\begin{CD}\mathbb{Z}_{2}@>{\rho}>{}>\mathbb{Z}_{2}\\\
@V{\phi}V{}V@V{\phi}V{}V\\\ <EMAIL_ADDRESS>
The above theorem may be read that
$(Y^{\prime},\mathbb{B}_{Y^{\prime}},\nu,\widetilde{T})$ gives a one-sided
variant of numeration system in the sense of Kamae [15].

###### Proof.

First we confirm that $\widetilde{T}$ is measure preserving. Denote by
$[d_{m_{1}},d_{m_{2}},\dots,d_{m_{\ell}}]$ the cylinder set:

(10)
$d_{m_{1}}+\frac{\zeta^{m_{1}}}{\omega^{2}}\left(d_{m_{2}}+\frac{\zeta^{m_{2}}}{\omega^{2}}\left(d_{m_{3}}+\frac{\zeta^{m_{3}}}{\omega^{2}}\dots\left(d_{m_{\ell}}+\frac{\zeta^{m_{\ell}}}{\omega^{2\ell}}Y^{\prime}\right)\dots\right)\right)$

By the OSC, we have $\nu([d_{m_{1}},d_{m_{2}},\dots,d_{m_{\ell}}])=4^{-\ell}$.
From Figure 10, we see that $\widetilde{T}^{-1}([d_{3}])=[d_{5}]$,
$\widetilde{T}^{-1}([d_{2}])=[d_{3}]$, $\widetilde{T}^{-1}([d_{0}])=[d_{2}]$
but $\widetilde{T}^{-1}([d_{5}])$ intersects both $A$ and $B$. Hence if
$m_{1}=0,2,3$, then
$\nu(\widetilde{T}^{-1}([d_{m_{1}},d_{m_{2}},\dots,d_{m_{\ell}}]))=4^{-\ell}$.
By using the self-inducing structure in Figure 10, we also have
$\widetilde{T}^{-1}([d_{5}d_{3}])=[d_{0}d_{5}]$,
$\widetilde{T}^{-1}([d_{5}d_{2}])=[d_{0}d_{3}]$ and
$\widetilde{T}^{-1}([d_{5}d_{0}])=[d_{0}d_{2}]$. Thus if $m_{2}=0,2,3$, then
$\nu(\widetilde{T}^{-1}([d_{5},d_{m_{2}},\dots,d_{m_{\ell}}]))=4^{-\ell}$.
Repeating this, we can show that

$\nu(\widetilde{T}^{-1}([d_{m_{1}},d_{m_{2}},\dots,d_{m_{\ell}}]))=4^{-\ell}$

holds for all $m_{i}\in\\{0,2,3,5\\}$ but a single exception
$m_{1}=m_{2}=\dots=m_{\ell}=5$. Since $\ell$ is arbitrary chosen, a simple
approximation argument shows that $\widetilde{T}$ is measure preserving and
$(Y^{\prime},\mathbb{B}_{Y^{\prime}},\nu,\widetilde{T})$ forms a measure
dynamical system.

Let us define a map $\eta$ from $X_{\sigma_{0}}$ to $Y^{\prime}$. Take an
element $z=x_{1}x_{2}\dots\in X_{\sigma_{0}}$. Then each prefix
$x_{1}x_{2}\dots x_{\ell}$ with $\ell>3$ is a subword of the fix point $v$ of
$\sigma_{0}$ starting with $a$. Therefore there is a word
$y\in\\{\lambda,a,ba,aba\\}$ and $z_{1}\in X_{\sigma_{0}}$ such that
$x=y_{1}\sigma_{0}(z_{1})$. It is easy to see from (6) that this $y$ and
$z_{1}$ are unique.

Iterating this we have $z_{i}=y_{i+1}\sigma(z_{i+1})$ with
$y_{i}\in\\{\lambda,a,ba,aba\\}$, $z_{i+1}\in X_{\sigma_{0}}$ and $z_{0}=z$.
Thus we have for any $\ell$,

$\displaystyle z$ $\displaystyle=$ $\displaystyle
y_{1}\sigma_{0}(y_{2}\sigma_{0}(y_{3}\sigma_{0}\dots
y_{\ell}(\sigma_{0}(z_{\ell}))))$ $\displaystyle=$ $\displaystyle
y_{1}\sigma_{0}(y_{2})\sigma_{0}^{2}(y_{3})\dots\sigma_{0}^{\ell-1}(y_{\ell})\sigma_{0}^{\ell}(z_{\ell}).$

Define a map from $\\{\lambda,a,ba,aba\\}$ to $\mathbb{Z}$ by

$\kappa(\lambda)=0,\kappa(a)=1,\kappa(ba)=2,\kappa(aba)=3.$

Then $z_{i}=y_{i+1}\sigma(z_{i+1})$ is equivalent to
$z_{i}=\sigma_{\kappa(y_{i+1})}(z_{i+1})$ and $z$ is represented as an
$S$-adic limit:

$z=\lim_{\ell\rightarrow\infty}\sigma_{\kappa(y_{1})}\circ\sigma_{\kappa(y_{2})}\circ\dots\circ\sigma_{\kappa(y_{\ell})}(z_{\ell}).$

for $\ell=1,2,\dots$. This gives a multiplicative coding
$\mathbf{d^{\prime}}:X_{\sigma_{0}}\rightarrow\\{\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{3}\\}^{\mathbb{N}}$.
Let $\mathbf{A}^{\prime}$ be the points of $X_{\sigma_{0}}$ whose
multiplicative coding does not end up in an infinite word produced by reading
the vertex labels of Figure 15.

$\textstyle{\sigma_{1}}$$\textstyle{\sigma_{0}}$$\textstyle{\sigma_{0},\sigma_{1}}$$\textstyle{\sigma_{2},\sigma_{3}}$$\textstyle{\sigma_{3}}$$\textstyle{\sigma_{2}}$
Figure 15. Forbidden suffix of $\mathbf{A}^{\prime}$

Let us associate to $z$ a $2$-adic integer
$\iota(z)=-\sum_{i=0}\kappa(y_{i})2^{2i}\in\mathbb{Z}_{2}$. The map $\iota$ is
clearly bijective bi-continuous and the value $\iota(z)$ is also called the
multiplicative coding of $z$. We write down first several iterates of $V$ on
the fix point of $\sigma_{0}$, to illustrate the situation:

$\displaystyle\sigma_{0}\sigma_{0}\sigma_{0}\sigma_{0}\dots$
$\displaystyle\stackrel{{\scriptstyle\iota}}{{\rightarrow}}$
$\displaystyle-0000\dots$
$\displaystyle\sigma_{3}\sigma_{3}\sigma_{3}\sigma_{3}\dots$
$\displaystyle\stackrel{{\scriptstyle\iota}}{{\rightarrow}}$
$\displaystyle-3333\dots$
$\displaystyle\sigma_{2}\sigma_{3}\sigma_{3}\sigma_{3}\dots$
$\displaystyle\stackrel{{\scriptstyle\iota}}{{\rightarrow}}$
$\displaystyle-2333\dots$
$\displaystyle\sigma_{1}\sigma_{3}\sigma_{3}\sigma_{3}\dots$
$\displaystyle\stackrel{{\scriptstyle\iota}}{{\rightarrow}}$
$\displaystyle-1333\dots$
$\displaystyle\sigma_{0}\sigma_{3}\sigma_{3}\sigma_{3}\dots$
$\displaystyle\stackrel{{\scriptstyle\iota}}{{\rightarrow}}$
$\displaystyle-0333\dots$
$\displaystyle\sigma_{3}\sigma_{2}\sigma_{3}\sigma_{3}\dots$
$\displaystyle\stackrel{{\scriptstyle\iota}}{{\rightarrow}}$
$\displaystyle-3233\dots$

One can see that the following commutative diagram (11) holds.

(11) $\begin{CD}X_{\sigma_{0}}@>{V}>{}>X_{\sigma_{0}}\\\
@V{\iota}V{}V@V{\iota}V{}V\\\ \mathbb{Z}_{2}@>{+1}>{}>\mathbb{Z}_{2}\end{CD}$

Therefore $(X_{\sigma_{0}},V)$ is topologically conjugate to the $2$-adic
odometer $(\mathbb{Z}_{2},x\mapsto x+1)$. Here the consecutive digits
$\\{0,1\\}$ in $\mathbb{Z}_{2}$ are glued together to give
$\\{0,1,2,3\\}=\\{0,1\\}+2\\{0,1\\}$. Indeed, $\sigma_{0}$ satisfies the
coincidence condition of height one in the sense of Dekking [25, 9] and above
conjugacy is a consequence of this. $(\mathbb{Z}_{2},x\mapsto x+1)$ is a
translation of a compact group $\mathbb{Z}_{2}$ which is minimal and uniquely
ergodic with the Haar measure of $\mathbb{Z}_{2}$. Moreover one can confirm
that $\iota$ preserves the measure and $(X_{\sigma_{0}},V)$ and
$(\mathbb{Z}_{2},x\mapsto x+1)$ are isomorphic through $\iota$ as measure
dynamical systems. In view of (6), we define

$\xi(i)=\begin{cases}0&i=\lambda\\\ 2&i=a\\\ 3&i=ba\\\ 5&i=aba\end{cases}$

and the map $\eta:X_{\sigma_{0}}\rightarrow Y^{\prime}$ by

(12)
$\eta(x)=d_{\xi(y_{1})}+\frac{\zeta^{\xi(y_{1})}}{\omega^{2}}\left(d_{\xi(y_{2})}+\frac{\zeta^{\xi(y_{2})}}{\omega^{2}}\left(d_{\xi(y_{3})}+\frac{\zeta^{\xi(y_{3})}}{\omega^{2}}\dots.\right.\right.$

Then $\eta$ is clearly surjective, continuous, and measurable because both
$\tau$ and $\nu$ are Borel probability measures. Since the set of points with
double addresses is on the open edge, the map $\eta$ is bijective from
$\mathbf{A^{\prime}}$ to $\mathbf{A}\cap T(\mathcal{Z})$. Since
$\widetilde{\mathbf{d}}(T(x))=V(\widetilde{\mathbf{d}}(x))$, we have a
commutative diagram:

(13) $\begin{CD}\mathbf{A^{\prime}}@>{V}>{}>\mathbf{A^{\prime}}\\\
@V{\eta}V{}V@V{\eta}V{}V\\\ \mathbf{A}\cap
T(\mathcal{Z})@>{\widetilde{T}}>{}>\mathbf{A}\cap T(\mathcal{Z}).\end{CD}$

From Figure 14, it is easy to see that the set $\mathcal{P}$ of
$\widetilde{T}$-periodic points in $Y^{\prime}$ is measure zero by $\nu$,
i.e., $\nu(\mathbf{A}\cap T(\mathcal{Z}))=\nu(Y^{\prime}\cap
T(\mathcal{Z}))=1$, because the number of words of length $n$ in Figure 14 is
$O(2^{n})$. Similarly as the Perron-Frobenius root of the substitution
$\sigma_{0}$ is $4$ and the number of words of lengths $n$ in Figure 15 are
$O(2^{n})$, we see that $\tau(\mathbf{A^{\prime}})=\tau(X_{\sigma_{0}})=1$.
From (13) the pull back measure $\nu\circ\eta^{-1}$ of $X_{\sigma_{0}}$ is
invariant by $V$, we have $\tau=\nu\circ\eta^{-1}$ by unique ergodicity.
Therefore by taking $\phi=\eta\circ\iota$, we have the commutative diagram (8)
with measure zero exceptions. Let $V^{\prime}$ be a map from $X_{\sigma_{0}}$
to itself which acts as the shift operator on the multiplicative coding
$\mathbf{d^{\prime}}$, i.e.,
$(\mathbf{d^{\prime}}(V^{\prime}(z))=\sigma_{n_{2}}\sigma_{n_{3}}\dots$ for
$\mathbf{d^{\prime}}(z)=\sigma_{n_{1}}\sigma_{n_{2}}\dots$. Then we see that

(14) $\begin{CD}\mathbf{A^{\prime}}@>{V^{\prime}}>{}>\mathbf{A^{\prime}}\\\
@V{\eta}V{}V@V{\eta}V{}V\\\ \mathbf{A}\cap
T(\mathcal{Z})@>{S}>{}>\mathbf{A}\cap T(\mathcal{Z}).\end{CD}$

and the commutative diagram (9) is valid but for measure zero exceptions. ∎

######  Corollary 4.

Each aperiodic point $x\in\mathbf{A}\cap T(\mathcal{Z})$, the
$\widetilde{T}$-orbit of $x$ is uniformly distributed in $Y^{\prime}$ with
respect to the self similar measure $\nu$.

###### Proof.

In the proof of Theorem 3 the map $\eta$ is bijective form
$\mathbf{A^{\prime}}$ to $\mathbf{A}\cap T(\mathcal{Z})$. Therefore if
$x\in\mathbf{A}\cap T(\mathcal{Z})$, then there exists a unique element in
$z\in X_{\sigma_{0}}$ with $\eta(z)=x$. Therefore there exist an element
$z_{0}\in\mathbb{Z}_{2}$ such that $\phi(z_{0})=x$. The Haar measure $\mu_{2}$
on $Z_{2}$ is given by the values on the semi-algebra:

$\mu_{2}([c_{0},c_{1},\dots,c_{\ell-1}])=4^{-\ell}$

for each cylinder set $[c_{0},c_{1},\dots,c_{\ell-1}]=\\{y\in\mathbb{Z}_{2}\
|\ y\equiv\sum_{i=0}^{\ell-1}c_{i}4^{i}\pmod{4^{\ell}}\\}$. Since
$(\mathbb{Z}_{2},x\mapsto x+1)$ is uniquely ergodic, the assertion follows
immediately from the commutative diagram (8). ∎

Not all points in $Y^{\prime}$ gives a dense orbit as we already mentioned
that $\mathbf{A}\cap Y^{\prime}$ is a proper dense subset of $Y^{\prime}$.
There are many periodic points in $Y^{\prime}$ as well. This gives a good
contrast to usual minimal topological dynamics given by a continuous map
acting on a compact metrizable space.

######  Corollary 5.

Each aperiodic point $x\in\mathbf{A}$, the $T$-orbit of $x$ is dense in the
set $X$.

###### Proof.

It is clear from the fact that $(T(\mathcal{Z}),\widetilde{T})$ is the induced
system of $(L,T)$. ∎

One can construct a dual expansion of the non-invertible dynamics
$(Y^{\prime},S)$ by the conjugate map $\phi:\zeta\rightarrow\zeta^{2}$ in
${\rm Gal}(\mathbb{Q}(\zeta)/\mathbb{Q})$ and then make a natural extension:
an invertible dynamics which contains $(Y^{\prime},S)$. The idea comes from
symbolic dynamics. We wish to construct the reverse expansion of (12) to the
other direction. To this matter, we compute in the following way:

$\frac{\omega^{2}(\eta(x)-d_{\xi(y_{1})})}{\zeta^{\xi(y_{1})}}=d_{\xi(y_{2})}+\frac{\zeta^{\xi(y_{2})}}{\omega^{2}}\left(d_{\xi(y_{3})}+\frac{\zeta^{\xi(y_{3})}}{\omega^{2}}\left(d_{\xi(y_{4})}+\frac{\zeta^{\xi(y_{4})}}{\omega^{2}}\left(\dots\right.\right.\right.$

and

$\frac{\omega^{2}}{\zeta^{\xi(y_{2})}}\left(\frac{\omega^{2}(\eta(x)-d_{\xi(y_{1})})}{\zeta^{\xi(y_{1})}}-d_{\xi(y_{2})}\right)=d_{\xi(y_{3})}+\frac{\zeta^{\xi(y_{3})}}{\omega^{2}}\left(d_{\xi(y_{4})}+\frac{\zeta^{\xi(y_{4})}}{\omega^{2}}\left(\dots\right.\right.$

Therefore it is natural to introduce a left ‘expansion’:

$\frac{\omega^{2}}{\zeta^{i_{1}}}\left(\frac{\omega^{2}}{\zeta^{i_{2}}}\left(\frac{\omega^{2}}{\zeta^{i_{3}}}\left(\left(\dots\right)-d_{i_{3}}\right)-d_{i_{2}}\right)-d_{i_{1}}\right)$

with $i_{k}\in\\{0,2,3,5\\}$. As this expression does not converge, we take
the image of $\phi$ because $\phi(\omega)=-1/\omega$. Let us denote by
$u_{i_{k}}=\phi(d_{i_{k}})$. Then the expansion

$\frac{\zeta^{-2i_{1}}}{\omega^{2}}\left(\frac{\zeta^{-2i_{2}}}{\omega^{2}}\left(\frac{\zeta^{-2i_{3}}}{\omega^{2}}\left(\left(\dots\right)-u_{i_{3}}\right)\right)-u_{i_{2}}\right)-u_{i_{1}}$

converges and the closure of the set of such expansions gives a compact set
$\mathcal{Y}$ depicted in figure 16.

Figure 16. The dual attractor $\mathcal{Y}$

Of course the set is an attractor of the IFS:

$\mathcal{Y}=\frac{1}{\omega^{2}}(\mathcal{Y}-u_{0})\cup\frac{\zeta}{\omega^{2}}(\mathcal{Y}-u_{2})\cup\frac{\zeta^{-1}}{\omega^{2}}(\mathcal{Y}-u_{3})\cup\frac{1}{\omega^{2}}(\mathcal{Y}-u_{5}).$

Combining $\mathcal{Y}$ we can construct a natural extension of
$(Y^{\prime},S)$ as:

$Y^{\prime}\times\mathcal{Y}\ni\left(\eta,\theta\right)\stackrel{{\scriptstyle\hat{S}}}{{\mapsto}}\left(\frac{(\eta-
d_{i})\omega^{2}}{\zeta^{i}},\frac{\zeta^{-2i}(\theta-\phi(d_{i}))}{\omega^{2}}\right)\in
Y^{\prime}\times\mathcal{Y}$

On the other hand $(\mathbb{Z}_{2},\rho)$ have a natural extension:

$\mathbb{Z}_{2}\times[0,1)\ni(x,y)\stackrel{{\scriptstyle\hat{\rho}}}{{\mapsto}}\left(\frac{x-(x\bmod{4})}{4},\frac{y+(x\bmod{4})}{4}\right)\in\mathbb{Z}_{2}\times[0,1)$

and two systems are isomorphic both as topological and measure theoretical
dynamics:

(15)
$\begin{CD}\mathbb{Z}_{2}\times[0,1)@>{\hat{\rho}}>{}>\mathbb{Z}_{2}\times[0,1)\\\
@V{\phi\times\phi^{\prime}}V{}V@V{\phi\times\phi^{\prime}}V{}V\\\
<EMAIL_ADDRESS>
where $\phi^{\prime}$ is given as:

$\sum_{i=1}^{\infty}x_{i}4^{-i}\mapsto g_{x_{1}}(g_{x_{2}}(g_{x_{3}}(\dots)))$

where
$g_{0}(x)=(x-u_{0})/\omega^{2},g_{1}(x)=(x-u_{2})\zeta/\omega^{2},g_{2}(x)=(x-u_{3})\zeta^{-1}/\omega^{2}$
and $g_{3}(x)=(x-u_{5})/\omega^{2}$.

From this ‘algebraic’ natural extension construction, we can characterize
purely $S$-periodic points in $Y^{\prime}\cap\mathbb{Q}(\zeta)$.

######  Theorem 6.

A point $y$ in $Y^{\prime}\cap\mathbb{Q}(\zeta)$ has purely periodic
multiplicative coding with four digits
$\sigma_{0},\sigma_{2},\sigma_{3},\sigma_{5}$ if and only if $(y,\phi(y))\in
Y^{\prime}\times\mathcal{Y}$.

This is an analogy of the results [14] for $\beta$-expansion. The proof below
is on the same line.

###### Proof.

As $\omega$ is an algebraic unit and $d_{i}\in\mathbb{Z}[\zeta]$, the
denominator of $g_{i}(y)$ is the same as that of $y$ for $i=0,1,2,3$.
Therefore the module $y\in\mathcal{M}=\frac{1}{M}\mathbb{Z}[\zeta]$ is stable
by $g_{i}$ for some positive integer $M$. Note that points $y\in\mathcal{M}$
with $(y,\phi(y))\in Y^{\prime}\times\mathcal{Y}$ is finite, because
$y,\phi(y)$ and their complex conjugates are bounded in $\mathbb{C}$. One can
confirm that the map $\hat{S}$ becomes surjective from $\mathcal{M}$ to
itself. For a finite set, surjectivity implies bijectivity. Therefore a point
$y\in\mathcal{M}$ with $(y,\phi(y))\in Y^{\prime}\times\mathcal{Y}$ produces a
purely periodic orbit. On the other hand if $x$ has purely periodic
multiplicative coding, it is easy to see $(y,\phi(y))\in
Y^{\prime}\times\mathcal{Y}$. ∎

##  3\. Other self-similar systems

Pisot scaling constants appear in several important dynamics. For irrational
rotations (2IET), it is well known that scaling constants of self-inducing
systems must be quadratic Pisot units. A typical example Figure 2 was shown in
the introduction. They are computed by the continued fraction algorithm as
fundamental units of quadratic number fields. Poggiaspalla-Lowenstein-Vivald
[24] showed that the scaling constant must be an algebraic unit for self-
inducing uniquely ergodic IET. When the scaling constant of self-inducing IET
is a cubic Pisot unit, we have further nice properties [4, 19, 20].

A necessary condition that $1$-dimensional substitutive point sets give point
diffraction is that the scaling constant is a Pisot number [7]. Suspension
tiling dynamics of such substitution is conjectured to have pure discrete
spectrum if the characteristic polynomial of its substitution matrix is
irreducible. For higher dimensional tiling dynamics the Pisot (or Pisot
family) property is essential to have relatively dense point spectra, see for
e.g. [27, 17].

Pisot scaling properties seem to extend to the case of piecewise isometries.
To conclude we present some examples, though we do not make a systematic
study.

It is already observed in [16, 3] that Pisot scaling constants appear in our
problem if $\theta$ is the $n$-th root of unity for $n=4,6,8,10,12$ in the
same way as we did in $n=5$ but in a more involved manner. In each case they
are quadratic Pisot units. What about if $\lambda=-2\cos(\theta)$ is cubic? In
this case, the dynamics of Conjecture 1 are embedded into the piecewise affine
mapping acting on $(\mathbb{R}/\mathbb{Z})^{4}$ which is harder to visualize.
Instead let us consider formal analogies of piecewise isometries generated by
cubic $n$-th fold rotation in the plane. At the expense of losing connection
to Conjecture 1, we find many Pisot unit scaling constants! Being an algebraic
unit is natural and may be explained from invertibility of dynamics. However
we have no idea why the Pisot numbers turn up or even how to formulate these
phenomena as a suitable conjecture.

###  3.1. Seven-fold

We start with 7-fold case. Both pieces are rotated clockwise by $4\pi/7$ as in
Figure 17. The triangle is rotated around A and the trapezium around B. The
first return map to a region and a smaller region with the same first return
map (up to scaling) are described. Unlike the five fold case, returning to the
subregion does not cover the full region. A simple consequence is that there
are infinitely many possible orbit closures for non-periodic orbits in the
system. The scaling constant $\alpha\approx 5.04892$ is a Pisot number whose
minimal polynomial is $x^{3}-6x^{2}+5x-1$. Figure 18 shows how this remaining
space can be filled in. As this region is already a little small we will zoom
in and now consider just this induced sub-system in Figure 19. The smaller
substitutions are easier to see as there are two scalings giving the same
dynamics (A and B). The scaling constant $\beta\approx 16.3937$ for these
subregions is the Pisot number associated to $x^{3}-17x^{2}+10x-1$. The proof
that the remaining substitutions work is shown in Figure 20. The first return
map to the two lower triangles is shown. The same dynamics occur on a smaller
region. The orbit of the smaller region covers all the regions left out of
Figure 19 and so the substitution rule from that figure is now complete. The
scaling constant for this triangle is $\alpha$. This gives an example of
recursive tiling structure by Lowenstein-Kouptsov-Vivaldi [18]. Knowing that
every aperiodic orbits are in one of the above self-inducing structures, we
can show that

######  Theorem 7.

Almost all points of this $7$-fold lozenge have periodic orbits.

The argument is similar to that given around Figure 7(a). We easily find
decreasing series $X_{n}$ of union of polygons satisfying $\mu(\alpha
X_{n+1})<\alpha^{2}\mu(X_{n})$ (or $\mu(\beta X_{n+1})<\beta^{2}\mu(X_{n})$)
which cover all self-inducing structures.

The fundamental units of the maximal real subfield $\mathbb{Q}(\cos(4\pi/7))$
of the cyclotomic field $\mathbb{Q}(\zeta_{7})$ are given by $b$ and $b-1$
where $b=1/(2\cos(3\pi/7))\approx 2.24698$. Here $b$ is the Pisot number
satisfying $x^{3}-2x^{2}-x+1$. We see that $\alpha=b^{2}$ and
$\beta=b^{4}/(b-1)^{2}$ and thus $\alpha$ and $\beta$ generates a subgroup of
fundamental units of $\mathbb{Q}(\cos(4\pi/7))$. Note that both
$\sqrt{\alpha}=b$ and $\sqrt{\beta}=b^{2}/(b-1)$ are Pisot numbers but $b-1$
is not. Our piecewise isometry somehow selects Pisot units out of the unit
group!

Figure 17. A seven-fold piecewise isometry. Figure 18. The regions remaining
from the self-similarity shown in Figure 17 Figure 19. The substitution rule
of the induced subsystem shown in Figure 18. Figure 20. The final pieces of
the structure of the piecewise isometry found in Figure 17.

###  3.2. Nine-fold

The next example is 9-fold case in Figure 21. Both pieces are rotated anti-
clockwise by $4\pi/9$, the triangle around A and the trapezium around B. The
first return map ($\triangle$) to the triangle is also shown. In addition the
same dynamics are found on a smaller piece of the map. Like the 7-fold shown
in Figure 17 this does give a full description of the dynamics, but it is
$\triangle^{2}$ not $\triangle$. The scaling constant $\gamma\approx 8.29086$
is a Pisot unit defined by $x^{3}-9x^{2}+6x-1$. Unfortunately in this case we
were not able to find a complete description of the scaling structure.

The fundamental units of $\mathbb{Q}(\cos(4\pi/9))$ are $b$ and $b^{2}-2b-1$
where $b=1/(2\cos(4\pi/9))\approx 2.87939$ is a Pisot number given by
$x^{3}-3x^{2}+1$. We have $\gamma=b^{2}$ and are expecting to find another
Pisot unit $b^{2}/(b^{2}-2b-1)\approx 5.41147$ (or its square) as a scaling
constant in this dynamics, which would give an analogy to the seven-fold case.

Figure 21. A nine fold piecewise isometry.

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# A Partial Reproduction of A Guided Genetic Algorithm for Automated Crash
Reproduction

Philip Oliver, Michael Homer, Jens Dietrich, and Craig Anslow  School of
Engineering and Computer Science
Victoria University of Wellington
Wellington, New Zealand
Email: {philip.oliver, michael.homer, jens.dietrich<EMAIL_ADDRESS>
###### Abstract

This paper is a partial reproduction of work by Soltani _et al._ which
presented EvoCrash, a tool for replicating software failures in Java by
reproducing stack traces. EvoCrash uses a guided genetic algorithm to generate
JUnit test cases capable of reproducing failures more reliably than existing
coverage-based solutions. In this paper, we present the findings of our
reproduction of the initial study exploring the effectiveness of EvoCrash and
comparison to three existing solutions: STAR, JCHARMING, and MuCrash. We
further explored the capabilities of EvoCrash on different programs to check
for selection bias. We found that we can reproduce the crashes covered by
EvoCrash in the original study while reproducing two additional crashes not
reported as reproduced. We also find that EvoCrash was unsuccessful in
reproducing several crashes from the JCHARMING paper, which were excluded from
the original study. Both EvoCrash and JCHARMING could reproduce 73% of the
crashes from the JCHARMING paper. We found that there was potentially some
selection bias in the dataset for EvoCrash. We also found that some crashes
had been reported as non-reproducible even when EvoCrash could reproduce them.
We suggest this may be due to EvoCrash becoming stuck in a local optimum.

###### Index Terms:

Automated crash reproduction, empirical software engineering, genetic
algorithms, reproduction, search-based software testing.

##  I Introduction

When software failures occur, developers must manually investigate stack
traces and other post-crash information to understand and then replicate the
behaviour. Several tools aim to automate reproducing crashes; Tools such as
STAR, JCHARMING, and MuCrash leverage information produced from a crash to
create new unit tests to reproduce the crashes [1, 2, 3]. However, there are
issues with these tools: STAR cannot handle cases that have external
environment dependencies and is affected by the path explosion problem [1];
MuCrash mutates an existing test suite, so has some reliance on existing tests
exploring method sequences of interest [3]; and JCHARMING applies
computationally expensive model checking [2].

Soltani _et al._ presented EvoCrash, a tool using an evolutionary approach
that leverages a stack trace to reduce the search space [4].
EvoCrash111http://www.evocrash.com uses the automatic test generation tool,
EvoSuite222http://www.evosuite.org, to generate tests. EvoCrash is an altered
version of EvoSuite, which incorporates a novel fitness function developed by
Soltani _et al._ This fitness function is a piece-wise function that checks:
the target line number is reached, the correct exception is thrown, and the
generated stack trace is similar enough to the original trace [5]. The
function is a measure of error and gives a value of 0 when the stack traces
match. This fitness function is used in a guided genetic algorithm to generate
tests to replicate stack traces from software crashes.

The guided genetic algorithm uses three genetic operators developed by the
original authors. The first generates an initial population of tests, while
the remaining two are altered crossover and mutation operations. These
operators ensure a call to a method within the stack trace contained in each
unit test in the search population.

We looked to evaluate the effectiveness of EvoCrash based upon the original
paper presented by Soltani  _et al._ [4]. We further extended the suite of
crashes used for evaluation to check for selection bias. Finally, we present
some evaluation of discrepancies in the results. A package containing the
supporting data from the original study and our experiments can be found at
https://doi.org/10.5281/zenodo.5139193.

##  II Original Study

The authors of the original paper [4] use EvoCrash to conduct an empirical
study with the following two Original Research Questions:

  * •

ORQ1: In which cases can EvoCrash successfully reproduce the targeted crashes,
and under what circumstances does it fail to do so?

  * •

ORQ2: How does EvoCrash perform compared to state-of-the-art reproduction
approaches based on stack traces?

The initial study was conducted over 50 bugs from Apache Commons
Collections333https://commons.apache.org/proper/commons-collections/ (ACC),
Apache Ant444http://ant.apache.org (ANT), and Apache
Log4j555http://logging.apache.org/log4j/1.2 (LOG) [4]. The generation of tests
for each bug was repeated 50 times to account for the random nature of the
guided genetic algorithm. Soltani _et al._ selected widely used parameter
values for the evolutionary component of EvoCrash:

  * •

Population size: Initially set to 50, but increased by 25 iteratively up to
300 if the fitness value does not reach 0.0.

  * •

Crossover probability: Set to 0.75.

  * •

Mutation probability: Set to 1/n, with n being the length of the current test
case.

  * •

Search timeout: Set to 30 minutes with early stopping if the fitness value
reaches 0.0.

Regarding the mutation probability, we cross-referenced the paper referenced
in the original study. Fraser and Arcuri state the mutation probability in
EvoSuite is 1/n, with n being the size of the _test suite_ [6]. This
probability results in one test case in the suite being mutated on average,
rather than one statement in a test case being altered, as reported by Soltani
_et al._ It is not clear if Soltani _et al._ have altered the mutation
probability as such in EvoCrash.

Soltani _et al._ selected two metrics for ORQ1 proposed by Chen and Kim [1].
_Crash Coverage_ ensures that the crash has been successfully replicated by
comparing the exception type thrown and the source line from which it is
thrown. The original authors consider a crash to be covered when a fitness
value of 0.0 is reached. _Test Case Usefulness_ concludes that a test case is
useful if it reveals the bug which caused the crash. Two of the original
authors independently performed manual validation to decide if the test cases
produced by EvoCrash successfully reveal the bug. In the case of
disagreements, the conclusions were discussed. We do not assess the test case
usefulness as a part of our reproduction. This omission is primarily due to
the subjective nature of this metric.

ORQ2 was investigated through comparison with three other crash reproduction
technologies: STAR, MuCrash, and JCHARMING. Soltani _et al._ used published
data from these tools, as the artifacts for the tools were unavailable at the
time of writing. The comparison to STAR was completed using 50 of the 52 bugs
collected by Chen and Kim [1]. EvoCrash was compared to JCHARMING using 8 of
the 20 bugs collected by Nayrolls  _et al._ [2]. Finally, the comparison to
MuCrash was performed using the 12 ACC bugs collected for testing STAR _et
al._ [3]. Several bugs were excluded from the original study.

EvoCrash Performance ORQ1. The original paper presents results for ORQ1, with
EvoCrash successfully replicating 41 of the 50 (82%) bugs [4]. EvoCrash
reproduced 10 out of 12 bugs for ACC, 14 out of 20 for ANT, and 17 out of 18
for LOG. EvoCrash does not support the six unreproducible cases for ANT due to
dependencies on missing external build.xml files. One of the cases from LOG is
unsupported due to a call to a static class initialiser. The two
unreproducible cases for ACC are due to the complexity of the bugs. Using the
_Test Case Usefulness_ criteria from Chen and Kim, the original authors
conclude that 34 of the 39 generated test cases were useful. The other 5 test
cases mainly were found to have dependencies on external files, which were not
available. In this study, we do not explore the usefulness of the test cases
generated by EvoCrash.

It is unclear what threshold the original authors have used to discern whether
a bug has been replicated. They state that “of the replicated cases, the crash
LOG-509 had the lowest rate of replications - 39 out of 50,” with these
numbers being 39 replications of the crash over 50 runs [4]. However, they
also state that for one of the non-reproducible cases (ACC-104), “EvoCrash
could replicate the case 4 times out of 50.” While this is a complex bug that
requires a specific order of method calls to trigger the crash, it would
appear that EvoCrash can successfully replicate the behaviour, albeit
occasionally.

Comparison to Other Tools ORQ2. Compared with STAR, EvoCrash has almost
identical results, except for ACC-104 (discussed above) [4]. EvoCrash is also
capable of replicating three additional cases which are prone to the path
explosion problem. Compared with MuCrash, EvoCrash can replicate all the
crashes replicated by MuCrash and an additional 3 cases, with one of these
cases marked as not useful. EvoCrash covers all the crashes successfully
reproduced by JCHARMING (6 out of 8) and can reproduce the two crashes
JCHARMING cannot. However, 3 of the test cases from EvoCrash are marked as not
useful, with two being crashes JCHARMING could reproduce. Nayrolles _et al._
do not identify if crashes reproduced by JCHARMING are useful; therefore, it
could be that the non-useful tests generated by EvoCrash are also not useful
when generated with JCHARMING [2].

##  III Reproduction

For the reproduction in this study, we performed two experiments using the
publicly available reproduction package666https://github.com/STAMP-
project/EvoCrash/releases/tag/evocrash-refactored for the original paper. The
first experiment was run using the parameters and configuration as-is from the
package. Following the further investigation into the parameters used in the
package, we found that some did not match what was reported in the original
paper for population sizes. Many population sizes were initialised at 80,
which does not follow the experiment procedure outlined in the initial study.
We increased these to the next largest population size that fit the procedure
for population sizes that did not conform to the experimental procedure. The
experiment was rerun using these updated parameters. In the second experiment,
we followed the initial study’s guidelines to increase the population sizes by
25 repeatedly up to 300 for crashes which cannot be reproduced. All other
parameters used match the experiment procedure from the initial study.

There were a few issues when beginning the reproduction. Firstly, the website
for the package location in the original paper no longer exists. This issue
was circumvented by finding the publicly available release package on GitHub.
The second issue was that the scripts used to run EvoCrash for the 50 crashes
were not OS-agnostic. Classpath separators had been hardcoded as semicolons
(;) for use on a Windows machine. These separators were changed to run the
experiment on Arch linux successfully. Thirdly, some of the paths for the
binaries for the targeted programs were incorrect. For example, there were a
few cases of the LOG4jb-1.0.4/ directory being referenced as Log4jb-1.0.4/.
These paths were fixed for the experiment. Another issue was that some of the
results from the original study were missing from the reproduction package.
The 30th run is missing most of the results, while some other runs do not have
the results for some crashes. Finally, ACC-377 was missing from the crashes
and results in the reproduction package. This crash was added to the
experiment to ensure similarity between the original experiment and the
reproduction.

After replicating the main results from the study, we looked to evaluate
EvoCrash on some other crashes, including those from the STAR and JCHARMING
papers which were excluded from the original study. We also selected 7 crashes
from Apache Commons Lang777https://commons.apache.org/proper/commons-lang/
(ACL) and 6 crashes from Apache Commons
BeanUtils888https://commons.apache.org/proper/commons-beanutils/ (BEAN) to
check for selection bias in the initial dataset.

TABLE I: Results from original paper and reproduction. Percentages of 100% are
not reported for brevity

Project | Bug ID | Original | Experiment 1 | Experiment 2
---|---|---|---|---
ACC | 4 | Y | Y | Y
28 | Y | Y | Y
35 | Y | Y | Y
48 | Y | Y | Y
53 | Y | Y | Y
68 | N (0%) | N (0%) | N (0%)
70 | Y | Y (100%) | Y (98%)
77 | Y | Y | Y
104 | N (8%) | Y (2%) | Y (8%)
331 | Y (82%) | Y (52%) | Y (88%)
377 | Y | Y (90%) | Y (60%)
441 | Y | Y | Y
ANT | 28820 | N (0%) | N (0%) | N (0%)
33446 | Y | Y | Y
34722 | Y | Y | Y
34734 | Y | Y | Y
36733 | Y | Y | Y
38458 | Y (92%) | Y (90%) | Y (90%)
38622 | Y (80%) | Y (86%) | Y (82%)
42179 | Y | Y | Y
43292 | N (94%) | Y (96%) | Y
44689 | Y | Y | Y
44790 | Y | Y | Y
46747 | N (0%) | N (0%) | N (0%)
47306 | N (0%) | N (0%) | N (0%)
48715 | N (0%) | N (0%) | N (0%)
49137 | Y | Y | Y
49755 | Y (94%) | Y | Y
49803 | Y | Y | Y (98%)
50894 | Y | Y | Y
51035 | N (0%) | N (0%) | N (0%)
53626 | Y | Y | Y
LOG | 29 | Y (88%) | Y (90%) | Y (96%)
43 | N (0%) | N (0%) | N (0%)
509 | Y (74%) | Y (50%) | Y (78%)
10528 | Y | Y | Y
10706 | Y | Y | Y
11570 | Y | Y | Y
31003 | Y | Y | Y
40212 | Y | Y | Y
41186 | Y | Y | Y
44032 | Y | Y | Y
44899 | Y | Y | Y
45335 | Y (94%) | Y (94%) | Y (96%)
46144 | Y (82%) | Y (78%) | Y (86%)
46271 | Y (94%) | Y | Y
46404 | Y | Y | Y
47547 | Y | Y | Y
47912 | Y | Y | Y
47957 | Y | Y | Y

Y - Crash has been replicated at least once
N - Crash has not been replicated
Percentage values are the number of successful replications from 50 runs

###  III-A Experimental Results

Table I presents the original study’s results alongside the results we have
achieved over our two runs of the experiment. It can be seen that our results
are mainly similar to those in the original study, with two notable
exceptions: ACC-104 and ANT-43292. As previously discussed, ACC-104 is
successfully reproduced by EvoCrash in the original study, albeit at a rate of
8%. In our experimental runs, we achieved success rates of 2% and 8%. The
original authors were looking to answer the research question of whether
EvoCrash could reproduce a crash. We argue that even a single success means
EvoCrash can reproduce the crash. We further argue that a low reproduction
rate could indicate issues within the initialisation of the genetic
programming parameters. It could be possible that EvoCrash becomes stuck in a
local optimum with not enough mutation occurring to allow the program to find
a better test case.

In the case of ANT-43292, the original study marked this crash as not
reproduced. We found 96% and 100% success rates for this crash in our
experiments. On closer inspection of the data from the original study, we
found that the crash was successfully reproduced. In the underlying data, we
found 47 successful reproductions, with two failures and one unreported
result. This data gives a success rate of 94% for ANT-43292 in the original
study. It could be that the original authors meant to mark this crash as _not
useful_. However, we do not confirm that this is the case.

TABLE II: Results from crashes excluded from original study, including
comparison to STAR and JCHARMING

Project | Bug ID | Results | STAR | JCHARMING
---|---|---|---|---
DnsJava | 38 | N (0%) | - | Y
Jfreechart | 434 | Y (98%) | - | Y
664 | N (0%) | - | Partial
916 | N (0%) | - | Y
Pdfbox | 1359 | N (0%) | - | N
1412 | Y (94%) | - | Partial
ANT | 41422 | Y (100%) | Y | N

Y - Crash has been replicated at least once
N - Crash has not been replicated
Percentage values are the number of successful replications from 50 runs

Table II presents the results of crashes from
DnsJava999https://github.com/dnsjava/dnsjava,
Jfreechart101010https://www.jfree.org/jfreechart/,
Pdfbox111111https://pdfbox.apache.org/, and ANT which were used in the STAR
and JCHARMING papers [1, 2]. In the JCHARMING paper there were also crashes
used from ArgoUML121212https://github.com/argouml-tigris-org and Open Mission
Control Software131313https://nasa.github.io/openmct/ [2]. However, we could
not find the stack traces for these crashes and thus have not included them.
The crashes excluded from the original study do not have a high success rate
of reproduction by EvoCrash, with 4 of the 13 crashes reproduced (43%). If
these crashes were included in the original study, EvoCrash would have
reproduced 44 out of 57 crashes (77%), rather than the 82% reported [4].

Table II also shows the comparison of the crashes excluded from the original
study to STAR and JCHARMING. The main comparisons here are between EvoCrash
and JCHARMING for the DnsJava, Jfreechart, Pdfbox, and ANT crashes. Of these
seven crashes, JCHARMING reproduced 5, while EvoCrash reproduced 3. Of the 15
crashes shared by EvoCrash and JCHARMING, JCHARMING successfully reproduced 11
(73%), while EvoCrash also reproduced 11 (73%). It is of particular interest
that JCHARMING is capable of reproducing DnsJava-38, Jfreechart-664, and
Jfreechart-916 where EvoCrash cannot. The original study found a significant
difference between the performances of EvoCrash and JCHARMING [4]. However, it
is clear that with other crashes from the JCHARMING paper, the performance is
similar.

Table III presents the results of our evaluation on crashes from Apache
Commons Lang and Apache Commons BeanUtils. We have selected these crashes to
identify any potential for selection bias in the original study. Of the ACL
crashes, EvoCrash successfully reproduced 4 out of 7 (57%). The three failing
tests use date formats or message formats, which require specifically
formatted strings as input. It is therefore unsurprising that EvoCrash cannot
reproduce these crashes, as it has not been created with the capability of
consistently generating strings that match the complex formats required by
these classes. Finally, given the complexity of configuration required to use
BeanUtils in a program, the 0% success rate is unsurprising. As most of these
BEAN crashes arise due to configuration issues, EvoCrash struggles to generate
a test case to initialise such a configuration.

##  IV Discussion

The authors of the original paper set out to evaluate the tool, EvoCrash, on
several crashes and to compare these results with the existing tools: STAR,
MuCrash, and JCHARMING [4]. The original study successfully reproduced 41 of
50 (82%) crashes. We found that EvoCrash can successfully reproduce all the
crashes presented in the original study through our two main experiments. We
also found two crashes (ACC-104 and ANT-43292) which we reproduced with
EvoCrash, but are not reported as reproduced in the original study. We found
in the data underlying the original study that ANT-43292 has a 94%
reproduction rate, while our experiments have 96% and 100% reproduction rates.
This misidentified result in the original study likely occurred due to human
error. We consider a crash to be reproduced if it can be successfully
reproduced in at least one run. Crashes with low reproduction rates could
point to issues in the genetic parameters for EvoCrash, as there may not be
enough variability introduced to allow EvoCrash to escape local optima.

TABLE III: Results from additional crashes used in this study

Project | Bug ID | Results
---|---|---
ACL | 948 | N (0%)
1186 | N (0%)
1192 | N (0%)
1276 | Y (100%)
1292 | Y (100%)
1310 | Y (86%)
1385 | Y (100%)
BEAN | 276 | N (0%)
302 | N (0%)
351 | N (0%)
421 | N (0%)
541 | N (0%)
547 | N (0%)

Y - Crash has been replicated at least once
N - Crash has not been replicated
Percentage values are the number of successful replications from 50 runs

We present a comparison between EvoCrash, STAR, and JCHARMING for crashes
excluded from the original study. For ANT-41422, EvoCrash and STAR could both
successfully reproduce this crash; however, JCHARMING could not. For the other
crashes from DnsJava, Jfreechart, and Pdfbox, we found that JCHARMING
outperforms EvoCrash, contrasting with the original result that EvoCrash
outperformed JCHARMING. We find that EvoCrash and JCHARMING both reproduce 73%
of crashes once the full JCHARMING dataset is used. This result could
potentially point to some selection bias in the original study, as these
crashes were excluded. As JCHARMING, STAR, and MuCrash are not publicly
available, selection bias could be present in the dataset chosen for those
studies.

While we do not analyse the usefulness of the test cases generated by
EvoCrash, we did consider the suitability of the metric for this. The metric
requires that the buggy stack frame exists in the reproduced stack trace. A
number of the crashes reproduced by EvoCrash are attempting to reproduce only
one stack frame in a larger stack trace. A potential question for future work
is raised: whether this metric is suitable and if the crashes can be
considered reproduced if this metric is met. Furthermore, this metric is
subjective and cannot be easily reproduced. Comparisons between the crashes
reproduced by EvoCrash and the actual bug fixes committed to the source code
could be drawn to clarify that the tests generated correctly identify a bug
and relate to the bug-fix in the main project.

We conclude that EvoCrash is a tool that can be used to reproduce several
crashes in Java successfully. However, we are not sure the data presented in
the original paper is representative of the capabilities of EvoCrash. Several
low-performing crashes appear to have been excluded from the original study,
including those which contribute significantly to the original paper’s
conclusion that EvoCrash performs significantly better than JCHARMING. We also
suggest there may be issues with the parametric setup of the genetic part of
EvoCrash, leading to low variability and the system becoming stuck in local
optima. Future work could look into these issues and the usefulness of the
test cases produced by EvoCrash.

## References

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