# A convex optimization approach to the Lyapunov exponents

Christoph Kawan111Email<EMAIL_ADDRESS>
###### Abstract

The aim of this paper is to shed more light on some recent ideas about
Lyapunov exponents and clarify the formal structures behind these ideas. In
particular, we show that the vector of (averaged) Lyapunov exponents of a
smooth measure-preserving dynamical system can be regarded as the solution of
a vector-valued optimization problem on a space of Riemannian metrics. This
result was first formulated and proved by Jairo Bochi in the language of
linear cocycles and their conjugacies. We go a step further and prove that the
optimization problem is geodesically convex with respect to the
$L^{2}$-metric. Moreover, we derive some consequences of this fact.

Keywords— Lyapunov exponents; Multiplicative Ergodic Theorem; Convex
optimization; Space of Riemannian metrics

##  1 Introduction

Originally introduced as stability indicators for ordinary differential
equations, Lyapunov exponents have become a cornerstone of the theory of
smooth dynamical systems. The main reason is that they are related in numerous
ways to other important characteristics of dynamical systems such as several
notions of entropy and dimensions of invariant sets as well as invariant
measures. The study of their properties and methods for their computation is
an ongoing endeavor. Already in dimension two, examples of dynamical systems
are known whose Lyapunov exponents up to this date cannot be effectively
estimated on large subsets of the state space. For instance, for the Chirikov
standard map, an area-preserving diffeomorphism of the $2$-torus, it is a
long-standing open problem to determine whether the maximal Lyapunov exponent
is positive on a set of positive Lebesgue measure, see e.g. [2].

Lyapunov’s first method for proving the asymptotic stability of a system
consists in showing that the maximal Lyapunov exponent is negative. His second
method uses a non-negative real-valued function on the state space with values
decreasing along trajectories. Once a candidate for such a _Lyapunov function_
has been found, the decrease condition can be checked by a simple computation
involving only the candidate function and the right-hand side of the equation
which defines the system. One benefit of this method thus consists in the
reduction of the proof of a long-term property to a short-term computation.

Now, asymptotic stability is a comparatively simple property. Other
properties, studied mainly in chaos theory, such as various forms of
hyperbolicity, can most likely not be checked by evaluating a single scalar-
valued function along trajectories. Since these properties are by nature
multi-dimensional as their definitions involve a distinction of different
state space directions, a structure which is itself multi-dimensional can help
to verify them via simple short-term computations. The main example of such a
structure is a Riemannian metric. Usually, a Riemannian metric which allows to
verify a long-term property by a short-term computation is called an _adapted
metric_. For instance, a classical result states that any uniformly hyperbolic
set of a discrete-time system admits an adapted metric (also called a
_Lyapunov metric_) in which contraction on the stable bundle and expansion on
the unstable bundle can be seen in one time step, see e.g. [18, Ch. 6]. An
analogous result for the more general property of supporting a dominated
splitting is proved in [17]. In the context of non-uniformly hyperbolic
systems, measurable adapted metrics can be constructed, see [19, p. 668].
Finally, in the context of stability analysis, besides Lyapunov functions,
also Riemannian metrics are used to prove contractivity properties (implying
asymptotic stability), which are known as _contraction metrics_ , see, e.g.,
the recent survey [14].

In [3, 4], Bochi and Navas introduced a novel method to construct Riemannian
metrics adapted to continuous linear cocycles over topological dynamical
systems. In this general context, they are not Riemannian metrics in the
actual sense (as there is no smooth manifold involved), but rather continuous
mappings assigning a positive-definite matrix to each point in the state
space. They can then be interpreted as conjugacies between cocycles. In the
special case of the derivative cocycle of a smooth map, they become Riemannian
metrics in the actual sense, which allow to approximate the average Lyapunov
vector (with respect to an invariant probability) by integration over the
vector of (log-) singular values of the right-hand side Jacobian. In this
paper, we provide a self-contained proof of Bochi’s result in this context. An
important interpretation of the result is that the averaged Lyapunov vector is
the infimum of a vector-valued function $\mathcal{F}$ defined on the space of
all continuous Riemannian metrics. Here, the infimum needs to be understood
with respect to the so-called majorization order. It is important to note that
it is not only a weak, i.e. a Pareto infimum, but an infimum in all
components. Hence, the averaged Lyapunov vector appears as the solution of a
vector-valued optimization problem.

We further prove that the optimization problem posed for $\mathcal{F}$ is
convex in a proper sense. First, one can restrict $\mathcal{F}$ to the space
$\mathcal{M}$ of _smooth_ Riemannian metrics which is a Fréchet manifold
modeled on the Fréchet space of smooth symmetric $(0,2)$-tensor fields. The
most-studied Riemannian metric on this manifold is the $L^{2}$-metric, which
is defined by integration over the fiberwise trace metric (the metric of the
symmetric space of positive-definite matrices). The $L^{2}$-metric on
$\mathcal{M}$ has been studied for a long time in different contexts; the main
references include [12, 13, 16, 9, 10, 7]. Though $\mathcal{M}$ is only a weak
Riemannian manifold, the central objects studied in Riemannian geometry
(including geodesics) can be shown to exist and explicit formulas for them are
available. An important submanifold of $\mathcal{M}$ is the space
$\mathcal{M}_{\omega}$ of metrics inducing the same (arbitrary) volume form
$\omega$. It was proved in [13] that $\mathcal{M}_{\omega}$ is a symmetric
space with the inherited $L^{2}$-metric; in particular, it is geodesically
complete. The function $\mathcal{F}$ has the property that its infimum is
preserved under restriction to $\mathcal{M}_{\omega}$. Hence, the search for a
(nearly) optimal metric can be restricted to $\mathcal{M}_{\omega}$. We prove
that the restriction of $\mathcal{F}$ to $\mathcal{M}_{\omega}$ is
geodesically convex with respect to the $L^{2}$-geodesics on its domain and
the majorization order on its co-domain. In fact, this proof was already given
in [20], but without putting it in the proper context and relating it to the
geometry of $\mathcal{M}$.

In summary, we prove that the averaged Lyapunov vector is the infimum of a
geodesically convex vector-valued function. The associated optimization
problem has some serious disadvantages, however. It is not guaranteed that a
minimizer exists, the problem is infinite-dimensional, defined on a Riemannian
manifold (rather than a vector space) and non-smooth. We elaborate on some of
these aspects and make suggestions how to mitigate the corresponding problems.

Several other works have proposed convex optimization to compute Lyapunov
exponents or associated quantities. In particular, we point the reader to the
paper [27], where SOS programming is used to find tight bounds on the maximal
Lyapunov exponent of a system given by an ordinary differential equation whose
right-hand side is polynomial. The optimization there is not over Riemannian
metrics, but over auxiliary real-valued functions. We also mention the papers
[15, 20, 24], in which the groundwork has been laid for the approach pursued
in this paper, and numerical algorithms were developed to use it for the
computation of contraction metrics [15], the estimation of Lyapunov dimension
[24] and the estimation of restoration entropy [20].

Organization of the paper: In Section 2, we provide the necessary background
on Lyapunov exponents. Section 3 is devoted to the proof of Bochi’s result on
the approximation of the Lyapunov vector by singular values of the time-one
Jacobian, see Theorem 2. In Section 4, we introduce the $L^{2}$-metric on the
space of Riemannian metrics and present some results on the structure of this
space. The subsequent Section 5 contains the convexity proof of the function
$\mathcal{F}$, Theorem 3. In Theorem 4, we also prove the global Lipschitz
continuity of the components of $\mathcal{F}$. In Section 6, we provide a
formula for the directional derivatives of the components of $\mathcal{F}$ in
an idealized case, where the existence of these derivatives is guaranteed, see
Theorem 5. The final Section 7 provides a summary and an outlook.

##  2 Preliminaries

###  2.1 Notation

By $\log(\cdot)$, we denote the natural logarithm. If $A$ is a square matrix,
we write $A^{\top}$ and $\mathrm{tr}(A)$ for the transpose and the trace of
$A$, respectively. If $x\in\mathbb{R}^{d}$, then $\|x\|_{p}$ denotes its
$p$-norm for $p\in\\{1,2,\infty\\}$. The components of $x$ will always be
denoted by $x_{1},\ldots,x_{d}$. By $\mathds{1}$, we denote the vector in
$\mathbb{R}^{d}$ whose components are all equal to $1$. We write
$\mathrm{GL}(d,\mathbb{R})$ for the general linear group of $\mathbb{R}^{d}$.
If $f:M\hookleftarrow$ is a differentiable self-map of a smooth manifold $M$,
we write $fx$ and $\mathrm{d}f_{x}$ for its value and its derivative at $x\in
M$, respectively. By $I$, we denote the identity matrix of any dimension. By a
_metric_ $g$ on a smooth manifold $M$, we always mean a Riemannian metric,
i.e. the assignment of an inner product $g_{x}$ to each tangent space
$T_{x}M$, which is at least continuous. We use the notation $\mathcal{M}^{0}$
($\mathcal{M}$) for the space of all continuous (smooth) Riemannian metrics on
a given manifold $M$.

###  2.2 Setup

We consider a discrete dynamical system given by a $C^{1}$-diffeomorphism
$f:M\hookleftarrow$ on a compact orientable222The orientability of $M$ is a
technical assumption which guarantees that we can work with volume forms.
smooth manifold $M$ of finite dimension $d$. We further assume that $f$
preserves a Borel probability measure $\mu$, i.e. $\mu(f^{-1}(A))=\mu(A)$ for
all Borel sets $A\subseteq M$.

###  2.3 Background on Lyapunov exponents

Our objects of interest are the Lyapunov exponents associated with the data
$(M,f,\mu)$. The existence of these numbers is guaranteed by the famous
_Multiplicative Ergodic Theorem (MET)_ , also known as _Oseledets Theorem_.
This theorem includes quite a number of statements and can be formulated on
different levels of generality. Here, we formulate a version that is adapted
to our setup and our needs. First, we define the _Lyapunov exponent_ at a
point $x\in M$ in direction $0\neq v\in T_{x}M$ as the limit

$\lambda(x,v):=\lim_{n\rightarrow\infty}\frac{1}{n}\log\|\mathrm{d}f^{n}_{x}v\|,$

provided that this limit exists. Here, $\|\cdot\|$ denotes the norm (on each
tangent space) induced by a given Riemannian metric. The value of
$\lambda(x,v)$, however, is independent of the chosen metric as a consequence
of compactness.

###### Theorem 1

(MET) In the given setup, for $\mu$-almost every $x\in M$ there exist numbers

$\lambda_{1}(x)>\lambda_{2}(x)>\cdots>\lambda_{r(x)}(x)$

with multiplicities $m_{1}(x),m_{2}(x),\ldots,m_{r(x)}(x)\in\mathbb{N}$,
respectively, such that the following statements hold:

  1. (i)

For every tangent vector $0\neq v\in T_{x}M$, $\lambda(x,v)=\lambda_{i}(x)$
for some $i$.

  2. (ii)

$\sum_{i}m_{i}(x)=d$.

  3. (iii)

$\sum_{i}m_{i}(x)\lambda_{i}(x)=\lim_{n\rightarrow\infty}(1/n)\log|\det\mathrm{d}f^{n}_{x}|$.

  4. (iv)

The functions $r(\cdot)$, $m_{i}(\cdot)$ and $\lambda_{i}(\cdot)$ are
$\mu$-measurable and $f$-invariant.

  5. (v)

Let
$\alpha_{1}(\mathrm{d}f^{n}_{x})\geq\alpha_{2}(\mathrm{d}f^{n}_{x})\geq\cdots\geq\alpha_{d}(\mathrm{d}f^{n}_{x})$
denote the singular values of the linear operator
$\mathrm{d}f^{n}_{x}:T_{x}M\rightarrow T_{f^{n}x}M$. Then the limits

$\lim_{n\rightarrow\infty}\frac{1}{n}\log\alpha_{i}(\mathrm{d}f^{n}_{x}),\quad
i=1,2,\ldots,d,$

exist and each of them equals one of the numbers
$\lambda_{1}(x),\ldots,\lambda_{r(x)}(x)$.

If $\mu$ is ergodic, then the functions $r(\cdot)$, $m_{i}(\cdot)$ and
$\lambda_{i}(\cdot)$ are constant almost everyhwere.

We base the analysis in this paper on statement (v) above, i.e. on the
relation between the Lyapunov exponents and the growth rates of singular
values. In contrast to eigenvalues, singular values are geometric quantities;
they depend on the inner products on domain and co-domain of the linear
operator in question. In our setup, this means that they depend on the
Riemannian metric imposed on $M$, while their exponential growth rates, the
Lyapunov exponents, do not. Hence, one can ask whether there exists a metric
adapted to the given system such that in this metric the Lyapunov exponents
can be computed (at least approximately) as the expansions rates in the first
iterate, i.e.333Here, we allow multiple repeated values in the enumeration of
the Lyapunov exponents.

$\lambda_{i}(x)\approx\log\alpha_{i}(\mathrm{d}f_{x}).$

The existence of such metrics follows from a result proved in [3, Prop. 4.1]
about linear cocycles over topological dynamical systems. In the language of
linear cocycles, switching to another metric corresponds to switching to a
conjugate cocycle. For the convenience of the reader, we provide a complete
proof of the result within the setup introduced above.

##  3 Bochi’s result

We first need to introduce some technical concepts and the associated
notation. Given an invertible linear operator $L:V\rightarrow W$ between
finite-dimensional inner product spaces of dimension $d$, we write

$\alpha_{1}(L)\geq\alpha_{2}(L)\geq\cdots\geq\alpha_{d}(L)$

for the singular values of $L$. Geometrically, these are the lengths of the
semi-axes of the ellipsoid $\mathcal{E}:=\\{Lv:\langle v,v\rangle_{V}=1\\}$
measured in the norm induced by the inner product on $W$. We then write

$\vec{\sigma}(L):=(\log\alpha_{1}(L),\log\alpha_{2}(L),\ldots,\log\alpha_{d}(L)).$

By our assumption that $L$ is invertible, the numbers $\alpha_{i}(L)$ are
strictly positive, and hence $\vec{\sigma}(L)\in\mathbb{R}^{d}$. We can be
more precise. Because of the ordering of the singular values, $\vec{\sigma}$
takes values in the closed convex cone444The notation $\mathfrak{a}^{+}$ is
adopted from the theory of semisimple Lie groups, where it is used for the so-
called positive Weyl chamber.

$\mathfrak{a}^{+}:=\\{\xi\in\mathbb{R}^{d}:\xi_{1}\geq\xi_{2}\geq\cdots\geq\xi_{d}\\}.$

A partial order on $\mathfrak{a}^{+}$, which is very useful for our purposes,
is defined by

$\xi\preceq\eta\quad:\Leftrightarrow\quad\left\\{\begin{array}[]{rl}\xi_{1}+\cdots+\xi_{k}\leq\eta_{1}+\cdots+\eta_{k}&\mbox{for
}k=1,\ldots,d-1,\\\ \xi_{1}+\cdots+\xi_{k}=\eta_{1}+\cdots+\eta_{k}&\mbox{for
}k=d.\end{array}\right.$

This order is called the _majorization order_. It can be shown that it is the
partial order induced by the dual cone of $\mathfrak{a}^{+}$. If we relax the
equality in the case $k=d$ to an inequality, we obtain the _weak majorization
order_ that we denote by $\preceq_{w}$. We refer to the book [26] for more
information about majorization.

In terms of the order $\preceq$, we can formulate _Horn’s inequality_ (see [5,
Prop. I.7.4.3]) equivalently as

$\vec{\sigma}(L_{1}L_{2})\preceq\vec{\sigma}(L_{1})+\vec{\sigma}(L_{2})$ (1)

for any linear operators $L_{1}$ and $L_{2}$ that can be composed in the
suggested way.

Now, consider again our dynamical system $(M,f)$. Given _any_ Riemannian
metric $g$ on $M$, the linear operator $\mathrm{d}f^{n}_{x}:T_{x}M\rightarrow
T_{f^{n}x}M$ has unique singular values with respect to the inner products
$g_{x}$ and $g_{f^{n}x}$. To highlight the dependence of these singular values
on the metric $g$, we write
$\vec{\sigma}^{g}(\mathrm{d}f^{n}_{x})=\vec{\sigma}(\mathrm{d}f^{n}_{x})$. By
the MET, for $\mu$-almost every $x\in M$, the _Lyapunov vector_

$\vec{\lambda}(x):=\lim_{n\rightarrow\infty}\frac{1}{n}\vec{\sigma}^{g}(\mathrm{d}f^{n}_{x})\in\mathfrak{a}^{+}$

exists as a limit and equals the corresponding vector of Lyapunov exponents:

$\vec{\lambda}(x)=(\underbrace{\lambda_{1}(x),\ldots,\lambda_{1}(x)}_{m_{1}(x)},\underbrace{\lambda_{2}(x),\ldots,\lambda_{2}(x)}_{m_{2}(x)},\ldots,\underbrace{\lambda_{r(x)}(x),\ldots,\lambda_{r(x)}(x)}_{m_{r(x)}}).$

Observe that the almost-everyhwere existence of $\vec{\lambda}(x)$ already
follows from Kingman’s subadditive ergodic theorem, the subadditivity being a
consequence of inequality (1). We also introduce the average over the Lyapunov
vector

$\vec{\lambda}(f):=\int\vec{\lambda}(x)\,\mathrm{d}\mu(x).$

Observe that in the ergodic case the integral equals the integrand almost
everywhere. As the Lyapunov exponents depend measurably on $x$, and the
compactness of $M$ together with the continuity of $\mathrm{d}f$ easily
implies their boundedness, the integral exists.

We now prove our first result.

###### Proposition 1

For every $g\in\mathcal{M}^{0}$, the following inequality holds:

$\vec{\lambda}(f)\preceq\int\vec{\sigma}^{g}(\mathrm{d}f_{x})\,\mathrm{d}\mu(x).$

###### Proof.

The inequality to be proven is mainly a consequence of Fekete’s subadditivity
lemma. For simplicity in notation, we write

$\lambda_{1}(x)\geq\lambda_{2}(x)\geq\cdots\geq\lambda_{d}(x)$

for the Lyapunov exponents of $f$ at $x$, i.e. we allow repetitions in this
enumeration. Then, we also write

$\lambda_{i}(f):=\int\lambda_{i}(x)\,\mathrm{d}\mu(x),\quad i=1,2,\ldots,d.$

With this notation at hand, the inequality to be proven is equivalent to

$\lambda_{1}(f)+\cdots+\lambda_{k}(f)\leq\int\sum_{i=1}^{k}\log\alpha^{g}_{i}(\mathrm{d}f_{x})\,\mathrm{d}\mu(x)$

for $k=1,2,\ldots,d$ with equality if $k=d$. For a fixed $k$, consider the
sequence

$a_{n}:=\int\sum_{i=1}^{k}\log\alpha^{g}_{i}(\mathrm{d}f^{n}_{x})\,\mathrm{d}\mu(x),\quad
n=1,2,\ldots$

The sequence is subadditive, which easily follows from Horn’s inequality and
the $f$-invariance of $\mu$. Hence, Fekete’s lemma guarantees that $a_{n}/n$
converges to $\inf\\{a_{n}/n:n\in\mathbb{N}\\}$. This implies

$\displaystyle a_{1}$
$\displaystyle=\int\sum_{i=1}^{k}\log\alpha^{g}_{i}(\mathrm{d}f_{x})\,\mathrm{d}\mu(x)\geq\lim_{n\rightarrow\infty}\frac{1}{n}\int\sum_{i=1}^{k}\log\alpha^{g}_{i}(\mathrm{d}f^{n}_{x})\,\mathrm{d}\mu(x)$
$\displaystyle=\sum_{i=1}^{k}\int\lim_{n\rightarrow\infty}\frac{1}{n}\log\alpha^{g}_{i}(\mathrm{d}f^{n}_{x})\,\mathrm{d}\mu(x)=\sum_{i=1}^{k}\int\lambda_{i}(x)\,\mathrm{d}\mu(x)=\sum_{i=1}^{k}\lambda_{i}(f).$

Interchanging the order of limit and integral is possible by the dominated
convergence theorem, using that

$\min_{z\in
M}\log\alpha_{d}^{g}(\mathrm{d}f_{z})\leq\frac{1}{n}\log\alpha^{g}_{i}(\mathrm{d}f^{n}_{x})\leq\max_{z\in
M}\log\alpha_{1}^{g}(\mathrm{d}f_{z})$ (2)

for all $i$, $n$ and $x$. It remains to show the equality in the case $k=d$:

$\displaystyle\sum_{i=1}^{d}\lambda_{i}(f)$
$\displaystyle=\lim_{n\rightarrow\infty}\frac{1}{n}\int\log|\mathrm{det}_{g}\mathrm{d}f^{n}_{x}|\,\mathrm{d}\mu(x)$
$\displaystyle=\lim_{n\rightarrow\infty}\frac{1}{n}\int\log\prod_{i=0}^{n-1}|\mathrm{det}_{g}\mathrm{d}f_{f^{i}x}|\,\mathrm{d}\mu(x)$
$\displaystyle=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}\int\log|\mathrm{det}_{g}\mathrm{d}f_{x}|\,\mathrm{d}\mu(x)$
$\displaystyle=\int\log|\mathrm{det}_{g}\mathrm{d}f_{x}|\,\mathrm{d}\mu(x)=\int\sum_{i=1}^{d}\log\alpha^{g}_{i}(\mathrm{d}f_{x})\,\mathrm{d}\mu(x).$

The first of these equalities follows from statement (iii) of the MET, the
second is the multiplicativity of the determinant together with the chain
rule, the third uses the invariance of the measure $\mu$, the fourth is
obvious, and the last one uses that the absolute determinant equals the
product of the singular values. ∎

Our next goal is to prove a converse result showing that $\vec{\lambda}(f)$
can be approximated by vectors of the form
$\int\vec{\sigma}^{g}(\mathrm{d}f_{x})\,\mathrm{d}\mu(x)$. This result
involves the construction of Riemannian metrics used to achieve the desired
approximation. For this construction, we use geometric features of the space
$\mathcal{S}^{+}$ of positive-definite matrices. A good introduction to
positive-definite matrices and the geometry of $\mathcal{S}^{+}$ is the book
[1]. In the following, we present some basic facts.

The space of symmetric $(d\times d)$-matrices is
$\mathcal{S}_{d}:=\\{s\in\mathbb{R}^{d\times d}:s=s^{\top}\\}$. It is a real
vector space of dimension $\frac{d}{2}(d+1)$ and contains the open cone of
positive-definite matrices
$\mathcal{S}_{d}^{+}:=\\{p\in\mathcal{S}_{d}:p>0\\}$. As an open subset,
$\mathcal{S}_{d}^{+}$ is a trivial smooth manifold of the same dimension as
$\mathcal{S}_{d}$. Its tangent space at any point $p\in\mathcal{S}_{d}^{+}$
can be identified canonically with $\mathcal{S}_{d}$. A very popular
Riemannian metric on $\mathcal{S}_{d}^{+}$, useful in many different contexts,
is the _trace metric_ , given by

$\langle v,w\rangle_{p}:=\mathrm{tr}(p^{-1}vp^{-1}w)\mbox{\quad for all\
}p\in\mathcal{S}_{d}^{+},\ v,w\in T_{p}\mathcal{S}^{+}_{d}=\mathcal{S}_{d}.$

Equipped with this metric, $\mathcal{S}_{d}^{+}$ becomes a complete Riemannian
manifold. By [1, Form. (6.14)], an explicit expression for the induced
distance function is

$d(p,q)=\Bigl{(}\sum_{i=1}^{d}\log^{2}\rho_{i}(p^{-1}q)\Bigr{)}^{\frac{1}{2}},$
(3)

where $\rho_{i}(p^{-1}q)$ are the eigenvalues of $p^{-1}q$ (which is similar
to $p^{-\frac{1}{2}}qp^{-\frac{1}{2}}\in\mathcal{S}_{d}^{+}$).

For each two points $p,q\in\mathcal{S}^{+}_{d}$, there is a unique geodesic
joining $p$ and $q$, denoted by $p\\#_{t}\,q$ and parameterized on $[0,1]$. An
explicit formula is

$p\\#_{t}\,q=p^{\frac{1}{2}}\bigl{(}p^{-\frac{1}{2}}qp^{-\frac{1}{2}}\bigr{)}^{t}p^{\frac{1}{2}},\quad
t\in[0,1].$

If we instead specify a geodesic in $\mathcal{S}^{+}_{d}$ by its initial point
$p$ and its initial direction $v$, an explicit formula is

$\gamma_{p,v}(t)=p^{\frac{1}{2}}\exp(tp^{-\frac{1}{2}}vp^{-\frac{1}{2}})p^{\frac{1}{2}},\quad
t\in\mathbb{R}.$

The group $\mathrm{GL}(d,\mathbb{R})$ acts transitively on
$\mathcal{S}^{+}_{d}$ by isometries via

$g\ast p:=gpg^{\top},\quad g\in\mathrm{GL}(d,\mathbb{R}),\
p\in\mathcal{S}^{+}_{d}.$

Another isometry of $\mathcal{S}^{+}_{d}$ is the matrix inversion. Using the
fact that isometries map geodesics to geodesics, the following identities
become obvious:

$\displaystyle g\ast(p\\#_{t}\,q)=(g\ast p)\\#_{t}\,(g\ast q)\mbox{\quad and
\quad}(p\\#_{t}\,q)^{-1}=p^{-1}\\#_{t}\,q^{-1}.$ (4)

Another useful concept for the analysis on $\mathcal{S}^{+}_{d}$ is a
vectorial distance given by555Caution: There is an overload of notation here.
We use the letter $d$ for the dimension of $M$, the distance function on $M$
and the vectorial distance on $\mathcal{S}^{+}_{d}$. It is hopefully always
clear from the context what is meant.

$\vec{d}(p,q):=2\vec{\sigma}(p^{-\frac{1}{2}}q^{\frac{1}{2}}),\quad
p,q\in\mathcal{S}^{+}_{d}.$

The vectorial distance is closely related to the distance (3) induced by the
trace metric. The next proposition summarizes some of its properties, from
which we will only use (i) and (ii). See also [3, Sec. 4].

###### Proposition 2

The vectorial distance
$\vec{d}(\cdot,\cdot):\mathcal{S}^{+}_{d}\times\mathcal{S}^{+}_{d}\rightarrow\mathfrak{a}^{+}$
has the following properties:

  1. (i)

$\vec{d}(I,p)=\vec{\sigma}(p)$ and $\|\vec{d}(p,q)\|_{2}=d(p,q)$ for all
$p,q\in\mathcal{S}^{+}_{d}$.

  2. (ii)

$\vec{d}(p_{1},q_{1})=\vec{d}(p_{2},q_{2})$ if and only if there exists
$g\in\mathrm{GL}(d,\mathbb{R})$ with $g\ast p_{1}=p_{2}$ and $g\ast
q_{1}=q_{2}$.

  3. (iii)

$\vec{d}(p,q)\preceq\vec{d}(p,r)+\vec{d}(r,q)$ for all
$p,q,r\in\mathcal{S}^{+}_{d}$.

  4. (iv)

$\vec{d}(q,p)=i(\vec{d}(p,q))$ with
$i(\xi):=-(\xi_{d},\xi_{d-1},\ldots,\xi_{1})$ for all
$p,q\in\mathcal{S}^{+}_{d}$.

Another important ingredient for the proof of the converse of Proposition 1 is
the following concept. The _barycenter_ of $l$ matrices
$p_{1},\ldots,p_{l}\in\mathcal{S}^{+}_{d}$ ($l\in\mathbb{N}$) is defined as

$\mathrm{bar}(p_{1},\ldots,p_{l}):=\operatorname*{arg\,min}_{p\in\mathcal{S}^{+}_{d}}\sum_{i=1}^{l}d(p,p_{i})^{2}.$

It can be shown that the minimizer exists and is unique, so
$\mathrm{bar}(p_{1},\ldots,p_{l})$ is well-defined. The following properties
are well-known, see e.g. [21]:

  * •

Symmetry:
$\mathrm{bar}(p_{1},\ldots,p_{l})=\mathrm{bar}(p_{\sigma(1)},\ldots,p_{\sigma(l)})$
for any permutation $\sigma$.

  * •

For any $g\in\mathrm{GL}(d,\mathbb{R})$,

$g\ast\mathrm{bar}(p_{1},\ldots,p_{l})=\mathrm{bar}(g\ast p_{1},\ldots,g\ast
p_{l}).$

  * •

If $u=\mathrm{bar}(p_{1},\ldots,p_{l-1},p_{l})$ and
$v=\mathrm{bar}(p_{1},\ldots,p_{l-1},p_{l}^{\prime})$, then

$\vec{d}(u,v)\preceq\frac{1}{l}\vec{d}(p_{l},p_{l}^{\prime}).$ (5)

  * •

The mapping $(p_{1},\ldots,p_{l})\mapsto\mathrm{bar}(p_{1},\ldots,p_{l})$ is
continuous.

We need to lift some of the constructions introduced above from
$\mathcal{S}^{+}_{d}$ to the space $\mathcal{M}^{0}$ of continuous metrics on
$M$. If $F:M\hookleftarrow$ is a $C^{1}$-diffeomorphism and
$g\in\mathcal{M}^{0}$, then $F$ acts on $g$ by pulling back:

$(F\ast g)_{x}(v,w):=g_{Fx}(\mathrm{d}F_{x}v,\mathrm{d}F_{x}w)\mbox{\quad for
all\ }v,w\in T_{x}M.$

Since $F$ is $C^{1}$, we have $F\ast g\in\mathcal{M}^{0}$. We will see below
that this can be regarded an an extension of the action of
$\mathrm{GL}(d,\mathbb{R})$ on $\mathcal{S}^{+}_{d}$.

Next, we extend the barycenter map to Riemannian metrics. Given $l$ metrics
$g^{1},\ldots,g^{l}$ in $\mathcal{M}^{0}$, we can define a new metric, denoted
by $\mathrm{bar}(g^{1},\ldots,g^{l})$, and defined pointwise via the
barycenter of positive-definite matrices. This is done as follows. Let
$(\phi,U)$ be any chart of $M$ around $x\in M$, i.e. $U\subset M$ is an open
neighborhood of $x$ and $\phi:U\rightarrow\phi(U)\subset\mathbb{R}^{d}$ a
diffeomorphism. Then we identify an element $p\in\mathcal{S}^{+}_{d}$ with an
inner product on $T_{x}M$ via666This identification is unusual because of the
involved inversion of $p$. However, for our purposes it is more convenient as
it allows to obtain more direct relations between the introduced operations on
$\mathcal{S}^{+}_{d}$ and corresponding operations on $\mathcal{M}^{0}$.

$\beta_{\phi}(p)(v,w):=\langle
p^{-1}\mathrm{d}\phi_{x}v,\mathrm{d}\phi_{x}w\rangle\mbox{\quad for all\
}v,w\in T_{x}M,$ (6)

where $\langle\cdot,\cdot\rangle$ denotes the standard Euclidean inner product
on $\mathbb{R}^{d}$. The map $\beta_{\phi}$ is obviously invertible. If
$(\phi_{1},U)$ and $(\phi_{2},V)$ are two charts around $x$, then the two
matrix representations of the inner product $g_{x}$ are related by

$\mathrm{d}(\phi_{2}\circ\phi_{1}^{-1})_{\phi_{1}(x)}\ast\beta_{\phi_{1}}^{-1}(g_{x})=\beta_{\phi_{2}}^{-1}(g_{x}),$
(7)

which is an easy consequence of the definition in (6). From this and
Proposition 2(ii), it follows that the vectorial distance of two inner
products $g^{1}_{x}$ and $g^{2}_{x}$ on $T_{x}M$ can be defined independently
of the chosen chart:

$\vec{d}(g^{1}_{x},g^{2}_{x}):=\vec{d}(\beta_{\phi}^{-1}(g^{1}_{x}),\beta_{\phi}^{-1}(g^{2}_{x})).$

The barycenter of $l$ metrics $g^{1},\ldots,g^{l}$ is now defined by

$\mathrm{bar}(g^{1},\ldots,g^{l})_{x}:=\beta_{\phi}(\mathrm{bar}(\beta_{\phi}^{-1}(g^{1}_{x}),\ldots,\beta_{\phi}^{-1}(g^{l}_{x}))),$
(8)

where for each $x\in M$ we choose an appropriate chart $(\phi,U)$ around $x$.
Using (7), it can easily be shown that this definition is independent of the
chosen chart. From (8), it follows that $\mathrm{bar}(g^{1},\ldots,g^{l})$ is
continuous in the domain of every chart, since all involved functions on the
right-hand side are continuous. Hence,
$\mathrm{bar}(g^{1},\ldots,g^{l})\in\mathcal{M}^{0}$.

The proof of the following lemma is trivial, since all statements follow from
a direct application of the definitions. Hence, we leave its proof to the
reader.

###### Lemma 1

For all $g,g^{i}\in\mathcal{M}^{0}$ and $C^{1}$-diffeomorphisms
$F,F_{i}:M\hookleftarrow$, the following properties hold:

  1. (i)

$F\ast\mathrm{bar}(g^{1},\ldots,g^{l})=\mathrm{bar}(F\ast g^{1},\ldots,F\ast
g^{l})$.

  2. (ii)

$F_{2}\ast(F_{1}\ast g)=(F_{1}\circ F_{2})\ast g$.

  3. (iii)

Let $(\phi,U)$ be a chart around $x$ and $(\psi,V)$ one around $Fx$. Then

$\beta_{\phi}^{-1}((F\ast g)_{x})=\mathrm{d}(\psi\circ
F\circ\phi^{-1})_{\phi(x)}^{-1}\ast\beta_{\psi}^{-1}(g_{Fx}).$

In this sense, the pullback of a metric via a diffeomorphism is an extension
of the action of $\mathrm{GL}(d,\mathbb{R})$ on $\mathcal{S}^{+}_{d}$.

  4. (iv)

$\vec{d}(g^{1}_{Fx},g^{2}_{Fx})=\vec{d}((F\ast g^{1})_{x},(F\ast g^{2})_{x})$
for all $x\in M$.

###### Lemma 2

Let $F:M\hookleftarrow$ be a $C^{1}$-diffeomorphism and $g\in\mathcal{M}^{0}$.
Then for any $x\in M$ we have

$\vec{\sigma}^{g}(\mathrm{d}F_{x})=\vec{\sigma}(p_{2}^{-\frac{1}{2}}A_{x}p_{1}^{\frac{1}{2}}),$

where $p_{1}=\beta_{\phi}^{-1}(g_{x})$, $p_{2}=\beta_{\psi}^{-1}(g_{Fx})$ and
$A_{x}=\mathrm{d}(\psi\circ F\circ\phi^{-1})_{\phi(x)}$ for charts $(\phi,U)$
and $(\psi,V)$ around $x$ and $Fx$, respectively.

###### Proof.

The singular values of $\mathrm{d}F_{x}:T_{x}M\rightarrow T_{Fx}M$ with
respect to the metric $g$ are the square roots of the eigenvalues of the self-
adjoint operator

$(\mathrm{d}F_{x})(\mathrm{d}F_{x})^{*}:T_{Fx}M\rightarrow T_{Fx}M.$

Using the charts $(\phi,U)$ and $(\psi,V)$, we have for any $v\in T_{x}M$,
$w\in T_{Fx}M$ that

$\displaystyle
g_{Fx}(\mathrm{d}F_{x}v,w)=\langle\beta_{\psi}^{-1}(g_{Fx})^{-1}\mathrm{d}\psi_{Fx}\mathrm{d}F_{x}v,\mathrm{d}\psi_{Fx}w\rangle$
$\displaystyle=\langle\mathrm{d}\psi_{Fx}\mathrm{d}F_{x}v,[\beta_{\psi}^{-1}(g_{Fx})^{-1}]^{\top}\mathrm{d}\psi_{Fx}w\rangle$
$\displaystyle=\langle\mathrm{d}\phi_{x}v,[\mathrm{d}\psi_{Fx}\mathrm{d}F_{x}(\mathrm{d}\phi_{x})^{-1}]^{\top}[\beta_{\psi}^{-1}(g_{Fx})^{-1}]^{\top}\mathrm{d}\psi_{Fx}w\rangle$
$\displaystyle=\langle\beta_{\phi}^{-1}(g_{x})^{-1}\mathrm{d}\phi_{x}v,\beta_{\phi}^{-1}(g_{x})[\mathrm{d}\psi_{Fx}\mathrm{d}F_{x}(\mathrm{d}\phi_{x})^{-1}]^{\top}\beta_{\psi}^{-1}(g_{Fx})^{-1}\mathrm{d}\psi_{Fx}w\rangle.$

In the last identity, we used, in particular, that the matrix
$\beta_{\phi}^{-1}(g_{x})$ is symmetric. On the other hand,

$\displaystyle
g_{x}(v,(\mathrm{d}F_{x})^{*}w)=\langle\beta_{\phi}^{-1}(g_{x})^{-1}\mathrm{d}\phi_{x}v,\mathrm{d}\phi_{x}(\mathrm{d}F_{x})^{*}w\rangle.$

When we compare the two results, we see that

$(\mathrm{d}F_{x})^{*}=\mathrm{d}\phi_{x}^{-1}\beta_{\phi}^{-1}(g_{x})[\mathrm{d}\psi_{Fx}\mathrm{d}F_{x}(\mathrm{d}\phi_{x})^{-1}]^{\top}\beta_{\psi}^{-1}(g_{Fx})^{-1}\mathrm{d}\psi_{Fx}.$

In the middle of the right-hand side term, we find the transpose of the local
representation of $\mathrm{d}F_{x}$, that we denote by $A_{x}$. Then

$\displaystyle\mathrm{d}F_{x}(\mathrm{d}F_{x})^{*}=\mathrm{d}\psi_{Fx}^{-1}A_{x}\beta_{\phi}^{-1}(g_{x})A_{x}^{\top}\beta_{\psi}^{-1}(g_{Fx})^{-1}\mathrm{d}\psi_{Fx}.$

It follows that the singular values of $\mathrm{d}F_{x}$ with respect to the
metric $g$ are the square roots of the eigenvalues of the matrix
$C_{x}:=A_{x}\beta_{\phi}^{-1}(g_{x})A_{x}^{\top}\beta_{\psi}^{-1}(g_{Fx})^{-1}$
which is similar to

$[p_{2}^{-\frac{1}{2}}A_{x}p_{1}^{\frac{1}{2}}][p_{1}^{\frac{1}{2}}A_{x}^{\top}p_{2}^{-\frac{1}{2}}]=B_{x}B_{x}^{\top},\quad
B_{x}:=p_{2}^{-\frac{1}{2}}A_{x}p_{1}^{\frac{1}{2}}.$

It follows that the singular values of $\mathrm{d}F_{x}$ in the metric $g$ are
the eigenvalues of $(B_{x}B_{x}^{\top})^{\frac{1}{2}}$, or equivalently, the
ordinary singular values of $B_{x}$, which directly implies the statement of
the lemma. ∎

Now, we can finally prove our converse result.

###### Proposition 3

For any $\varepsilon>0$, there exists a metric
$g^{\varepsilon}\in\mathcal{M}^{0}$ such that

$\int\vec{\sigma}^{g^{\varepsilon}}(\mathrm{d}f_{x})\,\mathrm{d}\mu(x)\preceq_{w}\vec{\lambda}(f)+\varepsilon\mathds{1}.$

###### Proof.

Using a fixed reference metric $g^{0}\in\mathcal{M}^{0}$, for each
$N\in\mathbb{N}$ we define the new metric

$g^{N}:=\mathrm{bar}(g^{0},f\ast g^{0},f^{2}\ast g^{0},\ldots,f^{N-1}\ast
g^{0})$

via the iterates of $f$ (in particular, $g^{1}=g^{0}$). Then, using Lemma 1(i)
and (ii) together with the symmetry of the barycenter map, we find that

$\displaystyle f^{-1}\ast g^{N}$ $\displaystyle=\mathrm{bar}(f^{-1}\ast
g^{0},g^{0},\ldots,f^{N-2}\ast g^{0})$ $\displaystyle=\mathrm{bar}(g^{0},f\ast
g^{0},\ldots,f^{N-2}\ast g^{0},f^{-1}\ast g^{0}).$

We now compare the two metrics $g^{N}$ and $f^{-1}\ast g^{N}$ at the point
$fx$ (for an arbitrarily chosen $x\in M$), using inequality (5):

$\displaystyle\vec{d}(g^{N}_{fx},(f^{-1}\ast g^{N})_{fx})$
$\displaystyle=\vec{d}(\mathrm{bar}(\beta_{\phi}^{-1}(g^{0}_{fx}),\beta_{\phi}^{-1}([f\ast
g^{0}]_{fx}),\ldots,\beta_{\phi}^{-1}([f^{N-1}\ast g^{0}]_{fx})),$
$\displaystyle\qquad\qquad\mathrm{bar}(\beta_{\phi}^{-1}(g^{0}_{fx}),\beta_{\phi}^{-1}([f\ast
g^{0}]_{fx}),\ldots,\beta_{\phi}^{-1}([f^{-1}\ast g^{0}]_{fx})))$
$\displaystyle\preceq\frac{1}{N}\vec{d}(\beta_{\phi}^{-1}([f^{N-1}\ast
g^{0}]_{fx}),\beta_{\phi}^{-1}([f^{-1}\ast g^{0}]_{fx}))$
$\displaystyle=\frac{1}{N}\vec{d}((f^{N-1}\ast g^{0})_{fx},(f^{-1}\ast
g^{0})_{fx})=\frac{1}{N}\vec{d}(g^{0}_{f^{N}x},(f^{-N}\ast g^{0})_{f^{N}x}).$

For the last identity, we used Lemma 1(iv). Choose charts $(\phi,U)$ and
$(\psi,V)$ around the points $f^{N}x$ and $x$, respectively. Let $p(f^{N}x)$
be the matrix representation of $g^{0}$ at $f^{N}x$ and $p(x)$ the matrix
representation of $g^{0}$ at $x$. Then, using Lemma 1(iii), we find that

$\displaystyle\vec{d}(g^{0}_{f^{N}x},(f^{-N}\ast
g^{0})_{f^{N}x})=\vec{d}(p(f^{N}x),\mathrm{d}(\psi\circ
f^{-N}\circ\phi^{-1})^{-1}_{\phi(f^{N}x)}\ast p(x)).$

We observe that $\mathrm{d}(\psi\circ f^{-N}\circ\phi^{-1})_{\phi(f^{N}x)}$ is
the inverse of the local representation of $\mathrm{d}f^{N}$ at $x$, for which
we write $A^{N}_{x}$. Hence, using Proposition 2(ii), we obtain

$\displaystyle\vec{d}(g^{0}_{f^{N}x},(f^{-N}\ast g^{0})_{f^{N}x})$
$\displaystyle=\vec{d}(p(f^{N}x),A^{N}_{x}\ast p(x))$
$\displaystyle=\vec{d}(I,[p(f^{N}x)^{-\frac{1}{2}}A^{N}_{x}p(x)^{\frac{1}{2}}][p(f^{N}x)^{-\frac{1}{2}}A^{N}_{x}p(x)^{\frac{1}{2}}]^{\top}).$

Writing $B^{N}_{x}:=p(f^{N}x)^{-\frac{1}{2}}A^{N}_{x}p(x)^{\frac{1}{2}}$,
Proposition 2(i) and Lemma 2 yield

$\displaystyle\vec{d}(g^{0}_{f^{N}x},(f^{-N}\ast g^{0})_{f^{N}x})$
$\displaystyle=\vec{\sigma}(B^{N}_{x}(B^{N}_{x})^{\top})=2\vec{\sigma}(B^{N}_{x})$
$\displaystyle=2\vec{\sigma}(p(f^{N}x)^{-\frac{1}{2}}A^{N}_{x}p(x)^{\frac{1}{2}})=2\vec{\sigma}^{g^{0}}(\mathrm{d}f^{N}_{x}).$

Using the same arguments for $\vec{d}(g^{N}_{fx},(f^{-1}\ast g^{N})_{fx})$, we
have thus proven that

$\vec{\sigma}^{g^{N}}(\mathrm{d}f_{x})\preceq\frac{1}{N}\vec{\sigma}^{g^{0}}(\mathrm{d}f^{N}_{x})\mbox{\quad
for all\ }x\in M.$ (9)

To prove the statement of the proposition, observe that

$\vec{\lambda}(f)=\int\lim_{n\rightarrow\infty}\frac{1}{n}\vec{\sigma}^{g^{0}}(\mathrm{d}f^{n}_{x})\,\mathrm{d}\mu(x)=\lim_{n\rightarrow\infty}\int\frac{1}{n}\vec{\sigma}^{g^{0}}(\mathrm{d}f^{n}_{x})\,\mathrm{d}\mu(x),$

where we use again the dominated convergence theorem. Hence, for the given
$\varepsilon>0$, we can find $N$ such that

$\int\vec{\sigma}^{g^{N}}(\mathrm{d}f_{x})\,\mathrm{d}\mu(x)\stackrel{{\scriptstyle\eqref{eq_intermediate_statement}}}{{\preceq}}\int\frac{1}{N}\vec{\sigma}^{g^{0}}(\mathrm{d}f^{N}_{x})\,\mathrm{d}\mu(x)\preceq_{w}\vec{\lambda}(f)+\varepsilon\mathds{1},$

which completes the proof. ∎

###### Remark 1

A careful inspection of the proof shows that the assumption of invertibility
of $f$ is actually not needed. It suffices to assume that the derivative
$\mathrm{d}f_{x}$ is an invertible linear operator for each $x\in M$. This is
because, even though we are pulling back the metric $g^{N}$ by $f^{-1}$, we
are only evaluating the resulting metric at $fx$, and it holds that

$(f^{-1}\ast
g)_{fx}=g_{x}(\mathrm{d}f^{-1}_{fx}v,\mathrm{d}f^{-1}_{fx}w)=g_{x}((\mathrm{d}f_{x})^{-1}v,(\mathrm{d}f_{x})^{-1}w).$

Also in [3], only the invertibility of the values of the linear cocycle is
required.

Proposition 1 and 3 together show that the Lyapunov vector $\vec{\lambda}(f)$
can be regarded as a _strong infimum_ 777We use the term _strong infimum_ in
contrast to a _weak ( $=$ Pareto) infimum_. of the vector-valued function

$\vec{\mathcal{S}}_{f,\mu}:g\mapsto\int\vec{\sigma}^{g}(\mathrm{d}f_{x})\,\mathrm{d}\mu(x),\quad\vec{\mathcal{S}}_{f,\mu}:\mathcal{M}^{0}\rightarrow\mathfrak{a}^{+}.$

The infimum needs to be understood with respect to the order $\preceq$ that
was defined on the convex cone $\mathfrak{a}^{+}$. Of course, the space
$\mathcal{M}^{0}$ is huge, so it would be convenient if we could reduce this
minimization problem to a smaller subspace. A first observation is that the
multiplication with a positive scalar function does not change the value of
$\vec{\mathcal{S}}_{f,\mu}$:

$\vec{\mathcal{S}}_{f,\mu}(\gamma\cdot
g)=\vec{\mathcal{S}}_{f,\mu}(g)\mbox{\quad for all\ }\gamma\in
C^{0}(M,\mathbb{R}_{>0}),\ g\in\mathcal{M}^{0}.$

Indeed, Lemma 2 implies

$\log\alpha_{i}^{\gamma\cdot
g}(\mathrm{d}f_{x})=\frac{1}{2}(\log\gamma(fx)-\log\gamma(x))+\log\alpha_{i}^{g}(\mathrm{d}f_{x}),$

and the integral over $\log\gamma(fx)-\log\gamma(x)$ vanishes because $\mu$ is
$f$-invariant. It follows that we can restrict the search for an optimal
metric to the space of metrics inducing the same (arbitrary) volume form on
$M$. To see this, let a volume form $\omega_{0}$ be fixed and pick an
arbitrary $g\in\mathcal{M}^{0}$. The metric $g$ induces a volume form
$\omega_{g}=\sqrt{\det g}\,\mathrm{d}x^{1}\wedge\ldots\wedge\mathrm{d}x^{d}$.
Then, there exists a positive function $\gamma$ such that
$\omega_{g}=\gamma\cdot\omega_{0}$, and the metric
$\tilde{g}:=\gamma^{-2/d}\cdot g$ induces the volume form
$\gamma^{-1}\cdot\omega_{g}=\omega_{0}$. In particular, in dimension $d=1$
there is no need to optimize, because here any metric is just a positive
scalar function.

Furthermore, we can restrict $\vec{\mathcal{S}}_{f,\mu}$ to the space of
smooth metrics on $M$, since every continuous metric can be approximated
uniformly (by compactness of $M$) by smooth metrics and the singular values
depend continuously on the metric in the uniform topology.

Recall that $\mathcal{M}$ denotes the space of all smooth Riemannian metrics
on $M$. Given a smooth volume form $\omega$, we let
$\mathcal{M}_{\omega}\subset\mathcal{M}$ denote the subspace of metrics
inducing $\omega$. We then have the following theorem which will be the basis
of our further investigations.

###### Theorem 2

Let $f:M\hookleftarrow$ be a $C^{1}$-diffeomorphism on a compact Riemannian
manifold, preserving the probability measure $\mu$. Then, for every smooth
volume form $\omega$ on $M$, the following statements hold:

  1. (i)

For every $g\in\mathcal{M}_{\omega}$, it holds that

$\vec{\lambda}(f)\preceq\vec{\mathcal{S}}_{f,\mu}(g).$

  2. (ii)

For every $\varepsilon>0$, there exists
$g^{\varepsilon}\in\mathcal{M}_{\omega}$ such that

$\vec{\mathcal{S}}_{f,\mu}(g^{\varepsilon})\preceq_{w}\vec{\lambda}(f)+\varepsilon\mathds{1}.$

###### Remark 2

It is a simple exercise to show that the inequality

$a\preceq_{w}b\preceq_{w}a+\varepsilon\mathds{1}$ (10)

for vectors $a,b\in\mathfrak{a}^{+}$ implies
$\|a-b\|_{\infty}\leq\varepsilon$. Hence, Theorem 2 really implies that
$\vec{\lambda}(f)$ can be approximated by vectors of the form
$\vec{\mathcal{S}}_{f,\mu}(g)$ with $g\in\mathcal{M}_{\omega}$. However, the
sequence of metrics employed in the proof is not very helpful for concrete
computations, because it involves the derivatives of the iterates of $f$ just
like the definition of the Lyapunov exponents.

##  4 The space of Riemannian metrics

To explore deeper properties of the optimization problem for
$g\mapsto\vec{\mathcal{S}}_{f,\mu}(g)$, we need to gain a better understanding
of the spaces $\mathcal{M}$ and $\mathcal{M}_{\omega}$ and their natural
geometric structures. In this section, we gather some facts from the
literature about these spaces, mainly taken from [12, 13, 9]. We also refer to
[28] for a comprehensive general treatment of manifolds of smooth mappings.

A Riemannian metric $g$ assigns to each $x\in M$ an inner product on the
tangent space $T_{x}M$, i.e. a positive-definite symmetric $(0,2)$-tensor
$g_{x}:T_{x}M\times T_{x}M\rightarrow\mathbb{R}$. We write $T^{0}_{2}M$ for
the tensor bundle of $(0,2)$-tensors on $M$ and $S^{0}_{2}M\subset T^{0}_{2}M$
for the subbundle of symmetric tensors. This subbundle attaches to each $x\in
M$ the vector space of all symmetric $(0,2)$-tensors on $T_{x}M$. In this
formal setup, a Riemannian metric is a special section of the bundle
$S^{0}_{2}M$, namely one that is positive-definite. We write
$\mathcal{S}^{0}_{2}M$ for the space of smooth sections of $S^{0}_{2}M$, i.e.,
of symmetric $(0,2)$-tensor fields of class $C^{\infty}$. Obviously,
$\mathcal{S}^{0}_{2}M$ is an infinite-dimensional vector space over the reals
with the pointwise addition and scalar multiplication. Indeed,
$\mathcal{S}^{0}_{2}M$ is a Fréchet space with the $C^{\infty}$-topology, of
which $\mathcal{M}$ is an open subset. The fact that $\mathcal{M}$ is open in
the Fréchet space $\mathcal{S}^{0}_{2}M$ implies that $\mathcal{M}$ is a
trivial Fréchet manifold whose tangent space at any point can be identified
canonically with $\mathcal{S}^{2}_{0}M$.

We define an inner product on each tangent space of $\mathcal{M}$ as

$\langle
h,k\rangle_{g}:=\int\mathrm{tr}(g^{-1}_{x}h_{x}g^{-1}_{x}k_{x})\,\mathrm{d}\omega_{g}(x)\mbox{\quad
for all\ }h,k\in T_{g}\mathcal{M}=\mathcal{S}^{2}_{0}M,$

where $\omega_{g}$ is the volume form induced by $g$, and $g_{x}$, $h_{x}$,
$k_{x}$ have to be understood as local representations. With similar arguments
as used in the previous section, one shows that the trace is independent of
the charts used to obtain these representations. In terms of the trace metric
$\langle\cdot,\cdot\rangle_{g_{x}}$, we can also write

$\langle h,k\rangle_{g}=\int\langle
h_{x},k_{x}\rangle_{g_{x}}\,\mathrm{d}\omega_{g}(x).$

For obvious reasons, this Riemannian metric is called the _$L^{2}$ -metric_ on
$\mathcal{M}$. It is only a _weak_ metric, meaning that the induced topology
on the tangent space does not coincide with the inherited topology of the
manifold. Nevertheless, as shown in [9], the geodesic distance induced by the
$L^{2}$-metric turns $\mathcal{M}$ into a (non-complete) metric space.888As on
a finite-dimensional manifold, the geodesic distance of two points
$g^{1},g^{2}$ is defined as the infimum over the lengths of all piecewise
differentiable curves connecting $g^{1}$ and $g^{2}$. On general weak
Riemannian manifolds, this is only a pseudo-metric. It can also be seen easily
(using Lemma 1(iii)) that the $L^{2}$-metric is invariant under the
diffeomorphism group of $M$ acting by pullback. An explicit expression for the
distance function on $\mathcal{M}$, induced by the $L^{2}$-metric, was given
in [10] (see also [8, App. B] for a more elegant proof):

$d_{L^{2}}(g^{1},g^{2})=\Bigl{(}\int_{M}d_{x}(g^{1}_{x},g^{2}_{x})^{2}\,\mathrm{d}\omega_{g^{0}}(x)\Bigr{)}^{\frac{1}{2}},$
(11)

where $g^{0}$ is an arbitrary reference metric with $\mathrm{vol}(M,g^{0})=1$,
and $d_{x}(\cdot,\cdot)$ is the fiber metric on the space of inner products on
$T_{x}M$, induced by the Riemannian metric $\langle
a,b\rangle_{p}=\mathrm{tr}(p^{-1}ap^{-1}b)\sqrt{\det(g^{0}(x)^{-1}p)}$.

The space $\mathcal{M}$ admits the global product structure
$\mathcal{M}\cong\mathcal{V}\times\mathcal{M}_{\omega}$, where $\mathcal{V}$
denotes the space of smooth volume forms on $M$. The space $\mathcal{V}$ is a
Fréchet manifold, in fact an open subset of $\Omega^{d}(M)$, the Fréchet space
of order-$d$ differential forms on $M$. Given $\omega\in\mathcal{V}$ and
$\nu\in\Omega^{d}(M)$, there exists a unique smooth function, denoted by
$\frac{\nu}{\omega}$, such that

$\nu=\left(\frac{\nu}{\omega}\right)\omega.$

In case that $\nu$ is also a volume form, the function $\frac{\nu}{\omega}$ is
strictly positive. The space $\mathcal{M}_{\omega}$ is a smooth submanifold of
$\mathcal{M}$ with tangent space

$T_{g}\mathcal{M}_{\omega}=\\{h\in\mathcal{S}^{0}_{2}M:\mathrm{tr}(g_{x}^{-1}h_{x})=0\mbox{\
for all\ }x\in M\\}.$

The following result is proven in [13, Prop. 1.13 and Prop. 2.2].999Actually,
the result says much more, namely that $\mathcal{M}_{\omega}$ is a symmetric
space.

###### Proposition 4

For each $\omega\in\mathcal{V}$, the submanifold $\mathcal{M}_{\omega}$ is
geodesically complete and the geodesic starting at
$g^{0}\in\mathcal{M}_{\omega}$ with initial tangent $h$ is given by

$g^{t}=g^{0}\exp(t(g^{0})^{-1}h).$ (12)

###### Remark 3

The formula (12) needs to be understood pointwise. That is,

$g^{t}_{x}=g^{0}_{x}\exp(t(g^{0}_{x})^{-1}h_{x})\mbox{\quad for all\ }x\in M,$

where we can use local representations of the involved objects $g^{0}$ and $h$
with respect to a fixed chart, and $\exp(\cdot)$ is the usual matrix
exponential.

###### Remark 4

The statement that $\mathcal{M}_{\omega}$ is geodesically complete does not
imply that $\mathcal{M}_{\omega}$ is complete as a metric space, when equipped
with the geodesic distance. In finite dimensions, this implication holds by
the Hopf-Rinow theorem, but in infinite dimensions there are counter-examples.
To the best of my knowledge, it is not known if $\mathcal{M}_{\omega}$ is also
complete as a metric space (but most probably it is not, because $\mathcal{M}$
is not complete).

###### Remark 5

The submanifold $\mathcal{M}_{\omega}$ is not totally geodesic in
$\mathcal{M}$. Indeed, the formula for geodesics in $\mathcal{M}$ is
different, and will not be used in this paper.

###### Corollary 1

For each pair of metrics $g^{a},g^{b}\in\mathcal{M}_{\omega}$, there exists a
unique geodesic on $[0,1]$, connecting $g^{a}$ and $g^{b}$. This geodesic is
given by

$g^{t}_{x}:=(g^{a}_{x}\\#_{t}\,g^{b}_{x})\mbox{\quad for all\ }x\in M,\
t\in[0,1],$ (13)

where the operation $\\#_{t}$ carries over from $\mathcal{S}^{+}_{d}$ to the
space of symmetric positive-definite $(0,2)$-tensors on $T_{x}M$ via local
representations.

###### Proof.

The proof consists of two parts. First, we show that the curve given by (13)
is a geodesic in $\mathcal{M}_{\omega}$ which connects $g^{a}$ and $g^{b}$.
Second, we show that each such geodesic is of the given form.

(1) The curve $t\mapsto g^{t}$ is well-defined as a curve in $\mathcal{M}$
because of (7) and (4). Obviously, $g^{0}=g^{a}$ and $g^{1}=g^{b}$. To show
that $t\mapsto g^{t}$ is a geodesic in $\mathcal{M}_{\omega}$, we define (in
terms of local representations)

$h_{x}:=(g^{a}_{x})^{\frac{1}{2}}\log((g^{a}_{x})^{-\frac{1}{2}}g^{b}_{x}(g^{a}_{x})^{-\frac{1}{2}})(g^{a}_{x})^{\frac{1}{2}}\mbox{\quad
for all\ }x\in M.$

Here, $\log(\cdot)$ denotes the matrix logarithm, which is well-defined and
unique as an operator on $\mathcal{S}^{+}_{d}$. It is clear that $h_{x}$ is
symmetric for each $x$ and easy to show that $h_{x}$ is well-defined, i.e.
independent of the representation. Hence, $h$ is a section of $S^{0}_{2}M$. It
is a smooth section, because it is constructed from smooth sections via smooth
operations. Hence, $h\in\mathcal{S}^{0}_{2}M$. Finally, we have

$\displaystyle\mathrm{tr}((g^{1}_{x})^{-1}h_{x})$
$\displaystyle=\mathrm{tr}(\log((g^{a}_{x})^{-\frac{1}{2}}g^{b}_{x}(g^{a}_{x})^{-\frac{1}{2}}))=\log\det((g^{a}_{x})^{-\frac{1}{2}}g^{b}_{x}(g^{a}_{x})^{-\frac{1}{2}}))$
$\displaystyle=\log[\det(g^{a}_{x})^{-1}\det(g^{b}_{x})]=\log(1)=0,$

because the fact that $g^{a}$ and $g^{b}$ induce the same volume form implies
$\det(g^{a}_{x})=\det(g^{b}_{x})$ for all $x\in M$. This shows that $h\in
T_{g^{a}}\mathcal{M}_{\omega}$. To see that $t\mapsto g^{t}$ is a geodesic, it
remains to show that $g^{t}=g^{a}\exp(t(g^{a})^{-1}h)$ and invoke Proposition
4. This, however, is a simple computation, left to the reader.

(2) To show the uniqueness, let $t\mapsto\tilde{g}^{t}$ be any geodesic in
$\mathcal{M}_{\omega}$ with $\tilde{g}^{0}=g^{a}$ and $\tilde{g}^{1}=g^{b}$.
By Proposition 4, it must be of the form
$\tilde{g}^{t}=g^{a}\exp(t(g^{a})^{-1}h)$ for some $h\in
T_{g^{a}}\mathcal{M}_{\omega}$, implying
$g^{a}_{x}\exp((g^{a}_{x})^{-1}h_{x})=g^{b}_{x}$ for all $x\in M$. Thus,

$\displaystyle h_{x}$
$\displaystyle=(g^{a}_{x})\log((g^{a}_{x})^{-1}g^{b}_{x})=(g^{a}_{x})\log((g^{a}_{x})^{-\frac{1}{2}}(g^{a}_{x})^{-\frac{1}{2}}g^{b}_{x}(g^{a}_{x})^{-\frac{1}{2}}(g^{a}_{x})^{\frac{1}{2}})$
$\displaystyle=(g^{a}_{x})^{\frac{1}{2}}\log((g^{a}_{x})^{-\frac{1}{2}}g^{b}_{x}(g^{a}_{x})^{-\frac{1}{2}})(g^{a}_{x})^{\frac{1}{2}}.$

Hence, by the first part of the proof, it follows that $\tilde{g}^{t}\equiv
g^{t}$. ∎

##  5 Convexity

In this section, we prove that the function
$\vec{\mathcal{S}}_{f,\mu}:\mathcal{M}_{\omega}\rightarrow\mathfrak{a}^{+}$ is
convex with respect to the structures introduced on its domain and co-domain.
First, we need to prove continuity, which is implied by the following lemma.

###### Lemma 3

The function $(g,x)\mapsto\vec{\sigma}^{g}(\mathrm{d}f_{x})$ is continuous on
$\mathcal{M}\times M$.

###### Proof.

We denote the function under consideration by $\beta$. By Lemma 2, we can
write $\beta$ locally as

$\beta(g,x)=\vec{\sigma}(\mathrm{ev}(g,fx)^{-\frac{1}{2}}\cdot\mathrm{d}f_{x}\cdot\mathrm{ev}(g,x)^{\frac{1}{2}}),$

where $\mathrm{ev}(g,x):=g_{x}$ is the evaluation map. By [28, Rem. 2.1.2],
the evaluation map is continuous, implying that
$(g,x)\mapsto\mathrm{ev}(g,fx)^{-\frac{1}{2}}\cdot\mathrm{d}f_{x}\cdot\mathrm{ev}(g,x)^{\frac{1}{2}}$
is continuous. Since the singular values depend continuously on the matrix,
then also $\beta$ is continuous. ∎

The notion of convexity to be used for $\vec{\mathcal{S}}_{f,\mu}$ is a
natural combination of the two well-studied notions of _geodesic convexity_
(see, e.g., [29, 6]) and _cone-convexity_ (see, e.g., [25]). Recall that a
function $\varphi:M\rightarrow\mathbb{R}$, defined on a finite-dimensional
Riemannian manifold $(M,g)$, is called _geodesically convex_ if the
composition $\varphi\circ\gamma$ with an arbitrary geodesic $\gamma$ in $M$ is
convex in the usual sense. This can be reformulated as

$\varphi(\gamma(t))\leq(1-t)\varphi(\gamma(0))+t\varphi(\gamma(1))\mbox{\quad
for all\ }t\in[0,1].$

If $\varphi$ instead takes values in a vector space equipped with a partial
order $\preceq$ induced by a convex cone, and the inequality

$\varphi(\gamma(t))\preceq(1-t)\varphi(\gamma(0))+t\varphi(\gamma(1))\mbox{\quad
for all\ }t\in[0,1]$

holds for any geodesic in $M$, we call $\varphi$ _geodesically cone-convex_.
It makes sense to extend this definition to functions on infinite-dimensional
Riemannian manifolds, which are well-behaved in terms of their geodesics.
Since this is certainly the case for the manifold $\mathcal{M}_{\omega}$ with
the $L^{2}$-metric, as Proposition 4 shows, we can ask whether
$\vec{\mathcal{S}}_{f,\mu}$ is geodesically cone-convex. The following
proposition shows that this is already the case for the integrand in the
definition of $\vec{\mathcal{S}}_{f,\mu}$.

###### Proposition 5

For each $x\in M$, the function $g\mapsto\vec{\sigma}^{g}(\mathrm{d}f_{x})$,
$\mathcal{M}_{\omega}\rightarrow\mathfrak{a}^{+}$, is geodesically cone-
convex.

###### Proof.

Let $g:[0,1]\rightarrow\mathcal{M}_{\omega}$, $t\mapsto g^{t}$, be a geodesic.
By Corollary 1, we have $g^{t}_{x}\equiv g^{0}_{x}\\#_{t}\,g^{1}_{x}$. We
prove the mid-point convexity of $g\mapsto\vec{\sigma}^{g}(\mathrm{d}f_{x})$,
i.e.

$\vec{\sigma}^{g^{\frac{1}{2}}}(\mathrm{d}f_{x})\preceq\frac{1}{2}\vec{\sigma}^{g^{0}}(\mathrm{d}f_{x})+\frac{1}{2}\vec{\sigma}^{g^{1}}(\mathrm{d}f_{x})\mbox{\quad
for all\ }x\in M.$ (14)

To prove this, we choose charts $(\phi,U)$ and $(\psi,V)$ around $x$ and $fx$,
respectively. Let $A_{x}$ be a local representation of $\mathrm{d}f_{x}$ with
respect to this pair of charts. Further, let $p_{0}$ and $p_{1}$ ($q_{0}$ and
$q_{1}$) represent $g^{0}_{x}$ and $g^{1}_{x}$ ($g^{0}_{fx}$ and
$g^{1}_{fx}$), respectively. According to Lemma 2, then (14) is equivalent
to101010Note that we are not using anymore the inversion of $p$ to define the
local representation as in (6).

$\vec{\sigma}([q_{0}\\#_{\frac{1}{2}}\,q_{1}]^{\frac{1}{2}}A_{x}[p_{0}\\#_{\frac{1}{2}}\,p_{1}]^{-\frac{1}{2}})\preceq\frac{1}{2}\vec{\sigma}(q_{0}^{\frac{1}{2}}A_{x}p_{0}^{-\frac{1}{2}})+\frac{1}{2}\vec{\sigma}(q_{1}^{\frac{1}{2}}A_{x}p_{1}^{-\frac{1}{2}}).$

A detailed proof of this inequality can be found in [20, Lem. 3.2]. The
continuity, proven in Lemma 3, together with the mid-point convexity implies
convexity by a standard argument in convex analysis, see also [20, Rem. 3.3].
∎

The main result of this section is the following theorem.

###### Theorem 3

For every $\omega\in\mathcal{V}$, the function
$\vec{\mathcal{S}}_{f,\mu}:\mathcal{M}_{\omega}\rightarrow\mathfrak{a}^{+}$ is
continuous and geodesically cone-convex.

###### Proof.

The continuity of $\vec{\mathcal{S}}_{f,\mu}$ is a simple consequence of the
continuity of $(g,x)\mapsto\vec{\sigma}^{g}(\mathrm{d}f_{x})$ together with
the compactness of $M$. The cone-convexity directly follows from the cone-
convexity of the integrand together with the fact that the order $\preceq$ is
induced by a closed cone. ∎

Observe that the geodesic cone-convexity of $\vec{\mathcal{S}}_{f,\mu}$ is
equivalent to the geodesic convexity of the functions

$g\mapsto
s_{k,f,\mu}(g):=\int\log\omega_{k}^{g}(\mathrm{d}f_{x})\,\mathrm{d}\mu(x),\quad
k=1,2,\ldots,d,$

where

$\omega_{k}^{g}(\mathrm{d}f_{x}):=\prod_{i=1}^{k}\log\alpha_{i}^{g}(\mathrm{d}f_{x}).$

Moreover, a minimizer of the vector-valued optimization problem is the same as
a common minimizer of the scalar optimization problems

$\min_{g\in\mathcal{M}_{\omega}}s_{k,f,\mu}(g),\quad k=1,2,\ldots.d.$

In particular, we obtain the following corollary.

###### Corollary 2

Let $f$ be a $C^{2}$-diffeomorphism and assume that $\mu$ is an ergodic SRB
measure. Let $k$ denote the number of positive Lyapunov exponents of
$(M,f,\mu)$. Then, the measure-theoretic entropy $h_{\mu}(f)$ can be written
as the solution of a geodesically convex optimization problem as follows:

$h_{\mu}(f)=\inf_{g\in\mathcal{M}_{\omega}}s_{k,f,\mu}(g).$

###### Proof.

If $\mu$ is an ergodic measure, the measure-theoretic entropy of $f$ with
respect to $\mu$ is given by

$h_{\mu}(f)=\sum_{i=1}^{d}\max\\{0,\lambda_{i}(f)\\}=\sum_{i=1}^{k}\lambda_{i}(f)$

according to the entropy-characterization of SRB, developed in [22] by
Ledrappier and Young. Together with Theorem 2, this immediately implies the
statement of the corollary. ∎

Convex optimization problems are more well-behaved than general (nonlinear)
optimization problems in many respects. In particular, every local optimum is
also a global optimum, the solution set is convex, there exist efficient
numerical algorithms to solve them, and in the case of differentiability there
exist first-order optimality criteria. However, the convex optimization
problem considered in this paper still has a number of disadvantages which
makes it hard to solve:

  * •

The existence of a minimizer is not guaranteed.

  * •

It is vector-valued.

  * •

It is defined on an infinite-dimensional space.

  * •

It is defined on a Riemannian manifold (rather than a vector space).

  * •

It cannot be expected to be differentiable, since singular value functions are
not differentiable, in general.

Despite these drawbacks, the optimization problem also has some properties
which could turn out to be advantageous. For instance, we can prove that the
functions $s_{k,f,\mu}$ satisfy global Lipschitz estimates on
$\mathcal{M}_{\omega}$ with respect to the $L^{2}$-metric.

###### Theorem 4

For each $k\in\\{1,2,\ldots,d\\}$, the function
$s_{k,f,\mu}:\mathcal{M}_{\omega}\rightarrow\mathbb{R}$ satisfies the global
Lipschitz-like estimate

$|s_{k,f,\mu}(g^{1})-s_{k,f,\mu}(g^{2})|\leq\sqrt{k}\left(\int
d(g^{1}_{x},g^{2}_{x})^{2}\,\mathrm{d}\mu(x)\right)^{\frac{1}{2}},$ (15)

where $d(\cdot,\cdot)$ is the distance function on $\mathcal{S}^{+}_{d}$. If
$\mu$ is a volume measure induced by some Riemannian metric
$g^{0}\in\mathcal{M}_{\omega}$, then

$|s_{k,f,\mu}(g^{1})-s_{k,f,\mu}(g^{2})|\leq\sqrt{k}\,d_{L^{2}}(g^{1},g^{2}).$
(16)

That is, $s_{k,f,\mu}$ satisfies a global Lipschitz estimate with respect to
the $L^{2}$-distance.

###### Proof.

First, observe that by Jensen’s inequality we have

$(s_{k,f,\mu}(g^{1})-s_{k,f,\mu}(g^{2}))^{2}\leq\int_{M}\log^{2}\frac{\omega_{k}^{g^{1}}(\mathrm{d}f_{x})}{\omega_{k}^{g^{2}}(\mathrm{d}f_{x})}\,\mathrm{d}\mu(x).$

We split $M$ into the two subsets

$M_{1}:=\left\\{x\in
M:\omega_{k}^{g^{1}}(\mathrm{d}f_{x})>\omega_{k}^{g^{2}}(\mathrm{d}f_{x})\right\\},\quad
M_{2}:=M\backslash M_{1}.$

Then we have the following identities and inequalities:

$\displaystyle\int_{M_{1}}\log^{2}\frac{\omega_{k}^{g^{1}}(\mathrm{d}f_{x})}{\omega_{k}^{g^{2}}(\mathrm{d}f_{x})}\,\mathrm{d}\mu(x)$
$\displaystyle=\int_{M_{1}}\log^{2}\frac{\omega_{k}((g^{1}_{fx})^{-\frac{1}{2}}\mathrm{d}f_{x}(g^{1}_{x})^{\frac{1}{2}})}{\omega_{k}((g^{2}_{fx})^{-\frac{1}{2}}\mathrm{d}f_{x}(g^{2}_{x})^{\frac{1}{2}})}\,\mathrm{d}\mu(x)$
$\displaystyle=\int\log^{2}\frac{\omega_{k}((g^{1}_{fx})^{-\frac{1}{2}}(g^{2}_{fx})^{\frac{1}{2}}(g^{2}_{fx})^{-\frac{1}{2}}\mathrm{d}f_{x}(g^{2}_{x})^{\frac{1}{2}}(g^{2}_{x})^{-\frac{1}{2}}(g^{1}_{x})^{\frac{1}{2}})}{\omega_{k}((g^{2}_{fx})^{-\frac{1}{2}}\mathrm{d}f_{x}(g^{2}_{x})^{\frac{1}{2}})}\,\mathrm{d}\mu(x)$
$\displaystyle\leq\int_{M_{1}}\left(\log\omega_{k}((g^{1}_{fx})^{-\frac{1}{2}}(g^{2}_{fx})^{\frac{1}{2}})+\log\omega_{k}((g^{2}_{x})^{-\frac{1}{2}}(g^{1}_{x})^{\frac{1}{2}})\right)^{2}\mathrm{d}\mu(x)$
$\displaystyle\leq
2\int_{M_{1}}\left(\log^{2}\omega_{k}((g^{1}_{fx})^{-\frac{1}{2}}(g^{2}_{fx})^{\frac{1}{2}})+\log^{2}\omega_{k}((g^{2}_{x})^{-\frac{1}{2}}(g^{1}_{x})^{\frac{1}{2}})\right)\,\mathrm{d}\mu(x).$

In the first equality, we simply use the definition of $s_{k,f,\mu}$. The
second equality identifies $\mathrm{d}f_{x}$, $g^{1}_{x}$ and $g^{2}_{x}$ with
their corresponding local representations and uses Lemma 2. The third equality
is trival. The subsequent inequality is Horn’s inequality together with the
fact that $\log^{2}(\cdot)$ is increasing on $(1,\infty)$. The last inequality
simply uses that $(x+y)^{2}\leq 2x^{2}+2y^{2}$ for any $x,y\in\mathbb{R}$.

Now, observe that for any $a,b\in\mathcal{S}^{+}_{d}$ we have

$\displaystyle\log^{2}\omega_{k}(a^{-\frac{1}{2}}b^{\frac{1}{2}})$
$\displaystyle=\left(\sum_{i=1}^{k}\log\alpha_{i}(a^{-\frac{1}{2}}b^{\frac{1}{2}})\right)^{2}$
$\displaystyle\leq
k\sum_{i=1}^{k}\log^{2}\alpha_{i}(a^{-\frac{1}{2}}b^{\frac{1}{2}})\leq
k\sum_{i=1}^{d}\log^{2}\alpha_{i}(a^{-\frac{1}{2}}b^{\frac{1}{2}})$
$\displaystyle=k\|\vec{\sigma}(a^{-\frac{1}{2}}b^{\frac{1}{2}})\|_{2}^{2}=\frac{k}{4}\|\vec{d}(a,b)\|_{2}^{2}=\frac{k}{4}d(a,b)^{2}.$

Altogether, we obtain the estimate

$\int_{M_{1}}\log^{2}\frac{\omega_{k}^{g^{1}}(\mathrm{d}f_{x})}{\omega_{k}^{g^{2}}(\mathrm{d}f_{x})}\,\mathrm{d}\mu(x)\leq\frac{k}{2}\int_{M_{1}}\left(d(g^{1}_{fx},g^{2}_{fx})^{2}+d(g^{1}_{x},g^{2}_{x})^{2}\right)\,\mathrm{d}\mu(x).$

In a similar fashion, we derive that

$\int_{M_{2}}\log^{2}\frac{\omega_{k}^{g^{1}}(\mathrm{d}f_{x})}{\omega_{k}^{g^{2}}(\mathrm{d}f_{x})}\,\mathrm{d}\mu(x)\leq\frac{k}{2}\int_{M_{2}}\left(d(g^{2}_{fx},g^{1}_{fx})^{2}+d(g^{2}_{x},g^{1}_{x})^{2}\right)\,\mathrm{d}\mu(x).$

Because of the symmetry of the distance and the $f$-invariance of $\mu$,
summing the two integrals gives

$\displaystyle(s_{k,f,\mu}(g^{1})-s_{k,f,\mu}(g^{2}))^{2}\leq
k\int_{M}d(g^{1}_{x},g^{2}_{x})^{2}\,\mathrm{d}\mu(x),$

which is equivalent to the claimed inequality (15).

To prove (16), assume that $\mu$ is the volume measure induced by some
$g^{0}\in\mathcal{M}_{\omega}$. According to (15) and (11), it suffices to
show that $d_{x}(g^{1}_{x},g^{2}_{x})=d(g^{1}_{x},g^{2}_{x})$ for all
$g^{1},g^{2}\in\mathcal{M}_{\omega}$ and $x\in M$. By definition
$d_{x}(\cdot,\cdot)$ is the distance function on $(S^{0}_{2}M)_{x}$ induced by
the Riemannian metric

$\langle
a,b\rangle_{p}=\mathrm{tr}(p^{-1}ap^{-1}b)\sqrt{\det((g^{0}_{x})^{-1}p)},$

while $d(\cdot,\cdot)$ is the distance function induced by

$\langle a,b\rangle_{p}^{\prime}=\mathrm{tr}(p^{-1}ap^{-1}b).$

We identify the space of inner products on $T_{x}M$ with
$\mathcal{S}_{d}^{+}$. The submanifold of $\mathcal{S}_{d}^{+}$ consisting of
all matrices with the same determinant $c>0$ is totally geodesic. Hence, if
$\gamma:[0,1]\rightarrow\mathcal{S}^{+}_{d}$ is a geodesic connecting
$g^{1}_{x}$ with $g^{2}_{x}$, then $\det(\gamma(t))$ is the same for all
$t\in[0,1]$ and equals $\det(g^{0}_{x})$. It follows that
$\det((g^{0}_{x})^{-1}\gamma(t))=1$ for all $t$, showing that
$\langle\cdot,\cdot\rangle^{\prime}=\langle\cdot,\cdot\rangle$, and hence
$d_{x}(\cdot,\cdot)=d(\cdot,\cdot)$. ∎

###### Remark 6

The proven fact that $s_{k,f,\mu}$ is globally Lipschitz on
$\mathcal{M}_{\omega}$ if $\mu$ is a volume measure implies that there exists
a unique extension of $s_{k,f,\mu}$ to the metric completion of
$\mathcal{M}_{\omega}$ which obeys the same Lipschitz estimate. The completion
of $\mathcal{M}$ was studied in [11] and, roughly speaking, characterized as
the space of measurable metrics inducing a finite volume. In fact, it is
isometric to the space of $L^{2}$-mappings from the manifold $M$ to the
completion of the space of positive-definite symmetric matrices, as proven in
[7].

##  6 Towards a first-order optimality criterion

Since singular value functions do not depend smoothly on parameters, we cannot
expect the function $\vec{\mathcal{S}}_{f,\mu}$ (or $s_{k,f,\mu}$) to be
differentiable. However, the geodesic convexity implies that unilateral
directional derivatives exist (see [29, Ch. 3, Thm. 4.2]) and possibly we can
compute some generalized derivative, e.g., a subdifferential. This would lead
to a first-order criterion for optimality.

In this section, we only take the first step and compute directional
derivatives of the functions $s_{k,f,\mu}$. Moreover, we restrict ourselves to
the simplest situation in which all singular values have multiplicity one at
each point of the manifold. Although this assumption is not very realistic, it
will give us some idea of what we can expect a subgradient of $s_{k,f,\mu}$ to
look like.

###### Theorem 5

Let $g\in\mathcal{M}_{\omega}$ be a metric satisfying
$\alpha_{k}^{g}(\mathrm{d}f_{x})>\alpha_{k+1}^{g}(\mathrm{d}f_{x})$ for all
$x\in M$ and some $k\in\\{1,2,\ldots,d-1\\}$. Then, all directional
derivatives of $s_{k,f,\mu}$ at $g$ exist. For any $h\in
T_{g}\mathcal{M}_{\omega}$, the directional derivative of $s_{k,f,\mu}$ at $g$
in direction $h$ is given by

$\partial_{h}s_{k,f,\mu}(g)=\int\langle
Q^{\mathrm{l}}_{k,x}-Q^{\mathrm{r}}_{k,x},h_{x}\rangle_{g_{x}}\,\mathrm{d}\mu(x),$

where $Q^{\mathrm{l}}_{k,x}$ ($Q^{\mathrm{r}}_{k,x}$) denotes the orthogonal
projection in $(T_{x}M,g_{x})$ onto the subspace spanned by the first $k$ left
(right) singular vectors of $\mathrm{d}f_{f^{-1}x}$ ($\mathrm{d}f_{x}$).

###### Proof.

The proof is subdivided into six steps.

_Step 1:_ Let us write $s_{k,x}(g):=\log\omega_{k}^{g}(\mathrm{d}f_{x})$,
which is the integrand in the definition of $s_{k,f,\mu}(g)$. We first try to
differentiate $s_{k,x}$ in the hope that the order of differentiation and
integration can be interchanged. To this end, we choose a smooth curve
$\gamma$ in $\mathcal{M}_{\omega}$ with $\gamma(0)=g$, $\dot{\gamma}(0)=h$ and
decompose the function $s_{k,x}\circ\gamma$ as

$(s_{k,x}\circ\gamma)(t)=\ell_{k}\circ\alpha\circ\zeta_{x}(t),$

where

$\displaystyle\zeta_{x}(t)$
$\displaystyle=\gamma(t)_{fx}^{\frac{1}{2}}\mathrm{d}f_{x}\gamma(t)_{x}^{-\frac{1}{2}},\quad\zeta_{x}:\mathbb{R}\rightarrow\mathrm{GL}(d,\mathbb{R}),$
$\displaystyle\alpha(g)$
$\displaystyle:=(\alpha_{1}(g),\ldots,\alpha_{d}(g)),\quad\alpha:\mathrm{GL}(d,\mathbb{R})\rightarrow\mathbb{R}^{d},$
$\displaystyle\ell_{k}(\xi)$
$\displaystyle:=\sum_{i=1}^{k}\log\hat{\xi}_{i},\quad\ell_{k}:\mathbb{R}^{d}\rightarrow[-\infty,\infty).$

Here, some explanation is necessary. In the definition of $\zeta_{x}(t)$, we
treat $\gamma(t)_{x}$, $\gamma(t)_{fx}$ and $\mathrm{d}f_{x}$ as matrices,
where we think of their local representations with respect to appropriate
charts (with Lemma 2 in mind). The function $\alpha$ simply maps a matrix
$g\in\mathrm{GL}(d,\mathbb{R})$ to the vector of its ordinary singular values.
Finally, for a vector $\xi=(\xi_{1},\ldots,\xi_{d})\in\mathbb{R}^{d}$, we
write $\hat{\xi}_{1}\geq\cdots\geq\hat{\xi}_{d}$ for the absolute values of
its components, ordered from the largest to the smallest, and then define
$\ell_{k}(\xi)$ as above. Here, the value $-\infty$ is possible if at least
one component of $\xi$ is zero. Using Lemma 2, we then see that $s_{k,x}$ can
be decomposed in the suggested way (keeping mind that we are not using anymore
the inversion of $p$ to define the local representation as in (6)).

_Step 2:_ We compute the derivative of $\zeta_{x}$ at $t=0$. By the product
rule, we have

$\dot{\zeta}_{x}(t)=\left[\frac{\partial}{\partial
t}\Bigl{|}_{t=0}\gamma(t)_{fx}^{\frac{1}{2}}\right]\mathrm{d}f_{x}\gamma(0)_{x}^{-\frac{1}{2}}+\gamma(0)_{fx}^{\frac{1}{2}}\mathrm{d}f_{x}\left[\frac{\partial}{\partial
t}\Bigl{|}_{t=0}\gamma(t)_{x}^{-\frac{1}{2}}\right].$

With the notation ${\bf r}:p\mapsto p^{\frac{1}{2}}$ and ${\bf i}:p\mapsto
p^{-1}$ (both defined on $\mathcal{S}^{+}_{d}$), we can write this as

$\dot{\zeta}_{x}(t)=\left[\mathrm{d}{\bf
r}_{\gamma(0)_{fx}}\dot{\gamma}(0)_{fx}\right]\mathrm{d}f_{x}\gamma(0)_{x}^{-\frac{1}{2}}+\gamma(0)_{fx}^{\frac{1}{2}}\mathrm{d}f_{x}\left[\mathrm{d}({\bf
i}\circ{\bf r})_{\gamma(0)_{x}}\dot{\gamma}(0)_{x}\right].$

Using that $\mathrm{d}{\bf i}_{p}h=-p^{-1}hp^{-1}$ and $\mathrm{d}{\bf
r}_{p}h=L_{p}^{-1}(h)$, where $L_{p}$ is the Lyapunov operator111111Since
$p^{\frac{1}{2}}$ has only strictly positive eigenvalues, the Lyapunov
operator is invertible.

$L_{p}(X):=p^{\frac{1}{2}}X+Xp^{\frac{1}{2}},$

we can finally write the derivative of $\zeta_{x}$ as

$\dot{\zeta}_{x}(t)=L_{g_{fx}}^{-1}(h_{fx})\mathrm{d}f_{x}g_{x}^{-\frac{1}{2}}-g_{fx}^{\frac{1}{2}}\mathrm{d}f_{x}g_{x}^{-\frac{1}{2}}L_{g_{x}}^{-1}(h_{x})g_{x}^{-\frac{1}{2}}.$
(17)

Observe that $X:=L_{g_{x}}^{-1}(h_{x})$ is a symmetric matrix, since it solves
$g_{x}^{\frac{1}{2}}X+Xg_{x}^{\frac{1}{2}}=h_{x}$ and by symmetry of $p$ and
$h_{x}$, also $X^{\top}$ solves this equation. Hence,
$X=L_{g_{x}}^{-1}(h_{x})=X^{\top}$.

_Step 3:_ Now, we consider the composed function $\ell_{k}\circ\alpha$.
According to [23, Prop. 6.2], $\ell_{k}\circ\alpha$ is differentiable at
$X\in\mathbb{R}^{d\times d}$ if and only if $\ell_{k}$ is differentiable at
$\alpha(X)$. The latter is the case if $\alpha_{k}(X)>\alpha_{k+1}(X)$, which
holds in our case by assumption, since
$\alpha_{k}(X)=\alpha_{k}(\zeta_{x}(0))=\alpha_{k}^{g}(\mathrm{d}f_{x})$. In
this case,

$\nabla(\ell_{k}\circ\alpha)(X)=U^{\top}\mathrm{Diag}\left(\frac{1}{\alpha_{1}(X)},\ldots,\frac{1}{\alpha_{k}(X)},0,\ldots,0\right)V,$

where $X=U^{\top}\mathrm{Diag}(\alpha(X))V$ is a singular value decomposition.

_Step 4:_ To compute the derivative of $s_{k,x}\circ\gamma$, we need to
compose the two derivatives $\nabla(\ell_{k}\circ\alpha)(\zeta_{x}(0))$ and
$\dot{\zeta}_{x}(0)$ in the proper way. This is the Euclidean inner product on
$\mathbb{R}^{d\times d}$, which is given by $\langle
X,Y\rangle=\mathrm{tr}[X^{\top}Y]$ (also known as _Frobenius inner product_).
For brevity, let us write $S_{x}:=\nabla(\ell_{k}\circ\alpha)(\zeta_{x}(0))$
and $Z_{x}:=L_{g_{x}}^{-1}(h_{x})$. Then

$\displaystyle\frac{\partial}{\partial
t}\Bigl{|}_{t=0}(s_{k,x}\circ\gamma)(0)=\mathrm{tr}[S_{x}^{\top}\dot{\zeta}_{x}(0)]$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq_zeta_derivative}}}{{=}}\mathrm{tr}\left[S_{x}^{\top}\left(Z_{fx}\mathrm{d}f_{x}g_{x}^{-\frac{1}{2}}-g_{fx}^{\frac{1}{2}}\mathrm{d}f_{x}g_{x}^{-\frac{1}{2}}Z_{x}g_{x}^{-\frac{1}{2}}\right)\right]$
$\displaystyle=\mathrm{tr}\left[S_{x}^{\top}Z_{fx}\mathrm{d}f_{x}g_{x}^{-\frac{1}{2}}\right]-\mathrm{tr}\left[S_{x}^{\top}g_{fx}^{\frac{1}{2}}\mathrm{d}f_{x}g_{x}^{-\frac{1}{2}}Z_{x}g_{x}^{-\frac{1}{2}}\right]$
$\displaystyle=\mathrm{tr}\left[Z_{fx}\mathrm{d}f_{x}g_{x}^{-\frac{1}{2}}S_{x}^{\top}\right]-\mathrm{tr}\left[Z_{x}g_{x}^{-\frac{1}{2}}S_{x}^{\top}g_{fx}^{\frac{1}{2}}\mathrm{d}f_{x}g_{x}^{-\frac{1}{2}}\right]$
$\displaystyle=\mathrm{tr}\left[Z_{fx}g_{fx}^{-\frac{1}{2}}\zeta_{x}(0)S_{x}^{\top}\right]-\mathrm{tr}\left[Z_{x}g_{x}^{-\frac{1}{2}}S_{x}^{\top}\zeta_{x}(0)\right].$

From the definition of $S_{x}$, it follows that

$\displaystyle\zeta_{x}(0)S_{x}^{\top}=U_{\zeta_{x}(0)}^{\top}(I_{k\times
k}\oplus 0_{(d-k)\times(d-k)})U_{\zeta_{x}(0)},$

where
$\zeta_{x}(0)=U_{\zeta_{x}(0)}^{\top}\mathrm{Diag}(\alpha(\zeta_{x}(0)))V_{\zeta_{x}(0)}$
is the chosen singular value decomposition. Analogously,

$S_{x}^{\top}\zeta_{x}(0)=V_{\zeta_{x}(0)}^{\top}(I_{k\times k}\oplus
0_{(d-k)\times(d-k)})V_{\zeta_{x}(0)}.$

We can write these matrices in a more compact way as
$\bar{U}_{fx}\bar{U}_{fx}^{\top}$ and $\bar{V}_{x}\bar{V}_{x}^{\top}$, where
$\bar{U}_{fx}$ is the $(d\times k)$-matrix whose columns are the first $k$
left singular vectors of $\zeta_{x}(0)$, and $\bar{V}_{x}$ is defined in the
same way via the right singular vectors. Observe that
$\bar{U}_{fx}\bar{U}_{fx}^{\top}$ is the orthogonal projection onto the
subspace spanned by the first $k$ left singular vectors of $\zeta_{x}(0)$
(analogously for $\bar{V}_{x}\bar{V}_{x}^{\top}$). We thus have

$\frac{\partial}{\partial
t}\Bigl{|}_{t=0}(s_{k,x}\circ\gamma)(0)=\mathrm{tr}\left[Z_{fx}g_{fx}^{-\frac{1}{2}}\bar{U}_{fx}\bar{U}_{fx}^{\top}\right]-\mathrm{tr}\left[Z_{x}g_{x}^{-\frac{1}{2}}\bar{V}_{x}\bar{V}_{x}^{\top}\right].$

Now, recall that $g_{x}^{\frac{1}{2}}Z_{x}+Z_{x}g_{x}^{\frac{1}{2}}=h_{x}$,
implying
$Z_{x}g_{x}^{-\frac{1}{2}}+g_{x}^{-\frac{1}{2}}Z_{x}=g_{x}^{-\frac{1}{2}}h_{x}g_{x}^{-\frac{1}{2}}$.
Observe that for arbitrary symmetric matrices $a,p,x,y$ with $px+xp=y$, we
have
$\mathrm{tr}[apx]=\mathrm{tr}[(apx)^{\top}]=\mathrm{tr}[xpa]=\mathrm{tr}[axp]$,
and hence
$\mathrm{tr}[ay]=\mathrm{tr}[a(px)]+\mathrm{tr}[a(xp)]=2\mathrm{tr}[apx]$.
This yields

$\displaystyle\begin{split}&\frac{\partial}{\partial
t}\Bigl{|}_{t=0}(s_{k,x}\circ\gamma)(0)=\frac{1}{2}\mathrm{tr}\left[g_{fx}^{-\frac{1}{2}}h_{fx}g_{fx}^{-\frac{1}{2}}\bar{U}_{fx}\bar{U}_{fx}^{\top}\right]-\frac{1}{2}\mathrm{tr}\left[g_{x}^{-\frac{1}{2}}h_{x}g_{x}^{-\frac{1}{2}}\bar{V}_{x}\bar{V}_{x}^{\top}\right]\\\
&=\frac{1}{2}\mathrm{tr}\left[g_{fx}^{\frac{1}{2}}g_{fx}^{-1}h_{fx}g_{fx}^{-1}g_{fx}^{\frac{1}{2}}\bar{U}_{fx}\bar{U}_{fx}^{\top}\right]-\frac{1}{2}\mathrm{tr}\left[g_{x}^{\frac{1}{2}}g_{x}^{-1}h_{x}g_{x}^{-1}g_{x}^{\frac{1}{2}}\bar{V}_{x}\bar{V}_{x}^{\top}\right]\\\
&=\frac{1}{2}\langle
g_{fx}^{\frac{1}{2}}\bar{U}_{fx}\bar{U}_{fx}^{\top}g_{fx}^{\frac{1}{2}},h_{fx}\rangle_{g_{fx}}-\frac{1}{2}\langle
g_{x}^{\frac{1}{2}}\bar{V}_{x}\bar{V}_{x}^{\top}g_{x}^{\frac{1}{2}},h_{x}\rangle_{g_{x}}.\end{split}$
(18)

_Step 5:_ We have to understand how the matrices
$\bar{U}_{fx}\bar{U}_{fx}^{\top}$ and $\bar{V}_{x}\bar{V}_{x}^{\top}$ change,
when we change the charts. Recall that $\bar{U}_{fx}$ is a matrix built from
left singular vectors of $\zeta_{x}(0)$, i.e. the eigenvectors of
$\zeta_{x}(0)\zeta_{x}(0)^{\top}$. One can easily show that with respect to
another pair of charts, the corresponding symmetric matrix becomes

$\tilde{\zeta}_{x}(0)\tilde{\zeta}_{x}(0)^{\top}=o\zeta_{x}(0)\zeta_{x}(0)^{\top}o^{\top}$

with an orthogonal matrix $o$ of the form

$o=(b\ast g_{fx})^{-\frac{1}{2}}bg_{fx}^{\frac{1}{2}},$

where $b$ is the derivative of the coordinate change at $fx$. Then, in the new
pair of charts, $\bar{U}_{fx}\bar{U}_{fx}^{\top}$ becomes
$o\bar{U}_{fx}\bar{U}_{fx}^{\top}o^{\top}$. Hence,

$(b\ast g_{fx})^{\frac{1}{2}}o\bar{U}_{fx}\bar{U}_{fx}^{\top}o^{\top}(b\ast
g_{fx})^{\frac{1}{2}}=bg_{fx}^{\frac{1}{2}}\bar{U}_{fx}\bar{U}_{fx}^{\top}g_{fx}^{\frac{1}{2}}b^{\top}=b\ast
g_{fx}^{\frac{1}{2}}\bar{U}_{fx}\bar{U}_{fx}^{\top}g_{fx}^{\frac{1}{2}}.$

This computation shows that
$g_{fx}^{\frac{1}{2}}\bar{U}_{fx}\bar{U}_{fx}^{\top}g_{fx}^{\frac{1}{2}}$ is
the local representation of a global section of $S^{0}_{2}M$, i.e. a symmetric
(not necessarily smooth) $(0,2)$-tensor field. The same can be shown for
$g_{x}^{\frac{1}{2}}\bar{V}_{x}\bar{V}_{x}^{\top}g_{x}^{\frac{1}{2}}$. What
are these global sections? Recall from the proof of Lemma 2 that a local
representation of $\mathrm{d}f_{x}^{*}\mathrm{d}f_{x}$ is given by
$p_{x}^{-1}A_{x}^{\top}p_{fx}A_{x}$, where $A_{x}$ is a local representation
of $\mathrm{d}f_{x}$ and $p_{x}$ ($p_{fx}$) one of $g_{x}$ ($g_{fx}$). The
eigenvectors $v_{1},\ldots,v_{d}$ of $\mathrm{d}f_{x}^{*}\mathrm{d}f_{x}$ then
have local representations $\tilde{v}_{1},\ldots,\tilde{v}_{d}$, which are the
eigenvectors of $p_{x}^{-1}A_{x}^{\top}p_{fx}A_{x}=:C_{x}$. The matrix
$\zeta_{x}(0)^{\top}\zeta_{x}(0)$ is then given by
$p_{x}^{\frac{1}{2}}C_{x}p_{x}^{-\frac{1}{2}}$, and hence its eigenvectors are
$p_{x}^{\frac{1}{2}}\tilde{v}_{1},\ldots,p_{x}^{\frac{1}{2}}\tilde{v}_{d}$
(which form an orthogonal basis with respect to the standard Euclidean inner
product). If $Q_{x}$ is the orthogonal projection onto the linear subspace of
$\mathbb{R}^{d}$ spanned by the first $k$ eigenvectors of
$\zeta_{x}(0)^{\top}\zeta_{x}(0)$, then for $i=1,\ldots,k$ we have
$(p_{x}^{\frac{1}{2}}Q_{x}p_{x}^{\frac{1}{2}})\tilde{v}_{i}=p_{x}\tilde{v}_{i}$
and for $i=k+1,\ldots,d$ we have
$(p_{x}^{\frac{1}{2}}Q_{x}p_{x}^{\frac{1}{2}})\tilde{v}_{i}=0$. Hence,

$\displaystyle\langle(p_{x}^{\frac{1}{2}}Q_{x}p_{x}^{\frac{1}{2}})\tilde{v}_{i},\tilde{v}_{j}\rangle=\left\\{\begin{array}[]{cc}\langle(p_{x}^{\frac{1}{2}}\tilde{v}_{i}),(p_{x}^{\frac{1}{2}}\tilde{v}_{j})\rangle=\delta_{ij}&\mbox{for
}i=1,\ldots,k,\\\ 0&\mbox{otherwise}.\end{array}\right.$

It follows that
$g_{fx}^{\frac{1}{2}}\bar{U}_{fx}\bar{U}_{fx}^{\top}g_{fx}^{\frac{1}{2}}$ is
the local representation of the orthogonal projection $Q^{\mathrm{l}}_{k,fx}$
onto the subspace of $T_{fx}M$ which is spanned by the first $k$ left singular
vectors of $\mathrm{d}f_{x}$. Analogously,
$g_{x}^{\frac{1}{2}}\bar{V}_{x}\bar{V}_{x}^{\top}g_{x}^{\frac{1}{2}}$
represents the orthogonal projection $Q^{\mathrm{r}}_{k,x}$ onto the subspace
of $T_{x}M$ spanned by the first $k$ right singular vectors of
$\mathrm{d}f_{x}$. Here, of course, the orthogonality is defined in terms of
the metric $g=\gamma(0)$.

_Step 6:_ Using the invariance of the measure $\mu$, we obtain from (18) via
integration that

$\displaystyle\int\frac{\partial}{\partial
t}\Bigl{|}_{t=0}(s_{k,x}\circ\gamma)(0)\,\mathrm{d}\mu(x)=\frac{1}{2}\int\langle
Q^{\mathrm{l}}_{k,x}-Q^{\mathrm{r}}_{k,x},h_{x}\rangle_{g_{x}}\,\mathrm{d}\mu(x).$

It remains to prove that integration and derivative on the left-hand side can
be interchanged. To show this, first observe that our assumption about the
spectral gap carries over from $g$ to nearby metrics $\gamma(t)$,
$|t|\leq\varepsilon$, by Lemma 3. Hence, the above formula holds for all
$(t,x)\in[-\varepsilon,\varepsilon]\times M$. Thus, the derivative of
$s_{k,x}\circ\gamma$ at any $t\in[-\varepsilon,\varepsilon]$ has the same form
as above, but the orthogonal projections are then defined in terms of
$\gamma(t)$ instead of $\gamma(0)$:

$\frac{\partial}{\partial
t}\Bigl{|}_{t=0}(s_{k,x}\circ\gamma)(t)=\frac{1}{2}\langle
Q^{\mathrm{l}}_{k,x}(t)-Q^{\mathrm{r}}_{k,x}(t),\dot{\gamma}(t)_{x}\rangle_{\gamma(t)_{x}}.$

Here, we can estimate

$\displaystyle\left|\langle
Q^{\mathrm{l}}_{k,x}(t)-Q^{\mathrm{r}}_{k,x}(t),\dot{\gamma}(t)_{x}\rangle_{\gamma(t)_{x}}\right|$
$\displaystyle\leq\left(\|Q^{\mathrm{l}}_{k,x}(t)\|_{\gamma(t)_{x}}+\|Q^{\mathrm{r}}_{k,x}(t)\|_{\gamma(t)_{x}}\right)\cdot\|\dot{\gamma}(t)_{x}\|_{\gamma(t)_{x}}.$

The term $\|\dot{\gamma}(t)_{x}\|_{\gamma(t)_{x}}$ is continuous in $(t,x)$,
and thus can be estimated by a constant $C$ on a compact set of the form
$[-\varepsilon,\varepsilon]\times M$. The orthogonal projections have norm
$1$, hence we can estimate the complete term by a constant. We conclude that
the Leibniz rule can be applied to interchange the order of integration and
differentiation. The proof is complete. ∎

###### Remark 7

In the case when $\mu$ is a volume measure, coming from a metric which induces
the volume form $\omega$, we would expect (in the setup of the theorem) that
$s_{k,f,\mu}$ is differentiable at $g$ with gradient $x\mapsto
Q^{\mathrm{l}}_{k,x}-Q^{\mathrm{r}}_{k,x}$ (which may be non-smooth and live
in some metric space completion). Hence, a vanishing gradient would mean that
$Q^{\mathrm{l}}_{k,x}=Q^{\mathrm{r}}_{k,x}$, which is some sort of alignment
of derivatives. If this holds for all $k$, it can also be interpreted as a
symmetry property, since the left and right singular vectors of a real square
matrix are the same if and only if the matrix is symmetric. This is only an
analogy, however, because here we are not dealing with a single matrix but the
composition of different linear operators (except at the fixed points of $f$).

##  7 Summary and outlook

We have shown that the vector of averaged Lyapunov exponents of a smooth
measure-preserving system on a compact manifold $M$ is the infimum of a
geodesically cone-convex function, defined on the space of smooth Riemannian
metrics on $M$ which preserve a given volume form. Several research directions
are conceivable to make this a fruitful approach for a better theoretical
understanding or numerical computations. These include:

  1. (1)

_Finding criteria for the existence of minimizers_ ; if such minimizers can be
shown to exist only in the completion of $\mathcal{M}_{\omega}$, an important
question concerns their regularity.

  2. (2)

_Computation of subgradients for the derivation of a first-order optimality
criterion_. Maybe in some cases it is possible, to find an optimal metric by
solving the equation defined by the first-order criterion.

  3. (3)

_Finding ways to restrict the optimization problem to one or several finite-
dimensional convex problems_. The approach for doing this that comes first to
mind is probably restricting the function $\vec{\mathcal{S}}_{f,\mu}$ to
finite-dimensional totally geodesic submanifolds of $\mathcal{M}_{\omega}$.
However, it is very likely that there are none except for possibly low-
dimensional ones. One way to find such submanifolds is to consider fixed point
components of isometries. If we consider $\mathcal{M}$ instead of
$\mathcal{M}_{\omega}$, we know that the diffeomorphism group of $M$ acts on
$\mathcal{M}$ by isometries. A fixed point of the action of one diffeomorphism
$\varphi$ is a metric in which $\varphi$ is an isometry. So, one possible
construction of a totally geodesic submanifold is

$S:=\\{g\in\mathcal{M}:f\in\mathrm{Isom}(M,g)\mbox{\ for all\ }f\in F\\},$

where $F\subset\mathrm{Diff}(M)$ is any subset. As an example, consider a
manifold $M$ which is a Lie group and let $F$ be the group of left
translations. Then $S$ is the space of left-invariant metrics which is finite-
dimensional and can be identified with $\mathcal{S}^{+}_{d}$. In general, to
obtain a finite-dimensional $S$ by the above construction, one would expect
that the orbits $Fx$ all have to be dense in $M$. But then, any metric in $S$
is determined by its value at a single point. Another way to obtain a finite-
dimensional problem is to start with low-dimensional totally geodesic
submanifold, solve the problem there, then use the solution as the initial
guess for another finite-dimensional problem (in some other submanifold) and
continue like this in the hope that the process converges to a globally
(nearly) optimal metric.

## Acknowledgements

I express my sincere gratitude to Dr. Nicola Cavallucci for his continuing
interest in my paper and helping me to gain a better understanding of the
space $\mathcal{M}$ and its geometry.

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