#  Ulam’s method for computing stationary densities of invariant measures for
piecewise convex maps with countably infinite number of branches

Md Shafiqul Islam  School of Mathematical and Computational Sciences
University of Prince Edward Island
550 University Ave, Charlottetown, PE, C1A 4P3, Canada<EMAIL_ADDRESS>,
Paweł Góra  Department of Mathematics and Statistics, Concordia University,
1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada
<EMAIL_ADDRESS>and  A H M Mahbubur Rahman  Department of mathematics
and Statistics,
Concordia University, 1455 De Maisonneuve Blvd. W.
Montreal, Quebec, H3G 1M8, Canada<EMAIL_ADDRESS>
###### Abstract.

Let $\tau:I=[0,1]\to[0,1]$ be a piecewise convex map with countably infinite
number of branches. In [Góra et al.(2022)], the existence of absolutely
continuous invariant measure (ACIM) $\mu$ for $\tau$ and the exactness of the
system $(\tau,\mu)$ has been proven. In this paper, we develop an Ulam method
for approximation of $f^{*}$, the density of ACIM $\mu$. We construct a
sequence $\\{\tau_{n}\\}_{n=1}^{\infty}$ of maps $\tau_{n}:I\to I$ s. t.
$\tau_{n}$ has a finite number of branches and the sequence $\tau_{n}$
converges to $\tau$ almost uniformly. Using supremum norms and Lasota-Yorke
type inequalities, we prove the existence of ACIMs $\mu_{n}$ for $\tau_{n}$
with the densities $f_{n}$. For a fixed $n$, we apply Ulam’s method with $k$
subintervals to $\tau_{n}$ and compute approximations $f_{n,k}$ of $f_{n}$. We
prove that $f_{n,k}\to f^{*}$ as $n\to\infty,k\to\infty,$ both a.e. and in
$L^{1}$.
We provide examples of piecewise convex maps $\tau$ with countably infinite
number of branches, their approximations $\tau_{n}$ with finite number of
branches and for increasing values of parameter $k$ show the errors
$\|f^{*}-f_{n,k}\|_{1}$.

###### Key words and phrases:

Stationary densities; Absolutely continuous invariant measures; Ulam’s method;
Convex maps with countably infinite number of branches; Frobenius-Perron (F-P)
Operator

##  1\. Introduction

The existence and properties of absolutely continuous invariant measures
(ACIMs) of deterministic dynamical systems reflect their long time behaviour
and play an important role in understanding their chaotic nature [Boyarsky &
Góra(1997), Lasota & Mackey(1994), Lasota & Yorke(1973)]. Let $\mathcal{B}$ be
the Lebesgue $\sigma$-algebra of subsets of $I=[0,1]$ and let $\lambda$ be the
normalized Lebesgue measure on $I$. Let $\tau:I\rightarrow I$ be a non-
singular measurable transformation. A measure $\mu$ on $\mathcal{B}$ is
$\tau$-invariant if $\mu(\tau^{-1}(A))=\mu(A)$ for all $A\in\mathcal{B}.$ The
Frobenius-Perron (F-P) operator $P_{\tau}:L^{1}(I,\mathcal{B},\lambda)\to
L^{1}(I,\mathcal{B},\lambda)$ of $\tau$ plays an important role for the
existence, approximations and properties of ACIMs. The F-P operator $P_{\tau}$
is defined by

(1.1) $\int_{A}P_{\tau}f\hskip 4.2679ptd\lambda=\int_{\tau^{-1}(A)}f\hskip
4.2679ptd\lambda,\hskip 2.84526pt\forall A\in\mathcal{B},\forall f\in L^{1}.$

It can be shown that for $\tau$ with countable number of monotonic branches,
the F-P operator $P_{\tau}$ has the following representation [Boyarsky &
Góra(1997)] :

(1.2) $P_{\tau}f(x)=\sum_{i\geq
1}\frac{f(\tau_{i}^{-1}(x))}{|{\tau^{\prime}(\tau_{i}^{-1}(x))}|}\chi_{I_{i}}(x)=\sum_{z\in\\{\tau^{-1}(x)\\}}\frac{f(z)}{\lvert\tau^{\prime}(z)\rvert},$

where $\tau_{i}^{-1}$ are inverse branches of $\tau_{i}=\tau_{|I_{i}}$ ,
$i\geq 1$, and $\\{I_{i}\\}_{i\geq 1}$ are the subintervals of the
monotonicity partition. The F-P operator $P_{\tau}$ has a number of useful
properties including: (i) Linearity: $P_{\tau}:L^{1}\rightarrow L^{1}$ is a
linear operator; (ii) Positivity: If $f\in L^{1}$ and $f\geq 0$, then
$P_{\tau}f\geq 0$; (iii) Contractivity: $P_{\tau}:L^{1}\rightarrow L^{1}$ is a
contraction, i.e.,
$\parallel{P_{\tau}f}\parallel_{1}\leq\parallel{f}\parallel_{1}$ for any $f\in
L^{1}$. Moreover, $P_{\tau}:L^{1}\rightarrow L^{1}$ is continuous with respect
to norm topology; (iv) $P_{\tau}$ preserves integrals, i.e.,
$\int_{0}^{1}P_{\tau}fd\lambda=\int_{0}^{1}fd\lambda$; (v) If
$\tau_{1},\tau_{2}:I\rightarrow I$ are non-singular transformations, then
$P_{\tau_{1}\circ\tau_{2}}f=P_{\tau_{1}}\circ P_{\tau_{2}}f$. In particular,
$P_{\tau^{n}}f=P^{n}_{\tau}f$ ; (vi) Operator $P_{\tau}$ is conjugated to the
Koopman operator $Kg=g\circ\tau$ on $L^{\infty}$, i.e.,

$\int_{I}g\cdot P_{\tau}f\,d\lambda=\int_{I}g\circ\tau\cdot f\,d\lambda\ ,f\in
L^{1}\ ,\ g\in L^{\infty};$

(vii) $P_{\tau}$ has a fixed point $f^{*}$ if and only if the measure
$\mu=f^{*}\cdot\lambda$ defined by $\mu(A)=\int_{A}f^{*}d\lambda$ is
$\tau-$invariant. From the property (vii) above one can see that the fixed
point of $P_{\tau}$ is the density $f^{*}$ of the ACIM $\mu$ of $\tau$ (see
[Boyarsky & Góra(1997)]).

In [Lasota & Yorke(1973)], Lasota and Yorke proved the existence of ACIMs for
piecewise expanding maps. In this paper, we consider transformations that are
not necessarily expanding, i.e., their derivatives may be smaller than $1$,
but they possess another property which makes them very special, namely
piecewise convexity. In [Lasota & Yorke(1982)], the authors studied the
existence of ACIMs and the exactness for piecewise convex transformations with
a finite number of branches and with a strong repeller. The authors in [Lasota
& Yorke(1982)] considered the following properties as key properties for the
proof of the existence of ACIMs (i) the F-P operator $P_{\tau}$ maps non-
increasing functions to non-increasing functions; (ii) If
$f:[0,1]\to\mathbb{R}^{+}$ is a non-increasing function, then

$\parallel P_{\tau}f\parallel_{\infty}\leq A\parallel
f\parallel_{\infty}+B\parallel f\parallel_{1},$

where $A<1$ and $B>0$ are some constants. In [Rahman(2019)], the author
studied ACIMs for piecewise convex maps with infinite number of branches,
where $1$ is the limit point of partition points. The literature on piecewise
convex maps with a finite number of branches is quite rich, see for example
[Bose et al.(2002)], [Ding(1996)], [Inoue(1992)], [Inoue(2023)], [Jabłoński &
Malczak(1984)], [Jabłoński & Malczak(1983)].

In [Góra et al.(2022)], we have studied ACIMs of maps in two classes
$\mathcal{T}_{pc}^{\infty}(I),\mathcal{T}_{pc}^{\infty,0}(I)$ of piecewise
convex maps $\tau:I\to I$ with countably infinite number of branches. Any
$\tau$ in the first class $\mathcal{T}_{pc}^{\infty}(I)$ is a piecewise convex
map $\tau:I\to I$ with a countable number of branches and arbitrary countable
number of limit points of partition points separated from $0$. Let
$\\{0=a_{0},a_{1},a_{2},\dots,a_{n},\dots\\}$ be a countable partition of $I$
such that $a_{0}<a_{1}$ and all $a_{2},a_{3},\dots\in[a_{1},1].$ We do not
assume the sequence $\\{a_{2},\dots,a_{n},\dots\\}$ to be increasing or
decreasing. For any $i\in\\{0,1,2,\dots\\},$ let $n(i)$ be the index such that
the interval $[a_{i},a_{n(i)}]$ does not contain any other points of the
partition. If $a_{k}$ is the limit point of decreasing subsequence of
$a_{n}$’s, the $n(k)$ is not defined. For such points the interval
$[a_{k},a_{k}]$ is not considered an element of the partition. We allow any
countable number of the limit points of the points of the partition. We assume
that the union of intervals $[a_{i},a_{n(i)}]$ covers the whole interval $I$
except of a countable number of points. At any such point $x$ we assume
$\tau(x)=0$ and $\tau^{\prime}(x)=+\infty$. These points do not contribute to
the images of the $P_{\tau}$ operator.

Any $\tau$ in the second class $\mathcal{T}_{pc}^{\infty,0}(I)$ is a piecewise
convex map $\tau:I\to I$ with countably infinite number of branches such that
there exists a countable partition
$\\{0=a_{0}<\dots<a_{0,-n}<a_{0,-(n-1)}<\dots<a_{0,-2}<a_{0,-1}=a_{1},a_{2},a_{3},\dots,a_{n},\dots\\}$
of $I$ with $\lim_{n\to\infty}a_{0,-n}=0.$ It is proved in [Góra et al.(2022)]
that any
$\tau\in\mathcal{T}_{pc}^{\infty}(I)\cup\mathcal{T}_{pc}^{\infty,0}(I)$ has
the unique stationary density $f^{*}$ of the ACIM $\mu^{*}.$ In this paper, we
apply Ulam’s method for approximation of $f^{*}.$ Fixed points of the F-P
operator $P_{\tau}$ of
$\tau\in\mathcal{T}_{pc}^{\infty}(I)\cup\mathcal{T}_{pc}^{\infty,0}(I)$ are
the stationary densities of $\tau.$ The F-P operator $P_{\tau}$ is an infinite
dimensional operator. Except some simple cases (where $\tau$ is piecewise
linear Markov and the Frobenius-Perron operator has a matrix representation),
it is not easy to obtain an analytical solution of the F-P equation
$P_{\tau}f=f.$

A method for numerical computations of invariant measures of dynamical systems
was suggested by Ulam [Ulam(1960)]. For piecewise expanding deterministic
transformations [Lasota & Yorke(1973)], T-Y Li [Li(1976)] first proved the
convergence of Ulam’s approximation. Since then, Ulam’s method has been
applied to one and higher dimensional expanding deterministic transformations
(see, for example, [Boyarsky & Lou(1991), Bose & Murray(2001), Bollt &
Santitissadeekorn(2013), Ding & Zhou(1996)]. Ulam’s method is one of the most
used and the best understood methods among numerical methods for the
approximation of ACIMs for deterministic maps [Li(1976), Ulam(1960)]. However,
approximate invariant densities via Ulam’s methods are piecewise constant on
subintervals of the partition of the state space. In [Miller(1994)], Miller
proved the convergence of Ulam’s method for piecewise convex transformations
with a finite number of branches and a strong repeller. J. Ding [Ding(1996)]
developed and presented piecewise linear and piecewise quadratic Markov finite
approximation methods for such piecewise convex transformations.

In Section 2, we introduce notations and review the existence of stationary
densities of ACIMs for
$\tau\in\mathcal{T}_{pc}^{\infty}(I)\cup\mathcal{T}_{pc}^{\infty,0}(I).$ In
Section 3, we construct a sequence $\\{\tau_{n}\\}_{n=1}^{\infty}$ of maps
$\tau_{n}:I\to I$ s. t. $\tau_{n}$ has a finite number of branches and the
sequence $\tau_{n}$ converges to $\tau$ almost uniformly. Using supremum norms
and Lasota-Yorke type inequalities, we prove the existence of densities
$f_{n}$ of ACIMs $\mu_{n}$ for $\tau_{n}$. Moreover, we prove that $f_{n}$
converges to $f^{*}.$ In Section 3.2 we apply Ulam’s method with $k$
subintervals to $\tau_{n}$, compute an approximation $f_{n,k}$ of $f_{n}$ and
prove that $f_{n,k}\to f_{n}$ as $k\to\infty.$ In Theorem 5 we prove that
$f_{n,k}\to f^{*}$, as $n,k\to\infty$, both a.e. and in $L^{1}$.
Theoretically, we could apply Ulam’s method directly to the map $\tau$.
Practically, such method would be impossible since the calculation of the
Ulam’s matrix $M_{\tau}^{(k)}$ would require solving of infinitely many
equations. In Section 4, we present numerical examples.

##  2\.  ACIMs for piecewise convex maps in classes
$\mathcal{T}_{pc}^{\infty}(I)$ and $\mathcal{T}_{pc}^{\infty,0}(I)$

In this section, we review results on the existence of stationary densities of
ACIMs of piecewise convex maps with countably infinite number of branches. We
closely follow [Góra et al.(2022)].

###  2.1.  ACIMs for piecewise convex maps in $\mathcal{T}_{pc}^{\infty}(I)$

For the maps in $\mathcal{T}_{pc}^{\infty}(I)$ the limit points of the
partition points are separated from $0$.

We say that $\tau\in\mathcal{T}_{pc}^{\infty}(I)$ if

  1. (1)

$\tau_{0}=\tau|_{[0,a_{1})}$ is continuous and convex;
$\tau_{i}=\tau|_{[a_{i},a_{n(i)})}$ is continuous and convex, $i=1,2,\cdots;$

  2. (2)

$\tau(a_{i})=0,\tau^{\prime}(a_{i})>0,i=1,2,\dots;$

  3. (3)

$\tau(0)=0,\tau^{\prime}(0)=\alpha_{1}>1;$

  4. (4)

$\sum_{i=1}^{\infty}\frac{1}{\tau^{\prime}(a_{i})}<\infty.$

For $\tau\in\mathcal{T}_{pc}^{\infty}(I),f\in L^{1}(I),f\geq 0$ the F-P
operator $P_{\tau}$ can be represented as

(2.1)
$P_{\tau}f(x)=\frac{f(\tau_{0}^{-1}(x))}{\tau^{\prime}(\tau_{0}^{-1}(x))}\chi_{\tau[0,a_{1})}(x)+\sum_{i=1}^{\infty}\frac{f(\tau_{i}^{-1}(x))}{\tau^{\prime}(\tau_{i}^{-1}(x))}\chi_{\tau[a_{i},a_{n(i)})}(x).$

The following results are proved in [Góra et al.(2022)].

######  Lemma 1.

Let $\tau\in\mathcal{T}_{pc}^{\infty}(I),f\in L^{1}(I),f\geq 0,f$ be non-
increasing. Then

  1. (1)

$P_{\tau}(f)\in L^{1}(I).$

  2. (2)

$P_{\tau}(f)\geq 0.$

  3. (3)

$P_{\tau}(f)$ is non-increasing.

  4. (4)

$\parallel P_{\tau}(f)\parallel_{\infty}\leq C\parallel f\parallel_{\infty},$
where
$C=\left(\frac{1}{\alpha_{1}}+\sum_{i=1}^{\infty}\frac{1}{\tau^{\prime}(a_{i})}\right).$

######  Proposition 1.

If $f\geq 0$ and $f$ is non-increasing, then
$f(x)\leq\frac{1}{x}\lambda(f),\text{~{}}{for}~{}x\in[0,1],$ where

$\lambda(f)=\int_{0}^{1}f(x)d\lambda(x).$

######  Lemma 2.

If $f:I\to\mathbb{R}^{+}$ is non-increasing and
$\tau\in\mathcal{T}_{pc}^{\infty}(I),$ then

(2.2) $\parallel
P_{\tau}(f)\parallel_{\infty}\leq\frac{1}{\alpha_{1}}\parallel
f\parallel_{\infty}+D\parallel f\parallel_{1},$

where
$D=\left(\sum_{i=1}^{\infty}\frac{1}{a_{i}}\frac{1}{\tau^{\prime}(a_{i})}\right).$

######  Theorem 1.

Let $\tau\in\mathcal{T}_{pc}^{\infty}(I).$ Then $\tau$ admits an ACIM
$\mu=f^{*}\cdot\lambda$ with non-increasing density function $f^{*}.$ The ACIM
$\mu$ is unique and the system $(\tau,\mu)$ is exact.

###  2.2. ACIMs for piecewise convex maps in $\mathcal{T}_{pc}^{\infty,0}(I)$

For the maps in $\mathcal{T}_{pc}^{\infty,0}(I)$ the point $0$ is the limit
point of partition points.

We say that $\tau\in\mathcal{T}_{pc}^{\infty,0}(I)$ if

  1. (1)

$\tau_{-j}=\tau|_{[a_{0,-(j+1)},a_{0,-j})}$ is continuous and convex,
$j=1,2,\dots$ ;
$\tau_{i}=\tau|_{[a_{i},a_{n(i)})}$ is continuous and convex, $i=1,2,\cdots;$

  2. (2)

$\tau(a_{0,-j})=0,\tau^{\prime}(a_{0,-j})>0,j=1,2,\dots;$

$\tau(a_{i})=0,\tau^{\prime}(a_{i})>0,i=1,2,\dots;$

  3. (3)

$\sum_{i=1}^{\infty}\frac{1}{\tau^{\prime}(a_{i})}<\infty;$

  4. (4)

$D_{1}=\sum_{j=1}^{\infty}\frac{1}{\tau^{\prime}(a_{0,-j})}<1.$

######  Remark 1.

The Condition 3 and the Condition 4 can be replaced by the following
condition:

$(3^{+})\ \ \
\sum_{j=1}^{\infty}\frac{1}{\tau^{\prime}(a_{0,-j})}+\sum_{i=1}^{\infty}\frac{1}{\tau^{\prime}(a_{i})}<\infty.$

If $(3^{+})$ is satisfied, then we can find an integer $J\geq 1$ such that

$\sum_{j=J}^{\infty}\frac{1}{\tau^{\prime}(a_{0,-j})}<1,$

and after proper renaming of partition points, the Condition 3 and the
Condition 4 are satisfied.

The following results are proved in [Góra et al.(2022)].

######  Lemma 3.

Let $\tau\in\mathcal{T}_{pc}^{\infty,0}(I),f\in L^{1}(I),f\geq 0,f$ be non-
increasing. Then

  1. (1)

$P_{\tau}(f)\in L^{1}(I).$

  2. (2)

$P_{\tau}(f)\geq 0.$

  3. (3)

$P_{\tau}(f)$ is non-increasing.

  4. (4)

$\parallel P_{\tau}(f)\parallel_{\infty}\leq C\parallel f\parallel_{\infty},$
where
$C=\left(D_{1}+\sum_{i=1}^{\infty}\frac{1}{\tau^{\prime}(a_{i})}\right).$

######  Lemma 4.

If $f:[0,1]\to\mathbb{R}^{+}$ is non-increasing and
$\tau\in\mathcal{T}_{pc}^{\infty,0}(I),$ then

(2.3) $\parallel P_{\tau}(f)\parallel_{\infty}\leq D_{1}\parallel
f\parallel_{\infty}+D\parallel f\parallel_{1},$

where $D=\sum_{i=1}^{\infty}\frac{1}{a_{i}\tau^{\prime}(a_{i})}.$

######  Theorem 2.

Let $\tau\in\mathcal{T}_{pc}^{\infty,0}(I).$ Then $\tau$ admits an ACIM
$\mu=f^{*}\cdot\lambda$ with non-increasing density function $f^{*}.$ The ACIM
$\mu$ is unique and the system $(\tau,\mu)$ is exact.

##  3\. Ulam’s method for piecewise convex maps with countable number of
branches

In this section our main goal is to develop Ulam’s method for stationary
densities of ACIMs for piecewise convex maps with countably infinite number of
branches. We do this in the following two steps.

###  3.1. Approximation of maps with infinite number of branches

In this section we approximate piecewise convex maps with countably infinite
number of branches by piecewise convex maps with a finite number of branches.

We concentrate on a case of a map $\tau\in\mathcal{T}^{\infty,0}_{pc}(I)$. On
the example of point 0, we show how to perform an approximation of $\tau$ with
a point of a partition that is a limit point of other points of the partition.
The method can be applied to maps in $\mathcal{T}^{\infty}_{pc}(I)$, that have
points with this property without substantial changes.

Let $\tau\in\mathcal{T}^{\infty,0}_{pc}(I)$. We assume that there are no other
limit points of the partition points. For simplicity we rename the partition
points. The new partition points are $a_{n}$, $n=0,1,2,\dots$, with $a_{0}=1$,
$a_{n+1}<a_{n}$, $n\geq 0$ and $\lim_{n\to\infty}a_{n}=0$. Then, the
assumptions (3) and (4) of Section 2.2 are restated as:
(3’) There exists an $N\geq 1$ such that

(3.1) $D_{1}=\sum_{n=N+1}^{\infty}\frac{1}{\tau^{\prime}(a_{n})}<1.$

Then, the Lemma 3 and the Lemma 4 of Section 2.2 hold with changed constants

(3.2)
$\begin{split}C&=D_{1}+\sum_{n=1}^{N}\frac{1}{\tau^{\prime}(a_{n})}=\sum_{n=1}^{\infty}\frac{1}{\tau^{\prime}(a_{n})},\\\
D&=\sum_{n=1}^{N}\frac{1}{a_{n}\tau^{\prime}(a_{n})}.\end{split}$

For $n\geq N$, we construct a sequence $\\{\tau_{n}\\}_{n=N}^{\infty}$ of maps
$\tau_{n}:I\to I$ s.t. $\tau_{n}$ has a finite number of branches and the
sequence $\tau_{n}$ converges to $\tau$ almost uniformly. Using supremum norms
and Lasota-Yorke type inequalities, we prove the existence of stationary
densities $f_{n}$ of ACIMs $\mu_{n}$ for $\tau_{n}$. We approximate $\tau:I\to
I$ with the following sequence of maps $\tau_{n}:I\to I$, $n\geq N$, with a
finite number of branches:

(3.3) $\tau_{n}(x)=\begin{cases}x/a_{n}\ ,\ 0\leq x<a_{n};\\\ \tau(x)\ ,\
a_{n}\leq x\leq 1.\end{cases}$

In the following, we show that for each $n\geq 0,$ the map $\tau_{n}$ has an
ACIM. It can be easily shown that each $\tau_{n}$ is a piecewise convex maps
with the finite partition $\\{1=a_{0},a_{1},a_{2},\cdots,a_{n},a_{n+1}=0\\}$
and $\tau_{n}$ satisfies following conditions:

  1. (1)

$\tau_{n_{j}}=\tau_{n}|_{{(a_{j},a_{j-1}]}}$ is continuous and convex,
$j=1,2,\cdots,n+1;$

  2. (2)

$\tau_{n}(a_{j})=0,\tau_{n}^{\prime}(a_{j})>0,j=1,\cdots,n+1;$

  3. (3)

$\tau_{n}(0)=0,\tau_{n}^{\prime}(0)=\frac{1}{a_{n}}>1.$

  4. (4)

$\sum_{j=1}^{n+1}\frac{1}{\tau^{\prime}(a_{j})}<\infty.$

Let $f\in L^{1}(I),f\geq 0.$ Then, the F-P operator $P_{\tau_{n}}$ is defined
as

(3.4)
$P_{\tau_{n}}f(x)=\frac{f(\tau_{a}^{-1}(x))}{1/a_{n}}+\sum_{j=1}^{n}\frac{f(\tau_{n_{j}}^{-1}(x))}{\tau_{n}^{\prime}(\tau_{n_{j}}^{-1}(x))}\chi_{\tau_{n}(a_{{j}},a_{{j-1}}]}(x),$

where $\tau_{a}=x/a_{n}$ on the interval $[0,a_{n})$.

######  Lemma 5.

Let $f\in L^{1}(I),f\geq 0,f$ be a non-increasing function. Then

  1. (1)

$P_{\tau_{n}}(f)\in L^{1}(I)$ for each $n=1,2,\dots.$

  2. (2)

$P_{\tau_{n}}(f)\geq 0$ for each $n=1,2,\dots.$

  3. (3)

$P_{\tau_{n}}(f)$ is non-increasing for each $n=1,2,\dots.$

  4. (4)

$\parallel P_{\tau_{n}}(f)\parallel_{\infty}\leq C_{n}\parallel
f\parallel_{\infty}\leq\left(1+C\right)\parallel f\parallel_{\infty}$ for each
$n=1,2,\dots$ where
$C_{n}=\left(a_{n}+\sum_{j=1}^{n}\frac{1}{\tau_{n}^{\prime}(a_{{j}})}\right).$

###### Proof.

  1. (1)

Holds for the F-P operator of any map.

  2. (2)

$P_{\tau_{n}}f(x)$ is shown in formula (3.4) and each summand is clearly non-
negative.

  3. (3)

It can be easily shown that each summand in formula (3.4) is non-increasing.

  4. (4)

We have

$\displaystyle P_{\tau_{n}}f(x)$ $\displaystyle=$
$\displaystyle\frac{f(\tau_{a}^{-1}(x))}{1/a_{n}}+\sum_{j=1}^{n}\frac{f(\tau_{n_{j}}^{-1}(x))}{\tau_{n}^{\prime}(\tau_{n_{j}}^{-1}(x))}\chi_{\tau_{n}(a_{{j}},a_{{j-1}}]}(x)$
$\displaystyle\leq$ $\displaystyle a_{n}\parallel
f\parallel_{\infty}+\sum_{j=1}^{n}\frac{\parallel
f\parallel_{\infty}}{\tau_{n}^{\prime}(\tau_{n_{j}}^{-1}(x))}\leq
a_{n}\parallel f\parallel_{\infty}+\sum_{j=1}^{n}\frac{\parallel
f\parallel_{\infty}}{\tau_{n}^{\prime}(a_{{j}})}$ $\displaystyle=$
$\displaystyle\left(a_{n}+\sum_{j=1}^{n}\frac{1}{\tau_{n}^{\prime}(a_{{j}})}\right)\parallel
f\parallel_{\infty}\leq\left(1+C\right)\parallel f\parallel_{\infty}$

∎

######  Lemma 6.

If $f:[0,1]\to\mathbb{R}^{+}$ is non-increasing, then for each $n\geq N$,

(3.5) $\parallel P_{\tau_{n}}(f)\parallel_{\infty}\leq(a_{n}+D_{1})\parallel
f\parallel_{\infty}+D\parallel f\parallel_{1},$

###### Proof.

Since $f$ is non-increasing, $f(0)\geq\parallel f\parallel_{\infty},$ and by
Lemma 5, $P_{\tau_{n}}f(0)=\parallel P_{\tau_{n}}f\parallel_{\infty}.$ Now,

$\displaystyle P_{\tau_{n}}f(0)$ $\displaystyle=$
$\displaystyle\frac{f(\tau_{a}^{-1}(x))}{1/a_{n}}+\sum_{j=1}^{n}\frac{f(\tau_{n_{j}}^{-1}(0))}{\tau_{n}^{\prime}(\tau_{n_{j}}^{-1}(0))}\chi_{\tau_{n}(a_{{j}},a_{{j-1}}]}(0)$
$\displaystyle=$
$\displaystyle\frac{1}{\tau_{n}^{\prime}(0)}f(0)+\sum_{j=1}^{N}\frac{f(\tau_{n_{j}}^{-1}(0))}{\tau_{n}^{\prime}(\tau_{n_{j}}^{-1}(0))}+\sum_{j=N+1}^{n}\frac{f(\tau_{n_{j}}^{-1}(0))}{\tau_{n}^{\prime}(\tau_{n_{j}}^{-1}(0))}$
$\displaystyle\leq$ $\displaystyle
a_{n}f(0)+\sum_{j=1}^{N}\frac{f(a_{n_{j}})}{\tau_{n}^{\prime}(a_{n_{j}})}+\sum_{j=N+1}^{n}\frac{f(a_{n_{j}})}{\tau_{n}^{\prime}(a_{n_{j}})}$
$\displaystyle\leq$
$\displaystyle(a_{n}+D_{1})f(0)+\sum_{j=1}^{N}\frac{\lambda(f)}{a_{{j}}}\frac{1}{\tau_{n}^{\prime}(a_{{j}})}$
$\displaystyle\leq$ $\displaystyle(a_{n}+D_{1})\parallel
f\parallel_{\infty}+D\parallel f\parallel_{1}.$

∎

######  Theorem 3.

For each $n\in\mathbb{N},$ $\tau_{n}$ admits an ACIM
$\mu_{n}=f_{n}^{*}\cdot\lambda$ with non-increasing density function
$f_{n}^{*}.$

###### Proof.

Our proof works for all $n\geq n_{0}$, where we have $a_{n_{0}}+D_{1}<1$ and
gives a uniform estimate. For the $n<n_{0}$ the claim follows from Subsection
2.1.

Let $f\in L^{1}$ be non-negative and non-increasing. Consider the sequence
$\\{P_{\tau_{n}}^{k}f\\}_{k=0}^{\infty}.$ Since every $P_{\tau_{n}}^{k}f$ is
non-negative and non-increasing, by part (3) of Lemma 5 we can apply Lemma 6
iteratively and obtain

$\displaystyle\parallel
P_{\tau_{n}}^{k}f\parallel_{\infty}\leq(a_{n_{0}}+D_{1})^{k}\parallel
f\parallel_{\infty}+\frac{D}{1-(a_{n_{0}}+D_{1})}\parallel f\parallel_{1}.$

So the sequence $\\{P_{\tau_{n}}^{k}f\\}_{k=0}^{\infty}$ is uniformly bounded
and weakly compact. By Yosida-Kakutani Theorem [Yosida & Kakutani(1941)],
$\frac{1}{k}\sum_{j=1}^{k}P_{\tau_{n}}^{j}f$ converges in $L^{1}$ to a
$P_{\tau_{n}}$ invariant function $f_{n}^{*}.$ It is non-increasing since it
is the limit of non-increasing functions. ∎

######  Definition 1.

Let $\tau$ and $\tau_{n}$, $n=1,2,\dots$ be maps from $I$ into itself. We say
that $\tau_{n}$ converges to $\tau$ almost uniformly if for any $\epsilon>0,$
there exists a measurable set
$A_{\epsilon}\subset[0,1],\lambda(A_{\epsilon})>1-\epsilon,$ such that
$\tau_{n}\to\tau$ uniformly on $A_{\epsilon}.$

######  Lemma 7.

Let $\tau$ be a piecewise convex map with countably many branches and the
sequence $\tau_{n}$, $n=1,2,\dots$ be defined by formula (3.3). Then,
$\tau_{n}$ converges to $\tau$ almost uniformly.

###### Proof.

Let $\epsilon>0.$ Choose the decreasing partition
$\\{1=a_{0},a_{1},a_{2},\cdots,a_{n},a_{n+1}=0\\}$ of $[0,1]$ for $\tau_{n}$
such that $a_{n}<\epsilon.$ Let $A_{\epsilon}=(a_{n},1).$ Then,
$\lambda(A_{\epsilon})=1-a_{n}>1-\epsilon$ and we have $\tau_{n}=\tau$ on
$A_{\epsilon}.$
∎

###  3.2. Ulam’s method

In this section, first, we describe Ulam’s method for a finite dimensional
approximation $P_{n,k}$ of the Perron-Frobenius operator $P_{\tau_{n}}$ of
$\tau_{n}.$ Ulam’s method computes $f_{n,k}$ on a partition of $k$
subintervals of the state space as an approximation of the actual stationary
density function $f_{n}$ of $\tau_{n},~{}n\geq 1.$ Moreover, we show that
$f_{n,k}$ converges to $f_{n}$ as $k\to\infty.$ We follow [Li(1976)],
[Miller(1994)] and [Ding(1996)]. In general, the Ulam’s method is widely
studied and used, see for example [Bose & Murray(2001)], [Bollt &
Santitissadeekorn(2013)], [Ding & Zhou(1996)], [Froyland(1999)], [Góra &
Boyarsky(1989)] and the references within.

Let $\tau_{n}$ be an approximation of $\tau\in\mathcal{T}_{pc}^{\infty}(I).$
Then, by the Theorem 3, $\tau_{n}$ has an ACIM $\mu_{n}$ with a stationary
density function $f_{n}.$ Now, we describe Ulam’s method for approximating
$f_{n}.$ Let $k$ be a positive integer. Let
$\mathcal{P}^{(k)}=\\{J_{1},J_{2},\dots,J_{k}\\}$ be a partition of the
interval $[0,1]$ into $k$ equal subintervals. Now, construct the matrix

$\mathbb{M}^{(k)}_{n}=\left(\frac{\lambda\left(\tau_{n}^{-1}(J_{j})\cap
J_{i}\right)}{\lambda(J_{i})}\right)_{1\leq i,j\leq k}.$

Let $L^{(k)}\subset L^{1}([0,1],\lambda)$ be a subspace of $L^{1}$ consisting
of functions which are constant on elements of the partition
$\mathcal{P}^{(k)}.$ We will represent functions in $L^{(k)}$ as vectors:
vector $f=[f_{1},f_{2},\dots,f_{k}]$ corresponds to the function
$f=\sum_{i=1}^{k}f_{i}\chi_{J_{i}}.$ Let $Q^{(k)}$ be the isometric projection
of $L^{1}$ onto $L^{(k)}$:

(3.6)
$Q^{(k)}(f)=\sum_{i=1}^{k}\left(\frac{1}{\lambda(J_{i})}\int_{J_{i}}fd\lambda\right)\chi_{J_{i}}=\left[\frac{1}{\lambda(J_{1})}\int_{J_{1}}fd\lambda,\dots,\frac{1}{\lambda(J_{k})}\int_{J_{k}}fd\lambda\right]$

Let $f=[f_{1},f_{2},\dots,f_{k}]\in L^{(k)}.$ We define the operator
$P_{\tau_{n}}^{(k)}:L^{(k)}\to L^{(k)}$ by

(3.7) $\displaystyle P_{\tau_{n}}^{(k)}f$ $\displaystyle=$
$\displaystyle\left[f_{1},f_{2},\dots,f_{k}\right]\cdot\left(\mathbb{M}^{(k)}_{n}\right)^{\text{T}},$

which is a finite dimensional approximation to the operator $P_{\tau_{n}}.$
$A^{\text{T}}$ denotes the transpose of the matrix $A.$

The following Lemma will be used several times in the sequel.

######  Lemma 8.

Let $\\{g_{n}\\}_{n=1,2\dots}$ be a sequence of non-increasing functions
uniformly bounded in $L^{\infty}$. If $g_{n}\to h$, as $n\to\infty$, weakly in
$L^{1}$, then $g_{n}\to h$, as $n\to\infty$, in $L^{1}$ and almost everywhere
(a. e.).

###### Proof.

Since $g_{n}$’s are non-increasing and uniformly bounded in $L^{\infty}$, they
are also of uniformly bounded variation. By Helly’s Theorem [Rudin(1976)]
there is a subsequence $g_{n_{k}}$ convergent a.e. to some function $h_{1}$.
Since $g_{n_{k}}\to h_{1}$ weakly in $L^{1}$ we have $h_{1}=h$. Considering
all possible subsequences we prove that $g_{n}\to h$ a.e. Since $g_{n}$
converge to $h$ a.e. and they are uniformly bounded in $L^{\infty}$, the
convergence is also in $L^{1}$ (e.g. by Lebesgue Dominated Convergence
theorem). ∎

Since each map $\tau_{n}$ is exact [Lasota & Yorke(1982)], by Proposition 1.2
of [Hunt & Miller(1992)] we obtain that the invariant densities $f_{n,k}$ of
$P_{\tau_{n}}^{(k)}$ are unique.

######  Lemma 9.

The invariant density $f_{n,k}$ of $P_{\tau_{n}}^{(k)}$ is non-increasing for
any $n,k>1$.

###### Proof.

Let $f\in L^{(k)}$. Since $P_{\tau_{n}}^{(k)}f=Q^{(k)}P_{\tau_{n}}f$, and both
operators $P_{\tau_{n}}$ and $Q^{(k)}$ transform non-increasing functions into
non-increasing functions, the operators $P_{\tau_{n}}^{(k)}$ also have this
property. Let $f=1$ be a constant function understood as $[1,1,\dots,1]\in
L^{(k)}$. It is non-increasing. Thus, all the functions
$(P_{\tau_{n}}^{(k)})^{m}f$, $m=1,2,\dots$, are non-increasing. Similarly as
the estimate (3.5) was obtained, they can be shown to be uniformly bounded in
$L^{\infty}$ and thus weakly compact in $L^{1}$. Then, Yosida-Kakutani theorem
[Yosida & Kakutani(1941)] shows that the sequence
$1/s\sum_{m=1}^{s}(P_{\tau_{n}}^{(k)})^{m}f$ converges in $L^{1}$ to the
invariant density $f_{n,k}$. By Lemma 8 the convergence is also a.e. and
$f_{n,k}$ is non-increasing. ∎

Using Ulam’s method and corresponding convergence analysis described in
[Li(1976), Miller(1994), Ding(1996)], the following theorem can be proved.

######  Theorem 4.

Let $\tau\in\mathcal{T}_{pc}^{\infty,0}(I)$ be a piecewise convex map with
countably many branches. Let $\\{\tau_{n}\\}_{n=1}^{\infty}$ be the
approximating sequence of piecewise convex maps with a finite number of
branches where $\tau_{n}$ are defined in Equation (3.3) in Subsection 3.1. If
$f_{n,k}$ is a normalized fixed point of $P_{\tau_{n}}^{(k)},k=1,2,\dots,$
defined in (3.7), then the sequence $\\{f_{n,k}\\}_{k=1}^{\infty}$ is weakly
pre-compact in $L^{1}.$ Any limit point $f_{n}^{*}$ of the sequence
$\\{f_{n,k}\\}_{k=1}^{\infty}$ is a fixed point of $P_{\tau_{n}}.$

###### Proof.

Let $P_{\tau_{n}}^{(k)}$ be the Ulam’s approximation of the F-P operator
$P_{\tau_{n}}$ of $\tau_{n}.$ Let $Q^{(k)}$ be the isometric projection
defined in (3.6). It can be shown that (see: (4), page 3 [Miller(1994)]; def.
(2.1), page 5 [Li(1976)]):

(3.8) $P_{\tau_{n}}^{(k)}Q^{(k)}f=Q^{(k)}P_{\tau_{n}}Q^{(k)}f.$

By the definition of $Q^{(k)}$ it is obvious that

(3.9) $\parallel Q^{(k)}f\parallel_{\infty}\leq\parallel f\parallel_{\infty}.$

By equations (2.3), (3.8) and (3.9) $P_{\tau_{n}^{(k)}}$ satisfies the
following Lasota-Yorke type inequality

(3.10) $\parallel
P_{\tau_{n}}^{(k)}f\parallel_{\infty}\leq\left(a_{n}+D_{1}\right)\parallel
f\parallel_{\infty}+D\parallel f\parallel_{1},$

which implies the following uniform estimate

(3.11) $\parallel
f_{n,k}\parallel_{\infty}\leq\frac{D}{1-\left(a_{n}+D_{1}\right)}.$

Now, the proof continues exactly as in [Miller(1994)]. ∎

######  Theorem 5.

Let $\tau\in\mathcal{T}_{pc}^{\infty,0}(I)$ be a piecewise convex map with
countably many branches. As described at the beginning of Subsection 3.1, let
$\\{\tau_{n}\\}_{n=1}^{\infty}$ be the approximating sequence of piecewise
convex maps with finite numbers of branches. Let $P_{\tau_{n}}^{(k)}$ ,
$k=1,2,\dots$ be the sequence of Ulam’s operators approximating the operators
$P_{\tau_{n}}$. Let $f_{n,k}$ be the normalized (in $L^{1}$) fixed point of
$P_{\tau_{n}}^{(k)}$. Then, the family
$\\{f_{n,k}\\}_{n=1,2,\dots,k=1,2,\dots}$ is weakly compact in $L^{1}$ and
uniformly bounded in $L^{\infty}$. If $f_{n_{j},k_{j}}$, $j=1,2,\dots$ is a
weakly convergent subsequence, then it converges in $L^{1}$ (and almost
everywhere) to a function $f$ which is a fixed point of $P_{\tau}$,
$P_{\tau}f=f$.

###### Proof.

The stationary densities $\\{f_{n}\\}_{n\geq 1}$ of $\\{\tau_{n}\\}_{n\geq 1}$
are uniformly bounded in $L^{\infty}$ by Theorem 3. Moreover, the densities
$f_{n,k}$, the piecewise constant approximations of $f_{n}$’s are also
uniformly bounded in $L^{\infty}$ by formula (3.11). Thus, the set
$\\{f_{n,k}\\}_{k=1}^{\infty}$ is weakly compact in $L^{1}$. Assume that
$\\{f_{n,k}\\}_{k=1}^{\infty}$ has weakly convergent subsequence
$\\{f_{n_{j},k_{j}}\\}$ with limit $f.$ Since the functions
$\\{f_{n_{j},k_{j}}\\}$ are decreasing and uniformly bounded, by Lemma 8 they
converge to $f$ a.e. and in $L^{1}$.

It remains to show that $f$ is a fixed point of $P_{\tau}$, $P_{\tau}f=f.$ We
will show that the measures $fd\lambda$ and $(P_{\tau}f)d\lambda$ are equal.
It is enough to show that for any $g\in C(I)$, we have $\int
g(f-P_{\tau}f)d\lambda=0.$ To simplify the notation we assume that the whole
sequence $f_{n,k}$ converges to $f$. We have,

$\displaystyle\left|{\int g(f-P_{\tau}f)d\lambda}\right|\leq\left|{\int
g(f-f_{n,k})d\lambda}\right|+\left|{\int
g(f_{n,k}-P_{\tau_{n}}^{(k)}f_{n,k})d\lambda}\right|$
$\displaystyle+\left|{\int
g(P_{\tau_{n}}^{(k)}f_{n,k}-P_{\tau_{n}}f_{n,k})d\lambda}\right|+\left|{\int
g(P_{\tau_{n}}f_{n,k}-P_{\tau_{n}}f)d\lambda}\right|+\left|{\int
g(P_{\tau_{n}}f-P_{\tau}f)d\lambda}\right|.$

Since $f_{n,k}\to f$ in $L^{1},$ the first and the fourth term go to $0$ as
$n,k\to\infty.$ Since $f_{n,k}$ are the fixed points of $P_{\tau_{n}}^{(k)},$
the second term is $0.$ We have

$\displaystyle{\int
g\left[P_{\tau_{n}}^{(k)}f_{n,k}-P_{\tau_{n}}f_{n,k}\right]d\lambda}={\int
g\left[Q^{(k)}(P_{\tau_{n}}f_{n,k})-P_{\tau_{n}}f_{n,k}\right]d\lambda}=\int\left[Q^{(k)}g-g\right]P_{\tau_{n}}f_{n,k}d\lambda,$

by the self adjointness of $Q^{(k)}$. Since $Q^{(k)}g\to g$ in $L^{1}$ and the
densities $\\{f_{n,k}\\}$ are uniformly bounded, $\int
g\left[P_{\tau_{n}}^{(k)}f_{n,k}-P_{\tau_{n}}f_{n,k}\right]d\lambda$ converges
to 0 as $k\to+\infty$. By the property (vi) of the F-P operator we have

$\displaystyle\left|{\int
g(P_{\tau_{n}}f-P_{\tau}f)d\lambda}\right|=\left|{\int(g\circ\tau_{n}-g\circ\tau)fd\lambda}\right|=\left|{\int_{[0,a_{n}]}(g\circ\tau_{n}-g\circ\tau)fd\lambda}\right|\leq
a_{n}\cdot 2M\cdot\|f\|_{1},$

where $M=\sup|g|$. This shows that$\left|{\int
g(P_{\tau_{n}}f-P_{\tau}f)d\lambda}\right|$ converges to 0 as $n\to+\infty$. ∎

##  4\. Examples

######  Example 1.

Consider the piecewise expanding and piecewise linear map
$T:[0,1]\rightarrow[0,1]$ with countable number of branches defined as

(4.1) $T(x)=i(i+1)\left(x-\frac{1}{i+1}\right)\hskip 8.53581pt\ \text{ on}\
\hskip 5.69054pt\left[\frac{1}{i+1},\frac{1}{i}\right],i=1,2,\cdots.$

It is easy to show that the Lebesgue measure is invariant under $T$ and
$f=\mathbb{1}$ is the invariant density of $T$. Now, consider the conjugation
$h:[0,1]\to[0,1]$ defined by $h(x)=1-(1-x)^{2}.$ We construct the piecewise
convex map $\tau:[0,1]\to[0,1]$ with countable number of branches defined by
$\tau=h^{-1}\circ T\circ h.$ See Figure 1 for a graph of $\tau.$

Figure 1. The graph of the piecewise convex map $\tau$ with countably infinite
number of branches for Example 1.

The piecewise convex map $\tau$ is topologically conjugate to the piecewise
linear and piecewise expanding map $T$ via the conjugation $h$. Therefore, the
stationary density $g$ of $\tau$ is given by $g=f\circ h\times\lvert
h^{\prime}\rvert.$ Now, $h(x)=1-(1-x)^{2}.$ Hence, $g(x)=f(h(x))\times\lvert
h^{\prime}(x)\rvert=\lvert 2(1-x)\rvert.$

The map $\tau$ is defined on the partition

$\left\\{\left[1-\sqrt{\frac{i}{i+1}},1-\sqrt{\frac{i-1}{i}}\right]\right\\}_{i=1}^{\infty},$

i.e., the partition points are

$\left\\{\dots,1-\sqrt{\frac{4}{5}},1-\sqrt{\frac{3}{4}},1-\sqrt{\frac{2}{3}},1-\sqrt{\frac{1}{2}},1\right\\}.$

We have

$\tau(x)=1-\sqrt{1-i^{2}+i(i+1)(1-x)^{2}}\ \text{ for }\
x\in\left[1-\sqrt{\frac{i}{i+1}},1-\sqrt{\frac{i-1}{i}}\right]\ ,\
i=1,2,\dots$

The first few, starting from 1 to the left, branches of $\tau$ are

$\displaystyle\tau_{1}(x)$ $\displaystyle=1-\sqrt{2}(1-x)\ $
$\displaystyle\text{for}\ 1-\sqrt{1/2}\leq x\leq 1;$
$\displaystyle\tau_{2}(x)$ $\displaystyle=1-\sqrt{6(1-x)^{2}-3}\ $
$\displaystyle\text{for}\ 1-\sqrt{2/3}\leq x<1-\sqrt{1/2};$
$\displaystyle\tau_{3}(x)$ $\displaystyle=1-\sqrt{12(1-x)^{2}-8}\ $
$\displaystyle\text{for}\ 1-\sqrt{3/4}\leq x<1-\sqrt{2/3};$
$\displaystyle\tau_{4}(x)$ $\displaystyle=1-\sqrt{20(1-x)^{2}-15}\ $
$\displaystyle\text{for}\ 1-\sqrt{4/5}\leq x<1-\sqrt{3/4}.$

Now, consider the following sequence $\\{\tau_{n}\\}_{n\geq 0}$ of piecewise
convex maps $\tau_{n}:[0,1]\to[0,1]$ with a finite number of branches:

$\tau_{n}(x)=\begin{cases}\frac{1}{1-\sqrt{\frac{n}{n+1}}}x\ ,\ 0\leq
x<1-\sqrt{\frac{n}{n+1}};\\\ \tau(x)\ ,\ 0\leq x<1-\sqrt{\frac{n}{n+1}}\leq
x\leq 1.\end{cases}$

See Figure 2 for a graph of $\tau_{n}$ with $n=5.$

Figure 2.  The graph of the piecewise convex map $\tau_{n}$ with a finite
number of branches ($n=8$).

The sequence $\\{\tau_{n}\\}_{n\geq 0}$ of piecewise convex map
$\tau_{n}:[0,1]\to[0,1]$ with finite number of branches converges almost
uniformly to $\tau$ with the countable number of branches.

In Figure 3, we present the graph of the actual density $f^{*}$ (in red) of
the piecewise convex map $\tau$ with countable number of branches and graphs
of the approximate stationary densities $f_{n,k}$ (in blue) via Ulam’s method
for maps $\tau_{n}$ with a finite number of branches. Numerical computations
are performed for a number of cases. In the following table, we present the
$L^{1}$ norm error $\parallel f^{*}-f_{n,k}\parallel_{1}.$ Note that for each
$n$, $\tau_{n}$ is a map with a finite number of branches which approximates
the piecewise convex map $\tau$ in Figure 1 with countable number of branches.

Figure 3.  The graph of the actual invariant density $f^{*}$ of the piecewise
convex map $\tau$ with infinite number of branches (in red) and the graph of
the approximating density $f_{n,k}$ (in blue): $n=5,k=1000$ on the left and
$n=10,k=1000$ on the right hand side.

$n$ | $k$ | $\parallel f^{*}-f_{n,k}\parallel_{1}$
---|---|---
$5$ |  1$00$ | $0.2195470623$
$5$ | $1000$ | $0.2195243505$
$6$ | 100 | $0.1943541673$
$6$ | 1000 | $0.1943541673$
$7$ | 1000 | $0.1742407352$
$10$ | 1000 | $0.133493040$
$12$ | 1000 | $0.11551923$

.

The above table shows that as we increase $n$, the $L^{1}$ norm error
$\parallel f^{*}-f_{n,k}\parallel_{1}$ gets smaller. For the fixed $n$, the
increasing of $k$ is not effective. For example, the errors for $k=100$ and
$k=1000$ are almost the same. The main method to lower the error is to
increase the number of branches of $\tau_{n}$.

By Theorem 5, $L^{1}$ norm error can be made arbitrarily small by making $n$
and $k$ large enough.

######  Example 2.

Consider the piecewise convex map $\tau:[0,1]\rightarrow[0,1]$ with countable
number of branches defined as

(4.2) $\tau(x)=\frac{1}{\frac{2i+1}{i(i+1)}-x}-i\hskip
8.53581pt\text{on}\hskip 5.69054pt\left[\frac{1}{i+1},\frac{1}{i}\right],\
i=1,2,\cdots.$

See Figure 4. The map $\tau$ is defined on the countable partition of $[0,1]$,
with partition points
$a_{0}=1,a_{1}=\frac{1}{2},a_{2}=\frac{1}{3},\cdots,a_{n}=\frac{1}{n+1},\cdots.$

Figure 4. The graph of the piecewise convex map with countable number of
branches for Example 2.

It is shown in [Góra et al.(2022)] that
$\tau\in\mathcal{T}_{pc}^{\infty,0}(I)$ and hence by Theorem 2, $\tau$ has an
ACIM. The actual density of $\tau$ is not known. The sequence
$\\{\tau_{n}\\}_{n\geq 0}$ of piecewise convex maps $\tau_{n}:[0,1]\to[0,1]$
with finite number of branches, see Figure 5 for graphs of $\tau_{n}$ with
$n=8$ and $n=10$, converges almost uniformly to $\tau,$ where

$\tau_{n}(x)=\begin{cases}(n+1)x\ ,\ 0\leq x<\frac{1}{n+1};\\\ \tau(x)\ ,\
\frac{1}{n+1}\leq x\leq 1.\end{cases}$

Figure 5. The graphs of the piecewise convex maps $\tau_{n}$ with finite
number of branches $n=8$ and $n=10$.

In Figure 6, we present the graph of the Ulam’s approximation
$f_{n,k},n=10,k=1000$ of the actual invariant density of $f_{n},n=10$, of the
piecewise convex map $\tau_{n},n=10$, with finite number of branches. By
Theorem 5 it is also an approximation of the invariant density $f^{*}$ of the
map $\tau$. The same Figure 6 shows also the approximation
$f_{n,k},n=10,k=500$, as the densities $f_{10,1000}$ and $f_{10,500}$ are
indistinguishable at this scale. We have $\|f_{10,1000}-f_{10,500}\|_{1}\sim
0.00055$. In Figure 7 we show the enlargement of both graphs on a small
subinterval. The scales on $x$, $y$ axes are different.

Figure 6. The graph of the Ulam’s approximation $f_{n,k},n=10,k=1000$.

Similarly as in Example 1, the effect of increasing $k$ (after some threshold)
is negligible. The improvement of the approximation is achieved by increasing
the parameter $n$. The density $f_{11,1000}$ is indistinguishable from
$f_{10,1000}$ at the paper illustrations scale with
$\|f_{10,1000}-f_{11,1000}\|_{1}\sim 0.00035$. The error
$\|f_{11,1000}-f_{12,1000}\|_{1}\sim 0.00029.$

Figure 7. Enlargement of the approximating densities $f_{10,1000}$ (in red)
and $f_{10,500}$ (in blue) on $[0,0.02].$

Acknowledgment: The authors are very grateful to anonymous reviewers for very
detailed and useful comments and suggestions and also for stating a new
problem related with their note. The research was supported by the NSERC
Discovery Grant (DG) of the first author and second author. The third author
is grateful to Concordia University and University of Prince Edward Island for
financial support via NSERC DG grants of the first author and the second
author respectively.

## References

  * [Bollt & Santitissadeekorn(2013)] Bollt, E. M. & Santitissadeekorn, N. [2013], “Applied and computational measurable dynamics,” Math. Model. Comput., 18, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  * [Bose et al.(2002)] Bose, C., Maume-Deschamps, V., Schmitt, B. & Shinand, S. S. [2002], “Invariant measures for piecewise convex transformations of an interval,” Studia Mathematica, 152, 263–297.
  * [Bose & Murray(2001)] Bose, C. & Murray, R. [2001], “The exact rate of approximation in Ulam’s method,” Discr. Contin. Dyn. Syst. 7, 219–235.
  * [Boyarsky & Góra(1997)] Boyarsky, A. & Góra, P. [1997], Laws of chaos. Invariant measures and dynamical systems in one dimension, Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA.
  * [Boyarsky & Lou(1991)] Boyarsky, A. & Lou, Y.-S. [1991], “Approximating measures invariant under higher-dimensional chaotic transformations,” J. Approx. Theory 65, no. 2, 231– 244.
  * [Ding(1996)] Ding, J. [1996], “Computing Invariant Measures of Piecewise Convex Transformations,” Journal of Statistical Physics, 83 (3 – 4), 623 – 635.
  * [Ding & Zhou(1996)] Ding, J. & Zhou, A. [1996], “Finite approximations of Frobenius-Perron operators. A solution of Ulam’s conjecture to multi-dimensional transformations,” Phys. D 92, no. 1-2, 61–68.
  * [Froyland(1999)] Froyland, G. [1999], “Ulam’s method for random interval maps,” Nonlinearity 12, 1029–1052.
  * [Góra & Boyarsky(1989)] Góra, P. & Boyarsky, A. [1989], “Approximating the invariant densities of transformations with infinitely many pieces on the interval,” Proc. Amer. Math. Soc. 105, no. 4, 922–928.
  * [Góra et al.(2022)] Góra, P., Islam, M. S. & Rahman, AHM. M. [2022], “Absolutely continuous invariant measures for piecewise convex maps with countable number of branches,” Dyn. of Contin. Disc. and Impul. Systems Series A (under review, submitted on Nov. 10, 2022).
  * [Hunt & Miller(1992) ] Hunt, F. Y. & Miller, W. M. [1992],“On the approximation of invariant measures,” J. Statist. Phys. 66, no.1–2, 535–548.
  * [Inoue(1992)] Inoue, T. [1992], “Asymptotic stability of densities for piecewise convex maps,” Ann. Polon. Math. 57, no. 1, 83–90.
  * [Inoue(2023)] Inoue, T. [2023], “Invariant measures for random piecewise convex maps,” https://arxiv.org/abs/2303.10873.
  * [Jabłoński & Malczak(1984)] Jabłoński, M.& Malczak, J. [1984], “A central limit theorem for piecewise convex transformations of the unit interval,” Colloq. Math. 49, no. 1, 59–65.
  * [Jabłoński & Malczak(1983)] Jabłoński, M. & Malczak, J. [1983], “The rate of the convergence of iterates of the Frobenius-Perron operator for piecewise convex transformations,” Bull. Polish Acad. Sci. Math. 31, no. 5-8, 249–254.
  * [Lasota & Yorke(1973)] Lasota, A. & Yorke, J.A. [1973], “On the existence of invariant measures for piecewise monotonic transformations,” Trans. Amer. Math. Soc. 186 481 – 488.
  * [Lasota & Mackey(1994)] Lasota A. & Mackey, M. C. [1994], Chaos, fractals, and noise, Stochastic aspects of dynamics, Applied Mathematical Sciences 97, Springer-Verlag, New York.
  * [Lasota & Yorke(1982)] Lasota, A. & Yorke, J. A. [1982], “Exact dynamical systems and the Frobenius-Perron operator,” Trans. Amer. Math. Soc. 273 (1), pp. 375–384.
  * [Li(1976)] Li, T-Y. [1976], “Finite approximation for the Frobenius-Perron operator: A solution to Ulam’s conjecture,” J. Approx. Theory. 17, 177 – 186.
  * [Miller(1994)] Miller, W. M. [1994], “Stability and approximation of invariant measures for a class of non-expanding transformations,” Nonlinear Anal. 23 (8), 1013 – 1025.
  * [Rahman(2019)] Rahman, M. H. [2019], “Absolutely Continuous Invariant Measures for Piecewise Convex Maps of Interval with Infinite Number of Branches,” Master’s Thesis, Concordia University, Canada.
  * [Rudin(1976)] Rudin, Walter [1976], Principles of mathematical analysis, Third edition, Internat. Ser. Pure Appl. Math.,McGraw-Hill Book Co., New York-Auckland-Düsseldorf.
  * [Ulam(1960)] Ulam, S. M. [1960], “A collection of Mathematical problems Interscience Tracts in pure and Applied Math.,” 8, Interscience, New York, USA.
  * [Yosida & Kakutani(1941)] Yosida, K. & Kakutani, S. [1941], “Operator-Theoretical Treatment of Markoff’s Process and Mean Ergodic Theorem,” Annals of Mathematics, Second Series, Vol. 42, No. 1 , pp. 188–228.


# Half-metallicity and giant magneto-optical Kerr effect in N-doped NaTaO3

Y. Saeed1, N. Singh1,2, and U. Schwingenschlgl1 1Physical Science &
Engineering division, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia
2Solar and Photovoltaic Energy Research Center, KAUST, Thuwal 23955-6900,
Kingdom of Saudi Arabia

###### Abstract

We employ density functional theory using the modified Becke-Johnson (mBJ)
approach to investigate the electronic and magneto-optical properties of
N-doped NaTaO3. The mBJ results reveal a half metallic nature of NaTaO2N, in
contrast to results obtained by the generalized gradient approximation. We
find a giant polar Kerr rotation of 2.16∘ at 725 nm wave length (visible
region), which is high as compared to other half metallic perovskites as well
as to the prototypical half metal PtMnSb.

Density Functional Theory, MBJ, Ferromagnetic Half Metal, Magneto-optical
properties

##  I Introduction

Room temperature ferromagnetism has been reported for different doped oxides
such as C/N-doped ZnO Pan ; Shen ; yang , TiO2 YangDai-apl ; YangDai-cpl ; Tao
, SnO2 Rahman ; Xiao  and confirmed recently Nagare-zno ; Bao-tio2 ; Hong-sio2
. Room temperature ferromagnetism with half metallicity is reported for
N-doped SrTiO3 and BaTiO3 Liu-sto-n ; Tan-bto-n , in which the magnetic
interactions between the nearest and next-nearest N dopants result in a strong
ferromagnetic coupling Yang-2011-sto-n-ferro . Ferromagnetic half-metals have
potential applications in spintronics devices Engen ; Groot  and also show
unusual magneto-optical effects due to a metallic state for one spin channel
and an insulating state for the other. Yang _et al_. have reported that a high
concentration of N is required for achieving a magnetic long rang order in
perovskite oxide Yang-2011-sto-n-ferro .

The perovskite oxide NaTaO3 (NTO) is a ferroelectric material with high
permittivity and low dielectric loss, which suggests usage in microwave
devices Rabe ; Geyer ; Axelsson . Several ab-initio calculations have been
performed to described the electronic properties of bulk NTO Choi-2011  but a
detailed study of the electronic structure and magneto-optical properties of
N-doped NTO is missing in the literature. The magneto-optical Kerr effect of
doped NTO is interesting for magneto-optical reading and recording devices
Fiebig . The N-doped perovskite oxide NTO is a 5d system. Therefore, electron
correlation effects are expected to be small as compared to 3d systems such as
SrTiO3 and BaTiO3. In the following we establish a half metallic nature for
NaTaO1-xNx ($x=0.04-0.33$) and discuss the electronic structure in comparison
to the strongly correlated perovskites SrTiO3 and BaTiO3. We also address the
magneto-optical Kerr effect in N-doped NTO.

##  II computational method

Our calculations are based on density functional theory, using the full-
potential linearized augmented plane wave approach as implemented in the
WIEN2k code wien2k . We use the modified Becke-Johnson (mBJ) exchange
correlation potential mBJ . The popular generalized gradient approximation
(GGA) GGA-PBE  is employed to optimize the volume and the internal atomic
coordinates. In general, the unit cell is divided into non-overlapping atomic
spheres centered at the atomic sites and an interstitial region. The
convergence parameter RmtKmax, where Kmax is the plane-wave cut-off and Rmt is
the smallest muffin-tin radius, controls the size of the basis set. This
convergence parameter is set to 7 together with G${}_{max}=24$. We use 66
k-points in the irreducible wedge of the Brillouin zone for calculating the
electronic structure and a dense mesh of 480 k-points in the magneto-optical
calculations. The cubic phase _Pm $\overline{3}$m_ ($a=b=c=3.93$ Å and
$\alpha=\beta=\gamma=90^{\circ}$) of NTO cubic-nto  is used in the present
calculations for simplicity because the differences to the monoclinic phase
$P2/m$ ($a=3.8995$ Å, $b=3.8965$ Å, and $c=3.8995$ Å,
$\alpha=\gamma=90^{\circ}$, and $\beta=90.15^{\circ}$) are subordinate
monoclinc-nto . As a consequence, the electronic band structures and density
of states (DOS) are found to be very similar in both phases wang-JPCC2011 ;
Lin-APL-phases .

Figure 1: Calculated spin polarization and majority spin hole density _n_ as a
function of N-concentration, as obtained by the GGA+SOC approximation.

##  III Results and Discussion

Our optimized lattice parameter (using the GGA) of cubic NTO is 3.98 Å, which
is in good agreement with the experimental value of 3.93 Å cubic-nto . We
replace one O with one N to form the oxynitrate (NaTaO2N). The optimized
lattice parameters of NaTaO2N is slightly increased to 4.03 Å. In order to
find the magnetic ground state, we construct a $1\times 1\times 2$ supercell
using the optimized structure and replace two O atoms with N. We compare the
ground state energies of ferromagnetic (FM) and anti-ferromagnetic (AFM)
configurations. The magnetic energy $E_{mag}=E_{FM}-E_{AFM}=-51.3$ meV, and
the N-N distance is $\sim$4 Å with a total magnetic moment of 2 $\mu_{B}$ per
cell (or 1 $\mu_{B}$ per N atom) in FM case. The Curie temperature $T_{C}$ is
calculated using the mean-field Heisenberg model, i.e.,
$T_{C}=(2/3)E_{mag}/k_{B}$ Kudrnovsk ; Maca . The calculated $T_{C}$ for
NaTaO2N is 396 K, which is close to that of N-doped SrTiO3 and BaTiO3 at the
same N-N distance Yang-2011-sto-n-ferro . In order to observe the long range
FM order, we study a high N-doping of 33% by replacing one O by one N in a
single unit cell. A magnetic moment is induced as the delocalized N _p_ states
become polarized, where 0.15 $\mu_{B}$ come from the interstitial, 0.13
$\mu_{B}$ come from O and 0.61 $\mu_{B}$ come from N, summing upto 1 $\mu_{B}$
per N atom.

To explain the induced spin-polarization in NaTaO2N, we analyzes DOS and
electronic band structure obtained by GGA approximation (not shown here). The
DOS shows a half-metallic character with a metallic state for the minority
spin and an insulating state for the majority spin. To confirm the half-
metallicity, we include spin orbit coupling (SOC) along with GGA in the
calculations, finding that NaTaO2N becomes a metal since the majority spin
states crosses the Fermi level. The spin polarization
($=\frac{N\uparrow-N\downarrow}{N\uparrow+N\downarrow}$, where N is the number
of states at the Fermi level) of NaTaO2N is obtained $\sim$94%. In order to
find the exact N-concentration at which the character of the system changes
from a half-metal to metal, we construct a $3\times 3\times 3$ supercell and
vary the N-concentration from 4% to 33% (including SOC in the calculations).
In Fig. 1(a), we plot the spin-polarization as a function of the
N-concentration. Below 16% N-doped, NTO shows a $\sim$99.8% spin-polarization
which decreases sharply to $\sim 94\%$ at 33% N-doping. In Fig. 1(b), we plot
the hole density (holes per volume) for the majority spin channel. Similar to
the spin-polarization, the hole density increases rapidly upto 46.8$\times
10^{24}$ m-3 as the N-concentration increases to 33%, while the hole density
is almost constant for low N-concentration.

Figure 2: Band structure and DOS of NaTaO2N as obtained by the mBJ
approximation.

Recently, Guo _et al._Guo-mBJLDA 2011  have applied the mBJ approach
successfully to improve the half-metallic ferromagnetism in zincblende MnAs,
which turns into a half-metal without affecting the _d_ _t_ 2g bands. We apply
the same method to NaTaO2N. The calculated band structure and DOS in Fig. 2
show a truly half-metallic nature for NaTaO2N. The majority spin bands are
similar to pristine NTO with a gap of 3.96 eV, which is in excellent agreement
with experiments Lin-APL-phases  and the previous GW calculations wang-
JPCC2011 . The minority spin channel is metallic due to a non-zero DOS at the
Fermi level. The band splitting at the Fermi level along R-$\Gamma$ and
M-$\Gamma$ is very small, while along $\Gamma$-X-M, it is large. The
calculated plasma frequency $\omega_{p}$ from the minority spin channel due to
metallic nature, is 2.7 eV, which is smaller in the ferromagnetic half-metal
PtMnSb ($\omega_{p}=4.5$ eV) Picozzi , reflecting less dispersed bands. The
calculated DOS shows that the valence bands (majority spin) are a combination
of N 2 _p_ and O 2 _p_ states. The bottom of the conduction bands is composed
of Ta 5 _d_ states (see Fig. 2). For the minority spin channel, the N 2 _p_
bands cross the Fermi level (with small O 2 _p_ contributions). The N
2$p^{\uparrow\downarrow}$ states split into
$(p_{x}+p_{y})^{\uparrow\downarrow}$ and $p_{z}^{\uparrow\downarrow}$ bands.
There is no shifting of peak position with respect to energy is observed at
the Fermi level in N 2$(p_{x}+p_{y})^{\uparrow\downarrow}$ and N
2$p_{z}^{\uparrow\downarrow}$ states from the N-doped SrTiO3 and BaTiO3
Yang-2011-sto-n-ferro  where N 2$p_{y}+p_{z}$ and N 2$p_{x}$ have different
peak position at the Fermi level. There is a strong hybridization between the
N 2$p^{\uparrow\downarrow}$ and O 2$p^{\uparrow\downarrow}$ states for the
minority spin channel. The Ta 5 _$d^{\uparrow\downarrow}$_ bands do not change
with the N-concentration.

Figure 3: Calculated polar Kerr angle $\theta_{K}$ and Kerr ellipticity
$\varepsilon_{K}$ of NaTaO2N.

Intense search aim at materials with large magneto-optical peaks in the low
wave-length region to be used for high-density storage Grundy-high disk . Both
borates borates  and Zintl compounds Zintl ; Zintl1 , can shows a remarkable
Kerr signal in the low energy range. The Kerr rotation $\theta_{K}$ and Kerr
ellipticity $\varepsilon_{K}$ of half metallic NaTaO2N are shown in Fig. 3. We
find a value of $\theta_{K}=2.16^{\circ}$ at 1.71 eV ($\sim$725 nm), which is
higher than in BiNiO3 ($\theta_{K}=1.28^{\circ}$) M. Q. Cai-BiNiO3  and the
Heusler compound PtMnSb ($\theta_{K}=1.27^{\circ}$) van Engen-1983 ;
Lobove-2012-ptmnsb . The high Kerr angle is an intraband effect, and not due
to the SOC (which creates an imbalance in the optical transitions in PtMnSb
and NiMnSb van Ek-1997 , for example. For the minority spin channel, the band
structure of NaTaO2N shows a set of parallel bands across the Fermi level
(R-$\Gamma$, $\Gamma$-X-M, and $\Gamma$-M) which consist of N
2$(p_{x}+p_{y})^{\downarrow}$ states. These parallel bands give rise to
intraband transitions which contribute significantly to the Kerr spectrum in
the low energy range. In NaTaO2N, the separation between these bands is much
smaller than in PtMnSb van Ek-1997 . This past explain the higher magneto-
optical Kerr effect in NaTaO2N. The calculated Kerr ellipticity
$\varepsilon_{K}$ has a maximum of $\sim$1.7∘ at 1.6 eV.

##  IV Conclusion

In conclusion, we have presented first principles results of the band
structure, DOS, and magneto-optical properties of N-doped NaTaO3, as obtained
from density functional theory. Our results for NaTaO1-xNx ($x=0.04-0.33$)
show that the GGA+SOC approach gives a 99% spin-polarization at low
N-concentrations upto 16%. The mBJ+SOC approach results in a pure
ferromagnetic half-metal in contrast to the GGA+SOC. We observe a giant
magneto-optical Kerr signal of $\theta_{K}$=2.16∘ at $\sim$725 nm in NaTaO2N,
which is the highest Kerr angle among the ferromagnetic half-metals in UV-
visible region. The origin of the high Kerr angle is attributed to intraband
transitions involving the N 2$(p_{x}+p_{y})^{\downarrow}$ orbital due to
parallel bands around the Fermi level. The large Kerr rotation in NaTaO2N in
the visible region may find applications in red/infrared laser magneto-optical
devices and the half metallic nature of NaTaO2N is interesting for spintronics
devices.

## References

  * (1) H. Pan, J. B. Yi, L. Shen, R. Q. Wu, J. H. Yang, J. Lin, Y. P. Feng, J. Ding, L.H. Van, and J. H. Yin, Phys. Rev. Lett. 99, 127201 (2007).
  * (2) L. Shen, R. Q. Wu, H. Pan, G. W. Peng, M. Yang, Z. D. Sha, and Y. P. Feng, Phys. Rev. B 78, 073306 (2008).
  * (3) K. Yang, R. Wu, L. Shen, Y. P. Feng, Y. Dai, and B. Huang, Phys. Rev. B 81, 125211 (2010).
  * (4) K. Yang, Y. Dai, B. Huang, and M.-H. Whangbo, Appl. Phys. Lett. 93, 132507 (2008).
  * (5) K. Yang, Y. Dai, B. Huang, and M.-H. Whangbo, Chem. Phys. Lett 481, 99 (2009).
  * (6) J. G. Tao, L. X. Guan, J. S. Pan, C. H. A. Huan, L.Wang, J. L. Kuo, Z. Zhang, J. W. Chai, and S. J. Wang, Appl. Phys. Lett. 95, 062505 (2009).
  * (7) G. Rahman and V. M. García-Suárez, Appl. Phys. Lett. 96, 052508 (2010).
  * (8) W.-Z. Xiao, L.-L. Wang, L. Xua, Q. Wan, and B. S. Zou, Solid State Commun. 149, 1304 (2009).
  * (9) B. J. Nagare, S. Chack, and D. G. Kanhere, J. Phys. Chem. A 114, 2689 (2010).
  * (10) N. N. Bao, H. M. Fan, J. Ding, and J. B. Yi, J. Appl. Phys. 109, 07C302 (2011).
  * (11) N. H. Hong, J.-H. Song, A. T. Raghavender, T. Asaeda, and M. Kurisu, Appl. Phys. Lett. 99, 052505 (2011).
  * (12) C. M. Liu, X. Xiang, and X. T. Zu, Chin. J. Phys. 47, 893 (2009).
  * (13) X. Tan, C. Chen, K. Jin, and B. Luo, J. Alloy. Compd. 509, L311 (2011).
  * (14) K. Yang, Y. Dai, and B. Huang, Appl. Phys. Lett. 100, 062409 (2012).
  * (15) P. G. van Engen, K. H. J. Buschow, and R. Jongebreur, Appl. Phys. Lett. 42, 202 (1982).
  * (16) R. A. de Groot, F. M. Mueller, P. G. van Engen, and K. H. J. Buschow, Phys. Rev. Lett. 50, 2024 (1983).
  * (17) K. Rabe, C. H. Ahn, and J.-M. Triscone, Physics of Ferroelectrics: A Modern Perspective, Topics in Applied Physics (Springer, Berlin, 2007), Vol. 105.
  * (18) R. G. Geyer, B. Riddle, J. Krupka, and L. A. Boatner, J. Appl. Phys. 97, 104111 (2005).
  * (19) A.-K. Axelsson, Y. Pan, M. Valant, and N. Alford, J. Am. Ceram. Soc. 92, 1773 (2009).
  * (20) M. Choi, F. Oba, and I. Tanaka, Phys. Rev. B 83, 214107 (2011).
  * (21) M. Fiebig, J. Phys. D: Appl. Phys. 38, R123 (2005).
  * (22) P. Blaha, K. Schwarz, G. Madsen, D. Kvasicka, and J. Luitz, WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties (TU Vienna, Vienna, 2001).
  * (23) F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009).
  * (24) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * (25) International Center for Diffraction Data, JCPDS Card No. 742488 (2001).
  * (26) International Center for Diffraction Data, JCPDS Card No. 742479 (2001).
  * (27) H. Wang, F. Wu, and H. Jiang, J. Phys. Chem. C 115, 16180 (2011).
  * (28) W. H. Lin, C. Cheng, C. C. Hu, and H. Teng, Appl. Phys. Lett. 89, 211904 (2006).
  * (29) J. Kudrnovský, I. Turek, V. Drchal, F. Máca, P. Weinberger, and P. Bruno, Phys. Rev. B 69, 115208 (2004).
  * (30) F. Máca, J. Kudrnovsḱ, V. Drchal, and G. Bouzerar, Appl. Phys. Lett. 92, 212503 (2008).
  * (31) S.-D. Guo and B.-G. Liu, Euro. Phys. Lett. 93, 47006 (2011).
  * (32) S. Picozzi, A. Continenza, and A. J. Freeman, J. Phys. D: Appl. Phys. 39, 851 (2006).
  * (33) P. J. Grundy, in Electronic and Magnetic Properties of Metals and Ceramics, Materials Science and Technology, edited by K. H. J. Buschow (VCH, 1994), Vol. 3B, p. 575.
  * (34) Y. Saeed, N. Singh, and U. Schwingenschlögl, J. Appl. Phys. 110, 103512 (2011).
  * (35) N. Singh and U. Schwingenschlögl, Chem. Phys. Lett. 508, 29 (2011).
  * (36) N. Singh and U. Schwingenschlögl, Appl. Phys. Lett. 100, 151906 (2012).
  * (37) M. Q. Cai, X. Tan, G. W. Yang, L. Q. Wen, L. L. Wang, W. Y. Hu, and Y. G. Wang, J. Phys. Chem. C 112, 16638 (2008).
  * (38) P. G. van Engen, K. H. J. Buschow, R. Jongebreur, and M. Erman, Appl. Phys. Lett. 42, 202 (1983).
  * (39) I. D. Lobov, A. A. Makhnev, and M. M. Kirillova, Phys. Met. Metallogr. 113, 135 (2012), and references therein.
  * (40) J. van Ek and J. M. Maclaren, Phys. Rev. B 56, R2924 (1997).