# Constrained ergodic optimization for generic continuous functions

Shoya Motonaga  Research Organization of Science and Technology, Ritsumeikan
University, 1-1-1 Noji-higashi, Kusatsu, Shiga 525-8577, Japan
<EMAIL_ADDRESS>and  Mao Shinoda  Department of Mathematics,
Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo, 112-8610, Japan
<EMAIL_ADDRESS>
###### Abstract.

One of the fundamental results of ergodic optimization asserts that for any
dynamical system on a compact metric space with the specification property and
for a generic continuous function $f$ every invariant probability measure that
maximizes the space average of $f$ must have zero entropy. We establish the
analogical result in the context of constrained ergodic optimization, which is
introduced by Garibaldi and Lopes (2007).

###### Key words and phrases:

###### 2010 Mathematics Subject Classification:

Primary 37E45, 37B10, 37A99

##  1\. Introduction

Let $T:X\rightarrow X$ be a continuous map on a compact metric space and
$\mathcal{M}_{T}(X)$ be the space of Borel probability measures endowed with
the weak*-topology. Let $C(X,\mathbb{R})$ be the space of real-valued
continuous functions on $X$ with the supremum norm $\|\cdot\|_{\infty}$. For
each $f\in C(X,\mathbb{R})$ we consider the maximum ergodic average

(1) $\displaystyle\beta(f)=\sup_{\mu\in\mathcal{M}_{T}(X)}\int f\ d\mu$

and the set of all maximizing measures of $f$

(2) $\displaystyle\mathcal{M}_{{\rm
max}}(f)=\left\\{\mu\in\mathcal{M}_{T}(X):\int f\ d\mu=\beta(f)\right\\}.$

The functional $\beta$ and the set $\mathcal{M}_{{\rm max}}(f)$ are main
objects in ergodic optimization, which has been actively studied for a decade
(see for more details [Jen06survey, Jen2017]).

Constrained erogdic optimization, which is introduced by Garibaldi and Lopes
in [GarLop07], investigates the analogical objects as (1) and (2) under some
constraint. Introducing a continuous $\mathbb{R}^{d}$-valued function
$\varphi\in C(X,\mathbb{R}^{d})$ playing the role of a constraint, we define
the rotation set of $\varphi=(\varphi_{1},\ldots,\varphi_{d})$ by

$\displaystyle{\rm Rot}(\varphi)=\left\\{{\rm
rv}_{\varphi}(\mu)\in\mathbb{R}^{d}:\mu\in\mathcal{M}_{T}(X)\right\\},$

where ${\rm
rv}_{\varphi}(\mu)=(\int\varphi_{1}d\mu,\ldots,\int\varphi_{d}d\mu)$. For
$h\in{\rm Rot}(\varphi)$, the fiber ${\rm rv}_{\varphi}^{-1}(h)$ is called the
rotation class of $h$. A point $h\in{\rm Rot}(\varphi)$ is called a rotation
vector. The terminology comes from Poincaré’s rotation numbers for circle
homeomorphisms. Many authors study the properties of rotation sets and
characterize several dynamics in terms of rotation vectors [GM99,
Jen01rotation, K92, K95, KW14, KW16, KW19].

We define the maximum ergodic average for a continuous function $f$ with
constraint $h\in{\rm Rot}(\varphi)$ by

(3) $\displaystyle\beta^{\varphi}_{h}(f)=\sup_{\mu\in{\rm
rv}_{\varphi}^{-1}(h)}\int f\ d\mu$

and the set of all relative maximizing measures of $f$ with a rotation vector
$h\in{\rm Rot}(\varphi)$ by

$\displaystyle\mathcal{M}^{\varphi}_{h}(f)=\left\\{\mu\in\mathcal{M}_{T}(X):\beta^{\varphi}_{h}(f)=\int
f\ d\mu,\ {\rm rv}_{\varphi}(\mu)=h\right\\}.$

Note that this formulation is a generalization of (unconstrained) ergodic
optimization, since for a constant constraint $\varphi\equiv h$ and its unique
rotation vector $h$ the rotation set of $h$ is $\mathcal{M}_{T}(X)$.

We have much interest in the above constrained optimization because the
constraints provide information about the asymptotic behavior of orbits. Such
problem is originally studied by Mather [Mather91] and Mañé [Mane96] for
Euler-Lagrange flows, where the asymptotic homological position of the
trajectory in the configuation space is given as a constraint. Recently, Bochi
and Rams [BochiRams16] proved that the Lyapunov optimizing measures for one-
step cocycles of $2\times 2$ matrices have zero entropy on the Mather sets
under some conditions, which implies that a low complexity phenomena occurs in
noncommutative setting such as a Lyapunov-optimization problem for one-step
cocycles. A relative Lyapunov-optimization is mentioned for their future
research but to the authors’ knowledge the classical commutative counterpart,
typicality of zero entropy of relative maximizing measure, has not been
established yet. We remark that Zhao [Zhao19] studies constrained ergodic
optimization for asymptotically additive potentials for the application in the
study of multifractal analysis.

It is natural to extend the fundamental results of unconstrained ergodic
optimization to constrained one. In [GarLop07], several general results on
constrained ergodic optimization are provided, especially, uniqueness of
maximizing measures with any constraint for generic continuous functions is
asserted. Moreover, prevalent uniqueness of maximizing measures for continuous
functions in the constrained settings easily follows from [Mor21] (see
Appendix B). However, in these studies, the differences between constrained
ergodic optimization and unconstrained one are not mentioned explicitly. In
some cases, constraints prevent existence of relatively maximizing periodic
measures (see Remark 1.3), which should give rise to the problem of existence
of periodic measures in a given rotation class. Moreover, in contrast to
Morris’ theorem [Mor2010] which asserts that for any dynamical systems with
the specification property every maximizing measure for a generic continuous
function has zero entropy, we can easily verify that there exists a constraint
such that for a generic potential its unique relatively maximizing measure has
positive entropy (see Proposition A.1). Thus it is important to investigate
the condition that the statement of Morris’s theorem holds for constrained
ergodic optimization.

In this paper, we study the structure of rotation classes and the generic
property in constrained ergodic optimization for symbolic dynamics. Our first
main result is the density of periodic measures with some rational
constraints. This is an analogical result to Sigmund’s work for a dynamical
system with the specification [Sigmund]. The difficulty in the constrained
case comes from the existence of a measure which is a convex combination of
ergodic measures with different rotation vectors. This prevents us to use the
ergodic decomposition in a given rotation class. To overcome this difficulty,
our approach requires a certain finiteness for both of a subshift and a
constraint function. Moreover, a detailed analysis is needed to construct a
periodic measure approximating a given invariant one with the same rotation
vector. Let $(\Omega,\sigma)$ be an irreducible sofic shift with finite
alphabets (See §2.2 below). A function $\varphi\in C(\Omega,\mathbb{R}^{d})$
is said to be locally constant if for each $i=1,\ldots,d$ there exists
$k_{i}\geq 0$ such that

$\displaystyle\varphi_{i}(x)=\varphi_{i}(y)\quad\mbox{if}\quad x_{j}=y_{j}\
\mbox{for}\ j=0,\ldots,k_{i}.$

Denote by $\mathcal{M}_{\sigma}^{p}(\Omega)$ the set of invariant measures
supported on a single periodic orbit.

######  Theorem 1.

Let $(\Omega,\sigma)$ be an irreducible sofic shift. Let
$\varphi=(\varphi_{1},\ldots,\varphi_{d})\in C(\Omega,\mathbb{Q}^{d})$ be a
$\mathbb{Q}^{d}$-valued locally constant function and $h\in{\rm int}({\rm
Rot}(\varphi))\cap\mathbb{Q}^{d}$. Then the set ${\rm
rv}_{\varphi}^{-1}(h)\cap\mathcal{M}_{\sigma}^{p}(\Omega)$ is dense in ${\rm
rv}_{\varphi}^{-1}(h)$.

######  Remark 1.1.

Theorem 1 is motivated by Theorem 10 in [GarLop07]: It was shown that for a
Walters potential $A$ on a subshift of finite type $\Omega$, there exists a
periodic measure $\mu$ whose action $\int A\ d\mu$ approximates $\int A\ d\nu$
for $\nu\in{\rm rv}_{\varphi}^{-1}(h)$ with ${\rm rv}_{\varphi}(\mu)=h$ if
$\nu$ is ergodic and the locally constant constraint $\varphi\in
C(\Omega,\mathbb{Q}^{d})$ is joint recurrent in relation to $\nu$ (see
[GarLop07] for the definition of the joint recurrence and their precise
statement). Note that Theorem 10 in [GarLop07] was suggested by the fact that
a circle homeomorphism with rational rotation number has a periodic point and
a result of this kind for symbolic dynamics was studied by Ziemian (see
Theorem 4.2 of [Zie95]).

######  Remark 1.2.

Although the properties of rotation sets are well-studied in [Zie95], to the
authors’ knowledge, that of rotation classes have attracted little attention.
We emphasize that Theorem 1 provides a more detailed description of Theorem
4.2 in [Zie95] under weaker assumptions.

######  Remark 1.3.

If a constraint $\varphi\in C(\Omega,\mathbb{Q}^{d})$ is locally constant, it
is easy to see that the rotation vector of a periodic measure should be
rational, i.e.,

$\displaystyle\varphi(\mathcal{M}_{\sigma}^{p}(\Omega))\subset{\rm
Rot}(\varphi)\cap\mathbb{Q}^{d}.$

Hence in Theorem 1 we need to choose a rotation vector $h$ in ${\rm
Rot}(\varphi)\cap\mathbb{Q}^{d}$.

Applying Theorem 1, we next investigate the property of relative maximizing
measure for symbolic dynamics. Regarding (3) as a functional on
$C(\Omega,\mathbb{R})$, we can characterize a relative maximizing measure as a
“tangent” measure. This characterization allows us to adapt argument in
[Mor2010] for our constrained case (See §3 for more details). The following
generic property is our second main result.

######  Theorem 2.

Let $(\Omega,\sigma)$ be an irreducible sofic shift. Let
$\varphi=(\varphi_{1},\ldots,\varphi_{d})\in C(\Omega,\mathbb{Q}^{d})$ be a
$\mathbb{Q}^{d}$-valued locally constant function and $h\in{\rm int}({\rm
Rot}(\varphi))\cap\mathbb{Q}^{d}$. Then for generic $f\in
C(\Omega,\mathbb{R})$ every relative maximizing measure of $f$ with constraint
$h\in{\rm Rot}(\varphi)$ has zero entropy. In particular, setting
$K(f):=\bigcup_{\mu\in\mathcal{M}_{h}^{\varphi}(f)}{\rm supp}\mu$, we have
$h_{{\rm top}}(K)=0$ for generic $f\in C(\Omega,\mathbb{R})$.

The remainder of this paper is organized as follows. In §2, we study the
structure of rotation classes. In particular, we clarify our definitions and
notations for symbolic dynamics and prove Theorem 1 in §2.2. In §3 we will
illustrate that a relative maximizing measure is regarded as a tangent measure
of (3) and prove Theorem 2.

##  2\. Structure of rotation classes

###  2.1. Density of convex combinations of periodic measures

We first investigate the structure of rotation classes for a continuous map
$T:X\rightarrow X$ on a compact metric space $X$ such that
$\mathcal{M}_{T}^{p}(X)$ is dense in $\mathcal{M}_{T}(X)$. As mentioned in
Remark 1.3, a rotation class does not contain a periodic measure in some
cases. Nevertheless, we have the density of convex combinations of periodic
measures in a rotation class for every continuous constraint function and for
every rotation vector in the interior of the rotation set.

For $\nu\in\mathcal{M}_{T}(X)$, a finite set $F\subset C(X,\mathbb{R})$ and
$\varepsilon>0$ set

$\displaystyle
U_{(F,\varepsilon)}(\nu):=\left\\{\mu\in\mathcal{M}_{T}(X):\left|\int
fd\mu-\int fd\nu\right|<\varepsilon,f\in F\right\\}.$

Let

$\displaystyle\Delta^{d}$
$\displaystyle=\left\\{(\lambda_{1},\ldots,\lambda_{d})\in(0,1)^{d}:\sum_{i=1}^{d}\lambda_{i}=1\right\\},$
$\displaystyle\mathcal{N}_{T}$
$\displaystyle=\Bigg{\\{}\sum_{i=1}^{d+1}\lambda_{i}\mu_{i}:\
\mu_{i}\in\mathcal{M}_{T}^{p}(X),\quad(\lambda_{1},\ldots,\lambda_{d+1})\in\Delta^{d+1},$
$\displaystyle\qquad\quad\dim{\rm span}\\{{\rm rv}_{\varphi}(\mu_{1})-{\rm
rv}_{\varphi}(\mu_{d+1}),\ldots,{\rm rv}_{\varphi}(\mu_{d})-{\rm
rv}_{\varphi}(\mu_{d+1})\\}=d\Bigg{\\}},$

where ${\rm span}\\{h_{1},\ldots,h_{d}\\}$ is the vector space spanned by
$\\{h_{1},\ldots,h_{d}\\}$. We begin with the following proposition.

######  Proposition 2.1.

Let $T:X\rightarrow X$ be a continuous map on a compact metric space $X$ such
that $\mathcal{M}_{T}^{p}(X)$ is dense in $\mathcal{M}_{T}(X)$. Let
$\varphi=(\varphi_{1},\ldots,\varphi_{d})\in C(X,\mathbb{R}^{d})$ and
$h\in{\rm int}({\rm Rot}(\varphi))$. Then the set ${\rm
rv}_{\varphi}^{-1}(h)\cap\mathcal{N}_{T}$ is dense in ${\rm
rv}_{\varphi}^{-1}(h)$.

###### Proof.

Take $\nu\in{\rm rv}_{\varphi}^{-1}(h)$ and a open neighborhood $U_{\nu}$ of
$\nu$. Then there exists a finite set $F\subset C(X,\mathbb{R})$ and
$\varepsilon>0$ such that $U_{(F,\varepsilon)}(\nu)\subset U_{\nu}$.

Since $h\in{\rm int}({\rm Rot}(\varphi))$, there exists $\delta>0$ such that

$\displaystyle B_{\sqrt{d}\delta}(h)\subset{\rm Rot}(\varphi).$

where $B_{\sqrt{d}\delta}(h)$ is the open ball of radius $\sqrt{d}\delta$
centered at $h$. Let $h^{\prime}_{i}=h+\delta e_{i}$ for $i=1,\ldots,d+1$
where $\\{e_{i}\\}_{i=1}^{d}$ is the standard basis of $\mathbb{R}^{d}$ and
$e_{d+1}=-\sum_{i=1}^{d}e_{i}$. Then $\\{h^{\prime}_{i}\\}_{i=1}^{d+1}$ forms
a simplex with the barycenter $h$. Let $\xi_{i}\in{\rm
rv}_{\varphi}^{-1}(h^{\prime}_{i})$.

Fix

$\displaystyle 0<t<\frac{\varepsilon}{4\max_{f\in F}\|f\|_{\infty}}$

and define $\nu_{i}:=(1-t)\nu+t\xi_{i}$. Then for $f\in F$ we have

$\displaystyle\left|\int fd\nu_{i}-\int fd\nu\right|$ $\displaystyle\leq
t\left|\int fd\xi_{i}-\int fd\nu\right|\leq
2t\|f\|_{\infty}<\frac{\varepsilon}{2}$

and $\nu_{i}\in U_{(F,\varepsilon/2)}(\nu)$. By the definition of $\nu_{i}$ we
have

$\displaystyle{\rm rv}_{\varphi}(\nu_{i})=(1-t){\rm rv}_{\varphi}(\nu)+t{\rm
rv}_{\varphi}(\xi_{i})=(1-t)h+th^{\prime}_{i}=h+t\delta e_{i}.$

Thus $\frac{1}{d+1}\sum_{i=1}^{d+1}{\rm rv}_{\varphi}(\nu_{i})=h$ holds.

Since $\mathcal{M}_{T}^{p}(X)$ is dense in $\mathcal{M}_{T}(X)$, for $r>0$,
there exists

$\displaystyle\mu_{i}\in U_{(F,\varepsilon/2)}(\nu_{i})\cap
U_{(\\{\varphi\\},r/\sqrt{d})}(\nu_{i})\cap\mathcal{M}_{T}^{p}(X).$

Since ${\rm rv}_{\varphi}(\nu_{1})-{\rm rv}_{\varphi}(\nu_{d+1}),{\rm
rv}_{\varphi}(\nu_{2})-{\rm rv}_{\varphi}(\nu_{d+1}),\ldots,{\rm
rv}_{\varphi}(\nu_{d})-{\rm rv}_{\varphi}(\nu_{d+1})$ are linearly
independent, by the open property of linearly independence and the continuity
of the map ${\rm rv}_{\varphi}$, we see that ${\rm rv}_{\varphi}(\mu_{1})-{\rm
rv}_{\varphi}(\mu_{d+1}),{\rm rv}_{\varphi}(\mu_{2})-{\rm
rv}_{\varphi}(\mu_{d+1}),\ldots,{\rm rv}_{\varphi}(\mu_{d})-{\rm
rv}_{\varphi}(\mu_{d+1})$ are also linearly independent for all sufficiently
small $r>0$. Moreover, we obtain

$\displaystyle\left|\left|\frac{1}{d+1}\sum_{i=1}^{d+1}{\rm
rv}(\mu_{i})-h\right|\right|_{\mathbb{R}^{d}}=\left|\left|\frac{1}{d+1}\sum_{i=1}^{d+1}{\rm
rv}(\mu_{i})-\frac{1}{d+1}\sum_{i=1}^{d+1}{\rm
rv}(\nu_{i})\right|\right|_{\mathbb{R}^{d}}$
$\displaystyle\leq\frac{1}{d+1}\sum_{i=1}^{d+1}\left|\left|{\rm
rv}_{\varphi}(\mu_{i})-{\rm
rv}_{\varphi}(\nu_{i})\right|\right|_{\mathbb{R}^{d}}<\frac{d+1}{d+1}r=r,$

where $\|\cdot\|_{\mathbb{R}^{d}}$ is the standard norm of $\mathbb{R}^{d}$.
Hence $h$ is contained in the open ball of radius $r$ centered at
$\frac{1}{d+1}\sum_{i=1}^{d+1}{\rm rv}(\mu_{i})$. Taking $r>0$ sufficiently
small, we deduce that $h$ is an interior point of the simplex whose vertices
are ${\rm rv}_{\varphi}(\mu_{1}),\ldots,{\rm rv}_{\varphi}(\mu_{d+1})$, which
implies that there exists $(\lambda_{1},\ldots,\lambda_{d+1})\in\Delta^{d+1}$
such that $h=\sum_{i=1}^{d+1}\lambda_{i}{\rm rv}_{\varphi}(\mu_{i})$.

Let $\tilde{\nu}=\sum_{i=1}^{d+1}\lambda_{i}\mu_{i}$. Trivially, ${\rm
rv}_{\varphi}(\tilde{\nu})=h$ holds. Then for every $f\in F$ we have

$\displaystyle\left|\int fd\tilde{\nu}-\int fd\nu\right|$
$\displaystyle=\left|\sum_{i=1}^{d+1}\lambda_{i}\int
fd\mu_{i}-\sum_{i=1}^{d+1}\lambda_{i}\int fd\nu\right|$
$\displaystyle\leq\sum_{i=1}^{d+1}\lambda_{i}\left|\int fd\mu_{i}-\int
fd\nu\right|$ $\displaystyle\leq\sum_{i=1}^{d+1}\lambda_{i}\left(\left|\int
fd\mu_{i}-\int fd\nu_{i}\right|+\left|\int fd\nu_{i}-\int fd\nu\right|\right)$
$\displaystyle<\sum_{i=1}^{d+1}\lambda_{i}\left(\frac{\varepsilon}{2}+\frac{\varepsilon}{2}\right)=\varepsilon,$

i.e., $\tilde{\nu}=\sum_{i=1}^{d+1}\lambda_{i}\mu_{i}\in U_{(F,\varepsilon)}$,
and this is precisely the assertion of the proposition. ∎

######  Remark 2.2.

In the proof of Proposition 2.1, we have $\mu_{i}\in U_{(F,\varepsilon)}(\nu)$
for every $i\in\\{1,\ldots,d+1\\}$ since we have $\nu_{i}\in
U_{(F,\varepsilon/2)}(\nu)$ and $\mu_{i}\in U_{(F,\varepsilon/2)}(\nu_{i})$.
We will use this fact in the proof of Theorem 1.

######  Corollary 2.3.

Assume the hypotheses of Proposition 2.1. Let $H_{\mu}$ be the entropy map of
$T$. Then the set

$\displaystyle\mathcal{Z}=\\{\mu\in{\rm rv}_{\varphi}^{-1}(h):H_{\mu}=0\\}$

is a residual subset of ${\rm rv}_{\varphi}^{-1}(h)$.

###### Proof.

The argument is similar to [DGS][Proposition 22.16, p.223]. By Proposition
2.1, $\mathcal{N}_{T}\cap{\rm rv}_{\varphi}^{-1}(h)$ is dense in ${\rm
rv}_{\varphi}^{-1}(h)$ and thus $\mathcal{Z}$ is also. Moreover, upper semi-
continuity of the entropy map $\mu\mapsto H_{\mu}$ implies for every $n\geq 1$

$\displaystyle\mathcal{Z}_{n}:=\left\\{\mu\in{\rm rv}_{\varphi}^{-1}(h):0\leq
H_{\mu}<\frac{1}{n}\right\\}\supset\mathcal{N}_{T}\cap{\rm
rv}_{\varphi}^{-1}(h)$

is nonempty, open and dense in ${\rm rv}_{\varphi}^{-1}(h)$. Hence
$\mathcal{Z}=\bigcap_{n\geq 1}\mathcal{Z}_{n}$ is a residual set in ${\rm
rv}_{\varphi}^{-1}(h)$. ∎

######  Lemma 2.4.

Assume the hypotheses of Proposition 2.1. Let
$\mu_{1},\ldots,\mu_{d+1}\in\mathcal{M}_{T}^{p}(X)$ such that ${\rm
rv}_{\varphi}(\mu_{1}),\ldots,{\rm rv}_{\varphi}(\mu_{d+1})\in\mathbb{Q}^{d}$
and

(4) $\displaystyle\dim{\rm span}\\{{\rm rv}_{\varphi}(\mu_{1})-{\rm
rv}_{\varphi}(\mu_{d+1}),\ldots,{\rm rv}_{\varphi}(\mu_{d})-{\rm
rv}_{\varphi}(\mu_{d+1})\\}=d.$

Let $(\lambda_{1},\ldots,\lambda_{d+1})\in\Delta^{d+1}$ and
$\tilde{\nu}=\sum_{i=1}^{d+1}\lambda_{i}\mu_{i}\in\mathcal{M}_{T}(X)$. Then
$\lambda_{i}$ is rational for each $i=1,\ldots,d+1$ if ${\rm
rv}_{\varphi}(\tilde{\nu})$ belongs to $\mathbb{Q}^{d}$.

###### Proof.

Since $\tilde{\nu}=\sum_{i=1}^{d+1}\lambda_{i}\mu_{i}$ and
$\sum_{i=1}^{d+1}\lambda_{i}=1$, we have

(5) $\displaystyle{\rm rv}_{\varphi}(\tilde{\nu})={\rm
rv}_{\varphi}(\mu_{d+1})+\sum_{i=1}^{d}\lambda_{i}\left({\rm
rv}_{\varphi}(\mu_{i})-{\rm rv}_{\varphi}(\mu_{d+1})\right).$

Let $V$ be a $d\times d$-matrix given by

(6) $\displaystyle V=\Big{(}{\rm rv}(\mu_{d+1})-{\rm rv}(\mu_{1}),\ldots,{\rm
rv}(\mu_{d+1})-{\rm rv}(\mu_{d})\Big{)}$

and $\lambda$ be a column vector given by $(\lambda)_{i}=\lambda_{i}$ for
$i=1,\ldots,d$. It follows from (4) that the matrix $V$ is invertible.
Therefore, by (5), we obtain

(7) $\displaystyle\lambda=V^{-1}\left({\rm rv}_{\varphi}(\mu_{d+1})-{\rm
rv}_{\varphi}(\tilde{\nu})\right).$

Since ${\rm rv}_{\varphi}(\mu_{1}),\ldots,{\rm rv}_{\varphi}(\mu_{d+1})$ and
${\rm rv}_{\varphi}(\tilde{\nu})$ are in $\mathbb{Q}^{d}$, each component of
$V$ and that of its inverse $V^{-1}$ are rational. Therefore, by (7), we
deduce that $\lambda_{i}$ is rational for each $i=1,\ldots,d+1$. ∎

###  2.2. Symbolic dynamics and density of periodic measures

We next consider symbolic dynamics. In this particular case, under some
assumptions, we can prove the density of periodic measures in a given rotation
class. Denote by $\mathbb{N}_{0}$ the set of all non-negative integers. For a
finite set $\mathcal{A}$ we consider the one-sided infinite product
$\mathcal{A}^{\mathbb{N}_{0}}$ equipped with the product topology of the
discrete one.

Let $\sigma$ be the shift map on $\mathcal{A}^{\mathbb{N}_{0}}$ (i.e.,
$(\sigma(\underline{x}))_{i}=x_{i+1}$ for each $i\in{\mathbb{N}_{0}}$ and
$\underline{x}=(x_{i})_{i\in{\mathbb{N}_{0}}}\in\mathcal{A}^{\mathbb{N}_{0}}$).
When a subset $\Omega$ of $\mathcal{A}^{\mathbb{N}_{0}}$ is $\sigma$-invariant
and closed, we call it a subshift. Slightly abusing the notation we denote by
$\sigma$ the shift map restricted on $\Omega$.

For a subshift $\Omega$, let $[u]=\left\\{\underline{x}\in\Omega:u=x_{0}\cdots
x_{n-1}\right\\}$ for each $u\in\mathcal{A}^{n}$, $n\geq 1$ and set
$\mathcal{L}(\Omega)=\left\\{u\in\bigcup_{n\geq
1}\mathcal{A}^{n}:[u]\neq\emptyset\right\\}$. We also denote
$\mathcal{L}_{n}(\Omega):=\\{u\in\mathcal{L}(\Omega):|u|=n\\}$ for $n\geq 1$,
where $|u|$ denotes the length of $u$, i.e., $|u|=n$ if $u=u_{0}\cdots
u_{n-1}\in\mathcal{A}^{n}$. For $u,v\in\mathcal{L}(\Omega)$, we use the
juxtaposition $uv$ to denote the word obtained by the concatenation and
$u^{\infty}$ means a one-sided infinite sequence
$uuu\cdots\in\mathcal{A}^{\mathbb{N}_{0}}$. We say that $\Omega$ is
irreducible if for any $i,j\in A$, we can find $u\in\mathcal{L}(\Omega)$ such
that $iuj\in\mathcal{L}(\Omega)$ holds. For each $u\in\mathcal{L}(\Omega)$
define its follower set $F_{\Omega}(u)$ by
$F_{\Omega}(u)=\\{v\in\mathcal{L}(\Omega):uv\in\mathcal{L}(\Omega)\\}$. Note
that the follower sets of different words may coincide. A subshift $\Omega$ is
said to be sofic if the set of follower sets is finite:
$\\#\\{F_{\Omega}(u):u\in\mathcal{L}(\Omega)\\}<+\infty$. We refer to [LM] and
references therein for additional details on sofic shifts. It is easy to see
that the finiteness of the set of the follower sets yields the following:

######  Lemma 2.5.

Let $(\Omega,\sigma)$ be a sofic shift. Then there exists $\kappa\geq 1$ such
that for every $u\in\mathcal{L}_{\kappa}(\Omega)$ and
$v\in\mathcal{L}(\Omega)$, we have $F_{\Omega}(vu)=F_{\Omega}(u)$ provided
$u\in F_{\Omega}(v)$.

In order to prove Theorem 1 we shall show that on a sofic shift periodic
orbits which share the same word can be concatenated without extra gap words.

######  Lemma 2.6.

Let $(\Omega,\sigma)$ be a sofic shift and $\kappa\geq 1$ for which Lemma 2.5
holds. Let $u\in\mathcal{L}_{\kappa}(\Omega)$ and $v,w\in\mathcal{L}(\Omega)$
such that $(vu)^{\infty},(wu)^{\infty}\in\Omega$. Then for every $k\geq 1$ and
sequences $\\{m_{i}\\}_{i=1}^{k},\\{n_{i}\\}_{i=1}^{k}\subset\mathbb{N}$, we
have

$\displaystyle((vu)^{m_{1}}(wu)^{n_{1}}(vu)^{m_{2}}\cdots(vu)^{m_{k}}(wu)^{n_{k}})^{\infty}\in\Omega.$

###### Proof.

Let $k\geq 1$, $\\{m_{i}\\}_{i=1}^{k}$ and
$\\{n_{i}\\}_{i=1}^{k}\subset\mathbb{N}$. Since $(vu)^{\infty}\in\Omega$, we
see that $u\in F_{\Omega}((vu)^{m_{1}-1}v)$. By Lemma 2.5,
$F_{\Omega}((vu)^{m_{1}})=F_{\Omega}(u)$ holds. Moreover, it follows from
$(wu)^{\infty}\in\Omega$ that

$\displaystyle(wu)^{n_{1}}\in F(u)=F((vu)^{m_{1}}),$

which implies $(vu)^{m_{1}}(wu)^{n_{1}}\in\mathcal{L}(\Omega)$. We also have
$F((vu)^{m_{1}}(wu)^{n_{1}})=F(u)$ by Lemma 2.5. Similarly, we obtain

$\displaystyle(vu)^{m_{2}}(wu)^{n_{2}}\in F(u)=F((vu)^{m_{1}}(wu)^{n_{1}})$

and $(vu)^{m_{1}}(wu)^{n_{1}}(vu)^{m_{2}}(wu)^{n_{2}}\in\mathcal{L}(\Omega)$.
Repeating this argument,

$\displaystyle((vu)^{m_{1}}(wu)^{n_{1}}(vu)^{m_{2}}\cdots(vu)^{m_{k}}(wu)^{n_{k}})^{l}\in\mathcal{L}(\Omega)$

holds for every $\ell\geq 1$, which implies

$\displaystyle((vu)^{m_{1}}(wu)^{n_{1}}(vu)^{m_{2}}\cdots(vu)^{m_{k}}(wu)^{n_{k}})^{\infty}\in\Omega.$

∎

In addition, an irreducible sofic shift satisfies the specification property
and thus $\mathcal{M}_{\sigma}^{p}(\Omega)$ is dense in
$\mathcal{M}_{\sigma}(\Omega)$ (see for example [Weiss73] and [Sigmund]).
Hence we have the following by Proposition 2.1.

######  Corollary 2.7.

Let $(\Omega,\sigma)$ be an irreducible sofic shift. Let
$\varphi=(\varphi_{1},\ldots,\varphi_{d})\in C(\Omega,\mathbb{R}^{d})$ be a
locally constant function and $h\in{\rm int}({\rm Rot}(\varphi))$. Then the
set ${\rm rv}_{\varphi}^{-1}(h)\cap\mathcal{N}_{\sigma}$ is dense in ${\rm
rv}_{\varphi}^{-1}(h)$.

Using Lemma 2.6 and Corollary 2.7, we can prove Theorem 1.

###### Proof of Theorem 1.

Take $\nu\in{\rm rv}_{\varphi}^{-1}(h)$ and a open neighborhood $U_{\nu}$ of
$\nu$. There exists a finite set $F\subset C(\Omega,\mathbb{R})$ and
$\varepsilon>0$ such that $U_{(F,\varepsilon)}(\nu)\subset U_{\nu}$. Without
loss of generality we may assume every $f\in F$ is locally constant. Take
$\kappa$ for which Lemma 2.5 holds and $u=u_{0}\cdots
u_{\kappa-1}\in\mathcal{L}_{\kappa}(\Omega)$ such that $\nu([u])>0$. Replace
$F$ and $\varepsilon$ with $F\cup\\{\chi_{[u]}\\}$ and
$\min\\{\varepsilon,\nu([u])/2\\}$ respectively, where $\chi_{[u]}$ is the
characteristic function of $[u]$.

By Corollary 2.7, there is $\tilde{\nu}\in U_{(F,\varepsilon/2)}(\nu)$ of the
form $\tilde{\nu}=\sum_{i=1}^{d+1}\lambda_{i}\mu_{i}$ where
$\mu_{i}\in\mathcal{M}_{\sigma}^{p}(\Omega)$ and
$(\lambda_{1},\ldots,\lambda_{d+1})\in\Delta^{d+1}$ with (4). Note that ${\rm
rv}_{\varphi}(\mu_{i})\in\mathbb{Q}^{d}\ (i=1,\ldots,d+1)$ as stated in Remark
1.3. For each $i=1,\ldots,d+1$, we will denote by $a_{i}^{\infty}$ the
corresponding periodic orbits to $\mu_{i}\in\mathcal{M}_{\sigma}^{p}(\Omega)$
where $a_{i}$ is the word with lengths of periods. Moreover, by Lemma 2.4,
each $\lambda_{i}$ can be written as $\lambda_{i}=q_{i}/Q$ where
$q_{1},\ldots,q_{d+1},Q\in\mathbb{N}$ with $q_{1}+\ldots+q_{d+1}=Q$ since
$(\lambda_{1},\ldots,\lambda_{d+1})\in\Delta^{d+1}\cap\mathbb{Q}^{d+1}$.

Let $\ell$ be the maximum length of the words on which elements of
$F\cup\\{\varphi_{1},\ldots,\varphi_{d}\\}$ depend. We can assume

$\displaystyle\ell<\min\\{|a_{1}|,\ldots,|a_{d+1}|\\}$

by replacing $a_{i}\ (i=1,\ldots,d+1)$ with concatenations $a_{i}^{k}$ for
some $k\in\mathbb{N}$ if necessary. As stated in Remark 2.2, we have
$\mu_{i}\in U_{(F,\varepsilon)}(\nu)$ for every $i=1,\ldots,d+1$, which
implies $\mu_{i}([u])>\nu([u])-\varepsilon\geq\nu([u])/2>0$. Hence $u$ is a
subword of each $a_{i}$ and without loss of generality we may assume
$a_{i,0}\cdots a_{i,|u|-1}=u$. For $g\in F\cup\\{\varphi\\}$ define
$\delta_{g,k}\ (k=1,\ldots,d+1)$ by

$\displaystyle\delta_{g,k}=\sum_{i=0}^{\ell-1}\Big{\\{}$ $\displaystyle
g(\sigma^{|a_{k}|-\ell+i}(a_{k}a_{k+1}))-g(\sigma^{|a_{k}|-\ell+i}(a_{k}a_{k}))\Big{\\}}$

where $a_{d+2}=a_{1}$. Set $\delta_{g}=\delta_{g,1}+\ldots+\delta_{g,d+1}$.

Let $V$ be a $d\times d$-matrix given by (6). As stated in the proof of Lemma
2.4, the matrix $V$ is invertible by (4). Since each component of $V$ and
$\delta_{\varphi}$ is rational, we denote $V^{-1}\delta_{\varphi}=v/R$ where
$v\in\mathbb{Z}^{d}$ and $R\in\mathbb{N}$. Let $v_{i}=(v)_{i}\
(i=1,\ldots,d)$.

Now we construct a periodic measure near $\nu$ with the rotation vector $h$.
Since $a_{1},\ldots,a_{d+1}$ share the same word $u$, by Lemma 2.6, we can
concatenate them without extra gap words. Hence let

$\displaystyle A$ $\displaystyle=|a_{1}|\ldots|a_{d+1}|,\quad C=\max_{f\in F,\
i=1,\ldots,d+1}\left|\int f\ d\mu_{i}\right|,\quad\delta^{*}=\max_{f\in
F}|\delta_{f}|,$ $\displaystyle m_{i}^{\prime}$ $\displaystyle=Av_{i}/|a_{i}|\
(i=1,\ldots,d),\quad m_{d+1}^{\prime}=-A(\sum_{i=1}^{d}v_{i})/|a_{d+1}|,$
$\displaystyle M_{j}^{\prime}$ $\displaystyle=Aq_{j}/|a_{j}|,\quad
M_{j}=tARM_{j}^{\prime}+m_{j}^{\prime}\ (j=1,\ldots,d+1),$ $\displaystyle y$
$\displaystyle=a_{1}^{tM_{1}^{\prime}+m_{1}^{\prime}}a_{2}^{tM_{2}^{\prime}+m_{2}^{\prime}}\cdots
a_{d+1}^{tM_{d+1}^{\prime}+m_{d+1}^{\prime}},$ $\displaystyle z$
$\displaystyle=a_{1}^{tM_{1}^{\prime}}a_{2}^{tM_{2}^{\prime}}\cdots
a_{d+1}^{tM_{d+1}^{\prime}},\quad x=yz^{(AR-1)},$

where $t\in\mathbb{N}$ is large enough to satisfy

(8)
$\displaystyle\frac{AR}{|x|}\delta^{*}+C\sum_{i=1}^{d+1}\left|\frac{M_{i}|a_{i}|}{|x|}-\lambda_{i}\right|<\frac{\varepsilon}{2}$

and $tM_{d+1}^{\prime}-A(\sum_{i=1}^{d}v_{i})/|a_{d+1}|>0$. Note that such
$t\in\mathbb{N}$ exists since

$\displaystyle|x|$
$\displaystyle=\sum_{i=1}^{d+1}M_{i}|a_{i}|=tAR\sum_{i=1}^{d+1}M_{i}^{\prime}|a_{i}|,$
$\displaystyle\lambda_{i}$
$\displaystyle=\frac{q_{i}}{\sum_{j=1}^{d+1}q_{j}}=\frac{M_{i}^{\prime}|a_{i}|}{\sum_{j=1}^{d+1}M_{j}^{\prime}|a_{j}|}$

hold and the left hand side of (8) tends to $0$ as $t\to+\infty$. Let $\mu$ be
the periodic measure supported on $x^{\infty}$.

First we check ${\rm rv}_{\varphi}(\mu)=h(={\rm rv}_{\varphi}(\tilde{\nu}))$.
Since $\varphi$ is locally constant,

$\displaystyle{\rm rv}_{\varphi}(\mu)$
$\displaystyle=\frac{1}{|x|}S_{|x|}\varphi(x)$
$\displaystyle=\frac{1}{|x|}\left(AR\delta_{\varphi}+\sum_{i=1}^{d+1}M_{i}|a_{i}|\int\varphi
d\mu_{i}\right)$
$\displaystyle=\frac{1}{|x|}AR\delta_{\varphi}+\frac{1}{|x|}\sum_{i=1}^{d+1}m_{i}^{\prime}|a_{i}|{\rm
rv}_{\varphi}(\mu_{i})+\sum_{i=1}^{d+1}\lambda_{i}{\rm rv}_{\varphi}(\mu_{i})$
$\displaystyle=h+\frac{A}{|x|}Vv+\frac{1}{|x|}\sum_{i=1}^{d+1}m_{i}^{\prime}|a_{i}|{\rm
rv}_{\varphi}(\mu_{i})$
$\displaystyle=h+\frac{A}{|x|}\left\\{Vv+\sum_{i=1}^{d}v_{i}{\rm
rv}_{\varphi}(\mu_{i})-(\sum_{j=1}^{d}v_{j}){\rm
rv}_{\varphi}(\mu_{d+1})\right\\}=h.$

Next we check $\mu\in U_{(F,\varepsilon)}(\nu)$. Let $f\in F$. We compute

$\displaystyle\left|\frac{1}{|x|}S_{|x|}f-\int f\ d\nu\right|$
$\displaystyle\leq\left|\frac{1}{|x|}S_{|x|}f-\int f\
d\tilde{\nu}\right|+\left|\int f\ d\tilde{\nu}-\int f\ d\nu\right|$
$\displaystyle=\left|\frac{1}{|x|}\Big{(}AR\delta_{f}+\sum_{i=1}^{d+1}M_{i}|a_{i}|\int
fd\mu_{i}\Big{)}-\sum_{j=1}^{d+1}\lambda_{i}\int
fd\mu_{i}\right|+\frac{\varepsilon}{2}$
$\displaystyle\leq\sum_{i=1}^{d+1}\left|\left(\frac{M_{i}|a_{i}|}{|x|}-\lambda_{i}\right)\int
fd\mu_{i}\right|+\frac{AR}{|x|}|\delta_{f}|+\frac{\varepsilon}{2}$
$\displaystyle<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon,$

which completes the proof. ∎

######  Remark 2.8.

In the proof of Theorem 1, we think the word $y$ as a corrective one to attain
the desired rotation vector. A similar approach is used in Theorem 5 of
[Jen01rotation] in a different setting but our construction is more explicit
than it.

######  Remark 2.9.

Note that the error term $\delta_{\varphi}$ does not depend on
$m^{\prime}_{i}\ (i=1,\ldots,d+1)$ in our case. For a subshift with the
specification condition, we can concatenate the words
$a_{1}^{m^{\prime}_{1}},\ldots,a_{d+1}^{m^{\prime}_{d+1}}$ with some gap words
but the error term in such case depends on $m^{\prime}_{i}\ (i=1,\ldots,d+1)$,
which implies we cannot choose a suitable corrective word $y$ for the error
term $\delta_{\varphi}$ in such case. So we have to overcome this difficulty
to extend Theorem 1 to the case of a subshift with the specification
condition.

######  Remark 2.10.

For a rotation vector in the boundary of a rotation set, there may exist no
periodic measure in the rotation class. Let
$\Omega\subset\\{1,2,3\\}^{\mathbb{N}_{0}}$ be a Markov shift with an
adjacency matrix

$\displaystyle A=\begin{pmatrix}1&1&0\\\ 1&1&1\\\ 0&1&1\end{pmatrix}$

(i.e.,
$\Omega=\\{\underline{x}\in\\{1,2,3\\}^{\mathbb{N}_{0}}:A_{x_{i}x_{i+1}}=1\
\mbox{for all}\ i\in\mathbb{N}_{0}\\}$) and define
$\varphi=(\varphi_{1},\varphi_{2},\varphi_{3}):\Omega\rightarrow\mathbb{Q}^{3}$
by

$\displaystyle\varphi_{i}(\underline{x})=\left\\{\begin{array}[]{cc}1&x_{0}=i\\\
0&\mbox{else.}\end{array}\right.$

Then its rotation set ${\rm Rot}(\varphi)$ is the polyhedron whose extremal
points are $e_{1},e_{2}$ and $e_{3}$ , where $\\{e_{1},e_{2},e_{3}\\}$ is the
standard basis of $\mathbb{R}^{3}$. Take a rotation vector $h$ from the open
side whose vertices are $e_{1}$ and $e_{3}$, i.e.,
$h\in\\{te_{1}+(1-t)e_{3}:t\in(0,1)\\}$. If there exists a periodic measure
$\mu\in{\rm rv}^{-1}(h)$, the corresponding periodic orbit $u^{\infty}$ should
contain both of $1$ and $3$. Since there is no sequence including $13$ and
$31$ in $\Omega$, the word $u$ must contain the symbol $2$. Hence we have
${\rm rv}_{\varphi_{2}}(\mu)>0$ and $h={\rm
rv}_{\varphi}(\mu)\notin\\{te_{1}+(1-t)e_{3}:t\in(0,1)\\}$, which is a
contradiction.

##  3\. Generic property for constraint ergodic optimization

###  3.1. Characterization by tangency

In this subsection, we turn to a general dynamical system and state analogous
results in [Mor2010] for the constrained case. Let $T:X\rightarrow X$ be a
continuous map on a compact metric space. Denote by $\mathcal{M}_{T}^{e}(X)$
the set of all ergodic measures on $X$. By the Riesz representation theorem a
Borel probability measure on $X$ can be regarded as a bounded linear
functional on $C(X,\mathbb{R})$. Hence we use the operator norm
$\|\mu\|=\sup\\{|\mu(f)|:f\in C(X,\mathbb{R})\ \mbox{with}\
\|f\|_{\infty}=1\\}$ for an invariant measure $\mu\in\mathcal{M}_{T}(X)$.

First we characterize a relative maximizing measures by tangency to (3). Let
$\varphi\in C(X,\mathbb{R}^{d})$ and $h\in{\rm Rot}(\varphi)$.

######  Lemma 3.1.

$\mu\in\mathcal{M}^{\varphi}_{h}(f)$ iff $\mu$ is tangent to
$\beta^{\varphi}_{h}$ at $f$.

###### Proof.

Let $\mu\in\mathcal{M}^{\varphi}_{h}(f)$. Then for every $g\in
C(X,\mathbb{R})$ we have

$\beta^{\varphi}_{h}(f+g)-\beta^{\varphi}_{h}(f)\geq\int f+g\ d\mu-\int f\
d\mu\geq\int g\ d\mu.$

Let $\mu$ be tangent to $\beta^{\varphi}_{h}$ at $f$. For every $g\in
C(X,\mathbb{R})$ we have

$\displaystyle\int g\ d\mu$
$\displaystyle\leq\beta^{\varphi}_{h}(f+g)-\beta^{\varphi}_{h}(f)$
$\displaystyle=\beta^{\varphi}_{h}(g)+\beta^{\varphi}_{h}(f+g)-\beta^{\varphi}_{h}(f)-\beta^{\varphi}_{h}(g)$
$\displaystyle\leq\beta^{\varphi}_{h}(g).$

Then we can show that $\mu$ is an invariant probability measure in the same
way as unconstrained case (See for example [Shi2018, Bre08]). We now see that
$\mu$ takes the rotation vector $h$. For $i=1,\ldots,d$ we have

$\displaystyle\int\varphi_{i}\
d\mu\leq\beta^{\varphi}_{h}(\varphi_{i})=h_{i},\quad-\int\varphi_{i}\
d\mu\leq\beta^{\varphi}_{h}(-\varphi_{i})=-h_{i},$

which yields ${\rm rv}_{\varphi}(\mu)=h$.

Finally we check $\int f\ d\mu=\beta^{\varphi}_{h}(f)$. Indeed,

$\displaystyle\int-f\
d\mu\leq\beta^{\varphi}_{h}(f-f)-\beta^{\varphi}_{h}(f)=-\beta_{h}^{\varphi}(f).$

Multiplying $-1$, we have

$\displaystyle\int fd\mu\geq\beta^{\varphi}_{h}(f).$

∎

Checking that a relative maximizing measure is characterized by tangency of
(3), we can adopt techniques in [Mor2010].

######  Lemma 3.2.

Let $f\in C(X,\mathbb{R})$ and $\varepsilon>0$. If $\nu\in{\rm
rv}_{\varphi}^{-1}(h)\cap\mathcal{M}_{T}^{e}(X)$ such that
$\beta^{\varphi}_{h}(f)-\int f\ d\nu<\varepsilon$, then there exists $g\in
C(X,\mathbb{R})$ such that $\|f-g\|_{\infty}<\varepsilon$ and
$\nu\in\mathcal{M}^{\varphi}_{h}(g)$.

###### Proof.

Applying Bishop-Phelps’s theorem [Israel, Theorem V.1.1.] to $f$, $\nu$ and
$\varepsilon^{\prime}=1$ we have $g\in C(X,\mathbb{R})$, $\eta\in{\rm
rv}_{\varphi}^{-1}(h)$ such that $\eta$ is tangent to $\beta^{\varphi}_{h}$ at
$g$, $\|\nu-\eta\|<\varepsilon^{\prime}=1$ and

$\|f-g\|_{\infty}<\frac{1}{\varepsilon^{\prime}}\left(\beta^{\varphi}_{h}(f)-\int
fd\nu\right)<\varepsilon.$

By Lemma 3.1 we have $\eta\in\mathcal{M}^{\varphi}_{h}(g)$. If $\eta=\nu$, the
proof is complete. Let $\eta\neq\nu$. We can conclude $\nu$ and $\eta$ are not
mutually singular by $\|\nu-\eta\|<1$ in the same way as in [Mor2010][Lemma
2.2]. Hence by the Lebesgue decomposition theorem, there exist
$\hat{\nu},\hat{\eta}\in\mathcal{M}$ and $\lambda\in(0,1)$ such that

$\eta=(1-\lambda)\hat{\eta}+\lambda\hat{\nu}$

where $\hat{\eta}\perp\nu$ and $\hat{\nu}\ll\nu$. By a standard argument using
the Radon-Nikodym theorem, it is easy to see $\hat{\nu}=\nu$ (See for example
[Walters]). Then for each $i=1,\ldots,d$

$\displaystyle h_{i}=\int\varphi_{i}\ d\eta$
$\displaystyle=(1-\lambda)\int\varphi_{i}\ d\hat{\eta}+\lambda\int\varphi_{i}\
d\nu$ $\displaystyle=(1-\lambda)\int\varphi_{i}\ d\hat{\eta}+\lambda h_{i}.$

Hence we have

$\displaystyle(1-\lambda)\left(\int\varphi_{i}\
d\hat{\eta}-h_{i}\right)=0\quad(i=1,\ldots,m),$

which yields $\hat{\eta}\in{\rm rv}_{\varphi}^{-1}(h)$. Since
$\hat{\eta},\nu\in{\rm rv}_{\varphi}^{-1}(h)$, we have

$\int g\ d\hat{\eta}\leq\beta^{\varphi}_{h}(g)\quad\mbox{and}\quad\int g\
d\nu\leq\beta^{\varphi}_{h}(g).$

They should be equality since $\int g\ d\eta=\beta^{\varphi}_{h}(g)$.
Therefore, we obtain $\nu\in\mathcal{M}^{\varphi}_{h}(g)$. ∎

Set $\mathcal{E}={\rm rv}_{\varphi}^{-1}(h)\cap\mathcal{M}^{e}_{T}(X)$. In the
reminder of this subsection we assume

(9) $\displaystyle\overline{\mathcal{E}}={\rm rv}_{\varphi}^{-1}(h)$

Since proofs of Lemma 3.3 and Lemma 3.4 are slight modifications of Lemma 3.1
and Lemma 3.2 in [Mor2010] respectively, we omit their proofs.

######  Lemma 3.3.

Let $\mathcal{U}\subset{\rm rv}_{\varphi}^{-1}(h)$ be an open set. Then

$U:=\\{f\in C(X,\mathbb{R}):\mathcal{M}^{\varphi}_{h}(f)\subset\mathcal{U}\\}$

is open in $C(X,\mathbb{R})$.

######  Lemma 3.4.

Let $U\subset C(X,\mathbb{R})$ be an open set. Then

$\mathcal{U}:=\mathcal{E}\cap\bigcup_{f\in U}\mathcal{M}^{\varphi}_{h}(f)$

is open in $\mathcal{E}$.

While proofs of the next two lemmas are also similar to [Mor2010], attention
should be paid to the constraint. Hence we give proofs.

######  Lemma 3.5.

Let $\mathcal{U}$ be a dense subset of $\mathcal{E}$. Then

$U:=\\{f\in C(X,\mathbb{R}):\mathcal{M}^{\varphi}_{h}(f)=\\{\mu\\}\ \mbox{for
some}\ \mu\in\mathcal{U}\\}$

is dense in $C(X,\mathbb{R})$.

###### Proof.

Let $V\subset C(X,\mathbb{R})$ be an open set. Set

$\mathcal{V}=\mathcal{E}\cap\bigcup_{f\in V}\mathcal{M}^{\varphi}_{h}(f).$

By Lemma 3.4, $\mathcal{V}$ is open in $\mathcal{E}$. Since $\mathcal{U}$ is
dense in $\mathcal{E}$, $\mathcal{U}\cap\mathcal{V}\neq\emptyset$. Take
$\mu\in\mathcal{U}\cap\mathcal{V}$ and let $f\in V$ such that
$\mu\in\mathcal{M}^{\varphi}_{h}(f)$. Note that $\mu\in{\rm rv}^{-1}(h)$.
Since $\mu\in\mathcal{E}\subset\mathcal{M}_{T}^{e}(X)$, by Jenkinson’s theorem
[Jen06][Theorem 1], there exists $g\in C(X,\mathbb{R})$ such that
$\mathcal{M}_{\rm max}(g)=\\{\mu\\}$. This implies
$\mathcal{M}^{\varphi}_{h}(g)=\\{\mu\\}$ since $\mu\in{\rm
rv}_{\varphi}^{-1}(h)$. Hence for $\delta>0$ we have
$\mathcal{M}^{\varphi}_{h}(f+\delta g)=\\{\mu\\}$ and $f+\delta g\in U$. For
sufficiently small $\delta$ we have $f+\delta g\in V$, which complete the
proof. ∎

######  Lemma 3.6.

Let $U\subset C(X,\mathbb{R})$ be a dense subset. Then

$\mathcal{U}:=\mathcal{E}\cap\bigcup_{f\in U}\mathcal{M}^{\varphi}_{h}(f)$

is dense in $\mathcal{E}$.

###### Proof.

Take a nonempty open subset $\mathcal{V}\subset\overline{\mathcal{E}}$. We
show $\mathcal{V}\cap\mathcal{U}\neq\emptyset$.

By Lemma 3.3,

$V:=\\{f\in
C(X,\mathbb{R}):\overline{\mathcal{E}}\cap\mathcal{M}^{\varphi}_{h}(f)\subset\mathcal{V}\\}$

is open in $C(X,\mathbb{R})$. Since $\mathcal{V}$ is nonempty, there exists
$\mu\in\mathcal{V}\cap\mathcal{E}\subset{\rm
rv}^{-1}(h)\cap\mathcal{M}_{T}^{e}(X)$. Since $\mu$ is ergodic, as stated in
Lemma 3.5, there exists $g\in C(X,\mathbb{R})$ such that $\mathcal{M}_{\rm
max}(g)=\\{\mu\\}$ and we have
$\mathcal{M}^{\varphi}_{h}(g)=\\{\mu\\}\subset\mathcal{V}$ by Jenkinson’s
Theorem [Jen06][Theorem 1]. Hence $V$ is nonempty. Since $U$ is dense in
$C(X,\mathbb{R})$, $U\cap V\neq\emptyset$ and
$\mathcal{U}\cap\mathcal{V}\neq\emptyset$. ∎

###  3.2. Generic zero entropy for a symbolic dynamics

In this subsection, we apply the lemmas in the previous subsection to the
symbolic case and give the proof of Theorem 2.

###### Proof of Theorem 2.

Suppose the hypotheses of Theorem 2 hold. As in Corollary 2.3, for each $n\geq
1$

$\displaystyle\mathcal{Z}_{n}:=\left\\{\mu\in{\rm rv}_{\varphi}^{-1}h:0\leq
h_{\mu}<\frac{1}{n}\right\\}\supset\mathcal{N}_{\sigma}\cap{\rm
rv}_{\varphi}^{-1}h$

is nonempty, open and dense subset in ${\rm rv}_{\varphi}^{-1}(h)$. Moreover,
by Theorem 1, we have

$\displaystyle\overline{{\rm
rv}_{\varphi}^{-1}(h)\cap\mathcal{M}_{\sigma}^{p}(\Omega)}={\rm
rv}_{\varphi}^{-1}(h),$

which implies (9).

Fix $n\geq 1$. Since (9) holds, we can apply Lemmas 3.3 and 3.5 to our
symbolic case. Therefore, we see that

$\displaystyle U_{n}:=\\{f\in
C(\Omega,\mathbb{R}):\mathcal{M}^{\varphi}_{h}(f)\subset\mathcal{Z}_{n}\\}$

is open in $C(\Omega,\mathbb{R})$ and

$\displaystyle\widehat{U}_{n}:=\\{f\in
C(\Omega,\mathbb{R}):\mathcal{M}_{h}^{\varphi}(f)=\\{\mu\\}\ \mbox{for some}\
\mu\in\mathcal{Z}_{n}\\}\subset U_{n}$

is dense in $C(\Omega,\mathbb{R})$. Hence $U_{n}$ is an open dense subset of
$C(\Omega,\mathbb{R})$.

Then the set

$\displaystyle R:=\bigcap_{n\geq 1}U_{n}=\left\\{f\in
C(\Omega,\mathbb{R}):\mathcal{M}_{h}^{\varphi}(f)\subset\bigcap_{n\geq
1}\mathcal{Z}_{n}\right\\}$

is a residual subset of $C(\Omega,\mathbb{R})$, and the proof is complete. ∎

##  Appendix A On positive entropy

In this appendix, we see that for generic continuous function every relative
maximizing measure has positive entropy under some constraints, which is a
trivial consequence of ergodic optimization.

######  Proposition A.1.

Let $T:X\rightarrow X$ be a continuous map on a compact metric space $X$ with
an ergodic invariant probability measure $\mu$ having positive entropy. Then
there exist $\varphi\in C(X,\mathbb{R})$ and $h\in{\rm Rot}(\varphi)$ such
that for generic $f\in C(X,\mathbb{R})$ every relative maximizing measure of
$f$ with the constraint $h\in{\rm Rot}(\varphi)$ has positive entropy.

###### Proof.

By Jenkinson’s theorem [Jen06][Theorem 1], there exists $\varphi\in
C(X,\mathbb{R})$ such that $\mathcal{M}_{\rm max}(\varphi)=\\{\mu\\}$. Let
$h:={\rm rv}_{\varphi}(\mu)$. Then $\mathcal{M}^{\varphi}_{h}(f)\subset{\rm
rv}_{\varphi}^{-1}(h)=\\{\mu\\}$ for all $f\in C(X,\mathbb{R})$. Since
$\mathcal{M}^{\varphi}_{h}(f)$ is not empty, we have
$\mathcal{M}^{\varphi}_{h}(f)=\\{\mu\\}$, which implies the unique relative
maximizing measure of $f$ with constraint $h\in{\rm Rot}(\varphi)$ has
positive entropy. ∎

##  Appendix B Prevalent uniqueness

In this paper we focus on generic property of continuous functions in
constrained setting. On the other hand, measure-theoretic “typicality” which
is known as prevalence is also important. A subset $\mathcal{P}$ of
$C(X,\mathbb{R})$ is prevalent if there exists a compactly supported Borel
probability measure $m$ on $C(X,\mathbb{R})$ for every $f\in C(X,\mathbb{R})$
the set $f+\mathcal{P}$ has full measure.

Prevalent uniqueness of maximizing measures for continuous functions in the
literature on (unconstrained) ergodic optimization is recently established by
Morris [Mor21, Theorem 1]. Moreover the result is generalized to more abstract
statement, which yields prevalent uniqueness in constrained setting.

######  Corollary B.1.

Let $T:X\rightarrow X$ be a continuous map on a compact metric space $X$. For
a continuous constraint $\varphi:X\rightarrow\mathbb{R}^{d}$ and a rotation
vector $h\in{\rm Rot}(\varphi)$, the set

$\displaystyle\\{f\in C(X,\mathbb{R}):\mathcal{M}^{\varphi}_{h}(f)\ \mbox{is a
singleton}\\}$

is prevalent.

###### Proof.

Since ${\rm rv}_{\varphi}^{-1}(h)\subset\mathcal{M}_{T}(X)$ is a nonempty
compact set in $C(X,\mathbb{R})^{*}$, the statement follows from Theorem 2 in
[Mor21]. ∎

Acknowledgement. The first author was partially supported by JSPS KAKENHI
Grant Number 22H01138 and the second author was partially supported by JSPS
KAKENHI Grant Number 21K13816.

Data Availability. Data sharing not applicable to this article as no datasets
were generated or analyzed during the current study.

## References


Analysissec:analysis

Since execution and emulation of the Xtensa instruction set is demonstrated
through successfully running the flash binary via QEMU, the primary focus of
the analysis is networking. Emulating the Xtensa is a core feature of the Tiny
Code Generator within QEMU. However, networking does have different possible
code paths and configurations as well as a requirement to interface with the
host network stack. Network emulation in QEMU qemu_net can take 2 forms: TAP
and user mode network stack. The former adds a virtual network device on the
host (called tapN) and can then be configured as a real ethernet card. The
latter was used in this project and will now be described.

As shown in Section sub:emulate_app, the networking interface was configured
via launching QEMU with the

> -nic user,model=open_eth,id=lo0,hostfwd=tcp::8000-:80

options. These options configure the user mode network stack without root
privileges. Figure  shows the resulting virtual network configuration. The
QEMU Virtual Machine (VM) behaves as if it was behind a firewall which blocks
all incoming connections. The DHCP server assigns addresses to the guests
starting from 10.0.2.15.

[width=1.0keepaspectratio]user_mode_stack.png

width=.8

Figure : The virtual network configuration created by the user mode network
stack options.

In order for the host to access the IP ports listening on the guest OS, port
forwarding must be configured. The hostfwd=tcp::8000-:80 takes care of this by
forwarding the guest OS port 80 (our HTTP server) to the host port 8000. The
forwarding is not specific to any one of the host’s network interfaces. For
instance, both lo0 and en0 on the host are listening on port 8000. From
Section LABEL:sub:emulate_app, we stated that the ESP32 load was built with
support for the OpenCores Ethernet MAC driver [eth_mac]. This driver,
implemented in QEMU, provides the Media Access Control (MAC) layer in the
emulator allowing the guest OS to transmit and receive ethernet frames which
are subsequently forwarded to the host MAC. In the QEMU project,
opencores_eth.c implements a set of functions providing the emulated MAC layer
interface. The relevant functions used in transmitting and receiving ethernet
frames to the guest OS are:

  * •

open_eth_desc_write()

  * •

open_eth_desc_read()

  * •

open_eth_reg_read()

  * •

open_eth_reg_write()

  * •

open_eth_mii_read()

  * •

open_eth_mii_write()

  * •

open_eth_start_xmit()

  * •

open_eth_receive()

  * •

open_eth_receive_desc()

  * •

open_eth_receive_mcast()

  * •

open_eth_update_irq()

##  GDB debugging

It is possible to start the QEMU session listening on a debug port. The “-s”
option allows this. Thus, the new launch command on native macOS: qemu-system-
xtensa -nographic -machine esp32 -nic user,model=open_eth,
id=lo0,hostfwd=tcp::8000-:80 -drive file=merged_qemu.bin, if=mtd,format=raw -s
Now, we can connect and debug the guest OS and application with: xtensa-
esp32-elf-gdb esp_http_server.elf -ex ”target remote :1234” -ex ”monitor
system_reset” -ex ”tb app_main” -ex ”c” This GDB session will provide an
interactive prompt and will temporarily break at the application entry point.
The FreeRTOS kernel, drivers and HTTP server application is in scope with this
debug session. The xtensa-esp32-elf-gdb is a ESP32-specific build of GDB that
is provided through the ESP-IDF SDK. In theory, GDB stepping through the ESP32
binary will not show anything unique in the emulated session since the guest
OS implements a network stack and simply binds to the MAC address at the lower
layer. A more relevant option is to follow the networking code path in the
QEMU emulator, specifically the OpenCores Ethernet MAC driver. Unfortunately,
a binary with symbols file (i.e. .elf) is not generated during a custom build
which prevents productive GDB stepping through the opencores_eth.c driver.
This is the entity that provides networking emulation in our experiments.
Also, due to the high transaction rate and asynchronous nature of the network
stack operation, GDB stepping will not provide the best analysis of the
emulated code path. A better option is to explore function tracing.

##  QEMU function tracing

QEMU provides a function tracing framework which can be enabled at runtime
[tracing]. The

> –trace ”open_eth*”

option will enable tracing on all opencores_eth.c driver functions. Thus, we
launch QEMU with the following options: qemu-system-xtensa -nographic -machine
esp32 -nic user,model=open_eth, id=lo0,hostfwd=tcp::8000-:80 -drive
file=merged_qemu.bin, if=mtd,format=raw –trace ”open_eth*” The new option
results in tracing messages for all functions prefixed with “open_eth”. These
messages print to stderr. Log options are also available. The trace framework
is realized by a “trace()” function call in each of the subsystem API
functions. The trace function calls outside open_eth generate a nop such that
there is no performance penalty or extraneous messages. The full stderr trace
is too much data to display here. Listing LABEL:lst:trace is an abbreviation
with many in-between open_eth messages removed. [basicstyle=, numbers=left,
label=lst:trace, caption=Standard console messages along with trace of
open_eth calls.] trace.log All lines with the “open_eth” prefix represent
functions calls to the OpenCores Ethernet MAC driver. Lines 1-35 show the
ESP32 load initializing the network stack through a sequence of calls to
discover the underlying ethernet interface. The OpenCores driver services
these requests and passes through to the host, eventually responding back to
the guest OS. Reads and writes to the MAC register are handled via the driver,
followed by data transmit (line 20), receive (line 21) and interrupt masking
(line 23). The next steps in lines 36-49 show the binding of the IP stack to
the ethernet interface. Again, the ethernet interface is serviced by the
OpenCores driver utilizing the same open_eth functions calls to read, write
the MAC register and transmit data frames. Lines 50-56 show the application
HTTP server starting and binding port 80 with the IP stack. As IP packet
headers are stripped and forwarded to the MAC layer, the OpenCores driver
services these requests and passes to the host. The same API is used with
function calls to read the MAC register (line 52) and write the MAC (line 53),
The final set is after we hit the host port 8000 with an HTTP request. These
are shown lines 57-73. The host command curl http://localhost:8000/hello sends
an HTTP (IP) request to the guest. When this is broken down into ethernet
frames, we see the OpenCores driver receive the request data (line 58) and set
the interrupt mask (line 60). The HTTP server then sends its response. The
server itself writes a message on line 66 that it has responded to the request
from localhost:8000. The data payload contains “Hello World!” and its transmit
is handled by OpenCores on line 69. For a graphical representation of the
above analysis, see Figure .

[width=1.0keepaspectratio]emulation_layers.png

width=.8

Figure : Components of the guest OS and the network data flow to the host. The
Tiny Code Generator (TCG) emulates the Xtensa-native instructions while the
OpenCores Ethernet MAC driver forwards ethernet traffic to the host MAC.


figure table

# Polarization properties of X-ray tubes used for Imaging X-ray Polarimetry
Explorer calibration

Ajay Ratheesh  Istituto di Astrofisica e Planetologia Spaziali, Via Fosso del
Cavaliere 100, I-00133 Rome, Italy  John Rankin  Istituto di Astrofisica e
Planetologia Spaziali, Via Fosso del Cavaliere 100, I-00133 Rome, Italy
Enrico Costa  Istituto di Astrofisica e Planetologia Spaziali, Via Fosso del
Cavaliere 100, I-00133 Rome, Italy  Ettore Del Monte  Istituto di Astrofisica
e Planetologia Spaziali, Via Fosso del Cavaliere 100, I-00133 Rome, Italy
Alessandro Di Marco  Istituto di Astrofisica e Planetologia Spaziali, Via
Fosso del Cavaliere 100, I-00133 Rome, Italy  Sergio Fabiani  Istituto di
Astrofisica e Planetologia Spaziali, Via Fosso del Cavaliere 100, I-00133
Rome, Italy  Fabio La Monaca  Istituto di Astrofisica e Planetologia Spaziali,
Via Fosso del Cavaliere 100, I-00133 Rome, Italy  Fabio Muleri  Istituto di
Astrofisica e Planetologia Spaziali, Via Fosso del Cavaliere 100, I-00133
Rome, Italy  Alda Rubini  Istituto di Astrofisica e Planetologia Spaziali, Via
Fosso del Cavaliere 100, I-00133 Rome, Italy  Paolo Soffitta  Istituto di
Astrofisica e Planetologia Spaziali, Via Fosso del Cavaliere 100, I-00133
Rome, Italy  Luca Baldini  Università di Pisa, Dipartimento di Fisica Enrico
Fermi, Largo B. Pontecorvo 3, I-56127 Pisa, Italy  Istituto Nazionale di
Fisica Nucleare, Sezione di Pisa, Largo B. Pontecorvo 3, I-56127 Pisa, Italy
Massimo Minuti  Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo
B. Pontecorvo 3, I-56127 Pisa, Italy  Michele Pinchera  Istituto Nazionale di
Fisica Nucleare, Sezione di Pisa, Largo B. Pontecorvo 3, I-56127 Pisa, Italy
Carmelo Sgrò  Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo B.
Pontecorvo 3, I-56127 Pisa, Italy

###### Abstract

In this work, we measured the polarization properties of the X-rays emitted
from the X-ray tubes, which were used during the calibration of the instrument
onboard Imaging X-ray Polarimetry Explorer (IXPE). X-ray tubes are used as a
source of unpolarized X-rays to calibrate the response of the gas pixel
detectors to unpolarized radiation. However, even though the characteristic
fluorescent emission lines are unpolarized, continuum bremsstrahlung emission
can be polarized based on the geometry of the accelerated electrons and
emitted photons. Hence, characterizing the contribution of polarized X-rays
from bremsstrahlung emission is of interest, also for future measurements. We
find that when accelerated electrons are parallel to the emitted photons, the
bremsstrahlung emission is unpolarized, and when they are perpendicular, the
polarization increases with energy, as expected from the theoretical
predictions. A comparison with the theoretical predictions is also shown.

###### keywords:

X-rays, X-ray tube, detectors, polarization, bremsstrahlung

<EMAIL_ADDRESS>
##  1 Introduction

X-ray tubes are widely used in various scientific fields as a source of X-rays
of known energy, depending on the material. When an electron beam strikes the
anode, the characteristic X-ray lines generated in the X-ray tube can be used
to calibrate X-ray detectors. Such X-ray tubes were used to calibrate the
response of the Gas Pixel Detector (GPD) onboard the Imaging X-ray Polarimetry
Explorer (IXPE) to unpolarized photons [1] [2, 3]. Even though IXPE is
equipped with radioactive sources for onboard in-orbit calibration purposes
[4], a comprehensive and extensive calibration had to be conducted on the
ground at different energies using X-ray tubes [3, 5]. For this reason, even
though the spectral properties of the X-rays arising from the X-ray tube are
well known, their polarization properties are poorly studied because of a lack
of suitable polarimeters, which is, in turn, necessary for the calibration of
X-ray polarimeters like the GPD onboard IXPE. Hence it is crucial to constrain
the polarization from these X-ray tubes while using them to calibrate
polarimeters like the GPD onboard IXPE.

Measuring the response of the detector to unpolarized radiation is an
essential procedure in the calibration of an X-ray polarimeter, because they
are not exempt from spurious modulation that has to be carefully studied,
calibrated, and fitered-out [6]. The spurious modulation observed in a
detector can be attributed to systematic effects inherent to the detector.
Hence, the source of X-rays needs to be unpolarized to measure the spurious
modulation independent of the source modulation. Even though fluorescent
k-shell emission from X-ray tubes is unpolarized, the continuum Bremsstrahlung
emission can be polarized based on the geometry of the X-ray tube [7]. The
contribution of the partially polarized continuum to the unpolarized
characteristic line must be addressed due to the finite energy resolution of
the detector. Therefore, it is crucial to understand the contribution of
polarization from the continuum emission, while using these X-ray tubes for
measuring the response to unpolarized radiation of any X-ray polarimeter. The
manufacturer does not provide the polarization properties of the X-ray tubes
used for IXPE calibration since calculating the polarization of X-ray tubes
theoretically from the first principle is difficult as it depends on the
details of the geometry of the emission. Even if a procedure to decouple the
intrinsic response of the instrument and the signal generated by the genuine
partial polarization of the X-ray tube has been used for the calibration of
detectors onboard IXPE [6], it is important to measure its intrinsic
polarization as a cross-check and future reference. Moreover, in general, we
notice that in X-ray Astronomy, after 60 years and around 50 space missions, a
good reference of cross-calibrations has been achieved [8, 9] so that any new
mission can benefit from ground facilities and a large sample of celestial
sources sometimes observed simultaneously with more than one satellite. In the
domain of polarimetry, IXPE is measuring the polarization of tens (potentially
hundreds) of X-ray sources and is building the first catalog that will be the
reference for any future experiment. It is, therefore, imperative to make a
substantial effort to the best knowledge of the absolute values of the
published results, including the level of systematics.

The experimental verification of X-ray tube polarization properties was only
done now, as measuring the polarization in wide X-ray bands was difficult
until the recent development of highly sensitive and wide-band photoelectric
X-ray polarimeters. Moreover, measurements involving polarization for material
science are currently performed in Synchrotron facilities. However, acquiring
sufficient allocation of time from Synchrotron facilities for polarimetric
calibrations of GPD like detectors, which demand prolonged exposure
measurements, is impractical. For example, calibration of a detector unit of
IXPE took 40 days of measurements [3]. On the other hand, the constant
availability of X-ray tubes renders them a convenient option for prolonged
usage and enables adherence to the IXPE mission launch timeline.

In this work, we outline the analysis of the measurements performed to
understand the polarization properties of X-ray tubes. Section 2 describes the
method of measuring X-ray polarization with photoelectric polarimeters, and
section 3 gives a theoretical background of X-ray tubes and expected
polarization from Bremsstrahlung. Section 4 shows the measurements and
results, and in section 5, we conclude by discussing the observations.

##  2 Measuring polarization with IXPE

IXPE is the first dedicated mission with focusing optics in space to measure
the polarization of X-rays [2, 1]. IXPE is a NASA Astrophysics Small Explorer
(SMEX) mission developed in collaboration with the Italian Space Agency (ASI)
and was launched on December 9, 2021. The IXPE focal plane instrument,
comprising three flight Detector Units (DU), each hosting a GPD and a spare
unit, were developed by Istituto Nazionale di Astrofisica/Istituto di
Astrofisica e Planetologia Spaziali (INAF-IAPS) in Rome and Istituto Nazionale
di Fisica Nucleare (INFN) in Pisa. The GPD is a photoelectric polarimeter,
which can image the photoelectron track and reconstruct the photoelectron
emission direction and the absorption point.

IXPE opens a new window in X-ray astronomy by providing additional information
on the polarization degree (PD) and polarization angle (PA) of X-rays along
with the energy, time of arrival, and interaction point in the detector. With
the addition of polarimetry, our understanding of the geometry and emission
mechanism of various X-ray sources can be substantially improved.
Specifically, polarization can help break the degeneracy of some geometrical
and physical models developed based on spectroscopy alone.

The sensitivity of any X-ray polarimeter can be estimated by a quantity known
as the minimum detectable polarization at 99$\%$ confidence ($MDP_{99}$).
$MDP_{99}$ is defined as [10, 11] :

$MDP_{99}=\frac{4.29}{\mu}\times\frac{1}{R_{S}}\times\sqrt{\frac{R_{S}+R_{B}}{T}},$
(1)

Where $\mu$ is the modulation factor (measured without background), which is
the response of a polarimeter to fully polarized X-rays, $T$ is the exposure
time, $R_{S}$, and $R_{B}$ is the source and background count rate. When the
$MDP_{99}$ is smaller, the sensitivity of a polarimeter is higher. Hence when
$\mu$ is larger the sensitivity of the polarimeter is larger.

In photoelectric absorption, the photoelectron’s emission direction is
parallel to the electric field of the incoming photon. Hence, the azimuthal
distribution would be sinusoidally modulated depending on the degree and angle
of polarization. Because the GPD is a photoelectric polarimeter, given the
cross-section for K-shell electrons, the azimuthal angle distribution for
photoelectrons (modulation curve) has a cosine dependence, well described by
the following function [11]:

$S(\phi)=A+B\cos 2(\phi-\phi_{0})$ (2)

where $\phi_{0}$ is the polarization angle and the amplitude of modulation can
then be described by [11]:

$m=\frac{S_{max}-S_{min}}{S_{max}+S_{min}}=\frac{B}{2A+B}$ (3)

However, the amplitude of modulation is not the polarization degree of the
observation and has to be normalized by the modulation factor of the detector
to obtain the source polarization degree [11]:

$p=\frac{m}{\mu}.$ (4)

The polarization degree and angle can also be expressed in terms of the Stokes
parameters. Stokes parameters are generally used to express the polarization
of electromagnetic radiation (I, Q, U, V). The parameter I represents the
intensity of the beam, Q represents the intensity in the $+90$ and $0$
directions. U represents the intensity in the $+45$ and $-45$ directions and V
represents clockwise and anticlockwise circular polarization. The circular
polarization arises when the electric field vector of X-rays oscillates in a
circular pattern as it travels through space. However, in astronomical X-ray
polarimetry, we do not consider circular polarization due to difficulty in
measurement. If an X-ray polarimeter is designed to be sensitive only to
linear polarization, the presence of circular polarization can potentially
reduce the measured degree of linear polarization. An example of a modulation
curve of polarized flux is shown in Figure. 1. In terms of the modulation
curve parameters, Stokes parameters can be expressed as[11]:

$I=A+B/2$ (5) $Q=(B/2)\cos(2\phi_{0})$ (6) $U=(B/2)\sin(2\phi_{0})$ (7)

The modulation curve can be expressed in terms of Stokes parameters[11] as:

$S(\phi)=I+Q\cos(2\phi)+U\sin(2\phi)$ (8)

Now the modulation amplitude ($m$) and angle ($\phi_{0}$) of modulation can be
expressed in terms of the Stokes parameters as:

$m=\frac{(Q^{2}+U^{2})^{0.5}}{I}$ (9) $\phi_{0}=1/2\tan^{-1}(U/Q)$ (10)

However, the approach followed in this work and, in general, the case of IXPE
is based on measuring the stokes parameters in an event-by-event approach to
account for the subtraction of systematic effects [12, 6]. For each photon,
the Stokes parameters are calculated as:

$q_{i}=2\;cos\;(2\phi_{i})$ (11) $u_{i}=2\;sin\;(2\phi_{i})$ (12)

where $\phi_{i}$ is the initial photo-electron direction, estimated by the
photo-electron track reconstruction algorithm [13, 14]. And for a total number
of N events

$q=\frac{\Sigma_{i}q_{i}}{N}$ (13) $u=\frac{\Sigma_{i}u_{i}}{N}$ (14)

and the modulation and angle can be computed by

$m=(q^{2}+u^{2})^{0.5}$ (15) $\phi_{0}=1/2\tan^{-1}(u/q)$ (16)

##  3 X-ray tubes and expected polarization

X-ray tubes emit radiation with the following principle. Electrons are
generated from a heated cathode and are accelerated towards an anode by a high
voltage. The electrons hit the target anode, get decelerated, and produce
X-rays due to bremsstrahlung emission. Bremsstrahlung emission is due to the
deflection of the electrons by an atomic nucleus, and the energy lost by the
electron is emitted as a photon. The maximum energy of the bremsstrahlung
emission depends on the peak accelerating potential of the X-ray tube, and the
bremsstrahlung peaks at approximately 1/3rd of the maximum energy [15]. The
intensity of the bremsstrahlung emission is directly proportional to the
atomic number of the target material and the charge of the particle and
inversely proportional to the mass of the particle, in our case, the electron.
The spectrum of bremsstrahlung emission in an X-ray tube can be described by
Kramers law [16]:

$d\;I(\lambda)\;=\;K\;\biggl{(}\frac{\lambda}{\lambda_{min}}-1\biggr{)}\;\frac{1}{\lambda^{2}}\;d\lambda$
(17)

where K is a constant depending on the electron beam current, $\lambda$ is the
wavelength of the emitted photons, and, $\lambda_{min}$ is the minimum
wavelength of the emitted photons from an X-ray tube for a given applied
voltage ($V$), given by the Duane–Hunt law [17]:

$\lambda_{min}\;=\;\frac{hc}{eV}$ (18)

Other than the bremsstrahlung emission, the X-ray tubes also emit
characteristic fluorescent X-rays when the accelerated electrons hit the
target material and kick an electron from the K-shell, and get refilled by an
outer shell electron. The energy of the discrete fluorescent lines depends on
the binding energy of the electron in the target material. About one percent
of the energy of the accelerated electrons is radiated through X-rays. The
high voltage applied to the X-ray tube will change the spectral shape, while
the applied current will change the intensity or normalization.

###  3.1 Theoretical prediction of polarization from bremsstrahlung emission

Theoretical calculations of the cross-section and polarization from the
bremsstrahlung emission were already published in the middle of 20th century
[18, 19, 20, 21, 7]. It was found that the bremsstrahlung emission is supposed
to be partially polarized. The primary step in determining the polarization is
to determine the cross-section. The bremsstrahlung cross-section gives the
probability of an electron transiting from one to another state with the
emission of a photon. The electron-nucleus collisions can also be elastic, and
hence the probability that a photon will be emitted depends on the cross-
section of the bremsstrahlung, which is 137 times smaller than the cross-
section of electron-nucleus elastic scattering. The differential cross-section
($d\sigma$) as a function of the photon energy for an electron emerging
through the solid angle $d\Omega$ in the non-relativistic limit, which is the
case for electron kinetic energies smaller than 10 keV, can be expressed as:

$d\sigma\;=\;\biggl{(}\frac{Z^{2}e^{6}}{\pi^{2}}\biggr{)}\;\biggl{(}\frac{p}{p_{0}}\biggr{)}\;\biggl{(}\frac{dk}{kq^{4}}\biggr{)}\;d\Omega\;d\Omega_{0}\;(p_{l}-p_{0l})^{2}$
(19)

where Z is the atomic number or nuclear charge, $e$ is the elementary charge,
$k$ is the energy of the emitted photon, $q$ is the momentum transferred to
the nucleus, $p$ and $p_{0}$ are the final and initial momentum of the
electron, and $p_{l}$ and $p_{0l}$ are the components of momentum of the
electron in the direction of the polarization. Hence the differential cross
section for a specified photon energy further depends on the atomic number of
the material, the direction of the initial and emerging electron, and the
momentum of the initial and emerging electron.

Given the energy and momentum of the initial and final electron, along with
the photon energy and momentum, linear polarization can be calculated by

$P\;=\;\frac{d\sigma_{\@slowromancap iii@}-d\sigma_{\@slowromancap
ii@}}{d\sigma_{\@slowromancap iii@}+d\sigma_{\@slowromancap ii@}}$ (20)

Here, $d\sigma_{\@slowromancap iii@}$ is the differential cross-section of the
polarization vector perpendicular to the plane of scattering, containing the
initial electron momentum and photon momentum, and $d\sigma_{\@slowromancap
ii@}$ is the differential cross-section of the polarization vector in the
plane of scattering and parallel to $p$ [7]. It is to be noted that the
notations used here are the same as that of Gluckstern et al. 1953[7] . The
expressions for the $d\sigma_{\@slowromancap ii@}$ and
$d\sigma_{\@slowromancap iii@}$:

$\displaystyle d\sigma_{\@slowromancap
ii@}\;=\>\frac{Z^{2}e^{6}}{8\pi}\frac{dk}{k}\;\frac{p}{p_{0}}\;d\Omega_{0}\;\bigg{\\{}\frac{8m^{2}sin^{2}\theta_{0}(2E^{2}_{0}+m^{2})}{p_{0}^{2}\Delta_{0}^{2}}-\frac{5E_{0}^{2}+2EE_{0}+5m^{2}}{p_{0}^{2}\Delta_{0}^{2}}-\frac{p_{0}^{2}-k^{2}}{T^{2}\Delta_{0}^{2}}+\frac{2(E+E_{0})}{p_{0}^{2}\Delta_{0}}$
(21)
$\displaystyle+(\frac{L}{pp_{0}})\big{[}\frac{4E_{0}m^{2}sin^{2}\theta_{0}(3km^{2}-p_{0}^{2}E)}{p_{0}^{2}\Delta_{0}^{4}}+\frac{2E_{0}^{2}(E_{0}^{2}+E^{2})-m^{2}(9E_{0}^{2}-4EE_{0}+E^{2})+2m^{4}}{p_{0}^{2}\Delta_{0}^{2}}+k\frac{E_{0}^{2}+EE_{0}}{p_{0}^{2}\Delta_{0}}\big{]}$
$\displaystyle(\frac{\epsilon^{T}}{pT})\big{[}\frac{4m^{2}}{\Delta_{0}^{2}}-\frac{7k}{\Delta_{0}}-k\frac{p_{0}^{2}-k^{2}}{T^{2}\Delta_{0}}-4\big{]}-\frac{4\epsilon}{p\Delta_{0}}+(\frac{1}{p_{0}^{2}sin^{2}\theta_{0}^{2}})\big{[}(\frac{2L}{pp_{0}})(2E_{0}^{2}-EE_{0}-m^{2}-\frac{m^{2}k}{\Delta_{0}})$
$\displaystyle-\frac{4\epsilon^{T}(\Delta_{0}-E)^{2}}{pT}-\frac{2\epsilon(\Delta_{0}-E)}{p}\big{]}\bigg{\\}}$

$\displaystyle d\sigma_{\@slowromancap
iii@}\;=\>\frac{Z^{2}e^{6}}{8\pi}\frac{dk}{k}\;\frac{p}{p_{0}}\;d\Omega_{0}\;\bigg{\\{}-\frac{5E_{0}^{2}+2EE_{0}+m^{2}}{p_{0}^{2}\Delta_{0}^{2}}-\frac{p_{0}^{2}-k^{2}}{T^{2}\Delta_{0}^{2}}-\frac{2k}{p_{0}^{2}\Delta_{0}}$
(22)
$\displaystyle+(\frac{L}{pp_{0}})\big{[}\frac{2E_{0}^{2}(E_{0}^{2}+E^{2})-m^{2}(5E_{0}^{2}-2EE_{0}+E^{2})}{p_{0}^{2}\Delta_{0}^{2}}+\frac{k(E_{0}^{2}+EE_{0}-2m^{2})}{p_{0}^{2}\Delta_{0}}\big{]}$
$\displaystyle+(\frac{\epsilon^{T}}{pT})\big{[}\frac{k}{\Delta_{0}}-\frac{k(p_{0}^{2}-k^{2})}{T^{2}\Delta_{0}}+4\big{]}-(\frac{1}{p_{0}^{2}sin^{2}\theta_{0}^{2}})\big{[}(\frac{2L}{pp_{0}})(2E_{0}^{2}-EE_{0}-m^{2}-\frac{m^{2}k}{\Delta_{0}})$
$\displaystyle-\frac{4\epsilon^{T}(\Delta_{0}-E)^{2}}{pT}-\frac{2\epsilon(\Delta_{0}-E)}{p}\big{]}\bigg{\\}}$

Where,

$\Delta_{0}=E_{0}-p_{0}cos\theta_{0},$ (23)
$T=p_{0}^{2}+k^{2}-2p_{0}kcos\theta_{0},$ (24)
$L=ln\bigg{(}\frac{EE_{0}-m^{2}+pp_{0}}{EE_{0}-m^{2}-pp_{0}}\bigg{)},$ (25)
$\epsilon=ln\bigg{(}\frac{E+p}{E-p}\bigg{)},$ (26)
$\epsilon^{T}=ln\bigg{(}\frac{T+p}{T-p}\bigg{)},$ (27)

and $\theta_{0}$ is the angle between $p_{0}$ and $k$. $E$ and $E_{0}$
represents the initial and final energy of the incoming electron. The
constants $\hbar$ and c are taken to be 1. The effects of shielding are
complicated. However, they can be approximated by replacing $k^{2}$ by
$k^{2}+\alpha^{2}p_{0}^{2}\Delta_{0}^{-2}$ in the logarithmically divergent
terms $L$ and $\epsilon^{T}$. Hence,

$L=ln\;\bigg{(}\;\frac{(EE_{0}-m^{2}+pp_{0})^{2}}{m^{2}k^{2}+m^{2}\alpha^{2}p_{0}^{2}\Delta_{0}^{-2}}\;\bigg{)},$
(28)

$\epsilon^{T}\;=\;\frac{1}{2}ln\bigg{(}\frac{(T+p)^{4}}{4k^{2}\Delta_{0}^{2}+4\alpha^{2}p_{0}^{2}}\bigg{)},$
(29)

where,

$\alpha=Z^{\frac{1}{3}}\frac{m}{108}$ (30)

Figures 2 and 3 shows the degree of polarization and cross-section with
respect to photon energy for a beam of electron of energy 0.1 MeV in aluminum
[7]. It is seen that the polarization is high around the lower and upper ends
of the photon spectrum. A jump in the polarization angle by 90$\degree$ is
seen as a change in the sign of the polarization degree at medium energies of
approximately 20 keV. The shielding effect decreases the polarization degree,
especially at lower energies.

##  4 Measurements and Results

We now present the polarization measured for the X-ray tubes used to calibrate
the GPDs onboard IXPE. There are different X-ray tubes used for the
calibration of IXPE [3], and are primarily two kinds, right angle X-ray tubes
from Oxford series 5000 (Fig. 4) and head-on X-ray tubes like the Calcium and
Tungsten anode ones from Hamamatsu, model N1335 (Fig. 5). The experimental
setup of the measurements are the same as what is mentioned in [3, 5]. The
source and detector were mounted to the instrument calibration equipment
(ICE), an equipment specifically designed for calibrating IXPE DUs [3]. This
equipment encompasses the mechanical frameworks employed to secure the
detector and sources, calibration sources, stages for aligning the detector
and source beam, test detectors and their corresponding mechanical assembly
(Figure 3 in [3]). The alignment process involves two stages. The first stage,
named ALIGN, aims to synchronize the source to the detector. The second stage,
referred to as MEAS, facilitates movement of the detector along the detector
plane, orthogonal shifts, azimuthal rotations, and tilting of the detector to
align the beam with the detector plane. In the case of right-angle X-ray
tubes, the generated X-ray photons are perpendicular to the accelerated beam
of electrons. In contrast, in the case of head-on X-ray tubes, the X-ray
photons are parallel to the accelerated beam of electrons. During the
calibration of the IXPE Instrument, X-ray filters were used to increase the
ratio between the fluorescence and the continuum emission, as only the former
component is unpolarized and of interest for calibration. For our study, such
filters were not used, as our primary aim is to measure the polarization of
the bremsstrahlung continuum. For measuring the polarization we used an IXPE
flight detector unit called as DU FM1 (Detector Unit Flight Model 1), which is
the spare unit of the IXPE Instrument and was fully calibrated as the other
flight units. The measurements were undertaken in the ICE after mounting DU
FM1 and the different X-ray tubes on the ICE. Figure 6 shows the sketch of the
geometrical configuration of the X-ray tubes and the detector.

Measurements were conducted at two arbitrary angles ($\epsilon 1$ and
$\epsilon 2$) that are mutually perpendicular to each other in the detector
plane ((ASIC X-Y Axis)) and with respect to the laboratory frame of reference.
The measurements at two angles are utilized to ensure that the polarization
that is measured originates from the source itself, as the polarization vector
of the source will be rotated with respect to the reference frame. If the
polarization indeed arises from the source and not from any detector
systematics, then the measured polarization angle must undergo a 90∘ rotation
upon rotating the source. Table 1 outlines the details of the X-ray tubes and
measurements used for this work. In the case of the Fe and W X-ray tubes, the
high voltage was chosen deliberately to be below the K-shell electron binding
energy to avoid the bright K fluorescence line from the material of the tube
target.

For the analysis of the measurements, the spurious modulation due to the
detector systematics has to be taken care of and subtracted carefully as in
the case of celestial sources. A method to subtract the contribution from the
detector on an event-by-event basis was developed for the IXPE flight
detectors [6]. We followed this method for subtracting the spurious
modulation. A spurious modulation database developed for calibrating IXPE[6],
which contains the spurious modulation as a function of spatial and energy
bins, was used to subtract the detector contribution while extracting the
polarization from the source.

The data calibrated for spurious modulation were then used for further
analysis. We used all events within the circular spatial region of radius 2 mm
from the center of the detector, as this is within the regions of the detector
that is calibrated with the highest sensitivity. The data from the selected
spatial region were then grouped into four different energy bins, with larger
bin sizes at energy ranges with lower count-rate, to allow sufficient
statistics for measuring the polarization. The normalized and spurious
modulation corrected Stokes parameters from all events in each bin are then
added to get the Stokes parameters and modulation amplitude of the total
measurement. The total Stokes parameters and the modulation degree are then
divided by the modulation factor of DU FM1 in that particular energy bin to
get the final Stokes parameters and the polarization from the X-ray tubes [5].
The modulation factor in each energy bin is weighted by the spectral counts in
that bin.

In the case of the measurements with Rh and Ca X-ray tube, we could correct
the gain variation during the measurement caused by charging of the gas
electron multiplier (GEM), which multiplies the primary charges before
collection [22], using the fluorescent line at the 2.7 and 3.69 keV. All the
events were re-scaled by a correction factor: the ratio of the fluorescent
line energy and the observed Gaussian line peak energy at different times. For
the case of Fe and W X-ray tubes, there were no fluorescent lines as the
applied high voltage was lower than the line energy; hence, these measurements
are slightly affected by charging. This means that the measured energy in
those measurements can have uncertainties of upto 10$\%$.

Figure 7 shows the spectra, while Figures 8, 9, 10, and 11 shows the contours
of polarization degree and angle. The corresponding data for these
measurements are noted down in Tables 2, 3, 4, and 5. For plotting the
spectrum, the 2-8 keV is divided into 100 energy bins and normalized by the
peak of the spectrum.

The polarization degree was seen to increase with energy for the right angle
X-ray tubes (Fe and Rh), while for head-on X-ray tubes (Ca and W), the X-rays
are unpolarized, while the measured polarization angle in both cases seems to
be parallel to the direction of the incoming electrons, as what was expected
in the energy range of the GPD. In the case of Rh and Fe X-ray tube, the
polarization increases from around $5\%$ to $22\%$ and around $15\%$ to $35\%$
across the IXPE energy band of 2 to 8 keV (see Table 4 and Table 5). Even
though the increasing trend of polarization is as expected, the absolute
values of the measured polarization are lower than what is expected for an
ideal case of single scattering of the electron in a crystal. However, in our
case, the mean free path of the electron with an energy of few keV in Rh and
Fe (of the order of nm) would be much less than the thickness of the anode
target (a few tens of microns), increasing the probability of multiple
scattering of the single electron giving rise to multiple Bremsstrahlung
photons. This effect would, in turn, dilute and decrease the polarization
degree. It was previously seen that even for very thin targets and at larger
energies the measured polarization was smaller than the theoretically expected
value [23]. In the case of Ca and W X-ray tubes, we obtained an upper limit
closer to $1\%$ or lesser (see Table 2 and Table 3). At some energies, we have
a measurement slightly above the MDP, but smaller than $1\%$, and it could be
due to some unknown systematic like the multiple scattering mentioned above.
The shift in the polarization angle for $\epsilon 2$ by 90$\degree$, with
respect to $\epsilon 1$ further indicates that the polarization is coming from
the source rather than a detector systematic like the spurious modulation
(which indeed we subtract). If the polarization was a detector effect, the
polarization angle should not have rotated by 90$\degree$ while rotating the
illuminating source. It can also be seen that the highest energy in the photon
spectrum matches with the applied high voltage of the tube.

The polarization measured at $\epsilon 1$ is fitted with the theoretical
function (Equations 20 to 22), allowing the applied voltage (V) and angle
between the incoming electron and the outgoing photon ($\theta$) left for the
fit to determine. The results of the fit are shown in Fig. 12. The solid red
line is the fit to the data, while the green lines indicate the expected
theoretical value based on V and $\theta$. For the case of the perpendicular
X-ray tubes, the opening angle of the anode crystal with respect to the
aperture towards the detector is 11∘, and hence to account for it within the
theoretical estimate, we shade the space around the line assuming an error of
11∘ from the tube. However, the deviation is well beyond this limit. Table. 6
outlines the fit results for all the X-ray tubes. In the case of the parallel
X-ray tubes, the fitted $\theta$ is closer to zero, while the fitted V is 3-4
times higher than the real applied voltage. This is mostly because of the non-
zero values of polarization. In the case of the perpendicular X-ray tubes,
both the $\theta$ and V are different from the real applied values. V is
higher by a factor of 2, while $\theta$ is smaller by a factor of 3. This
mismatch of values could be due to the multiple electron scattering effects.

##  5 Conclusions

In this work, we calculated the polarization of the continuum Bremsstrahlung
emission from X-ray tubes used to calibrate the GPD onboard IXPE. The
theoretical predictions of polarization from Bremsstrahlung depend on the
angle between the incoming electron and the emitted photon ($\theta_{0}$).
When $\theta_{0}$=0$\degree$, the polarization is supposed to be constant at
$0\degree$, and when $\theta_{0}$=90$\degree$, the polarization is supposed to
decrease until a certain energy and then increase. In the case of X-ray tubes,
the $\theta_{0}$ depends on the geometrical configuration of the X-ray tube.
We used 2 X-ray tubes where $\theta_{0}$=0$\degree$ and 2 X-ray tubes where
$\theta_{0}$=90$\degree$. For perpendicular X-ray tubes, we find that the
measured polarization degrees are significantly lower than the ones predicted
by theory, despite a good match between the observed polarization angle and
the overall trend of the polarization degree with energy in the 2-8 keV range.
This mismatch of the absolute value could be due to the multiple electron
scattering in the anode crystal, which is relevant in 2-8 keV. This could not
be tested as the manufacturer does not provide the dimensions of the anode
crystal. Nonetheless, this scenario is plausible, considering that the mean
free path of electrons with energies in the range of a few keVs in Ca, W, Rh
and Fe is of the order of nanometers, whereas the thickness of the anode could
exceed micrometers. For parallel X-ray tubes, we obtained upper-limits
consistent with the predictions.

Our work experimentally shows the differences in the continuum polarization of
two different configurations of X-ray tubes. The fact that the general trend
of polarization degree and polarization angle is coherent with what is
expected is positive. However, our work aimed to quantitatively fix the values
of practical implementation to be used to calibrate detectors of IXPE or any
other detector. These measurements can now form a base while using these X-ray
tubes in the future to understand the response of any X-ray polarimeter to
unpolarized X-rays.

### Acknowledgments

The Imaging X-ray Polarimetry Explorer (IXPE) is a joint US and Italian
mission. The US contribution is supported by the National Aeronautics and
Space Administration (NASA) and led and managed by its Marshall Space Flight
Center (MSFC), with industry partner Ball Aerospace (contract NNM15AA18C). The
Italian contribution is supported by the Italian Space Agency (Agenzia
Spaziale Italiana, ASI) through contract ASI-OHBI-2017-12-I.0, agreements ASI-
INAF-2017-12-H0 and ASI-INFN-2017.13-H0, and its Space Science Data Center
(SSDC) with agreements ASI-INAF-2022-14-HH.0 and ASI-INFN 2021-43-HH.0, and by
the Istituto Nazionale di Astrofisica (INAF) and the Istituto Nazionale di
Fisica Nucleare (INFN) in Italy. This research used data products provided by
the IXPE Team (MSFC, SSDC, INAF, and INFN) and distributed with additional
software tools by the High-Energy Astrophysics Science Archive Research Center
(HEASARC), at NASA Goddard Space Flight Center (GSFC).

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Ajay Ratheesh is a Post-Doctoral researcher affiliated with INAF-IAPS,
situated in Rome. In 2021, he earned his PhD in Astronomy and Astrophysics
through the joint PhD program offered by Sapienza University of Rome,
University of Rome ”Tor Vergata”, and INAF in Italy. Currently, he is a member
of the IXPE collaboration.
Biographies and photographs of the other authors are not available.

##  6 Tables

Table 1: The table outlining the details of the measurement configurations used in this work. X-ray tube | Anode | Fluorescent line energy | Source configuration | Angle
---|---|---|---|---
Hamamatsu, model N1335 | Ca | 3.69 | 5.4 kV, 0.003 mA |  $\epsilon 1$, $\epsilon 2$
Hamamatsu | W | 8.34 | 8.0 kV, $<$ 0.001 mA | $\epsilon 1$
Oxford series | Fe | 6.40 | 6.1 kV, 0.003 mA |  $\epsilon 1$, $\epsilon 2$
Oxford series 5000 | Rh | 2.70 | 6.2 kV, 0.001 mA |  $\epsilon 1$, $\epsilon 2$
Table 2: Polarization results from the Ca X-ray tube for $\epsilon 1$ and $\epsilon 2$.  Energy (keV) | Q/I ($\%$) | U/I ($\%$) | Pol deg ($\%$) | Pol angle | MDP ($\%$))
---|---|---|---|---|---
| $\epsilon 1$ |
$2.0$-$3.0$ | $0.51\pm 0.7$ | $-0.32\pm 0.7$ | $<1.3$ | $-15.99^{\circ}\pm 33.44^{\circ}$ | $2.11$
$3.0$-$3.7$ | $-0.03\pm 0.27$ | $0.19\pm 0.27$ | $<0.38$ | $49.05^{\circ}\pm 40.95^{\circ}$ | $0.83$
$3.7$-$4.2$ | $0.25\pm 0.24$ | $0.49\pm 0.24$ | $<0.79$ | $31.3^{\circ}\pm 12.63^{\circ}$ | $0.74$
$4.2$-$8.0$ | $-0.7\pm 0.31$ | $0.38\pm 0.31$ | $<1.11$ | $75.74^{\circ}\pm 11.08^{\circ}$ | $0.94$
| $\epsilon 2$ |
$2.0$-$3.0$ | $0.34\pm 0.42$ | $-0.54\pm 0.42$ | $<1.05$ | $-28.89^{\circ}\pm 18.97^{\circ}$ | $1.27$
$3.0$-$3.7$ | $0.58\pm 0.16$ | $-0.04\pm 0.16$ | $0.59\pm 0.16$ | $-2.19^{\circ}\pm 8.07^{\circ}$ | $0.5$
$3.7$-$4.2$ | $0.6\pm 0.15$ | $0.02\pm 0.15$ | $0.6\pm 0.15$ | $1.12^{\circ}\pm 7.0^{\circ}$ | $0.44$
$4.2$-$8.0$ | $-0.08\pm 0.19$ | $-0.12\pm 0.19$ | $<0.33$ | $-61.22^{\circ}\pm 37.93^{\circ}$ | $0.56$
Table 3: Polarization results from the W X-ray tube for $\epsilon 1$.  Energy (keV) | Q/I ($\%$) | U/I ($\%$) | Pol deg ($\%$) | Pol angle | MDP ($\%$))
---|---|---|---|---|---
| $\epsilon 1$ |
$2.0$-$3.25$ | $-0.86\pm 0.5$ | $-0.31\pm 0.5$ | $<1.42$ | $-80.01^{\circ}\pm 15.65^{\circ}$ | $1.52$
$3.25$-$4.25$ | $-0.51\pm 0.23$ | $0.22\pm 0.23$ | $<0.78$ | $78.17^{\circ}\pm 11.95^{\circ}$ | $0.7$
$4.25$-$5.25$ | $-0.43\pm 0.21$ | $0.55\pm 0.21$ | $0.7\pm 0.21$ | $63.9^{\circ}\pm 8.58^{\circ}$ | $0.64$
$5.25$-$8.0$ | $0.27\pm 0.19$ | $0.24\pm 0.19$ | $<0.56$ | $20.89^{\circ}\pm 14.92^{\circ}$ | $0.58$
Table 4: Polarization results from the Rh X-ray tube for $\epsilon 1$ and $\epsilon 2$.  Energy (keV) | Q/I ($\%$) | U/I ($\%$) | Pol deg ($\%$) | Pol angle | MDP ($\%$))
---|---|---|---|---|---
| $\epsilon 1$ |
$2.0$-$2.6$ | $-2.3\pm 0.79$ | $4.42\pm 0.79$ | $4.98\pm 0.79$ | $58.73^{\circ}\pm 4.53^{\circ}$ | $2.39$
$2.6$-$3.2$ | $-1.49\pm 0.46$ | $5.12\pm 0.46$ | $5.33\pm 0.46$ | $53.13^{\circ}\pm 2.47^{\circ}$ | $1.4$
$3.2$-$4.0$ | $-5.4\pm 0.42$ | $14.13\pm 0.42$ | $15.12\pm 0.42$ | $55.45^{\circ}\pm 0.79^{\circ}$ | $1.27$
$4.0$-$7.0$ | $-7.38\pm 0.29$ | $20.93\pm 0.29$ | $22.2\pm 0.29$ | $54.71^{\circ}\pm 0.37^{\circ}$ | $0.88$
| $\epsilon 2$ |
$2.0$-$2.6$ | $-0.85\pm 1.66$ | $-0.53\pm 1.66$ | $<2.66$ | $-73.99^{\circ}\pm 47.41^{\circ}$ | $5.04$
$2.6$-$3.2$ | $1.74\pm 0.94$ | $-5.96\pm 0.94$ | $6.21\pm 0.94$ | $-36.86^{\circ}\pm 4.34^{\circ}$ | $2.85$
$3.2$-$4.0$ | $6.16\pm 0.86$ | $-13.63\pm 0.86$ | $14.95\pm 0.86$ | $-32.83^{\circ}\pm 1.64^{\circ}$ | $2.6$
$4.0$-$7.0$ | $8.93\pm 0.59$ | $-20.61\pm 0.59$ | $22.46\pm 0.59$ | $-33.29^{\circ}\pm 0.75^{\circ}$ | $1.79$
Table 5: Polarization results from the Fe X-ray tube for $\epsilon 1$ and $\epsilon 2$.  Energy (keV) | Q/I ($\%$) | U/I ($\%$) | Pol deg ($\%$) | Pol angle | MDP ($\%$))
---|---|---|---|---|---
| $\epsilon 1$ |
$2.0$-$3.0$ | $-5.82\pm 0.66$ | $13.28\pm 0.66$ | $14.5\pm 0.66$ | $56.82^{\circ}\pm 1.3^{\circ}$ | $1.99$
$3.0$-$4.0$ | $-5.53\pm 0.3$ | $15.44\pm 0.3$ | $16.4\pm 0.3$ | $54.86^{\circ}\pm 0.52^{\circ}$ | $0.9$
$4.0$-$5.0$ | $-8.59\pm 0.31$ | $23.06\pm 0.31$ | $24.61\pm 0.31$ | $55.22^{\circ}\pm 0.36^{\circ}$ | $0.93$
$5.0$-$8.0$ | $-11.93\pm 0.45$ | $32.5\pm 0.45$ | $34.62\pm 0.45$ | $55.08^{\circ}\pm 0.38^{\circ}$ | $1.38$
| $\epsilon 2$ |
$2.0$-$3.0$ | $5.93\pm 1.31$ | $-15.8\pm 1.31$ | $16.87\pm 1.31$ | $-34.72^{\circ}\pm 2.23^{\circ}$ | $3.98$
$3.0$-$4.0$ | $5.09\pm 0.6$ | $-14.29\pm 0.6$ | $15.17\pm 0.6$ | $-35.2^{\circ}\pm 1.13^{\circ}$ | $1.82$
$4.0$-$5.0$ | $8.81\pm 0.62$ | $-24.09\pm 0.62$ | $25.65\pm 0.62$ | $-34.96^{\circ}\pm 0.69^{\circ}$ | $1.88$
$5.0$-$8.0$ | $13.31\pm 0.92$ | $-32.83\pm 0.91$ | $35.42\pm 0.91$ | $-33.97^{\circ}\pm 0.75^{\circ}$ | $2.8$
Table 6: The parameters of the fit (Applied voltage-V, angle between incoming electrons and outgoing photons-$\theta$) to the measured polarization as a function of photon energy. The real values are also shown to highlight the discrepancy.  X-ray tube | Vreal (kV) | $\theta_{real}$ | Vfit (kV) | $\theta_{fit}$
---|---|---|---|---
Ca | 5.4 | 0∘ | $19.1\pm 1.9$ | $-0.5^{\circ}\pm 6.5^{\circ}$
W | 8.0 | 0∘ | $23.8\pm 0.2$ | $0.05^{\circ}\pm 0.01^{\circ}$
Rh | 6.2 | 90∘ | $13.4\pm 9.0$ | $28.9^{\circ}\pm 23.6^{\circ}$
Fe | 6.1 | 90∘ | $12.7\pm 10.8$ | $35.3^{\circ}\pm 40.7^{\circ}$

Figure 1: An example modulation curve for linearly polarized light, decomposed
into stokes parameters as labeled (for modulation amplitude, m = 0.5, and
polarization angle ,$\phi_{0}$ = 60∘). Figure 2: Polarization degree of the
outgoing photons as a function of photon energy for incident electron energies
of 0.1 MeV for Z=13 (Aluminium) for different values of the angle between
incoming electrons and outgoing photons ($\theta_{0}$) as well as with and
without shielding effects.

Figure 3: The cross-section of the outgoing photons as a function of photon
energy for incident electron energies of 0.1 MeV for Z=13 (Aluminium) for
different values of the angle between incoming electrons and outgoing photons
($\theta_{0}$) as well as with and without shielding effects.

.

Figure 4: Photograph of the Iron X-ray tube used as a perpendicular X-ray tube
in this work. Figure 5: Photograph of the tungsten X-ray tube used as a
parallel X-ray tube in this work.  Figure 6: A rough sketch of the
experimental geometrical configuration of perpendicular (left) and parallel
(right) X-ray tubes with respect to the gas pixel detector (GPD). In the
former, the photon beam is perpendicular to the electron beam while in the
later both are parallel. Figure 7: The spectrum of all the X-ray tubes used in
this work as seen in the GPD for $\epsilon 1$. The spectrum of each X-ray tube
has been normalized by the corresponding peak of the spectrum. The vertical
lines indicate the maximum possible energy of the X-rays from the tube
corresponding to the applied voltage. For W and Rh tubes the spectrum has only
continuum Bremsstrahlung, as the applied voltage is below the characteristic
line energy. The spectrum of $\epsilon 2$ is identical to that of $\epsilon
1$, and thus it has not been presented to avoid redundancy. Figure 8: Polar
plots of the polarization degree (PD) and polarization angle (PA) for the
measurements of Ca X-ray tube at angles of $\epsilon 1$ and $\epsilon 2$.
Contours indicates 99.7% confidence levels. Colors black, red, blue, and
magenta indicate energy ranges 2.0-3.0 keV, 3.0-3.7 keV, 3.7-4.2 keV, and
4.2-8.0 keV. These contours indicate that there are no significant
polarization measurements but only constraints of upper-limits. Figure 9: Same
as Figure 8, but for W X-ray tube and only for the angle of $\epsilon 1$.
Contours indicates 99.7% confidence levels. Colors black, red, blue, and
magenta indicate energy ranges 2.0-3.25 keV, 3.25-4.25 keV, 4.25-5.25 keV, and
5.25-8.0 keV.  Figure 10:  Polar plots of the polarization degree (PD) and
polarization angle (PA) for the measurements of Rh X-ray tube at angles of
$\epsilon 1$ and $\epsilon 2$. Contours indicates 99.7% confidence levels.
Black, red, blue, and magenta indicate energy ranges of 2.0-2.6 keV, 2.6-3.2
keV, 3.2-4.0 keV, and 4.0-7.0 keV. The increase in polarization with respect
to energy can be seen clearly. The 90∘ phase shift observed in the
polarization angle (PA) between $\epsilon 1$ and $\epsilon 2$ confirms that
the PA measurement undergoes a 90∘ rotation when the source is rotated by 90∘,
and hence confirming the source origin of polarization than detector
systematics. Figure 11:  Same as Figure 10, but for Fe X-ray tube. Contours
indicates 99.7% confidence levels. Black, red, blue, and magenta indicate
energy ranges of 2.0-3.0 keV, 3.0-4.0 keV, 4.0-5.0 keV, and 5.0-8.0 keV.
Figure 12: The polarization fraction as a function of the photon energy for
all the X-ray tubes (at $\epsilon 1$). The black points show the measured
polarization. The solid green line indicates the expected value from the
theory, while the shaded green region in the case of Fe and Rh tubes shows the
range due to the non-zero opening angle subtended by the crystal on the
detector. The red line indicates the values of the best fit. The negative
polarization fraction means that the polarization angle is parallel to the
electron direction, while the positive indicates the polarization angle
perpendicular to the incoming electron and outgoing photon.