# Operator Growth in Open Quantum Systems

Thomas Schuster  Department of Physics, University of California, Berkeley,
California 94720 USA  Norman Y. Yao  Department of Physics, University of
California, Berkeley, California 94720 USA  Materials Science Division,
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

###### Abstract

The spreading of quantum information in closed systems, often termed
scrambling, is a hallmark of many-body quantum dynamics. In open systems,
scrambling competes with noise, errors and decoherence. Here, we provide a
universal framework that describes the scrambling of quantum information in
open systems: we predict that the effect of open-system dynamics is
fundamentally controlled by operator size distributions and independent of the
microscopic error mechanism. This framework allows us to demonstrate that open
quantum systems exhibit universal classes of information dynamics that
fundamentally differ from their unitary counterparts. Implications for
Loschmidt echo experiments and the classical simulability of open quantum
dynamics will be discussed.

Conventionally, the study of quantum many-body systems has focused on the
prediction of few-body observables, such as local correlation functions. More
recently, sparked by fundamental questions in quantum thermalization and chaos
Bohrdt  _et al._ (2017), the classical simulation of quantum systems
Verstraete  _et al._ (2004), and quantum gravity Harlow (2016), physicists
have turned to a complementary pursuit: quantifying the complexity of many-
body _dynamics_ itself.

At the heart of this pursuit is the notion of quantum information _scrambling_
; across nearly the entirety of interacting many-body quantum systems,
information encoded in initially local operators grows to become highly non-
local Lieb and Robinson (1972); Roberts  _et al._ (2015). Remarkably, recent
experimental advances have enabled the direct measurement of scrambling—a task
that, most commonly, utilizes backwards time-evolution Li  _et al._ (2017);
Gärttner  _et al._ (2017); Sánchez  _et al._ (2020, 2022); Domínguez  _et al._
(2021); Domínguez and Álvarez (2021); Mi  _et al._ (2021); Cotler  _et al._
(2022), but can also be performed using multiple copies of the system Islam
_et al._ (2015); Landsman  _et al._ (2019); Blok  _et al._ (2020) or
randomized measurements Brydges  _et al._ (2019); Joshi  _et al._ (2020). In
such systems, the interplay between scrambling dynamics, extrinsic
decoherence, and experimental noise motivates an essential question: What is
the nature of quantum information scrambling in _open_ quantum systems
Landsman  _et al._ (2019); Arute  _et al._ (2019); Alonso  _et al._ (2019); Mi
_et al._ (2021); Yoshida and Yao (2019); Swingle and Halpern (2018); Vermersch
_et al._ (2019); Zhang  _et al._ (2019); Agarwal and Bao (2020); Bao and
Kikuchi (2021); Touil and Deffner (2021); Andreadakis  _et al._ (2022); Harris
_et al._ (2022)?

In this Letter, we introduce a universal framework—based upon _operator size
distributions_ Nahum  _et al._ (2018); Roberts  _et al._ (2018); Qi and
Streicher (2019); Schuster  _et al._ (2021)—for capturing the effect of local
errors on scrambling dynamics. In particular, we conjecture that the
propagation of errors in chaotic many-body systems is fundamentally controlled
by the size distributions of time-evolved operators and independent of the
microscopic error mechanism. Our framework immediately offers predictions for
both the Loschmidt echo Loschmidt (1876); Goussev  _et al._ (2012) and out-of-
time-ordered correlation (OTOC) functions Larkin and Ovchinnikov (1969); Xu
and Swingle (2022). In particular, we predict that the decay of the Loschmidt
echo, which measures the fidelity associated with backwards time-evolution,
occurs at a rate proportional to the operator size. Meanwhile, we predict that
the decay of the OTOC, which measures the growth of local operators, is
inhibited by open-system dynamics (by an amount proportional to the width of
the operator size distribution).

Figure 1:  Left: Size distributions for three classes of systems under unitary
(blue) versus open (red) dynamics. Rightward arrows denote growth in time to
larger sizes, ticks denote a fixed size, and downward arrows denote loss of
probability at a given size. Right: Qualitative depiction of open-system
operator growth. In all cases, operators lose normalization due to open
dynamics (dark to light gray boxes). In the latter two classes, operators are
dominated by smaller size components compared to unitary evolution (smaller
boxes).

We leverage our framework to characterize operator growth in five distinct
classes of open quantum systems, which vary in their dimensionality, range of
interaction, conservation laws, and integrability (Table 1, Figs. 1–3). In
each class, our framework yields markedly distinct predictions for the
Loschmidt echo and OTOCs. We hypothesize that these results provide a
theoretical underpinning for recent nuclear magnetic resonance (NMR)
experiments Sánchez  _et al._ (2020, 2022); Domínguez  _et al._ (2021), and
also serve to resolve apparent disagreements between previous empirical
studies of open-system scrambling Vermersch  _et al._ (2019); Mi  _et al._
(2021); Zhang  _et al._ (2019). Finally, we propose and analyze a protocol for
measuring operator size distributions via engineered dissipation.

Table 1: Size distributions in various physical regimes System |  Unitary dynamics | Open dynamics
---|---|---
$(d\geq 1)$D, no | peaked-size | size unaffected
conservation law | $\overline{\mathcal{S}}\sim t^{d}$ | $\overline{\mathcal{S}}\sim t^{d}$
$(d\geq 1)$D, con- | bimodal | size decreases
served quantity | $\overline{\mathcal{S}}\sim t^{d}$ | $\overline{\mathcal{S}}\sim 1/t^{d}$
all-to-all | broad size | size plateaus
coupled (0D) | $\overline{\mathcal{S}}\sim e^{\lambda t}$ | $\overline{\mathcal{S}}\sim\lambda/\varepsilon$
$(d\geq 1)1$D, | broad size | size growth slowed
long-range | super-ballistic | $\overline{\mathcal{S}}\sim t^{d}$
free fermion | broad size | size decreases
integrable | $\overline{\mathcal{S}}\sim t$ | $\overline{\mathcal{S}}\sim 1/t$

_Operator size distributions._ —We begin with a simple example to build
intuition. Consider a lattice of qubits acted on by a series of local quantum
gates, each featuring some error $\varepsilon$, before measuring a local
operator $\hat{M}$. Noting that the measurement can only be influenced by
gates in its past light cone, a naive estimate of the measurement fidelity is
$\mathcal{F}\approx(1-\varepsilon)^{\mathcal{V}_{\text{LC}}}$, where
$\mathcal{V}_{\text{LC}}$ is the light cone volume, i.e. the number of gates
it contains Arute  _et al._ (2019); Mi  _et al._ (2021). This relation in fact
already contains the essential intuition underlying our work: a connection
between the measurement fidelity of a local operator and the operator’s growth
under Heisenberg evolution. By generalizing the light cone volume using
operator size distributions, we will show that this connection is
significantly richer and more universal than the above example suggests
Roberts  _et al._ (2015, 2018).

To introduce the notion of an operator size distribution, we first define the
size of a Pauli string, $\hat{R}$, as its number of non-identity elements; for
instance, $\hat{R}=Y\otimes\mathbbm{1}\otimes Z\otimes X$ has size
$\mathcal{S}_{R}=3$. From this, one can define the size superoperator:

$\mathcal{S}\\{\hat{\mathcal{O}}\\}\equiv-\sum_{P_{i}}(P_{i}\hat{\mathcal{O}}P_{i}^{\dagger}-\hat{\mathcal{O}})/4,$
(1)

which gives $\mathcal{S}\\{\hat{R}\\}=\mathcal{S}_{R}\hat{R}$, where
$\hat{P}_{i}\in\\{\hat{\mathbbm{1}}_{i},\hat{X}_{i},\hat{Y}_{i},\hat{Z}_{i}\\}$
are single-qubit Pauli operators Schuster  _et al._ (2021). More general
operators can be expressed as a sum of Pauli strings,
$\hat{\mathcal{O}}=\sum_{\hat{R}}c_{R}\hat{R}$, and thereby possess a _size
distribution_ ,
$P(\mathcal{S})=\sum_{\\{\mathcal{S}_{R}=\mathcal{S}\\}}|c_{R}|^{2}$, with
normalization
$\mathcal{N}=\langle\mathcal{O}^{\dagger}\mathcal{O}\rangle=\sum_{R}|c_{R}|^{2}$;
here, $\langle\cdot\rangle\equiv\Tr(\cdot)/\Tr(\mathbbm{1})$ represents the
infinite temperature expectation value. We note that the operator size
distribution is closely related to out-of-time-ordered correlation functions,
$\langle\hat{M}(t)\hat{V}_{j}\hat{M}(t)\hat{V}_{j}\rangle$. As an operator,
$\hat{M}(t)$, grows to have support on site $j$, the OTOC typically decays to
zero. From Eq. (1), one immediately sees that the average size of $\hat{M}(t)$
is directly proportional to unity minus the OTOC averaged over all single-
qubit Pauli operators:

$\overline{\mathcal{S}}=\frac{\langle\hat{M}(t)\mathcal{S}\\{\hat{M}(t)\\}\rangle}{\langle\hat{M}(t)\hat{M}(t)\rangle}=\frac{1}{4}\sum_{P_{i}}\left(1-\frac{\langle\hat{M}(t)P_{i}\hat{M}(t)P_{i}\rangle}{\langle\hat{M}(t)\hat{M}(t)\rangle}\right).$
(2)

_Open-system operator growth hypothesis._ —Let us now turn to open quantum
systems. Operator evolution is typically governed by the Lindblad master
equation:

$\partial_{t}\hat{M}=i[\hat{H},\hat{M}]-\sum_{\alpha}\varepsilon_{\alpha}\left(\hat{L}_{\alpha}^{\dagger}\hat{M}\hat{L}_{\alpha}-\frac{1}{2}\\{\hat{L}^{\dagger}_{\alpha}\hat{L}_{\alpha},\hat{M}\\}\right).$
(3)

The first term describes unitary time-evolution, while the second describes a
sum of local error processes, each characterized by a Lindblad operator,
$\hat{L}_{\alpha}$, and an associated error rate, $\varepsilon_{\alpha}$.

Our central conjecture is that the effects of local errors on operator growth
are in fact captured by a much simpler, effective Lindblad equation:

$\partial_{t}\hat{M}=i[\hat{H},\hat{M}]-\varepsilon\mathcal{S}\\{\hat{M}\\},$
(4)

where $\mathcal{S}$ is the size superoperator. In effect, this model replaces
the original Lindblad operators with isotropic decoherence at each qubit [Eq.
(1)].

This conjecture is rooted in the following intuition—higher size operators are
affected by a greater number of local error processes, and thus decohere at a
faster rate fn (3). More specifically, we expect large-size components of
$\hat{M}$ to typically involve exponentially many Pauli strings varying
rapidly in time. This serves to “average” the effect of Lindblad operators
such that their action depends solely on whether they are in the support of a
given size component of $\hat{M}$, independent of their precise microscopic
form. The number of Lindblad operators in the support is directly proportional
to the size.

Our framework predicts two effects of open-system dynamics on operator growth,
which are captured by the behavior of the Loschmidt echo and the average OTOC
(i.e. the average operator size), respectively. For the former, we note that
the Loschmidt echo fidelity with respect to a local operator is in fact equal
to the _normalization_ of the operator’s size distribution,
$\mathcal{N}(t)=\langle\hat{M}(t)\hat{M}(t)\rangle=\int
d\mathcal{S}P(\mathcal{S})$. Our framework predicts fn (5) that the Loschmidt
echo decays in time at a rate equal to the average size multiplied by the
error rate:

$\partial_{t}\log\mathcal{N}(t)=-2\varepsilon\overline{\mathcal{S}}(t).$ (5)

Turning to the OTOC, we note that errors decrease the amplitude of large-size
components of $\hat{M}(t)$ at a faster rate than small-size components. Thus,
compared to purely unitary evolution, open-system dynamics inhibit the growth
of operators. More specifically, we predict that the average size,
$\overline{\mathcal{S}}$ [related to the OTOC via Eq. (2)], evolves according
to:

$\partial_{t}\overline{\mathcal{S}}(t)=(\text{unitary})-2\varepsilon\delta\mathcal{S}(t)^{2}.$
(6)

Here, the first term captures the specific unitary dynamics of the system,
while the second term decreases the size at a rate proportional to the
variance of the size distribution, $\delta\mathcal{S}^{2}$.

_Open-system scrambling dynamics._ —We now apply our framework to five
distinct classes of scrambling dynamics (Table 1, Figs. 1–3): local and all-
to-all interacting systems without conservation laws, local systems _with_
conservation laws, long-range interacting systems, and free fermion integrable
systems (see supplemental materials for a detailed discussion of the latter
two cases sup ).

We begin by demonstrating that operator growth in two paradigmatic
scramblers—systems with no conserved quantities under local and all-to-all
interactions—are affected by open-system dynamics in drastically different
ways. For the former (focusing on 1D systems for specificity), one expects
operators to grow ballistically in time under unitary dynamics, with
$\overline{\mathcal{S}}\approx\frac{3}{2}v_{B}t$, where $v_{B}$ is the
butterfly velocity. Meanwhile, the width of the operator size distribution
grows “diffusively”, $\delta\mathcal{S}\approx c\sqrt{v_{B}t}$ where $c$ is a
constant Nahum  _et al._ (2018); Von Keyserlingk  _et al._ (2018); Schuster
_et al._ (2021).

Combining these expectations via Eq. (6), we arrive at a simple
phenomenological equation for operator growth under _open-system_ dynamics,
$\partial_{t}\overline{\mathcal{S}}\approx\frac{3}{2}v_{B}-\varepsilon(c\sqrt{v_{B}t})^{2}$,
whose solution yields the prediction:
$\overline{\mathcal{S}}(t)\approx\frac{3}{2}v_{B}t-\frac{c}{2}\varepsilon
v_{B}t^{2}$. From Eq. (5), the Loschmidt echo fidelity thus decays as a
Gaussian in time,
$\mathcal{N}(t)=\exp\big(-\varepsilon\int_{0}^{t}dt^{\prime}\,\overline{\mathcal{S}}(t^{\prime})\big{missing})\approx\exp(-\frac{3}{4}\varepsilon
v_{B}t^{2})$, to leading order in $\varepsilon$. To explore these predictions,
we numerically simulate the open-system dynamics of a 1D random unitary
circuit (RUC) Nahum  _et al._ (2017, 2018); sup . As depicted in Fig. 2(a), we
find that both the operator size and the Loschmidt echo fidelity (solid lines)
agree remarkably well with our phenomenological predictions (dashed lines)
across multiple orders of magnitude in the error rate.

In contrast, in all-to-all interacting systems, unitary dynamics typically
exhibit “fast scrambling” characterized by the exponential growth of operator
size in time, $\overline{\mathcal{S}}\sim\\!e^{\lambda t}$, where $\lambda$ is
the Lyapunov exponent Sachdev and Ye (1993); Kitaev (2015); Maldacena and
Stanford (2016); Kobrin  _et al._ (2020); Schuster  _et al._ (2021); Bentsen
_et al._ (2019). Unlike local systems, the size distribution is also extremely
broad, $\delta\mathcal{S}\approx b\mathcal{S}$ where $b$ is a constant, owing
to the exponential growth of early-time fluctuations Roberts  _et al._ (2018);
Qi and Streicher (2019); Schuster  _et al._ (2021). Solving Eq. (6), i.e.
$\partial_{t}\overline{\mathcal{S}}\approx\lambda\overline{\mathcal{S}}-\varepsilon
b^{2}\overline{\mathcal{S}}\hskip 0.28453pt^{2}$, then yields an intriguing
prediction: under open-system dynamics, the average operator size _plateaus_
to a system-size independent value,
$\overline{\mathcal{S}}_{p}\approx\lambda/(\varepsilon b^{2})$, after a time
$t_{p}\sim\log(\lambda/(\varepsilon b^{2}))$. This causes the Loschmidt echo
to approach a constant rate of decay, $\mathcal{N}(t)\sim\exp(-\lambda
t/b^{2})$. Notably, the decay rate, $\lambda/b^{2}$, is _independent_ of the
microscopic error rate, $\varepsilon$, echoing seminal results in single-
particle quantum chaos Jalabert and Pastawski (2001) and tantalizing recent
NMR experiments Sánchez  _et al._ (2020, 2022); Domínguez  _et al._ (2021). As
shown in Fig. 2(b), both of these predictions are indeed born out by RUC
simulations.

Figure 2: (a) Average operator size, $\overline{\mathcal{S}}$, and the
Loschmidt echo fidelity, $\mathcal{N}$, in a 1D RUC with $N=200$ sup . The
size grows ballistically with quadratic corrections due to open-system
dynamics (solid, data; dashed, theory). Inset: The Loschmidt echo fidelity,
$\mathcal{N_{*}}$, when
$\frac{d\overline{\mathcal{S}}}{dt}=0.9\frac{d\overline{\mathcal{S}}}{dt}\big{|}_{\varepsilon=0}$,
decays exponentially in the inverse error rate, $\varepsilon^{-1}$. (b) All-
to-all RUC with $N=1500$ sup . The size grows exponentially before plateauing
to a value which is independent of the system size. The decay rate of the
Loschmidt echo is independent of $\varepsilon$ after plateauing (solid, data;
dashed, theory). Inset: The Loschmidt echo fidelity, $\mathcal{N_{*}}$, when
$\frac{d\log\overline{\mathcal{S}}}{dt}=0.9\frac{d\log\overline{\mathcal{S}}}{dt}\big{|}_{\varepsilon=0}$,
is constant with respect to $\varepsilon$.

One can further sharpen the distinction between open-system dynamics for local
versus all-to-all interactions, by analyzing their behavior at asymptotically
small error rates. Specifically, consider the value of the Loschmidt echo,
$\mathcal{N_{*}}$, at a time when the open-system dynamics have substantially
deviated from the unitary dynamics. In all-to-all systems, this occurs shortly
after the plateau time, $t_{p}$, which gives an order one Loschmidt echo,
$\mathcal{N}(t_{p})\approx\exp\big(-\varepsilon\int_{0}^{t_{p}}dt^{\prime}\,e^{\lambda
t^{\prime}}\big{missing})\approx\exp(-1/b^{2})$, independent of the error rate
[inset, Fig. 2(b)].

In contrast, in 1D systems this occurs when $(\varepsilon
v_{B}t^{2}/v_{B}t)\sim 1$, at which point the Loschmidt echo has decayed to an
_exponentially_ small value, $\mathcal{N}(t)\sim\exp(-v_{B}/\varepsilon)$
[inset, Fig. 2(a)]. For small error rates, this implies that large deviations
in operator growth are in practice unobservable for locally-interacting
systems, since the signal is exponentially small in $1/\varepsilon$.
Physically, this is a direct consequence of the asymptotic separation,
$\delta\mathcal{S}\ll\mathcal{S}$.

_Effects of conservation laws._ —We now show that the above behaviors are
strikingly modified when an operator has overlap with a conserved quantity,
$\hat{Q}=\sum_{i}\hat{q}_{i}$ (e.g. the total spin, or the Hamiltonian). Such
systems feature an interplay between hydrodynamics and scrambling, which is
embodied by a ‘bimodal’ profile for unitary time-evolved operators Khemani
_et al._ (2018); Rakovszky  _et al._ (2018):

$\hat{M}(t)=\sum_{i}q(i,t)\,\hat{q}_{i}+\sum_{\hat{R}\neq\hat{q}_{i}}c_{R}(t)\hat{R}.$
(7)

The operator contains both small-size components, $\hat{q}_{i}$, representing
the dynamics of the conserved quantity, as well as large-size Pauli strings,
$\hat{R}$, representing scrambled information.

The small-size components arise because an operator’s overlap with $\hat{Q}$,
$\langle\hat{M}(t)\hat{Q}\rangle=\int dx\,q(x,t)$, is conserved in time. As an
example, in chaotic 1D systems, one expects the local overlap,
$q(i,t)=\langle\hat{M}(t)\hat{q}_{i}\rangle$, to spread diffusively, which
causes the total normalization of the small-size components to decay in time,
$\int dx\,|q(x,t)|^{2}\sim\\!1/\sqrt{t}$. This in turn, implies that the total
normalization of the large-size Pauli strings is increasing in time;
physically, this corresponds to the dynamics of $q(i,t)$ ‘emitting’ chaotic
components, which spread ballistically from thereon. In combination, this
leads to a size distribution [Fig. 1],
$P(\mathcal{S})\approx\frac{1}{\sqrt{Dt}}\delta(\mathcal{S}-1)+\frac{v_{B}}{\sqrt{D}}(\frac{3}{2}v_{B}t-\mathcal{S})^{-3/2}$,
where we have assumed that $\mathcal{S}_{q_{i}}=1$ Rakovszky  _et al._ (2020).

Figure 3: The OTOC as a function of time and space for an $N=12$ one-
dimensional spin chain. (a) Operators that do not overlap the Hamiltonian
exhibit an OTOC which follows a ballistic light cone. (b) For an operator that
overlaps with the Hamiltonian, the OTOC at a given site $i$ initially decays,
before increasing at later times. The specific OTOC we calculate takes the
form, $\frac{1}{4}\sum_{P}\langle
e^{-iH_{1}t}\hat{M}e^{iH_{1}t}\hat{P}_{i}e^{-iH_{2}t}\hat{M}e^{iH_{2}t}\hat{P}_{i}\rangle/\mathcal{N}(t)$,
where the forwards and backwards time-evolution are governed by two distinct
1D Hamiltonians, $H_{1}=H_{2}+\eta\,\delta H$ (for details see supplemental
materials sup ).

We expect open-system dynamics to damp the large-size components of $\hat{M}$
by a factor $\sim\\!e^{-\varepsilon\mathcal{S}^{2}/v_{B}}$, where
$\mathcal{S}^{2}/v_{B}$ characterizes the space-time volume of a chaotic
component fn (4). This effectively truncates the size distribution above
$\mathcal{S}_{\text{tr}}\\!\sim\\!\sqrt{v_{B}/\varepsilon}$ [Fig. 1]. At late
times, $v_{B}t\gtrsim\mathcal{S}_{\text{tr}}$, this suggests that the average
operator size will actually _shrink_ in time, since small-size components
decay more slowly $\sim\\!t^{-1/2}$, than large-size components,
$P(\mathcal{S}_{\text{tr}})\sim\\!t^{-3/2}$. This sharply contrasts with the
behavior of operators that do not overlap conserved quantities, where one
expects monotonic growth [Fig. 2(a)].

To explore this, we simulate the dynamics of a one-dimensional spin chain and
measure the OTOC as a proxy for operator growth. For an operator that does not
overlap with the Hamiltonian, we find that the OTOC decays monotonically
following a linear light-cone [Fig. 3(a)]. For an operator exhibiting overlap,
we find that the decay of the OTOC indeed _reverses_ as a function of time,
indicative of a decrease in the average operator size [Fig. 3(b)].
Interestingly, this insight immediately resolves an apparent disagreement
between previous studies of open-system operator growth. In particular,
certain studies found that OTOCs were only minimally affected by errors
Vermersch  _et al._ (2019); Mi  _et al._ (2021), while others found a dramatic
reversal of scrambling Swingle and Halpern (2018); Zhang  _et al._ (2019). We
attribute this difference to the presence or absence of conservation laws.

_Discussion and outlook._ —Our results lead to a number of implications.
First, we provide a new perspective on protocols which divide error-prone OTOC
measurements by an independent characterization of the error Swingle and
Halpern (2018); Vermersch  _et al._ (2019); Mi  _et al._ (2021); Sánchez  _et
al._ (2020). In our language, the latter is precisely the normalization,
$\mathcal{N}(t)=\langle\hat{M}(t)\,\hat{M}(t)\rangle$. To this end, these
protocols will only replicate unitary dynamics when the total error is small
($1-\mathcal{N}\approx\varepsilon\int_{0}^{t}dt\,\overline{\mathcal{S}}\ll 1$)
or when size distributions are tightly peaked,
$\delta\mathcal{S}\ll\mathcal{S}$.

Second, our results suggest a novel protocol for measuring operator size
distributions (Fig. 4), which circumvents the need to either perform
exponentially many measurements Qi  _et al._ (2019) or utilize two entangled
copies of the system Schuster  _et al._ (2021). Specifically, in order to
measure the _generating function_ of the size distribution,
$G_{\mathcal{S}}(\mu)=\sum_{\mathcal{S}}P(\mathcal{S})e^{-\mu\mathcal{S}}$, we
propose the following protocol (Fig. 4): (i) prepare an initial state,
$\rho=(\mathbbm{1}+\hat{M})\otimes\mathbbm{1}^{\otimes N-1}/2^{N}$, (ii) time-
evolve forward, e.g. via a unitary operation, $U$, (iii) apply a set of
single-qubit Pauli operators, $\\{P_{1},\ldots,P_{N}\\}$, (iv) time-evolve
backward via $U^{\dagger}$, and (v) measure $\hat{M}$. If the intervening
Pauli operators are fixed, this reduces to previous schemes for measuring
OTOCs Gärttner  _et al._ (2017). However, if one randomly samples each Pauli
matrix in each experimental shot, with probability $p=(1-e^{-\mu})/4$ to be
$\\{X,Y,Z\\}$ and probability $1-3p$ to be the identity, this in effect
implements a decoherence channel, $e^{-\mu\mathcal{S}}$, that explicitly
depends on the size superoperator. The fidelity to recover the initial state
then gives the generating function via
$\mathcal{F}=\frac{1}{2}[1+\mathcal{N}G_{\mathcal{S}}(\mu)]$, where
$\mathcal{N}$ can be measured by setting $\mu=0$.

Finally, we conjecture that our framework also applies to an alternate
scenario (often explored in experiments Sánchez  _et al._ (2020, 2022);
Domínguez  _et al._ (2021); Domínguez and Álvarez (2021)), where one evolves
forward via a Hamiltonian, $H$, and backward via a perturbed Hamiltonian,
$-H+\eta\,\delta H$ fn (6); Jalabert and Pastawski (2001). Naively, this
scenario features perturbations that are highly correlated in time and space,
and thus outside the Lindbladian framework. However, in a chaotic many-body
system, one expects such correlations to quickly decay outside of some
thermalization time- ($\tau_{\text{th}}$) and length-scale
($\xi_{\text{th}}$). This assumption leads to a Fermi’s golden rule Jalabert
and Pastawski (2001); Cucchietti  _et al._ (2003) estimate of an effective
decoherence rate,
$\partial_{t}\log(\mathcal{N})\sim\eta^{2}\tau_{\text{th}}\xi_{\text{th}}\overline{\mathcal{S}}$,
which scales linearly with the average operator size sup . Somewhat
intriguingly, recent NMR experiments Domínguez  _et al._ (2021) precisely
observe this linear scaling with operator size for $\eta\gtrsim 0.1$; this
transitions to a square root scaling at smaller $\eta$ and developing a
microscopic understanding of this regime remains an open question.

Looking forward, our results also have implications for the classical
simulability of open quantum systems—if operator sizes are bounded from above
by a constant, $\mathcal{S}_{\varepsilon}$, then time-evolution is in
principle efficiently simulable, since the dimension of the accessible
operator Hilbert space is polynomial in the system size,
$\sim\\!N^{\mathcal{S}_{\varepsilon}}$. A similar idea was recently proposed
in diffusive 1D spin chains Rakovszky  _et al._ (2020); our results suggest
that it may hold more broadly.

Figure 4:  Protocol to measure the generating function,
$G_{\mathcal{S}}(\mu)$, of the operator size distribution. Gray qubits are
initially random in the computational basis, $\ket{\psi}$ is an $\hat{M}$
eigenstate, and each site is acted upon randomly by either the identity (gray)
or a non-identity Pauli operator (red) in each experimental shot.

_Acknowledgements_ —We are grateful to Ehud Altman, Anish Kulkarni and Rahul
Sahay for illuminating discussions and to Bryce Kobrin and Francisco Machado
for detailed feedback on the manuscript. The numerical simulations performed
in this work used the dynamite Python frontend dyn , which supports a matrix-
free implementation of Krylov subspace methods based on the PETSc and SLEPc
packages Hernandez  _et al._ (2005). This work was supported by the U.S.
Department of Energy, Office of Science, National Quantum Information Science
Research Centers, Quantum Systems Accelerator (QSA) and by the U.S. Department
of Energy, Quantum Information Science Enabled Discovery (QuantISED) for High
Energy Physics (KA2401032). T.S. acknowledges support from the National
Science Foundation (QII-TAQS program and GRFP).

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pages 4 of supp.pdf See pages 5 of supp.pdf See pages 6 of supp.pdf See pages
7 of supp.pdf See pages 8 of supp.pdf See pages 9 of supp.pdf See pages 10 of
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supp.pdf See pages 14 of supp.pdf See pages 15 of supp.pdf See pages 16 of
supp.pdf See pages 17 of supp.pdf See pages 18 of supp.pdf


Common dynamics of two Pisot substitutions with the same incidence matrix

Tarek<EMAIL_ADDRESS>
Institut de Mathématiques de Luminy, CNRS U.M.R. 6206, 163, Avenue de Luminy,
Case 907, 13288 Marseille Cedex 09 France.
Département de Mathématiques, Faculté des Sciences de Sfax, BP 802, 3018 Sfax,
Tunisie.

Abstract. The matrix of a substitution is not sufficient to completely
determine the dynamics associated, even in simplest cases since there are many
words with the same abelianization.
In this paper we study the common points of the canonical broken lines
associated to two different Pisot irreducible substitutions $\sigma_{1}$ and
$\sigma_{2}$ having the same incidence matrix. We prove that if $0$ is inner
point to the Rauzy fractal associated to $\sigma_{1}$ these common points can
be generated with a substitution on an alphabet of so-called "balanced
blocks".

Résumé: On sait que la matrice d’une substitution ne suffit pas à déterminer
complètement le système dynamique associé; même dans les cas les plus simples,
il existe de nombreuses substitutions associées à une matrice : il existe de
nombreux mots ayant le même abélianisé.
Dans ce papier, on étudie les points commun de deux lignes brisées associé aux
deux substitutions $\sigma_{1}$ et $\sigma_{2}$ irreducible de type Pisot qui
ont la même matrice d’incidence. On montre que si $0$ est un point intérieur à
l’un des deux fractals de Rauzy associé à $\sigma_{1}$ ou $\sigma_{2}$ alors
ces points communs peut être générés par une substitution définit sur un
alphabet appelé "block balncé".

##  1 Introduction

Let $\sigma_{1}$ and $\sigma_{2}$ be two different Pisot substitutions having
the same incidence matrix. Although the fixed points of each substitution have
the same letter frequencies, they usually show different dynamical and
geometrical properties, e.g., their Rauzy fractals have different properties.
(The Rauzy fractals can give a geometric model of the dynamical system defined
by the substitution, for more detail see section $2$).

A classic example is given by the Tribonacci substitution and the flipped
Tribonacci substitution, i.e.,

$\sigma_{1}:\left\\{\begin{array}[]{ll}a\rightarrow ab\\\ b\rightarrow ac\\\
c\rightarrow a\end{array}\right.$ and
$\sigma_{2}:\left\\{\begin{array}[]{ll}a\rightarrow ab\\\ b\rightarrow ca\\\
c\rightarrow a\end{array}\right.$

The incidence matrix of $\sigma_{1}$ and $\sigma_{2}$ is
$\begin{pmatrix}1&1&1\\\ 1&0&0\\\ 0&1&0\\\ \end{pmatrix}$. The dominant
eigenvalue satisfies the relation $X^{3}-X^{2}-X-1=0$, hence the name
Tribonacci for the substitution.

The Rauzy fractal of the first substitution is a topological disc [1], simply
connected , while it is a well known fact that the second fractal is not
simply connected, compare Figure[1].

Figure 1:  The Rauzy fractals of $\sigma_{1}$ and $\sigma_{2}$

We consider another simple example of substitutions $\tau_{1}$ and $\tau_{2}$,
i.e.,

$\tau_{1}:\left\\{\begin{array}[]{ll}a\rightarrow aba\\\ b\rightarrow ab\\\
\end{array}\right.$ and $\tau_{2}:\left\\{\begin{array}[]{ll}a\rightarrow
aab\\\ b\rightarrow ba\\\ \end{array}\right.$

The Rauzy fractal of $\tau_{2}$ is the closure of a countable union of
disjoint intervals and the Rauzy fractal of $\tau_{1}$ is an interval, see [9]
and Figure[6].
We can deduce from one matrix we can obtain many different substitutions, so
many different Rauzy fractals. We are interested to studies commons dynamics
of these Rauzy fractals, we are interested to characterize their intersection,
for this we need to define a new object. prove that we can consider their
intersection as a substitutive set.

######  Definition 1.1.

A substitutive set is the closure of the projection of a canonical stepped
line associated to a primitive substitution on a contracting space associated
to the restriction of a positive integer matrix. For more detail see section
2.

The main result of this paper is the following:

######  Theorem 1.1.

Let $\sigma_{1}$ and $\sigma_{2}$ be two irreducible unimodular Pisot
substitutions with the same incidence matrix. Let $X_{\sigma_{1}}$ and
$X_{\sigma_{2}}$ the two associated Rauzy fractals; suppose that $0$ is inner
point to $X_{\sigma_{1}}$ .
Then the intersection of $X_{\sigma_{1}}$ and $X_{\sigma_{2}}$ has non-empty
interior, and it is substitutive.There is an algorithm to obtain the
substitution for intersection.

##  2 Substitutions and Rauzy fractals

###  2.1 General setting

Let $\mathcal{A}:=\\{a_{1},...,a_{d}\\}$ be a finite set of cardinal $d$
called alphabet. The free monoid $\mathcal{A}^{*}$ on the alphabet
$\mathcal{A}$ with empty word $\varepsilon$ is defined as the set of finite
words on the alphabet $\mathcal{A}$, this is
$\mathcal{A}^{*}:=\bigcup_{k\in\mathbb{N}}\mathcal{A}^{k}$, endowed with the
concatenation map. We denote by $\mathcal{A}^{\mathbb{N}}$ and
$\mathcal{A}^{\mathbb{Z}}$ the set of one and two-sided sequences on
$\mathcal{A}$, respectively. The topology of $\mathcal{A}^{\mathbb{N}}$ and
$\mathcal{A}^{\mathbb{Z}}$ is the product topology of discrete topology on
each copy of $\mathcal{A}$. Both spaces are metrizable.

The length of a word $w\in\mathcal{A}^{n}$ with $n\in\mathbb{N}$ is defined as
$|w|=n$. For any letter $a\in\mathcal{A}$, we define the number of occurrences
of $a$ in $w=w_{1}w_{2}\ldots w_{n-1}w_{n}$ by
$|w|_{a}=\sharp\\{i|w_{i}=a\\}$.

Let
$l:\mathcal{A}^{*}\mapsto\mathbb{Z}^{d}:w\mapsto(|w|_{a})_{a\in\mathcal{A}}\in\mathbb{N}^{d}$
be the natural homomorphism obtained by abelianization of the free monoid,
called the abelianization map.

A substitution over the alphabet $\mathcal{A}$ is an endomorphism of the free
monoid $\mathcal{A}^{*}$ such that the image of each letter of $\mathcal{A}$
is a nonempty word.
A substitution $\sigma$ is primitive if there exists an integer $k$ such that,
for each pair $(a,b)\in\mathcal{A}^{2}$, $|\sigma^{k}(a)|_{b}>0$. We will
always suppose that the substitution is primitive, this implies that for all
letter $j\in\mathcal{A}$ the length of the successive iterations
$\sigma^{k}(j)$ tends to infinity.

A substitution naturally extends to the set of two sided sequences
$\mathcal{A}^{\mathbb{Z}}$. We associate to every substitution $\sigma$ its
incidence matrix $M$ which is the $n\times n$ matrix obtained by
abelianization, i.e. $M_{i,j}=|\sigma(j)|_{i}$. It holds that
$l(\sigma(w))=Ml(w)$ for all $w\in\mathcal{A}^{*}$.

Remark. The incidence matrix of a primitive substitution is a primitive
matrix, so with the Perron-Frobenius theorem, it has a simple real positive
dominant eigenvalue $\beta$.

###  2.2 Rauzy fractals

######  Definition 2.1.

A Pisot number is an algebric integer $\beta>1$ such that each Galois
conjugate $\beta^{(i)}$ of $\beta$ satisfies $\mid\beta^{(i)}\mid<1$.

From now, we will suppose that all the substitutions that we consider are
irreducible of Pisot type and unimodular. This mean that the characteristic
polynomial of its incidence matrix is irreducible, its determinant is equal to
$\pm 1$ and its dominant eigenvalues is a Pisot number. We can prove that any
irreducible Pisot substitution is primitive (see [8]).

Remark. Note that there exist substitution whose largest eigenvalue is Pisot
but whose incidence matrix has eigenvalues that are not conjugate to the
dominant eigenvalue. Example is 1$\rightarrow$ 12, 2$\rightarrow$ 3,
3$\rightarrow$ 4, 4$\rightarrow$ 5, 5$\rightarrow$1\. The characteristic
polynomial is reducible. Such substitutions are called Pisot reducible.

######  Definition 2.2.

Let $\sigma$ a substitution and $u\in\mathcal{A}^{\mathbb{N}}$, $u$ is a fixed
point of $\sigma$ if $\sigma(u)=u$. The infinite word $u$ is a periodic point
of $\sigma$ if there exist $k\in\mathbb{N}$ such that $\sigma^{k}(u)=u$.

Let $\sigma$ be a primitive substitution, then there exist a finite number of
periodic points (see [7]). We associate to the fixed point $u$ of the
substitution a symbolic dynamical system $(\Omega_{u},S)$ where $S$ is the
shift map on $\mathcal{A}^{\mathbb{N}}$ given by
$S(a_{0}a_{1}...)=a_{1}a_{2}...$ and $\Omega_{u}$ is the closure of
$\\{S^{m}(u):m\geq 0\\}$ in $\mathcal{A}^{\mathbb{N}}$.

Remark. If $\sigma$ is a primitive substitution then the symbolic dynamical
system $(\Omega_{u},S)$ does not depend on $u$; we denote it by
$(\Omega_{\sigma},S$).

We say that a dynamical system $(X,f)$ is semiconjugate to another dynamical
system $(Y,g)$ if there exists a continuous surjective map
$\Theta:X\rightarrow Y$ such that $\Theta\circ f=g\circ\Theta$. An important
question is whether and how the symbolic dynamical system $(\Omega_{u},S)$
admit a geometric model. By geometrically realizable we mean there exists a
dynamical system $(X,f)$ defined on a geometrical structure, such that
$(\Omega_{u},S)$ is semiconjugate to $(X,f)$.

In [10], G.Rauzy proves that the dynamical system generated by the
substitution $\sigma(1)=12$, $\sigma(2)=13$, $\sigma(3)=1$, is measure-
theoretically conjugate to an exchange of domains in a compact set
$\mathcal{R}$ of the complex plane. This compact subset has a self-similar
structure : using method introduced by F.M.Dekking in [6], S.Ito and Pierre
Arnoux obtain in [1] an alternative construction of $\mathcal{R}$ and prove
that each of exchanged domains has fractal boundary. We will use the
projection method to obtain the Rauzy fractal.

######  Definition 2.3.

A stepped line $L=(x_{n})$ in $\mathbb{R}^{d}$ is a sequence (it could be
finite or infinite) of points in $\mathbb{R}^{d}$ such that $x_{n+1}-x_{n}$
belong to a finite set.
A canonical stepped line is a stepped line such that $x_{0}=0$ and for all
$n\geq 0$, $x_{n+1}-x_{n}$ belong to the canonical basis of $\mathbb{R}^{d}$.

Using the abelianization map, to any finite or infinite word $W$, we can
associate a canonical stepped line in $\mathbb{R}^{d}$ as a sequence
$(l(P_{n}))$, where $P_{n}$ are the prefix of length $n$ of $W$.

An interesting property of the canonical stepped line associated to a fixed
point of primitive Pisot substitution is that it remains within bounded
distance from the expanding direction (given by the right eigenvector of
Perron-Frobenius of $M_{\sigma}$). We denote by $E_{s}$ the stable space (or
contracting space) and $E_{u}$ the unstable space (or expanding direction). We
denote by $\pi_{s}$ the linear projection in the contracting plane, parallel
to the expanding direction and $\pi_{u}$ the projection in the expanding
direction parallel to the contracting plane. We will project the stepped line
on the contracting space in the direction of the right Perron-Frobenius
eigenvector. We obtain a bounded set in $(d-1)$-dimensional vector space.

######  Definition 2.4.

Let $\sigma$ an irreducible Pisot substitution. The Rauzy fractal associated
to $\sigma$ is the closure of the projection of the canonical stepped line
associated to any fixed point of $\sigma$ in the contracting plane parallel to
the expanding direction.

We note the projection $\pi$ of the orbit of the fixed point associated to a
Pisot irreducible substitution $\sigma$ on a contracting space associated to
its incidence matrix.

######  Proposition 2.1.

The projection $\pi$ of the symbolic dynamical system $\Omega_{\sigma}$
associated to a Pisot irreducible substitution $\sigma$ to the Rauzy fractal
is a continuous map.

###### Proof.

The proof is given in [7]. ∎

We denote by $X_{\sigma}$ the Rauzy fractal (Central tile) associated to
$\sigma$ : $X_{\sigma}:=\overline{\\{\pi_{s}(l(u_{0}\ldots
u_{k-1}),k\in{\mathbb{N}}\\}}$. with $u_{0}\ldots u_{k-1}$ is a prefix of the
fixed point of length $k$. Subtiles of the central tile $X_{\sigma}$ are
naturally defined, depending on the letter associated to the vertex of the
stepped line that is projected. On thus sets for $i\in\mathcal{A}$ :
$X_{\sigma}(i):=\overline{\\{\pi_{s}(l(u_{0}\ldots
u_{k-1}),k\in{\mathbb{N}},u_{k}=i\\}}$.

######  Proposition 2.2.

Let $\sigma$ a Pisot substitution and $X_{\sigma}$ its associated Rauzy
fractal. The boundary of $X_{\sigma}$ has zero measure.

###### Proof.

See [4] and [16]. ∎

Figure 2:  The projection method to get the Rauzy fractal.

###  2.3 Central tiles viewed as a graph directed iterated function

The tiles $X_{\sigma}(i)$ can be written as a so-called graph iterated
function system (GIFS).

######  Definition 2.5.

(GIFS)

Let G be a finite directed graph with set of vertices $\\{1,\ldots,q\\}$ and
set of edges $E$. Denote the set of edges leading from $i$ to $j$ by $E_{ij}$.
To each $e\in E$ associated a contractive mapping
$\tau_{e}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$. If for each $i$ there is
some outgoing edge we call $(G,\\{\tau_{e}\\})$ a GIFS.

######  Definition 2.6.

(Prefix-suffix automaton)

Let $\sigma$ be a substitution over the alphabet $\mathcal{A}$ and let
$\mathcal{P}$ be the finite set

${P}:=\\{(p,a,s)\in\mathcal{A}^{\ast}\times\mathcal{A}\times\mathcal{A}^{\ast};\exists
b\in\mathcal{A},\sigma(b)=pas\\}$.

The prefix-suffix automaton of sigma has $\mathcal{A}$ as a set of vertices
and $\mathcal{P}$ as a set of label edges : there is an edge labeled by
$(p,a,s)$ from $a$ to $b$ if and only if $pas=\sigma(b)$.

Example. For the Fibonacci substitution $1\mapsto 12$ and $2\mapsto 1$, one
gets:

${P}=\\{(e,1,2),(1,2,e),(e,1,e)\\}$.

The prefix-suffix automaton of the Fibonacci substitution is :

Figure 3:  Prefix-suffix automaton for the fibonacci substitution.

######  Theorem 2.1.

Let $\sigma$ be a primitive unit Pisot substitution over the alphabet
$\mathcal{A}$. The central tile $X_{\sigma}$ is a compact subset with nonempty
interior. Each subtile is the closure of its interior. The subtiles of
$X_{\sigma}$ are solution of the $GIFS$

$\forall
i\in\mathcal{A},X_{\sigma}(i)=\bigcup_{j\in\mathcal{A},i\xrightarrow[\text{}]{\text{(p,i,s)}}j}$
$MX_{\sigma}(j)+\pi_{s}l(p)$.

###### Proof.

The proof is given in [13].

∎

###  2.4 Disjointness of the subtiles of the central tile

To ensure that the subtiles are disjoint, we introduce the following
combinatorial condition in substitutions.

######  Definition 2.7.

(Strong coincidence condition).

A substitution $\sigma$ over the alphabet $\mathcal{A}$ satisfies the strong
coincidence condition if for every pair $(b_{1},b_{2})\in\mathcal{A}^{2}$,
there exist $k\in\mathbb{N}$ and $a\in\mathcal{A}$ such that
$\sigma^{k}(b_{1})=p_{1}as_{1}$ and $\sigma^{k}(b_{2})=p_{2}as_{2}$ with
$l(p_{1})=l(p_{2})$ or $l(s_{1})=l(s_{2})$.

Remark. The strong coincidence condition is satisfied by every unit Pisot
substitution over two letter alphabet [3]. It is conjectured that every
substitution of Pisot type satisfies the strong coincidence condition.

######  Theorem 2.2.

Let $\sigma$ be a primitive unit Pisot substitution. If $\sigma$ satisfies the
strong coincidence condition, then the subtiles of the central tiles have
disjoint interiors.

###### Proof.

The proof for the disjointness is given in [1]. ∎

Remark. If $0$ is inner point to the Rauzy fractal associated to a Pisot
substitution then the subtiles of the central tiles have disjoint interiors
(see [13]).

###  2.5 Substitutive sets

A substitutive set is the closure of the projection of a canonical stepped
line associated to substitution on a contracting space of a restriction of a
positive integer matrix. In particular a Rauzy fractal is a substitutive set
since it is the projection of canonical stepped line associated to a fixed
point on the contracting space associated to the matrix of substitution. So we
can expand the definition of Rauzy fractal to substitutive set. In particular
a substitutive set can be expressed as the attractor of some graph directed
iterated function system (IFS). See [2]

##  3 Intersection of Rauzy fractals

Let $\sigma_{1}$ and $\sigma_{2}$ two Pisot irreducible substitutions with the
same incidence matrix, we consider $X_{\sigma_{1}}$ and $X_{\sigma_{2}}$ their
associated Rauzy fractals respectively.
The intersection of $X_{\sigma_{1}}$ and $X_{\sigma_{2}}$ is non-empty since
it contains $0$, and it is a compact set (intersection of two compacts).

######  Proposition 3.1.

Let $\sigma_{1}$ and $\sigma_{2}$ be two Pisot irreducible substitutions with
the same incidence matrix. We consider $L_{1}$ and $L_{2}$ the canonical
broken lines associated to a fixed point of $\sigma_{1}$ and $\sigma_{2}$
respectively, let $P_{1}$ and $P_{2}$ two points from $L_{1}$ and $L_{2}$
respectively. Then $\pi_{s}(P_{1})=\pi_{s}(P_{2})$ implies $P_{1}=P_{2}$.

###### Proof.

The Perron Frobenius eigenvalues is irrational in the irreducible case. We
project $P_{1}$ and $P_{2}$ in the contracting space parallel to the expanding
space (the direction of the Perron Frobenius eigenvectors). If
$\pi_{s}(P_{1})=\pi_{s}(P_{2})$ then $(P_{1}P_{2})$ is parallel to the
expanding direction. This implies that expanding direction is rational.

∎

######  Proposition 3.2.

Let $\sigma_{1}$ and $\sigma_{2}$ be two substitutions with the same incidence
matrix, we consider $X_{\sigma_{1}}$ (resp. $X_{\sigma_{2}}$) the Rauzy
fractal associated to $\sigma_{1}$ (resp. $\sigma_{2}$) and $X_{\sigma}$ the
common point of $X_{\sigma_{1}}$ and $X_{\sigma_{2}}$.
Then the boundary of $X_{\sigma}$ is included in the union boundary of
$X_{\sigma_{1}}$ and $X_{\sigma_{2}}$ and has zero measure.

###### Proof.

Let $x$ a point from the boundary of $X_{\sigma}$. We suppose that $x$ is not
on the boundary of $X_{\sigma_{1}}$. Then there exist $r_{1}>0$ such that
$B(x,r_{1})\subset X_{\sigma_{1}}$. If $x$ is not in the boundary of
$X_{\sigma_{2}}$ then there exist $r_{2}>0$ such that $B(x,r_{2})\subset
X_{\sigma_{2}}$. Then there exist $r=\min(r_{1},r_{2})$ such that
$B(x,r)\subset X_{\sigma_{1}}\cap X_{\sigma_{2}}$, $x$ is in the boundary of
$X_{\sigma}$ then $x$ is in the boundary of $X_{\sigma_{2}}$. Then $\partial
X_{\sigma}\subset\partial X_{\sigma_{1}}\cup\partial X_{\sigma_{2}}$.
Since $\partial X_{\sigma_{1}}$ and $\partial X_{\sigma_{2}}$ have zero
measure then $\partial X_{\sigma}$ has zero measure. ∎

Figure 4: Sets of common points of the fractals of Tribonacci and the flipped
Tribonacci .

###  3.1 The main result: Morphism generating the common points of two Pisot
substitutions with the same incidence matrix

In this section we consider $\sigma_{1}$ and $\sigma_{2}$ be two unimodular
irreducible Pisot substitutions with the same incidence matrix. We denote
$X_{\sigma_{1}}$ and $X_{\sigma_{2}}$ their associated Rauzy fractals
respectively. We suppose that $0$ is an inner point to $X_{\sigma_{1}}$. We
note $X_{\sigma}$ the closure of the intersection of the interior of
$X_{\sigma_{1}}$ and the interior of $X_{\sigma_{2}}$. Let
$(\Omega_{\sigma_{1}},S)$ and $(\Omega_{\sigma_{2}},S)$ the symbolic dynamical
systems associated to $\sigma_{1}$ and $\sigma_{2}$ respectively. We consider
$\pi_{1}$ (resp. $\pi_{2}$) the projection map from the symbolic dynamical
system $(X_{\sigma_{1}},S)$ into the Rauzy fractal (rep.$\pi_{2}$).

We will prove that $X_{\sigma}$ is a substitutive set, and it can be generated
by a substitution obtained with algorithm generating the common point of the
interior of $X_{\sigma_{1}}$ and $X_{\sigma_{2}}$.

######  Definition 3.1.

For a dynamical system $(X,T)$ if $A$ is a subset of $X$, and $x\in A$, we
define the first return time of $x$ as
$n_{x}=\inf\\{n\in\mathbb{N}^{*}|T^{n}(x)\in A\\}$ (it is infinite if the
orbit of $x$ does not come back to $A$). If the first return time is finite
for all $x\in A$, we define the induced map of $T$ on $A$ (or first return
map) as the map $x\mapsto T^{n_{k}}(x)$, and we denote this map by $T_{A}$.

######  Definition 3.2.

A sequence $u=(u_{n})$ is minimal (or uniformly recurrent) if every word
occurring in $u$ occurs in an infinite number of positions with bounded gaps,
that is, if for every factor $W$, there exist $s$ such that for every $n$, $W$
is a factor of $u_{n}\ldots u_{n+s-1}$.

######  Lemma 3.1.

The closure of the intersection $X_{\sigma}$ has non empty interior and non-
zero Lebesque measure.

###### Proof.

We suppose that $0$ is an inner point to $X_{\sigma_{1}}$. Then there exist an
open set $U$ such that $0\in U\subset X_{\sigma_{1}}$. The Rauzy fractal is
the closure of its interior and $0$ is a point of $X_{\sigma_{2}}$, hence
there exist a sequence of points $(x_{n})_{n\in\mathbb{N}}$ from the interior
of $X_{\sigma_{2}}$ which converges to $0$. Then there exist open sets $V_{n}$
such that $x_{n}\in V_{n}\subset X_{\sigma_{2}}$. Since $(x_{n})$ converge to
$0$, there exists $N\in\mathbb{N}$ such that $x_{N}\in U$.
We denote by $W$ the open set $W=U\cap V_{N}$,$W$ is non-empty and $W\subset
X_{\sigma_{1}}\cap X_{\sigma_{2}}$. The intersection of $X_{\sigma_{1}}$ and
$X_{\sigma_{2}}$ contains a non empty open set, hence it has non-zero Lebesque
mesure. ∎

We define the subgroup $\Gamma$ of $\mathbb{Z}^{d}$ as :

$\Gamma=\\{\sum_{i=1}^{d}n_{i}e_{i}/\sum_{i=1}^{d}n_{i}=0,n_{i}\in\mathbb{Z}\\}$

with $e_{i}$ is the canonical bases of $\mathbb{R}^{d}.$

######  Lemma 3.2.

Let $\sigma$ be an irreducible Pisot substitution, and $X_{\sigma}$ its
associated Rauzy fractal. If $0$ is inner point to $X_{\sigma}$ then
$X_{\sigma}$ is a fundamental domain of $E_{s}$ for the projection of $\Gamma$
on the stable space.

###### Proof.

The proof is given in [13].

∎

######  Lemma 3.3.

Let $W$ be a non-empty open set in $X_{\sigma}$. Let $V_{1}=\pi_{1}^{-1}(W)$
and $V_{2}=\pi_{2}^{-1}(W)$ from $\Omega_{\sigma_{1}}$ and
$\Omega_{\sigma_{2}}$ respectively. If $n$ is a first return time in $V_{2}$
then $n$ is a return time in $V_{1}$.

###### Proof.

We consider $v_{1}\in V_{1}$ and $v_{2}\in V_{2}$ such that
$\pi_{1}(v_{1})=\pi_{2}(v_{2})$. Let $n$ the first return time of $v_{2}$ in
$V_{2}$. Then there exist $w\in\Gamma$ such that
$\pi_{1}(S^{n}(v_{1}))=\pi_{2}(S^{n}(v_{2}))+\pi_{2}(w)$.
$\pi_{2}(S^{n}(v_{2}))$ is a point from the interior of $X_{\sigma}$, then it
is a point from the interior of $X_{\sigma_{1}}$. And we have
$\pi_{1}(S^{n}(v_{1}))$ is a point from $X_{\sigma_{1}}$.
Since $0$ is an inner point to $X_{\sigma_{1}}$, from lemma $3.3$,
$X_{\sigma_{1}}$ is a fundamental domain. Then we obtain $\pi_{2}(w)=0$. So we
have $\pi_{1}(S^{n}(v_{1}))=\pi_{2}(S^{n}(v_{2}))$. Then if $n$ is a return
time in $V_{2}$ we deduce that $n$ is a return time in $V_{1}$. ∎

######  Definition 3.3.

Let $U$ and $V$ two finite words, we say that $\begin{pmatrix}U\\\ V\\\
\end{pmatrix}$ is balanced block if $l(U)=l(V)$, where $l$ is the
abelianization map from $\mathcal{A}^{*}$ in $\mathbb{Z}^{d}$.

######  Definition 3.4.

A minimal balanced block is a balanced block, such for every strict prefix
$U_{k}$, $V_{k}$ of $U$ and $V$ respectively of length $k$, $l(U_{k})\neq
l(V_{k})$.

######  Lemma 3.4.

Let $u$ and $v$ be tow fixed points of $\sigma_{1}$ and $\sigma_{2}$
respectively, then we can decompose $u$ and $v$ on a finite minimal balanced
blokcs.

###### Proof.

Let $u$ and $v$ be tow fixed points of $\sigma_{1}$ and $\sigma_{2}$
respectively. We have $0\in X_{\sigma}$ then there exist $v_{1}$ and $v_{2}$
two prefix of $u$ and $v$ respectively such that $x=\pi(v_{1})=\pi(v_{2})$ and
$l(v_{1})=l(v_{2})$. We obtain a balanced block: $\begin{pmatrix}v_{1}\\\
v_{2}\\\ \end{pmatrix}$, we can decompose it with minimal balanced blocks and
we consider the image of each new minimal balanced block with $\sigma_{1}$ and
$\sigma_{2}$. Then there exist new minimal balanced blocks which appear, we
consider the image of each new blocks by $\sigma_{1}$ and $\sigma_{2}$. Since
every word appears with a bounded distance, all the minimal balanced blocks
will appear after a finite time. Then we can obtain a decomposition of $u$ and
$v$ with a finite number of minimal balanced blocks. A simple case appears
when $u$ and $v$ begin with the same letter $i$, then the first minimal
balanced block is $\begin{pmatrix}i\\\ i\\\ \end{pmatrix}$.

∎

######  Theorem 3.1.

$X_{\sigma}$ is a substitutive set.

###### Proof.

We have $X_{\sigma}$ is the closure of the projection of points associated to
balanced blocks, from the two stepped lines associated to the fixed points of
$\sigma_{1}$ and $\sigma_{2}$. These common points can be obtained as a fixed
point of a new substitution defined on the set of the minimal balanced blocks.
There exist an algorithm to obtain this morphism (or substitution). Since $0$
is a point from $X_{\sigma}$ there exist two minimal initial word $v_{1}$ and
$v_{2}$ from the language of $\sigma_{1}$ and $\sigma_{2}$ respectively such
that $l(v_{1})=l(v_{2})$

We denote the block $\begin{pmatrix}v_{1}\\\ v_{2}\\\ \end{pmatrix}$ and we
consider $\sigma_{1}(v_{1})$ and $\sigma_{2}(v_{2})$ we obtain a second block
$\begin{pmatrix}\sigma_{1}(v_{1})\\\ \sigma_{2}(v_{2})\\\ \end{pmatrix}$ with
the property $l(\sigma_{1}(v_{1}))=l(\sigma_{2}(v_{2}))$ because $\sigma_{1}$
and $\sigma_{2}$ have the same matrix. These blocks have a finite length,
because the return time in $X_{\sigma}$ is bounded. We consider the
decomposition of this balanced block $\begin{pmatrix}\sigma_{1}(v_{1})\\\
\sigma_{2}(v_{2})\\\ \end{pmatrix}$ with minimal balanced blocks.
This mean we can write $\begin{pmatrix}\sigma_{1}(v_{1})\\\
\sigma_{2}(v_{2})\\\ \end{pmatrix}=\begin{pmatrix}u_{1}\ldots u_{k}\\\
w_{1}\ldots w_{k}\\\ \end{pmatrix}$ with the property $l(u_{1})=l(w_{1})$,
…,$l(u_{n})=l(v_{n})$.

With this method we obtain a finite numbers of blocks with the same
abelianization. We consider this set of blocks and we consider the image of
each block with the two substitutions $\sigma_{1}$ and $\sigma_{2}$ and we
obtain a morphism witch generate all the common points of the stepped lines.

∎

##  4 Examples

###  4.1 Algorithm to obtain the morphism of the common points of two Rauzy
fractals

####  4.1.1 Example 1

I will take the example of $\tau_{1}$ and $\tau_{2}$ to show how the algorithm
is working. In this example the first minimal balanced block that we consider
is the beginning of the two fixed points associated to $\tau_{1}$ and
$\tau_{2}$ it will be $\begin{pmatrix}a\\\ a\\\ \end{pmatrix}$.
And we consider the image of the first element of this block by $\tau_{1}$ and
the second one by $\tau_{2}$ so we obtain : $\begin{pmatrix}a\\\ a\\\
\end{pmatrix}\overset{\tau_{1},\tau_{2}}{\longrightarrow}\begin{pmatrix}aba\\\
aab\\\ \end{pmatrix}$.
We denote by $A$ the minimal balanced block $\begin{pmatrix}a\\\ a\\\
\end{pmatrix}$ and by $B$ the minimal balanced block $\begin{pmatrix}ba\\\
ab\\\ \end{pmatrix}$.
So we obtain $A\rightarrow AB.$
The second step is to consider the same thing with the new block
$\begin{pmatrix}ba\\\ ab\\\ \end{pmatrix}$.
We consider the image of this block with the two substitution $\tau_{1}$ and
$\tau_{2}$, and we obtain :

$\begin{pmatrix}ba\\\ ab\\\
\end{pmatrix}\overset{\tau_{1},\tau_{2}}{\longrightarrow}\begin{pmatrix}ababa\\\
aabba\\\ \end{pmatrix}$.

We obtain an other block $\begin{pmatrix}b\\\ b\\\ \end{pmatrix}$ and we
denote by $C$ the projection over this new block and we obtain the image of
$B$ is $ABCA$. We continuous with this algorithm and we obtain the image of
the block $\begin{pmatrix}b\\\ b\\\ \end{pmatrix}$ is the new block
$\begin{pmatrix}ab\\\ ba\\\ \end{pmatrix}$. So we obtain the image of the
letter $C$ is a new letter $D$. Finally the image of the letter $D$ is $DAAC$.
So, we obtain an alphabet $\mathcal{B}$ in $4$ letters and we can define the
morphism $\phi$ as :

$\phi:\left\\{\begin{array}[]{ll}A\rightarrow AB\\\ B\rightarrow ABCA\\\
C\rightarrow D\\\ D\rightarrow DAAC\\\ \end{array}\right.$

And we consider the projection $\pi$ of the letters $A,B,C,D$ in the sets of
blocks $\begin{pmatrix}a\\\ a\\\ \end{pmatrix}$, $\begin{pmatrix}ba\\\ ab\\\
\end{pmatrix}$, $\begin{pmatrix}b\\\ b\\\ \end{pmatrix}$ et
$\begin{pmatrix}ab\\\ ba\\\ \end{pmatrix}$
Then we have : $\begin{pmatrix}\tau_{1}^{n}(a)\\\ \tau_{2}^{n}(a)\\\
\end{pmatrix}$ $=\pi(\phi^{n}(A))$.
The morphism $\phi$ generate all the common points of the two Rauzy fractals
associated to $\tau_{1}$ and $\tau_{2}$.

Figure 5:  The Rauzy fractals of $\tau_{1}$ and $\tau_{2}$.

Figure 6:  Common points of $\tau_{1}$ and $\tau_{2}$ with distinction of
blocs defined with $\phi$

####  4.1.2 Example 2

For the two substitutions of Tribonacci and the flipped Tribonacci it is more
complicated see Figure[7], we can define the morphism $\phi$ which generate
all the common points as follows:

$\phi:\left\\{\begin{array}[]{ll}A\rightarrow AB\\\ B\rightarrow C\\\
C\rightarrow AD\\\ D\rightarrow AE\\\ E\rightarrow F\\\ F\rightarrow ADDGA\\\
G\rightarrow AH\\\ H\rightarrow ID\\\ I\rightarrow ADJ\\\ J\rightarrow AHK\\\
K\rightarrow IDGA\end{array}\right.$

and the projection map $\pi$ :

$\pi:\left\\{\begin{array}[]{ll}A\rightarrow\begin{pmatrix}a\\\ a\\\
\end{pmatrix}\\\ B\rightarrow\begin{pmatrix}b\\\ b\\\ \end{pmatrix}\\\
C\rightarrow\begin{pmatrix}ac\\\ ca\\\ \end{pmatrix}\\\
D\rightarrow\begin{pmatrix}ba\\\ ab\\\ \end{pmatrix}\\\
E\rightarrow\begin{pmatrix}cab\\\ bca\\\ \end{pmatrix}\\\
F\rightarrow\begin{pmatrix}aabac\\\ caaab\\\ \end{pmatrix}\\\
G\rightarrow\begin{pmatrix}cab\\\ abc\\\ \end{pmatrix}\\\
H\rightarrow\begin{pmatrix}abac\\\ bcaa\\\ \end{pmatrix}\\\
I\rightarrow\begin{pmatrix}abaca\\\ caaab\\\ \end{pmatrix}\\\
J\rightarrow\begin{pmatrix}cabaab\\\ ababca\\\ \end{pmatrix}\\\
K\rightarrow\begin{pmatrix}ababac\\\ bcaaab\\\
\end{pmatrix}\end{array}\right.$

Figure 7: Sets of common points of the Tribonacci substitution and the flipped
substitution. Each color stands for a different letter of $\mathcal{B}$ and
shows the dynamics of the morphism $\phi$.

######  Corollaire 4.1.

Let $\sigma_{1}$ and $\sigma_{2}$ be the two substitution Tribonacci and the
flipped Tribonacci defined as follows:

$\sigma_{1}:\left\\{\begin{array}[]{ll}a\rightarrow ab\\\ b\rightarrow ac\\\
c\rightarrow a\end{array}\right.$ et
$\sigma_{2}:\left\\{\begin{array}[]{ll}a\rightarrow ab\\\ b\rightarrow ca\\\
c\rightarrow a\end{array}\right.$

We consider $U$ and $V$ their two fixed points, then the letter $c$ doesn’t
occur in the same position in $U$ and $V$.

###### Proof.

Minimal balanced blocks represents a decomposition of the two fixed points
$U$and $V$. We remark that in these finite minimal blocks there is no $c$
which appears in the same position. One can then deduce that the letter $c$
does not appear in the same position in two fixed points $U$ and $V$. ∎

####  4.1.3 Example 3

Now we will consider more general example defined as follows :

$\delta_{i}^{1}:\left\\{\begin{array}[]{ll}a\rightarrow a^{i}b\\\ b\rightarrow
a^{i-1}c\\\ c\rightarrow a\end{array}\right.$ and
$\delta_{i}^{2}:\left\\{\begin{array}[]{ll}a\rightarrow aba^{i-1}\\\
b\rightarrow aca^{i-2}\\\ c\rightarrow a\end{array}\right.$

$\delta_{i}^{1}$ and $\delta_{i}^{2}$ have the same incidence matrix. We can
define the morphism of their common points for all $i\geq 3$ as :

Figure 8:  The Rauzy fractals of $\delta_{3}^{1}$ and $\delta_{3}^{2}$.

$\phi_{i}:\left\\{\begin{array}[]{ll}A\rightarrow AB\\\ B\rightarrow AC\\\
C\rightarrow(AAD)^{i-1}[AAE(AAD)^{i}]^{i-2}AAE(AAD)^{i-1}A\\\ D\rightarrow
AF\\\ E\rightarrow(AAD)^{i-3}A\\\
F\rightarrow(AAD)^{i-1}[AAE(AAD)^{i}]^{i-3}AAE(AAD)^{i-1}A.\end{array}\right.$

$\pi_{i}:\left\\{\begin{array}[]{ll}A\rightarrow\begin{pmatrix}a\\\ a\\\
\end{pmatrix}\\\ B\rightarrow\begin{pmatrix}a^{i-1}b\\\ ba^{i-1}\\\
\end{pmatrix}\\\ C\rightarrow\begin{pmatrix}a^{i-1}b(a^{i}b)^{i-2}a^{i-1}c\\\
ca^{i-1}(ba^{i})^{i-2}ba^{i-1}\\\ \end{pmatrix}\\\
D\rightarrow\begin{pmatrix}a^{i-2}b\\\ ba^{i-2}\\\ \end{pmatrix}\\\
E\rightarrow\begin{pmatrix}a^{i-3}c\\\ ca^{i-3}\\\ \end{pmatrix}\\\
F\rightarrow\begin{pmatrix}a^{i-1}b(a^{i}b)^{i-3}a^{i-1}c\\\
ca^{i-1}(ba^{i})^{i-3}ba^{i-1}\\\ \end{pmatrix}\\\ \end{array}\right.$

Figure 9: Sets of common points of $\delta_{3}^{1}$ and $\delta_{3}^{2}$.

Remark. The property $0$ is inner point is sufficient, and we have this
example of substitutions with the same incidence matrix but the intersection
is reduced to the origin.

We can give an example where the intersection is empty. We consider the two
substitutions $\chi_{1}$ and $\chi_{2}$ defined as follows :

$\chi_{1}:\left\\{\begin{array}[]{ll}a\rightarrow aab\\\ b\rightarrow ab\\\
\end{array}\right.$ and $\chi_{2}:\left\\{\begin{array}[]{ll}a\rightarrow
baa\\\ b\rightarrow ba\\\ \end{array}\right.$

Figure 10:  The Rauzy fractals of $\chi_{1}$ and $\chi_{2}$.

###### Proof.

We consider $u_{1}$ and $u_{2}$ the two fixed points associated to $\chi_{1}$
and $\chi_{2}$ respectively. If $a.x$ is a prefix of $u_{1}$ then $b.x$ is a
prefix of $u_{2}$.
We will reason by induction : for $x=a$ it is so verified for $n=1$.
We suppose now that $a.x$ is prefix of $u_{1}$ and $b.x$ is a prefix of
$u_{2}$ with $|x|=n$.
$\chi_{1}(a.x)=aab\chi_{1}(x)$ is prefix of $u_{1}$,
$\chi_{2}(b.x).b=ba\chi_{2}(x)b$ is a prefix of $u_{2}$ if and only if
$\chi_{1}(x)=\chi_{2}(x)b$.
We have for the two letter $a$ and $b$:

  * •

$x=a$ : $b.\chi_{1}(a)=baab=\chi_{2}(a)b.$

  * •

$x=b$ : $b.\chi_{1}(b)=bab=\chi_{2}(b)b.$

We consider now $x=x_{1}x_{2}\ldots x_{n}$ with $x_{i}\in\\{a,b\\}$
$b.\chi_{1}(x)=b.\chi_{1}(x_{1}x_{2}\ldots
x_{n})=b.\chi_{1}(x_{1})\ldots\chi_{1}(x_{n})$
$=\chi_{2}(x_{1}).b.\chi_{1}(x_{2})\ldots\chi_{1}(x_{n})$
⋮
$=\chi_{2}(x_{1})\chi_{2}(x_{2})\ldots\chi_{2}(x_{n}).b$
So we prove that there exist an infinite word $u$ such that $u_{1}=a.u$ and
$u_{2}=b.u$. ∎

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