# Lipschitz Carnot-Carathéodory structures and their limits

Gioacchino Antonelli  ,  Enrico Le Donne  and  Sebastiano Nicolussi Golo
Gioacchino Antonelli
Scuola Normale Superiore, Piazza dei Cavalieri, 7, Pisa, 56126
<EMAIL_ADDRESS>Enrico Le Donne
Department of Mathematics, University of Fribourg, Chemin du Musée 23, 1700
Fribourg, Switzerland & University of Jyväskylä, Department of Mathematics and
Statistics, P.O. Box (MaD), FI-40014, Finland<EMAIL_ADDRESS>Sebastiano Nicolussi-Golo
University of Jyväskylä, Department of Mathematics and Statistics, P.O. Box
(MaD), FI-40014, Finland<EMAIL_ADDRESS>
(Date: .)

###### Abstract.

In this paper we discuss the convergence of distances associated to converging
structures of Lipschitz vector fields and continuously varying norms on a
smooth manifold. We prove that, under a mild controllability assumption on the
limit vector-fields structure, the distances associated to equi-Lipschitz
vector-fields structures that converge uniformly on compact subsets, and to
norms that converge uniformly on compact subsets, converge locally uniformly
to the limit Carnot-Carathéodory distance.

In the case in which the limit distance is boundedly compact, we show that the
convergence of the distances is uniform on compact sets. We show an example in
which the limit distance is not boundedly compact and the convergence is not
uniform on compact sets.

We discuss several examples in which our convergence result can be applied.
Among them, we prove a subFinsler Mitchell’s Theorem with continuously varying
norms, and a general convergence result for Carnot-Carathéodory distances
associated to subspaces and norms on the Lie algebra of a connected Lie group.

###### Key words and phrases:

sub-Finsler geometry; sub-Riemannian geometry; Lipschitz vector fields;
Mitchell’s Theorem.

###### 2010 Mathematics Subject Classification:

28A75; 53C60; 53C17; 53C23; 49J52; 26A16.

###### Contents

  1. 1 Introduction
  2. 2 The End-point map for Lipschitz-vector-field structures
    1. 2.1 Lipschitz-vector-field structures on a manifold
    2. 2.2 Continuity of the End-point map
    3. 2.3 Essentially non-holonomic Lipschitz distributions
    4. 2.4 Uniform openness of the End-point map
  3. 3 Sub-Finsler distances for Lipschitz-vector-field structures
    1. 3.1 Continuous norms along a segment
    2. 3.2 Energy and length
    3. 3.3 Carnot-Carathéodory distances
    4. 3.4 CC distances and sub-Finsler lengths of curves
    5. 3.5 Limits of CC distances
  4. 4 Examples
    1. 4.1 Asymptotic cone of the Riemannian Heisenberg group
    2. 4.2 Tangents to sub-Finsler manifolds
    3. 4.3 Left-invariant CC distances on Lie groups
    4. 4.4 Limit of sub-Finsler distances on a manifold
  5. A Gronwall Lemma
  6. B Lemma about open mappings
  7. C A shorter proof of the convergence result for smooth vector fields

##  1\. Introduction

This paper deals with the following general problem. Let $M$ be a smooth
manifold endowed with a family of vector fields and a continuously varying
norm on the tangent spaces. Let us consider the length distance associated to
the trajectories that infinitesimally follow such a family. What are the
weakest notion of convergence and the most general assumptions on the family
of vector fields and the norm that ensure the uniform convergence of the
associated length distances?

Such a question is natural while studying metric geometry. For example, fixed
a continuously varying norm on the tangents, a sub-Finsler distance on a
manifold is defined as the length distance associated to a bracket-generating
family of smooth vector fields. It is a classical fact that a sub-Finsler
distance can be approximated from below by increasing sequences of Finsler
distances, see, e.g., [9, 10]. Nevertheless, understanding whether a specific
approximation of the vector fields (and, possibly, of the norm) gives raise to
the convergence of the associated length distances seems not to have been
deeply studied in the literature. A by-product of our study goes in this
direction, since it gives an effective tool to approximate subFinsler
distances with Finsler distances in a controlled way. We refer the reader to
the examples discussed in Theorem 4.5 and Theorem 1.6, which are consequences
of the main Theorem 1.4 below.

Another situation in which the convergence of the distance associated to
converging subFinsler structures emerge is while studying asymptotic or
tangent cones of subFinsler structures, see the celebrated works of Mitchell
and Bellaiche [12, 4] (and the account in [7]), and the work of Pansu [14]. We
refer the reader to the examples discussed in Proposition 4.1, and Theorem 1.5
below.

We now introduce the language that we will adopt in the paper.

Let us fix from now on a finite-dimensional Banach space $\mathbb{E}$, and a
smooth manifold $M$. A Lipschitz-vector-field structure
$\mathtt{f}:M\to\mathbb{E}^{*}\otimes TM$ on $M$ modelled by $\mathbb{E}$ is a
Lipschitz choice, for every point $p\in M$, of a linear map between
$\mathbb{E}$ and $T_{p}M$, see Definition 2.3. We say that a sequence of
Lipschitz-vector-field structure $\\{\mathtt{f}_{n}\\}_{n\in\mathbb{N}}$
converges to a Lipschitz-vector-field structure $\mathtt{f}_{\infty}$ if, on
every compact subset of $M$, $\\{\mathtt{f}_{n}\\}_{n\in N}$ is an equi-
Lipschitz family that converges to $\mathtt{f}_{\infty}$ uniformly, see
Definition 2.2 for details. Attached to a Lipschitz-vector-field structure
$\mathtt{f}$ we have the notion of an End-point map, as follows.

######  Definition 1.1 (End-point map).

Let $\mathbb{E}$ be a finite-dimensional Banach space, and $M$ be a smooth
manifold. Let $\mathtt{f}$ be a Lipschitz-vector-field structure on $M$
modelled by $\mathbb{E}$, and let $u\in L^{\infty}([0,1];\mathbb{E})$. For
$o\in M$, we define the _End-point map_

$\mathrm{End}^{\mathtt{f}}_{o}[u]:=\mathrm{End}(o,\mathtt{f},u):=\gamma(1),$

where $\gamma:[0,1]\to M$ is the solution of the Cauchy problem

(1) $\begin{cases}\gamma^{\prime}(t)&=\mathtt{f}|_{\gamma(t)}[u(t)],\\\
\gamma(0)&=o,\end{cases}$

whenever it exists. Notice that $\gamma(t)=\mathrm{End}^{\mathtt{f}}_{o}(tu)$
for every $t\in[0,1]$ whenever the End-point map is well-defined.

Let $\mathtt{f}$ be a Lipschitz-vector-field structure on $M$ modelled by
$\mathbb{E}$. We say that $N:M\times\mathbb{E}\to\mathbb{R}$ is a continuously
varying norm on $M\times\mathbb{E}$ if $N$ is continuous, and $N(p,\cdot)$ is
a norm on $\mathbb{E}$ for every $p\in M$. We aim now at describing how the
couple $(\mathtt{f},N)$ gives raise to a distance on $M$.

First, we define the energy and the length functionals working on the space
$L^{\infty}([0,1];\mathbb{E})$, which we sometimes call space of controls. We
stress that in our approach the energy and the length are not defined at the
level of the curves, but instead on the controls that define them. In
particular, if $o\in M$, $u\in L^{\infty}([0,1];\mathbb{E})$,
$(o,\mathtt{f},u)$ is in the domain of the End-point map, see Definition 1.1,
and $N$ is a continuously varying norm on $M\times\mathbb{E}$, we define the
length $\ell$ (resp., the energy $\mathtt{J}$) associated to
$(o,\mathtt{f},u,N)$ to be $\int_{0}^{1}N(\gamma(t),u(t))\,\mathrm{d}t$
(resp., $\operatorname*{{esssup}}_{t\in[0,1]}N(\gamma(t),u(t))$), where
$\gamma(t):=\mathrm{End}^{\mathtt{f}}_{o}(tu)$. Given the notion of energy, we
can define the distance as follows.

######  Definition 1.2 (CC distance).

Let $\mathbb{E}$ be a finite-dimensional Banach space, and $M$ be a smooth
manifold. Let $\mathtt{f}$ be a Lipschitz-vector-field structure on $M$
modelled by $\mathbb{E}$, and let $N:M\times\mathbb{E}\to[0,+\infty)$ be a
continuously varying norm. We define _the Carnot-Carathéodory distance_ , or
_CC distance_ , between $p$ and $q$ as follows

(2)
$d_{(\mathtt{f},N)}(p,q):=\inf\\{\mathtt{J}(p,\mathtt{f},u,N):\mathrm{End}^{\mathtt{f}}_{p}(u)=q\\},$

where $\mathtt{J}(p,\mathtt{f},u,N)$ is the above defined energy associated to
$(p,\mathtt{f},u,N)$.

In Lemma 3.5 we shall show that a constant-speed reparametrization of a curve
always exists, hence we can also equivalently take the infimum of the lengths
in (2). We remark that, without any further hypotheses on the couple
$(\mathtt{f},N)$, it might happen that $d_{(\mathtt{f},N)}(p,q)=+\infty$ for
some points $p,q\in M$.

In Corollary 3.19, we shall show that, if $\mathtt{f}$ is a Lipschitz-vector-
field structure on $M$ modelled by $\mathbb{E}$, and
$N:M\times\mathbb{E}\to[0,+\infty)$ is a continuously varying norm, then, for
every $p,q\in M$, we have also

$d_{(\mathtt{f},N)}(p,q)=\inf\left\\{\int_{0}^{1}|\gamma^{\prime}(t)|_{(\mathtt{f},N)}\,\mathrm{d}t:\begin{array}[]{l}\gamma:[0,1]\to
M\text{ absolutely continuous,}\\\ \text{with }\gamma(0)=p,\
\gamma(1)=q\end{array}\right\\},$

where

(3) $|v|_{(\mathtt{f},N)}:=\inf\\{N(p,u):u\in\mathbb{E},\
\mathtt{f}(p,u)=v\\},\quad\text{for $p\in M$ and $v\in T_{p}M$}.$

This means that the definition of the distance $d_{(\mathtt{f},N)}$ working
with the controls, i.e., the one given in (2), is equivalent to the definition
of the distance as the infimum of the lengths evaluated with respect to the
natural sub-Finsler metric associated to $(\mathtt{f},N)$, i.e., (3).

We now aim at understanding which kind of convergence is expected from the
sequence of distances $\\{d_{(\mathtt{f}_{n},N_{n})}\\}_{n\in\mathbb{N}}$ when
we have that the sequence $\\{(\mathtt{f}_{n},N_{n})\\}_{n\in\mathbb{N}}$
converges. The key hypothesis in order to have the local uniform convergence
of the distances is a kind of essential non-holonomicity of the limit vector-
fields structure $\mathtt{f}_{\infty}$, which we next introduce.

First of all we introduce the notion of essentially open map. We say that a
continuous map $f:M\to N$ between two topological manifolds of the same
dimension $k$ is essentially open at $p\in M$ at scale $U$ if $U$ is a
neighborhood of $p$ homeomorphic to the $k$-dimensional Euclidean ball, with
$\partial U$ homeomorphic to the sphere $\mathbb{S}^{k-1}$, and there exists
$V$ a neighborhood of $f(p)$ homeomorphic to the $k$-dimensional Euclidean
ball, such that $f(\partial U)\subset V\setminus f(p)$ and the map $f:\partial
U\to V\setminus f(p)$ induces a nonconstant map between the $(k-1)$-homology
groups, see Definition 2.10.

Notice that the definition of essential openness at $p\in M$ does depend on
the scale $U\ni p$. Notice also that if the map induced between the local
homology of $p$ and $f(p)$ is not trivial, i.e.,
$f_{\star}:H^{k}(M,M\setminus\\{p\\})\to H^{k}(N,N\setminus f(p))$ is nonzero,
then $f$ is essentially open at $p\in M$ at every sufficiently small
neighborhood $U$ of $p$. Essential opennes at a point $p$ at scale $U$ does
not imply openness at $p$, but only that $f(p)$ is in the interior of $f(U)$,
cf. Remark 2.11. Still a local homeomorphism at $p$ is indeed essentially open
at $p$ at some scale $U\ni p$. We further stress that

Then we are ready to give the following definition. We recall that we denote
by $\Phi_{X}^{t}$ the flow at time $t$ of a vector field $X$ on the smooth
manifold $M$, whenever it exists.

######  Definition 1.3 (Essentially non-holonomic).

Let $M$ be a smooth manifold of dimension $m$ and let $\mathscr{F}$ be a
family of Lipschitz vector fields on $M$. We say that $\mathscr{F}$ is
_essentially non-holonomic_ if for every $T>0$ and every $o\in M$, there are
$X_{1},\dots,X_{m}\in\mathscr{F}$ and $\hat{t}\in\mathbb{R}^{m}$ with
$|\hat{t}|_{1}<T$ such that there exists a neighborhood
$\Omega_{\hat{t}}\subseteq B(0,T)\subseteq\mathbb{R}^{m}$ of $\hat{t}$ for
which the map

$\phi(t_{1},\dots,t_{m}):=\Phi^{t_{m}}_{X_{m}}\circ\dots\circ\Phi^{t_{1}}_{X_{1}}(o),$

is defined on $\Omega_{\hat{t}}$ and, when restricted to $\Omega_{\hat{t}}$,
is an essentially open map at $\hat{t}$ in a neighborhood of it.

A Lipschitz-vector-field structure $\mathtt{f}$ on $M$ modelled by a finite-
dimensional Banach space $\mathbb{E}$ is said to be essentially non-holonomic
if there is a basis $(e_{1},\dots,e_{r})$ of $\mathbb{E}$ such that
$\mathscr{F}=\\{\mathtt{f}(\cdot,e_{1}),\dots,\mathtt{f}(\cdot,e_{r})\\}$ is
essentially non-holonomic.

Let us explain the definition above. Equivalently, a set $\mathscr{F}$ of
Lipschitz vector fields on a smooth manifold $M^{m}$ is essentially non-
holonomic at a point $p\in M$ whenever there exists a sequence of points
$p_{n}\in M$ that converges to $p$ such that $p_{n}$ is connected to $p$ with
the concatenation, starting at $p$, of line flows of $m$ vector fields in
$\mathscr{F}$ for times $(t_{1},\dots,t_{m})$, and moreover such concatenation
is essentially open around $(t_{1},\dots,t_{m})$. We stress that the latter
notion is weaker than the bracket-generating condition in the case the vector
fields are smooth, cf. Proposition 2.12.

We are now ready to give the main theorem of the paper. The following theorem
is proved at the end of Section 3.5.

######  Theorem 1.4.

Let $M$ be a smooth manifold, and let $\mathbb{E}$ be a finite-dimensional
Banach space. Let $\hat{\mathtt{f}}$ be an essentially non-holonomic
Lipschitz-vector-field structure modelled by $\mathbb{E}$, and let
$\hat{N}:M\times\mathbb{E}\to[0,+\infty)$ be a continuously varying norm. Then
the following hold.

  1. (i)

if $M$ is connected, then $d_{(\hat{\mathtt{f}},\hat{N})}(p,q)<\infty$ for
every $p,q\in M$;

  2. (ii)

$d_{(\hat{\mathtt{f}},\hat{N})}$ induces the manifold topology on $M$;

  3. (iii)

Let $\\{\mathtt{f}_{n}\\}_{n\in\mathbb{N}}$ be a sequence of Lipschitz-vector-
field structures on $M$ modelled by $\mathbb{E}$, and let
$\\{N_{n}\\}_{n\in\mathbb{N}}$ be a sequence of continuously varying norms on
$M\times\mathbb{E}$. Let us assume that $\mathtt{f}_{n}\to\hat{\mathtt{f}}$ in
the sense of Lipschitz-vector-field structures (see Definition 2.2), and
$N_{n}\to\hat{N}$ uniformly on compact subsets of $M\times\mathbb{E}$.

Then $d_{(\mathtt{f}_{n},N_{n})}\to d_{(\hat{\mathtt{f}},\hat{N})}$ locally
uniformly on $M$, i.e., every $o\in M$ has a neighborhood $U$ such that
$d_{(\mathtt{f}_{n},N_{n})}\to d_{(\hat{\mathtt{f}},\hat{N})}$ uniformly on
$U\times U$ as $n\to+\infty$.

  4. (iv)

If in the hypotheses of item (iii) we additionally have that
$d_{(\hat{\mathtt{f}},\hat{N})}$ is a boundedly compact (or equivalently
complete) distance, we conclude that

$\lim_{n\to+\infty}d_{(\mathtt{f}_{n},N_{n})}=d_{(\hat{\mathtt{f}},\hat{N})},$

uniformly on compact subsets of $M\times M$. Moreover, for every $x\in M$, we
have $(M,d_{(\mathtt{f}_{n},N_{n})},x)\to(M,d_{(\hat{\mathtt{f}},\hat{N})},x)$
in the pointed Gromov–Hausdorff topology as $n\to+\infty$.

Let us comment on the latter statements. The first item is a Chow-Rashevskii
type result for Lipschitz vector fields that satisfy the essentially non-
holonomic condition. It implies in particular the classical Chow-Rashevskii
theorem, [1]. The item (i) is easy since the fact that two sufficiently near
points can be connected by a finite-length curve is guaranteed by Definition
1.3. Item (ii) requires the essentially non-holonomic condition. Indeed, for
an arbitrary couple $(\mathtt{f},N)$, one only has that the topology induced
by $(\mathtt{f},N)$ is finer than the topology of the manifold, see Lemma 3.8
and the discussion after it. Item (iii) is the main convergence result, and it
only holds locally around every point. When one adds the hypothesis that the
limit distance is boundedly compact, the uniform convergence on compact sets
can be obtained, as stated in item (iv). Without the hypothesis on the
boundedly compactness of the limit distance, the convergence result might be
false, see the example in Remark 3.24.

Let us now describe the main steps of the proof of the Theorem 1.4. The first
nontrivial achievement to obtain the proof is that, when $\hat{\mathtt{f}}$ is
essentially non-holonomic, the topology of $M$ is finer than the topology
induced by $d_{(\hat{\mathtt{f}},\hat{N})}$; and thus equal taking into
account that the other inclusion, i.e., the one in Lemma 3.8, holds in
general. The nontrivial inclusion follows from the fact that when
$\hat{\mathtt{f}}$ is essentially non-holonomic, the End-point map associated
to $\hat{\mathtt{f}}$ is open, see Theorem 2.13, and Lemma 3.21. Such an
opennes property is a direct by-product of the essentially non-holonomic
condition, see Remark 2.11.

The latter described opennes property is stable along a sequence
$\\{\mathtt{f}_{n}\\}_{n\in\mathbb{N}}$ of Lipschitz-vector-field structures
that converge to $\hat{\mathtt{f}}$, and this is the key point to obtain item
(iii). Such a stability is the content of Theorem 2.13. Its proof builds on
the top of the joint continuity of the End-point map in the three variables
$o\in M$, $\mathtt{f}$ a Lipschitz-vector-field structure on $M$ modelled by
$\mathbb{E}$, and $u\in L^{\infty}([0,1],\mathbb{E})$ (with the weak*
topology), see Theorem 2.5 and Proposition 2.9, and on the topological Lemma
B.1 proved with the aid of degree theory, which tells us that continuous
functions that are near to an essentially open function are uniformly
surjective.

To conclude the proof of item (iii) one exploits the latter stability property
to prove that, on compact sets, the topology of $M$ is uniformly finer than
the topologies induced by the distances $d_{(\mathtt{f}_{n},N_{n})}$, see
Lemma 3.21. This directly implies that the functions
$\\{d_{(\mathtt{f}_{n},N_{n})}\\}_{n\in\mathbb{N}}$ are equicontinuous on
compact sets, see Proposition 3.22. To end the proof of item (iii) one then
finally uses the previous equicontinuity together with the fact that,
$d_{(\hat{\mathtt{f}},\hat{N})}$ is locally obtained as a relaxation of
$d_{(\mathtt{f}_{n},N_{n})}$, see Proposition 3.20.

Item (iv) is then proved by exploiting Item (iii) and the general metric
result in Lemma 3.25 according to which one can pass from the local uniform
convergence to the uniform converge on compact sets in a very general setting
under mild assumptions, i.e., the metric spaces are length spaces and the
limit distance is boundedly compact.

In Appendix C we offer a shorter proof of Theorem 1.4(iv) assuming that the
vector-fields structures are smooth. In this case we can argue directly by
using Gronwall’s Lemma A.1, and the quantitative open-mapping type result in
Lemma B.3, which does not need degree theory since we have enough regularity
of the flow maps.

Let us now discuss some corollaries of the general convergence result in
Theorem 1.4. In Section 4 we list some examples in which a direct application
of Theorem 1.4 gives nontrivial consequences. We record here a couple of them.
The first is Mitchell’s Theorem for subFinsler manifolds with continuously
varying norms.

Let us give some preliminary notation and definitions. Let $M^{m}$ be a smooth
manifold of dimension $m\in\mathbb{N}$, and let
$\mathcal{X}:=\\{X_{1},\dots,X_{k}\\}$ be a bracket-generating family of
smooth vector fields on $M$. Let $\mathbb{E}:=\mathbb{R}^{k}$, and let
$N:M\times\mathbb{E}\to[0,+\infty)$ be a continuously varying norm. For every
$p\in M$ and $v\in T_{p}M$, let $|\cdot|_{\mathcal{X},N}$ be the sub-Finsler
metric defined as follows

(4)
$|v|_{\mathcal{X},N}:=\inf\left\\{N(p,u):u\in\mathbb{E},\sum_{i=1}^{k}u_{i}X_{i}(p)=v\right\\}.$

The subFinsler distance between $p,q\in M$ is

(5)
$d_{(\mathcal{X},N)}(p,q):=\inf\left\\{\int_{0}^{1}|\gamma^{\prime}(t)|_{\mathcal{X},N}\,\mathrm{d}t:\gamma:[0,1]\to
M\,\text{is AC},\,\gamma(0)=p,\gamma(1)=q\right\\}.$

We recall that the definition of a regular point can be found, e.g., in [7,
Definition 2.4]. The following theorem is proved at the end of Section 4.2.

######  Theorem 1.5 (Mitchell’s Theorem for subFinsler manifolds with
continuously varying norm).

Let $M^{m},\mathcal{X},N$ be as above. Let us fix $o\in M$. There exists a
bracket-generating family of polynomial vector fields
$\hat{\mathcal{X}}:=\\{\hat{X}_{1},\dots,\hat{X}_{k}\\}$ on $\mathbb{R}^{m}$
such that the following holds.

Let $N_{o}$ be the continuously varying norm on
$\mathbb{R}^{m}\times\mathbb{E}$ such that $N_{o}(p,\cdot)\equiv N(o,\cdot)$
for every $p\in\mathbb{R}^{m}$. Let $|\cdot|_{\hat{\mathcal{X}},N_{o}}$ be the
subFinsler metric on $T\mathbb{R}^{m}$ defined as in (4), and let
$d_{(\hat{\mathcal{X}},N_{o})}$ be the subFinsler distance on
$\mathbb{R}^{m}$, defined as in (5), by using the subFinsler metric
$|\cdot|_{\hat{\mathcal{X}},N_{o}}$.

Then, the Gromov–Hausdorff tangent of $(M,d_{(\mathcal{X},N)})$ at $o\in M$ is
isometric to $(\mathbb{R}^{m},d_{(\hat{\mathcal{X}},N_{o})},0)$. Moreover,
$(\mathbb{R}^{m},d_{(\hat{\mathcal{X}},N_{o})})$ is isometric to a quotient of
a subFinsler Carnot group by one of its closed subgroups. If $o$ is regular,
$(\mathbb{R}^{m},d_{(\hat{\mathcal{X}},N_{o})})$ is isometric to a subFinsler
Carnot group.

We remark that the construction of $\hat{\mathcal{X}}$, which is usually
called the homogeneous nilpotent approximation of $\mathcal{X}$ at $o$, can be
made explicitely with respect to $\mathcal{X}$, compare with [7, Section 2.1],
and does not depend on $N$.

Giving for granted the construction of privileged coordinates, for which we
give precise references in Section 4.2, the proof of the Theorem 1.5 is a
direct consequence of the application of Theorem 1.4(iv), see Section 4.2.

The last consequence we want to discuss is in the setting of Lie groups. We
introduce some notation. Let $\mathbb{G}$ be a connected Lie group, and
$\mathfrak{g}$ its Lie algebra. Given a vector subspace
$\mathcal{H}\subseteq\mathfrak{g}$ of $\mathfrak{g}$, and a norm $b$ on
$\mathcal{H}$, we associate to $(\mathcal{H},b)$ a left-invariant subFinsler
structure $(\mathcal{D},b)$ as follows

(6) $\mathcal{D}_{p}:=\mathrm{d}L_{p}(\mathcal{H}),\,\,\quad
b_{p}(X):=b((\mathrm{d}L_{p})^{-1}X),\,\,\,\,\,\forall p\in\mathbb{G},\forall
X\in T_{p}\mathbb{G},$

where $L_{p}(q):=p\cdot q$ for $p,q\in\mathbb{G}$.

To each $(\mathcal{H},b)$, one attaches the (possibly infinite-valued)
distance between $p,q\in\mathbb{G}$ as

(7)
$\begin{split}d^{(\mathcal{H},b)}(p,q):=\inf\Big{\\{}&\int_{a}^{b}b_{\gamma(t)}(\gamma^{\prime}(t)):\gamma:[a,b]\to\mathbb{G}\,\,\text{AC},\gamma(a)=p,\gamma(b)=q,\\\
&\gamma^{\prime}(t)\in\mathcal{D}_{\gamma(t)}\,\text{for a.e.
$t\in[a,b]$}\Big{\\}}.\end{split}$

Notice that if $\mathcal{H}$ is a bracket-generating vector subspace of
$\mathfrak{g}$, the sub-Finsler structure $(\mathbb{G},\mathcal{D},b)$
satisfies the bracket-generating condition and thus $d^{(\mathcal{H},b)}$ is
finite-valued. Let us denote by $\mathrm{Gr}_{\mathfrak{g}}(k)$ the
Grassmannian of $k$-planes endowed with the usual topology of the Grassmannian
of a vector space. The proof of the following result is at the end of Section
4.3 and it is an immediate consequence of Theorem 1.4(iv).

######  Theorem 1.6 (Convergence of distances on Lie groups).

Let $\mathbb{G}$ be a connected Lie group with Lie algebra $\mathfrak{g}$, and
let $\mathcal{H}\subseteq\mathfrak{g}$ be a bracket-generating vector subspace
of dimension $k$.

Let $\\{\mathcal{H}_{n}\\}_{n\in\mathbb{N}}$ be a sequence of vector subspaces
of $\mathfrak{g}$ that converges in the topology of
$\mathrm{Gr}_{\mathfrak{g}}(k)$ to $\mathcal{H}$, and let
$\\{b_{n}\\}_{n\in\mathbb{N}}$ be a sequence of norms on $\mathfrak{g}$ that
converges uniformly on compact sets to a norm $b$ on $\mathfrak{g}$. Then,
being $d^{(\mathcal{H}_{n},b_{n})}$ and $d^{(\mathcal{H},b)}$ the distances
defined as in (7), we have

$d^{(\mathcal{H}_{n},b_{n})}\to d^{(\mathcal{H},b)},\qquad\text{uniformly on
compact subsets of $\mathbb{G}\times\mathbb{G}$}.$

Moreover, for any $p\in\mathbb{G}$,
$(\mathbb{G},d^{(\mathcal{H}_{n},b_{n})},p)\to(\mathbb{G},d^{(\mathcal{H},b)},p)$
in the pointed Gromov–Hausdorff topology as $n\to+\infty$.

We briefly describe the structure of the paper and we refer the reader to the
introductions of the sections for more details.

In Section 2 we introduce the main notation and definitions of the paper, we
study the continuity property of the End-point map, the notion of essentially
open map and essentially non-holonomic set of vector fields, and we finally
prove the open property of the End-point map associated to an essentially non-
holonomic structure.

In Section 3 we define and study the length, the energy and the
Carnot-Carathéodory distance associated to a Lipschitz-vector-field structure
and a continuously varying norm. We thus study how the distances behave under
taking limit of the corresponding structures, and we prove the main Theorem
1.4.

In Section 4 we discuss some examples in which Theorem 1.4 applies, namely the
fact that the asymptotic cone of the Riemannian Heisenberg group is the
subRiemannian Heisenberg group; the subFinsler Mitchell’s Theorem with
continuously varying norms; a general convergence result for Carnot-
Carathéodory distances on connected Lie groups; and a general convergence
result for Carnot-Carathéodory distances on manifolds.

In the Appendix we give a self-contained proof of the version of the Gronwall
Lemma that we need, we prove some ancillary lemmas about open maps, and we
finally give a shorter and more direct proof of Theorem 1.4(iv) in the case in
which the vector fields are smooth.

Acknowledgements. The authors are partially supported by the European Research
Council (ERC Starting Grant 713998 GeoMeG ‘ _Geometry of Metric Groups_ ’).
E.L.D. was partially supported by the Academy of Finland (grant 288501 ‘
_Geometry of subRiemannian groups_ ’ and by grant 322898 ‘ _Sub-Riemannian
Geometry via Metric-geometry and Lie-group Theory_ ’.

Part of this research was done while the first author was visiting the second
author at the University of Fribourg. The excellent work atmosphere is
acknowledged.

The authors express their gratitute to Emilio Lauret for his interest in this
work and his encouragement to add more material to a preliminary version of
this article.

##  2\. The End-point map for Lipschitz-vector-field structures

In this section we introduce the notion of Lipschitz-vector-field structure on
a smooth manifold modelled by a finite-dimensional Banach space, see
Definition 2.1 and Definition 2.3. We give the definitions of the convergence
of Lipschitz-vector-field structures, see Definition 2.2.

After the discussion of the basic definitions, we are going to prove the first
main theorem of this section, see Theorem 2.5, which says that the domain of
the End-point map associated to a Lipschitz-vector-field structure is open and
the End-point map is continuous. The proof of Theorem 2.5 heavily relies on
Proposition 2.9, according to which the curves satisfying the Cauchy system
(1) starting from a converging sequence $(o_{n},\mathtt{f}_{n},u_{n})\in
M\times\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)\times
L^{\infty}([0,1];\mathbb{E})$ uniformly converge. The latter result relies on
a general convergence criterion for flow lines of (not necessarily smooth)
vector fields, see Propostion 2.6.

In Section 2.3 we introduce the notion of essentially open map between two
topological manifolds of the same dimension, see Definition 2.10. We recall
that such a notion is used to give the notion of essentially non-holonomic
distribution of vector fields, see Definition 1.3.

In Section 2.4 we finally show the second main result of this section. Namely,
we show that whenever a Lipschitz-vector-field structure $\hat{\mathtt{f}}$
modelled by $\mathbb{E}$ is essentially non-holonomic, the End-point map of
$\hat{\mathtt{f}}$ enjoys a uniform openness property, see Theorem 2.13 for
further details. Such a uniform openness result, which follows both from the
continuity result proved in Theorem 2.5, and from the ancillary Lemma B.1,
strongly requires the essentially non-holonomicity property. Eventually, the
latter uniform openness result will be of key importance in proving Lemma
3.21, which is one of the main steps to prove the main result of the next
section, see Theorem 1.4(iv). We refer the reader to the introduction of the
next section for further details.

###  2.1. Lipschitz-vector-field structures on a manifold

In this subsection we study basic facts about Lipschitz-vector-field
structures.

######  Definition 2.1 (Uniformly locally Lipschitz sections
$\mathrm{Lip}\Gamma$).

Let $M$ be a smooth manifold and $E\to M$ a vector bundle. We say that a
family $\cal X$ of sections of $E$ is _uniformly locally Lipschitz_ if for
every $p\in M$ there exist a coordinate neighborhood of $p$ that trivializes
$E$ and a constant $L>0$ so that every element of $\cal X$ is $L$-Lipschitz in
this trivialization. We denote by $\mathrm{Lip}\Gamma(E)$ the collection of
all locally Lipschitz sections of $E$, i.e., sections $X$ so that the
singleton $\\{X\\}$ is uniformly locally Lipschitz.

It is a direct consequence of the definition that a family $\cal X$ of
sections is uniformly locally Lipschitz if and only if it is uniformly
Lipschitz on compact sets on every trivialization of $E$.

######  Definition 2.2 (Sequential topology on uniformly locally Lipschitz
sections).

We define a sequential topology on $\mathrm{Lip}\Gamma(E)$ as follows. We say
that a sequence
$\\{X_{n}\\}_{n\in\mathbb{N}}\subset\mathrm{Lip}\Gamma(E)$ converges to
$X_{\infty}\in\mathrm{Lip}\Gamma(E)$ if and only if
$\\{X_{n}\\}_{n\in\mathbb{N}}$ is uniformly locally Lipschitz and $X_{n}\to X$
uniformly on compact subsets of $M$, that is, if every $p\in M$ has a
coordinate neighborhood that trivializes $E$ on which $X_{n}$ converge to $X$
uniformly on compact sets.

######  Definition 2.3 (Lipschitz-vector-field structure).

Let $\mathbb{E}$ be a finite-dimensional Banach space and $M$ a smooth
manifold. We denote by $\mathbb{E}^{*}\otimes TM$ the vector bundle on $M$
whose fibers are $\mathbb{E}^{*}\otimes T_{p}M$. A section $\mathtt{f}$ of
$\mathbb{E}^{*}\otimes TM$ is a choice, for each $p\in M$, of a linear map
$\mathtt{f}|_{p}:\mathbb{E}\to T_{p}M$. An element
$\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ will be called a
Lipschitz-vector-field structre on $M$ modelled by $\mathbb{E}$.

The definition of the Banach spaces $L^{p}([0,1];\mathbb{E})$ does not depend
on the choice of a Banach norm on $\mathbb{E}$. The predual of
$L^{\infty}([0,1];\mathbb{E})$ is $L^{1}([0,1];\mathbb{E}^{*})$ with pairing

$\langle v|u\rangle:=\int_{0}^{1}\langle v(t)|u(t)\rangle\,\mathrm{d}t,$

for $u:[0,1]\to\mathbb{E}$ and $v:[0,1]\to\mathbb{E}^{*}$. So, a sequence
$\\{u_{n}\\}_{n\in\mathbb{N}}\subset L^{\infty}([0,1];\mathbb{E})$ weakly*
converges to $u_{\infty}\in L^{\infty}([0,1];\mathbb{E})$ if for every $v\in
L^{1}([0,1];\mathbb{E}^{*})$ it verifies $\lim_{n\to\infty}\langle
v|u_{n}\rangle=\langle v|u_{\infty}\rangle$.

We denote by $\mathscr{D}_{\mathrm{End}}\subset
M\times\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)\times
L^{\infty}([0,1];\mathbb{E})$ the domain of the End-point map, as defined in
Definition 1.1.

######  Remark 2.4 (Concatenation of flows of vector fields).

Let $X_{1},\dots,X_{\ell}$ be locally Lipschitz vector fields on $M$. Take
$\mathbb{E}=\mathbb{R}^{\ell}$ with the standard Euclidean norm, and with the
canonical basis $\\{e_{1},\dots,e_{\ell}\\}$, and define
$\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ by extending
linearly

$\mathtt{f}_{p}(e_{i}):=X_{i}(p),\qquad\text{for all $p\in M$ and $i\in
1,\dots,\ell$}.$

The Cauchy system in Equation (1), for any $u:[0,1]\to\mathbb{R}^{\ell}$,
becomes

$\begin{cases}\gamma^{\prime}(t)&=\sum_{i=1}^{\ell}u_{i}(t)X_{i}(\gamma(t)),\\\
\gamma(0)&=o.\end{cases}$

Fix $j\in\mathbb{N}$, $(t_{1},\dots,t_{j})\in\mathbb{R}^{j}$ and
$e_{i_{1}},\dots,e_{i_{j}}\in\\{e_{1},\dots,e_{\ell}\\}$. If we define the
controls

$u_{(t_{1},\dots,t_{j})}(s):=T\sum_{k=1}^{j}\chi_{\left[\frac{\sum_{i=1}^{k-1}|t_{i}|}{T},\frac{\sum_{i=1}^{k}|t_{i}|}{T}\right]}(s)\,\mathrm{sgn}(t_{k})\,e_{i_{k}},$

where $T:=\sum_{i=1}^{j}|t_{i}|$, and

$v_{(t_{1},\dots,t_{j})}(s):=j\sum_{k=1}^{j}\chi_{\left[\frac{k-1}{j},\frac{k}{j}\right]}(s)t_{k}e_{i_{k}},$

we have that

(8)
$\gamma_{(o,\mathtt{f},u_{(t_{1},\dots,t_{j})})}(1)=\gamma_{(o,\mathtt{f},v_{(t_{1},\dots,t_{j})})}(1)=\Phi_{X_{i_{j}}}^{t_{j}}\circ\dots\Phi_{X_{i_{1}}}^{t_{1}}(o),$

whenever one of the terms exists. We recall that $\Phi_{X}^{t}(o)$ is the flow
line at time $t$ of the vector field $X$ starting at $o$.

###  2.2. Continuity of the End-point map

In this section we shall prove the following continuity theorem for the End-
point map. On $\mathscr{D}_{\mathrm{End}}\subset
M\times\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)\times
L^{\infty}([0,1];\mathbb{E})$ we consider the topology that is the product of
the manifold topology on $M$, the sequential topology defined in Definition
2.2 on $\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$, and the weak* topology
on $L^{\infty}([0,1];\mathbb{E})$.

######  Theorem 2.5.

Let $M$ be a smooth manifold and let $\mathbb{E}$ be a finite-dimensional
Banach space. Then the domain $\mathscr{D}_{\mathrm{End}}$ of the End-point
map is open and the function $\mathrm{End}:\mathscr{D}_{\mathrm{End}}\to M$ is
continuous.

Moreover, given
$(\hat{o},\hat{\mathtt{f}},\hat{u})\in\mathscr{D}_{\mathrm{End}}$, and given a
weak*-compact neighborhood $\mathscr{U}$ of $\hat{u}$ such that
$(\hat{o},\hat{\mathtt{f}},u)\in\mathscr{D}_{\mathrm{End}}$ for every
$u\in\mathscr{U}$, we have that that the limit

(9)
$\lim_{(o,\mathtt{f})\to(\hat{o},\hat{\mathtt{f}})}\mathrm{End}^{\mathtt{f}}_{o}=\mathrm{End}^{\hat{\mathtt{f}}}_{\hat{o}}$

is uniform on $\mathscr{U}$.

Inspired by [11, Proposition 3.6], we prove the following ancillary
proposition.

######  Proposition 2.6.

Let $M\subset\mathbb{R}^{N}$ be a closed subset and for
$n\in\mathbb{N}\cup\\{\infty\\}$ let $X_{n}:[0,1]\times M\to\mathbb{R}^{N}$ be
measurable (thought as non-autonomous vector-fields) such that there are
$R,L\geq 0$ with $\|X_{n}\|_{L^{\infty}}\leq R$ and there exists a full
measure subset $\mathcal{T}\subset[0,1]$ such that
$|X_{n}(t,p)-X_{n}(t,q)|\leq L|p-q|$ for all $p,q\in M$, and for all
$t\in\mathcal{T}$. For each $n$, let $\gamma_{n}:[0,1]\to M$ be an integral
curve of $X_{n}$. Suppose that $\gamma_{n}(0)\to\gamma_{\infty}(0)$ and that,
for every $t\in[0,1]$,

(10)
$\lim_{n\to\infty}\int_{0}^{t}X_{n}(s,\gamma_{\infty}(s))\,\mathrm{d}s=\int_{0}^{t}X_{\infty}(s,\gamma_{\infty}(s))\,\mathrm{d}s.$

Then $\gamma_{n}\to\gamma_{\infty}$ uniformly.

###### Proof.

The family of curves $\gamma_{n}$ is equibounded and equicontinuous with
values in $\mathbb{R}^{N}$. By Ascoli-Arzelà, $\\{\gamma_{n}\\}_{n}$ is
precompact. Let $\xi:[0,1]\to M$ be a limit curve. We will show that
$\xi=\gamma_{\infty}$, thus proving that the whole sequence $\gamma_{n}$
uniformly converge to $\gamma_{\infty}$.

For each $n$, define $o_{n}:=\gamma_{n}(0)$ and

$\eta_{n}(t):=o_{n}+\int_{0}^{t}X_{n}(s,\gamma_{\infty}(s))\,\mathrm{d}s.$

By the assumption (10), $\eta_{n}(t)\to\gamma_{\infty}(t)$ for every
$t\in[0,1]$.

Next,

$\displaystyle|\gamma_{n}(t)-\gamma_{\infty}(t)|$
$\displaystyle\leq\begin{multlined}|o_{n}-o_{\infty}|+\int_{0}^{t}|X_{n}(s,\gamma_{n}(s))-X_{n}(s,\gamma_{\infty}(s))|\,\mathrm{d}s\\\
+\left|\int_{0}^{t}(X_{n}(s,\gamma_{\infty}(s))-X_{\infty}(s,\gamma_{\infty}(s)))\,\mathrm{d}s\right|\end{multlined}|o_{n}-o_{\infty}|+\int_{0}^{t}|X_{n}(s,\gamma_{n}(s))-X_{n}(s,\gamma_{\infty}(s))|\,\mathrm{d}s\\\
+\left|\int_{0}^{t}(X_{n}(s,\gamma_{\infty}(s))-X_{\infty}(s,\gamma_{\infty}(s)))\,\mathrm{d}s\right|$
$\displaystyle\leq
2|o_{n}-o_{\infty}|+L\int_{0}^{t}|\gamma_{n}(s)-\gamma_{\infty}(s)|\,\mathrm{d}s+|\eta_{n}(t)-\gamma_{\infty}(t)|.$

Letting $n\to\infty$, we obtain, for every $t\in[0,1]$,

$|\xi(t)-\gamma_{\infty}(t)|\leq
L\int_{0}^{t}|\xi(s)-\gamma_{\infty}(s)|\,\mathrm{d}s.$

Starting with $\|\xi-\gamma_{\infty}\|\leq C$ and iterating this inequality,
we get $\xi=\gamma_{\infty}$. The latter implication is actually an instance
of the Gronwall inequality $f^{\prime}\leq Lf$. ∎

######  Remark 2.7.

Suppose that the hypothesis of Proposition 2.6 except (10) are satisfied. We
claim that, if $X_{n}\to X_{\infty}$ pointwise a.e. on $[0,1]\times M$, then
(10) is satisfied on every curve $\gamma$. Indeed, first, for a.e.
$t\in[0,1]$, we have that $X_{n}(t,\cdot)\to X_{\infty}(t,\cdot)$ almost
everywhere on $M$. Since $X_{n}$ are uniformly Lipschitz and bounded in the
spatial variable, for a.e. $t\in[0,1]$, actually the convergence
$X_{n}(t,\cdot)\to X_{\infty}(t,\cdot)$ is uniform on compact subsets of $M$.
Therefore, since $\gamma([0,1])$ is compact, for almost every $s\in[0,1]$,
$\lim_{n\to\infty}X_{n}(s,\gamma(s))=X_{\infty}(s,\gamma(s))$. Since $X_{n}$
are uniformly bounded, by the Lebesgue dominated convergence theorem, we can
exchange the limit and the integral in the left-hand side of (10).

######  Lemma 2.8.

Let $\mathbb{E}_{1}$ and $\mathbb{E}_{2}$ be two finite-dimensional Banach
spaces. Suppose that $\\{u_{n}\\}_{n\in\mathbb{N}}\subset
L^{\infty}([0,1];\mathbb{E}_{1})$ is a sequence weakly* converging to
$u_{\infty}$ and that $A_{s}:\mathbb{E}_{1}\to\mathbb{E}_{2}$ is a continuous
family of linear maps, with $s\in[0,1]$. Then $v_{n}(s):=A_{s}[u_{n}(s)]$ is a
sequence in $L^{\infty}([0,1];\mathbb{E}_{2})$ weakly* converging to
$v_{\infty}$.

###### Proof.

We denote by $\langle\cdot|\cdot\rangle$ the pairing of a Banach space with
its dual space. We need to show that, for every $\alpha\in
L^{1}([0,1];\mathbb{E}_{2}^{*})$,

(11)
$\lim_{n\to\infty}\langle\alpha|v_{n}\rangle=\langle\alpha|v_{\infty}\rangle.$

Since $A$ is continuous in $s$, the operator norm of the adjoint operators
$A_{s}^{*}$ is bounded uniformly in $s\in[0,1]$, by $R$ say. Hence
$\int_{0}^{1}\|A^{*}_{s}\alpha(s)\|\,\mathrm{d}s\leq
R\|\alpha\|_{L^{1}}<\infty$, that is, $A^{*}\alpha:s\mapsto
A_{s}^{*}\alpha(s)$ is an element of $L^{1}([0,1];\mathbb{E}_{1}^{*})$. We
conclude that, for every $\alpha\in L^{1}([0,1];\mathbb{E}_{2}^{*})$,

$\displaystyle\lim_{n\to\infty}\langle\alpha|v_{n}\rangle$
$\displaystyle=\lim_{n\to\infty}\int_{0}^{1}\langle\alpha(s)|A_{s}[u_{n}(s)]\rangle\,\mathrm{d}s=\lim_{n\to\infty}\int_{0}^{1}\langle
A_{s}^{*}\alpha(s)|u_{n}(s)\rangle\,\mathrm{d}s$
$\displaystyle=\lim_{n\to\infty}\langle A^{*}\alpha|u_{n}\rangle=\langle
A^{*}\alpha|u_{\infty}\rangle=\langle\alpha|Au_{\infty}\rangle,$

hence (11) indeed holds for every $\alpha\in L^{1}([0,1];\mathbb{E}_{2}^{*})$.
∎

The next proposition will be at the core of the proof of Theorem 2.5.

######  Proposition 2.9.

Let $\\{(o_{n},\mathtt{f}_{n},u_{n})\\}_{n\in\mathbb{N}}\subset
M\times\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)\times
L^{\infty}([0,1];\mathbb{E})$ be a sequence converging to
$(o_{\infty},\mathtt{f}_{\infty},u_{\infty})\in\mathscr{D}_{\mathrm{End}}$.
Then the following hold:

  1. (1)

There is $N\in\mathbb{N}$ such that $\\{(o_{n},\mathtt{f}_{n},u_{n})\\}_{n\geq
N}\subset\mathscr{D}_{\mathrm{End}}$, that is, for every $n\geq N$ the Cauchy
system (1) has an integral curve $\gamma_{(o_{n},\mathtt{f}_{n},u_{n})}$
defined on $[0,1]$;

  2. (2)

The sequence of integral curves $\gamma_{(o_{n},\mathtt{f}_{n},u_{n})}$
uniformly converge to $\gamma_{(o_{\infty},\mathtt{f}_{\infty},u_{\infty})}$.

###### Proof.

The convergence in the assumptions means that $o_{n}\to o_{\infty}$ in $M$,
$u_{n}\to u_{\infty}$ weakly* in $L^{\infty}([0,1];\mathbb{E})$,
$\mathtt{f}_{n}\to\mathtt{f}_{\infty}$ uniformly on compact sets of every
trivialization of $\mathbb{E}^{*}\otimes TM$, and $\\{\mathtt{f}_{n}\\}_{n}$
is uniformly locally Lipschit on compact sets of every trivialization of
$\mathbb{E}^{*}\otimes TM$. For $n\in\mathbb{N}\cup\\{\infty\\}$, denote by
$\gamma_{n}:[0,T_{n})\to M$ the maximal integral curve of the Cauchy system
(1) with control $u_{n}$ and initial point $o_{n}$. Notice that $\gamma_{n}$
can be extended to $T_{n}$ if and only if $T_{n}=1$, if and only if
$\gamma_{n}([0,T_{n}))$ is contained in a compact subset of $M$.

Let $\rho$ be a complete Riemannian metric on $M$ and let
$\iota:M\hookrightarrow\mathbb{R}^{N}$ be a Riemannian isometric embedding.
The $\rho$-length of curves in $M$ is thus equal to their Euclidean length in
$\mathbb{R}^{N}$. We denote by $d_{\rho}$ and
$\operatorname{\mathrm{diam}}_{\rho}$ the Riemannian distance and the
corresponding diameter on $M$ defined by $\rho$. Notice that, for every $r\geq
0$ and $p\in M$, the closed $\rho$-ball of radius $r$ and center $p$ in $M$,
$\bar{B}_{\rho}(p,r)$, is a compact subset of $\mathbb{R}^{N}$.

By means of the isometric embedding $\iota$, we interpret the sections
$\mathtt{f}_{n}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ as locally
uniformly Lipschitz maps
$\mathtt{f}_{n}:M\to\mathbb{E}^{*}\otimes\mathbb{R}^{N}$. We will denote by
$\|\mathtt{f}_{n}(p)\|$ the operator norm of the linear map
$\mathtt{f}_{n}(p):\mathbb{E}\to\mathbb{R}^{N}$.

Let $D=\operatorname{\mathrm{diam}}_{\rho}(\gamma_{\infty}([0,1]))$,
$H=\sup\\{\|\mathtt{f}_{n}(p)\|:n\in\mathbb{N},p\in\bar{B}_{\rho}(o_{\infty},4D)\\}<\infty$,
and $R=\sup_{n}\|u_{n}\|_{L^{\infty}}<\infty$.

We claim that, for every $\hat{t}\in[0,1]$, if there is $N$ such that
$\hat{t}<T_{n}$ for every $n\geq N$ and if
$\gamma_{n}(\hat{t})\to\gamma_{\infty}(\hat{t})$, then there is $\hat{N}\geq
N$ such that, for all $n\geq\hat{N}$, the curve $\gamma_{n}$ is defined on
$[\hat{t},\hat{t}+\frac{D}{RH}]\cap[0,1]$ and
$\gamma_{n}([0,\hat{t}+\frac{D}{RH}]\cap[0,1])\subset\bar{B}_{\rho}(o_{\infty},4D)$
for all $n\geq\hat{N}$.

To prove the claim, first notice that there is $\hat{N}\in\mathbb{N}$ such
that $\gamma_{n}(\hat{t})\in\bar{B}(o_{\infty},2D)$ for all $n\geq\hat{N}$.
Next, for almost every $s\in[0,1]$ with
$\gamma_{n}(s)\in\bar{B}_{\rho}(o_{\infty},4D)$, we have

(12)
$|\gamma_{n}^{\prime}(s)|=|\mathtt{f}_{n}(\gamma_{n}(s))u_{n}(s)|\leq\|\mathtt{f}_{n}(\gamma_{n}(s))\||u_{n}(s)|\leq
HR,$

and so, if $\gamma_{n}(r)\in\bar{B}_{\rho}(o_{\infty},4D)$ for all
$r\in[\hat{t},s]$, then

(13)
$|\gamma_{n}(s)-\gamma_{\infty}(\hat{t})|\leq|\gamma_{n}(s)-\gamma_{n}(\hat{t})|+|\gamma_{n}(\hat{t})-\gamma_{\infty}(\hat{t})|\leq
HR|s-\hat{t}|+2D.$

Now, fix $n>\hat{N}$ and define

$\hat{s}=\min\left\\{\hat{t}+\frac{D}{HR},\
\inf\\{r\geq\hat{t}:\gamma_{n}(r)\in\partial\bar{B}_{\rho}(o_{\infty},4D)\\},\
T_{n}\right\\}.$

If $\hat{s}=T_{n}$, then
$\gamma_{n}([0,T_{n}))\subset\bar{B}_{\rho}(o_{\infty},4D)$, which is compact,
and thus $T_{n}=1$ and $\gamma_{n}$ is defined on $[0,1]$. Suppose
$\hat{s}<T_{n}$. From (13), we obtain
$|\gamma_{n}(\hat{s})-\gamma_{\infty}(\hat{t})|\leq 3D$ and thus we deduce
that $\hat{s}=\hat{t}+\frac{D}{HR}$. We conclude that $\gamma_{n}$ is defined
on $[0,\hat{s}]=[0,\hat{t}+\frac{D}{HR}]\cap[0,1]$ and that
$\gamma_{n}([\hat{t},\hat{s}])\subset\bar{B}_{\rho}(o_{\infty},4D)$ for all
$n\geq\hat{N}$.

Since $\frac{D}{HR}$ is a fixed positive quantity, the above claim readily
implies the first part of the lemma. For the second part of the proof, we
assume that $\gamma_{n}$ is defined on $[0,1]$ for all $n$.

We claim that, for every $\hat{t}\in[0,1]$, if
$\gamma_{n}(\hat{t})\to\gamma_{\infty}(\hat{t})$, then
$\gamma_{n}|_{[\hat{t},\hat{t}+\frac{D}{RH}]}$ uniformly converge to
$\gamma_{\infty}|_{[\hat{t},\hat{t}+\frac{D}{RH}]}$.

To prove the latter claim, notice that from the previous claim we have that
there is $\hat{N}\geq N$ such that, for all $n\geq\hat{N}$,
$\gamma_{n}([0,\hat{t}+\frac{D}{RH}]\cap[0,1])\subset\bar{B}_{\rho}(o_{\infty},4D)$.
We next apply Proposition 2.6, whose hypotheses must be met. Define the vector
fields $X_{n}:[0,1]\times M\to TM$,
$X_{n}(t,p)=\mathtt{f}_{n}|_{p}[u_{n}(t)]$. Since
$\bar{B}_{\rho}(o_{\infty},4D)$ is compact, there is $L$ so that
$\mathtt{f}_{n}$ is $L$-Lipschitz on $\bar{B}_{\rho}(o_{\infty},4D)$ for all
$n\in\mathbb{N}$. Thus, if $p,q\in\bar{B}_{\rho}(o_{\infty},4D)$, then

$|X_{n}(t,p)-X_{n}(t,q)|\leq\|\mathtt{f}_{n}(p)-\mathtt{f}_{n}(q)\|\|u_{n}\|_{L^{\infty}}\leq
LR|p-q|.$

If $\gamma:[0,1]\to M$ is a continuous curve, then

$\displaystyle\int_{0}^{t}X_{n}(s,\gamma(s))\,\mathrm{d}s$
$\displaystyle=\int_{0}^{t}\mathtt{f}_{\infty}|_{\gamma(s)}[u_{n}(s)]\,\mathrm{d}s+\int_{0}^{t}\mathtt{f}_{n}|_{\gamma(s)}[u_{n}(s)]-\mathtt{f}_{\infty}|_{\gamma(s)}[u_{n}(s)]\,\mathrm{d}s,$

where

$\lim_{n\to\infty}\left|\int_{0}^{t}\mathtt{f}_{n}|_{\gamma(s)}[u_{n}(s)]-\mathtt{f}_{\infty}|_{\gamma(s)}[u_{n}(s)]\,\mathrm{d}s\right|\leq\lim_{n\to\infty}\|\mathtt{f}_{n}-\mathtt{f}_{\infty}\|_{L^{\infty}}Rt,=0$

and $v_{n}(s):=\mathtt{f}_{\infty}|_{\gamma(s)}[u_{n}(s)]$ weakly* converge to
$v_{\infty}(s):=\mathtt{f}_{\infty}|_{\gamma(s)}[u_{\infty}(s)]$ by Lemma 2.8.
Since $v\mapsto\int_{0}^{t}v(s)\,\mathrm{d}s$ is a continuous linear operator
from $L^{\infty}([0,1];\mathbb{R}^{N})$ to $\mathbb{R}^{N}$, we then have

$\lim_{n\to\infty}\int_{0}^{t}\mathtt{f}_{\infty}|_{\gamma(s)}[u_{n}(s)]\,\mathrm{d}s=\int_{0}^{t}\mathtt{f}_{\infty}|_{\gamma(s)}[u_{\infty}(s)]\,\mathrm{d}s.$

We have thus shown that the non-autonomous vector fields $X_{n}$ satisfy all
conditions in Proposition 2.6 on $\bar{B}_{\rho}(o_{\infty},4D)$. Since
$\gamma_{n}([\hat{t},\hat{t}+\frac{D}{HR}])\subset\bar{B}_{\rho}(o_{\infty},4D)$
for all $n\geq\hat{N}$ and $\gamma_{n}(\hat{t})\to\gamma_{\infty}(\hat{t})$,
Proposition 2.6 ensures that $\gamma_{n}|_{[\hat{t},\hat{t}+\frac{D}{RH}]}$
uniformly converge to $\gamma_{\infty}|_{[\hat{t},\hat{t}+\frac{D}{RH}]}$. The
claim is proven.

Finally, since the constant $\frac{D}{RH}$ does not depend on $\hat{t}$, we
can subdivide $[0,1]$ into intervals of length less than $\frac{D}{RH}$ and
apply the above claim iteratively to each interval, concluding the proof of
the proposition. ∎

###### Proof of Theorem 2.5.

Proposition 2.9.(1) implies that $\mathscr{D}_{\mathrm{End}}$ is open, while
Proposition 2.9.(2) implies that $\mathrm{End}$ is continuous. The uniform
limit (9) is a direct consequence of the continuity of $\mathrm{End}$
simultaneously in the three variables $(o,\mathtt{f},u)$, and a standard
compactness argument. ∎

###  2.3. Essentially non-holonomic Lipschitz distributions

In this section we discuss the notions of essentially open maps and
essentially non-holonomic distributions of vector fields.

######  Definition 2.10 (Essentially open map).

A continuous map $f:M\to N$ between topological manifolds of the same
dimension $k$ is said to be essentially open at $p\in M$ at scale $U$ if

  1. (1)

$U$ is a neighborhood of $p$ such that $U$ is homeomorphic to the Euclidean
$k$-dimensional ball, and $\partial U$ is homeomorphic to the
$(k-1)$-dimensional Euclidean sphere $\mathbb{S}^{k-1}$;

  2. (2)

there exists a neighborhood $V$ of $f(p)$ such that $V$ is homeomorphic to the
Euclidean $k$-dimensional ball;

  3. (3)

$f(\partial U)\subseteq V\setminus f(p)$;

  4. (4)

The induced group homomorphism $f_{*}:H_{k-1}(\partial U)\cong\mathbb{Z}\to
H_{k-1}(V\setminus f(p))\cong\mathbb{Z}$ is not constant.

######  Remark 2.11.

We claim that if a map $f$ is essentially open at $p$, then $f(p)$ is
contained in the interior of $f(U)$. Indeed, up to homeomorphism, $f$ induces
by restriction a map $f:\overline{B}(0,r_{1})\subseteq\mathbb{R}^{k}\to
B(0,r_{2})\subseteq\mathbb{R}^{k}$, for some radii $r_{1},r_{2}$, such that
$f(0)=0$, $f(\partial B(0,r_{1}))\subseteq B(0,r_{2})\setminus\\{0\\}$, and
such that $f|_{\partial B(0,r_{1})}:\partial B(0,r_{1})\to
B(0,r_{2})\setminus\\{0\\}$ induces a non-constant group homomorphism between
the $(k-1)$-th homology groups. Hence, arguing as in the first few lines of
the proof of Lemma B.2, the map
$\widetilde{f}:\mathbb{S}^{k-1}\to\mathbb{S}^{k-1}$ defined as

$\widetilde{f}(x):=\frac{f(r_{1}x)}{|f(r_{1}x)|},$

has nonzero degree. Then we can apply Lemma B.1 with the constant sequence
$f_{n}:=f$ to obtain that $f(0)$ is contained in the interior of
$f(B(0,r_{1}))$.

We notice that by virtue of Lemma B.1, the essential openness at a point $p$
at some scale, implies a surjectivity property at $p$ which is stable with
respect to uniform convergence. This stability is of crucial importance in
Definition 1.3. Indeed, if we naïvely only require that $\phi$ in Definition
1.3 is open at $\hat{t}$, it is not possible to prove the uniform openness in
Theorem 2.13, because the property of being open is not stable under uniform
convergence. For example, consider the functions $f_{t}(x):=tx$ defined on
$[0,1]$ and let $t\to 0$.

We show in the next proposition that the condition of essential non-
holonomicity (as in Definition 1.3) is a weakening of the bracket-generating
condition for smooth vector fields.

######  Proposition 2.12.

Every bracket-generating family of smooth vector fields is essentially non-
holonomic.

###### Proof.

Let $\mathscr{F}\subset\Gamma(TM)$ be a bracket-generating family of smooth
vector fields. Due to [1, Lemma 3.33], cf. also [2, Section 5.4], we have that
for every $T>0$ and every $o\in M$, there are
$X_{1},\dots,X_{m}\in\mathscr{F}$ such that the map

$\phi(t_{1},\dots,t_{m}):=\Phi^{t_{m}}_{X_{m}}\circ\dots\circ\Phi^{t_{1}}_{X_{1}}(o)$

is a topological embedding into $M$ from an open subset $U$ of
$\mathbb{R}^{m}$ contained in the set $\\{t\in\mathbb{R}^{m}:|t|_{1}<T\\}$.
Then an application of Lemma B.2 gives the sought conclusion. ∎

###  2.4. Uniform openness of the End-point map

In this final part of Section 2, we prove that whenever a Lipschitz-vector-
field $\hat{\mathtt{f}}$ is essentially non-holonomic, and a point $o\in M$ is
given, then the End-point map $\mathrm{End}^{\mathtt{f}}_{o}(\cdot)$ is open
at $0$ uniformly as $\mathtt{f}$ varies in a neighborhood of
$\hat{\mathtt{f}}$.

######  Theorem 2.13.

Let $\hat{\mathtt{f}}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ be
essentially non-holonomic, let $o\in M$, and $R>0$. Then there are
neighborhoods $\mathscr{F}$ of $\hat{\mathtt{f}}$ and $U$ of $o$ such that

$\mathtt{f}\in\mathscr{F}\Rightarrow
U\subset\mathrm{End}^{\mathtt{f}}_{o}(B_{L^{\infty}}(0,R)).$

###### Proof.

Let $m$ be the dimension of the manifold $M$. Arguing by contradiction,
suppose that there are $\mathtt{f}_{n}\to\hat{\mathtt{f}}$ and $p_{n}\to o$
such that $p_{n}\notin\mathrm{End}^{\mathtt{f}_{n}}_{o}(B_{L^{\infty}}(0,R))$.

Let $e_{1},\dots,e_{r}$ be a basis of $\mathbb{E}$ as in Definition 1.3. Given
$\sigma:\\{1,\dots,m\\}\to\\{1,\dots,r\\}$ and
$\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$, define, for every
$j=1,\dots,m$,

$X^{\sigma,\mathtt{f}}_{j}(p)=\mathtt{f}(p,e_{\sigma(j)}),$

and, for every $(s_{1},\dots,s_{m},t_{1},\dots,t_{m})\in\mathbb{R}^{2m}$, let
us define

$\psi^{\sigma,\mathtt{f}}(s_{1},\dots,s_{m},t_{1},\dots,t_{m}):=\Phi^{s_{1}}_{-X^{\sigma,\mathtt{f}}_{1}}\circ\dots\circ\Phi^{s_{m}}_{-X^{\sigma,\mathtt{f}}_{m}}\circ\Phi^{t_{m}}_{X^{\sigma,\mathtt{f}}_{m}}\circ\dots\circ\Phi^{t_{1}}_{X^{\sigma,\mathtt{f}}_{1}}(o),$

whenever it exists. Notice that

$\psi^{\sigma,\mathtt{f}}(s_{1},\dots,s_{m},t_{1},\dots,t_{m})=\mathrm{End}^{\mathtt{f}}_{o}(u(s_{1},\dots,s_{m},t_{1},\dots,t_{m}))$

where, for every $t\in[0,1]$,

$u(s_{1},\dots,s_{m},t_{1},\dots,t_{m})(t)=2m\left(\sum_{j=1}^{m}t_{j}e_{\sigma(j)}\chi_{(\frac{j-1}{2m},\frac{j}{2m})}(t)-\sum_{j=1}^{m}s_{j}e_{\sigma(j)}\chi_{(\frac{2m-j}{2m},\frac{2m-j+1}{2m})}(t)\right),$

and there is $C>0$ such that
$\|u(s_{1},\dots,s_{m},t_{1},\dots,t_{m})\|_{L^{\infty}}\leq
C\sum_{j}(|t_{j}|+|s_{j}|)$.

Since $\hat{\mathtt{f}}$ is essentially non-holonomic, there are
$\hat{t}\in\mathbb{R}^{m}$, a neighborhood $\Omega$ of $\hat{t}$, and $\sigma$
such that $2C|t|_{1}<R$ for all $t\in\Omega$, and
$\hat{f}=\psi^{\sigma,\hat{\mathtt{f}}}(\hat{t},\cdot)$ restricted to $\Omega$
is essentially open at $\hat{t}$. We recall that by definition
$|(t_{1},\dots,t_{m})|_{1}:=\sum_{i=1}^{m}|t_{i}|$, for every
$(t_{1},\dots,t_{m})\in\mathbb{R}^{m}$.

Now, the maps $f_{n}=\psi^{\sigma,\mathtt{f}_{n}}(\hat{t},\cdot)$ are
continuous on $\Omega$ and converge uniformly to $\hat{f}$, by Theorem 2.5. By
reasoning as at the beginning of Remark 2.11, we have that at some scale
around $\hat{t}$, $\hat{f}$ satisfies the hypothesis on the degree required to
apply Lemma B.1. Hence, using Lemma B.1, there is an open neighborhood $U$ of
$o$ such that $U\subset f_{n}(\Omega)$ for large $n$. Since
$f_{n}(\Omega)\subset\mathrm{End}^{\mathtt{f}_{n}}_{o}(B_{L^{\infty}}(0,R))$,
we reach a contradiction with $p_{n}\to o$ and
$p_{n}\notin\mathrm{End}^{\mathtt{f}_{n}}_{o}(B_{L^{\infty}}(0,R))$. ∎

##  3\. Sub-Finsler distances for Lipschitz-vector-field structures

In this section we shall fix a smooth manifold $M$ and a Banach space
$\mathbb{E}$. In $M$ we shall fix a Lipschitz-vector-field structure
$\mathtt{f}$ modelled by $\mathbb{E}$, as in Definition 2.3.

In Section 3.2, given a continuously varying norm $N$ on $M\times\mathbb{E}$,
we are going to define the energy and the length functionals associated to
controls $u\in L^{\infty}([0,1];\mathbb{E})$, we prove that they are lower
semicontinuous, see Proposition 3.4, and we prove that every curve has a
constant speed reparametrization, see Lemma 3.5.

In Section 3.3, we define the distance $d_{(\mathtt{f},N)}(p,q)$ between
$p,q\in M$ to be the infimum of the energy (equivalently the length) of a
control that gives raise to a curve that connects the two points, see
Definition 3.6. We then prove the local existence of geodesics, see
Proposition 3.9, and in Proposition 3.10 we give a criterion, ultimately based
on the growth of $(\mathtt{f},N)$, to show that $(M,d_{(\mathtt{f},N)})$ is a
complete, boundedly compact, geodesic metric space. We prove that the topology
generated by $d_{(\mathtt{f},N)}$ is larger than the topology of $M$, see
Lemma 3.8. We shall then show that they are equal if $\mathtt{f}$ is
essentially non-holonomic, see Theorem 1.4(ii).

In Section 3.4 we shall address the problem of linking the definition of the
distance given in Section 3.2 with the Lagrangian one given directly on the
curves. In particular, having a couple $(\mathtt{f},N)$ on $M$, one can define
a sub-Finsler metric on $TM$, see (32). We show that a curve $\gamma:[0,1]\to
M$ is $d_{(\mathtt{f},N)}$-Lipschitz if and only if $\gamma^{\prime}$ has a
bounded sub-Finsler metric as in (32), see Lemma 3.16. As a consequence, on
$d_{(\mathtt{f},N)}$-Lipschitz curves, the length associated to
$d_{(\mathtt{f},N)}$ coincides with the one associated to the sub-Finsler
metric, see Proposition 3.17. We thus conclude that the distance
$d_{(\mathtt{f},N)}(p,q)$ is the infimum of the sub-Finsler lengths of the
curves connecting the points $p,q$, see Corollary 3.19.

In Section 3.5 we finally investigate what happens when we take the limit of
CC distances. We first prove a relaxation result which tells us that when
$(\mathtt{f}_{n},N_{n})\to(\mathtt{f}_{\infty},N_{\infty})$ the distance
$d_{(\mathtt{f}_{\infty},N_{\infty})}$ on compact sets can be recovered as the
relaxation of the distances $d_{(\mathtt{f}_{n},N_{n})}$, see Lemma 3.20.
Hence we prove that when
$(\mathtt{f}_{n},N_{n})\to(\mathtt{f}_{\infty},N_{\infty})$ and
$\mathtt{f}_{\infty}$ is essentially non-holonomic, then
$\\{d_{(\mathtt{f}_{n},N_{n})}\\}_{n\in\mathbb{N}}$ are equi-continuous
functions on compact sets, see Lemma 3.21 and Proposition 3.22. The equi-
continuity together with the relaxation property allows us to show that if
$(\mathtt{f}_{n},N_{n})\to(\mathtt{f}_{\infty},N_{\infty})$, and
$\mathtt{f}_{\infty}$ is essentially non-holonomic, then
$d_{(\mathtt{f}_{n},N_{n})}$ converges to
$d_{(\mathtt{f}_{\infty},N_{\infty})}$ locally uniformly, see Theorem
1.4(iii). Finally, when $(M,d_{(\mathtt{f}_{\infty},N_{\infty})})$ is
boundedly compact, the local uniform convergence can be upgraded to a uniform
convergence on compact sets by means of the metric Lemma 3.25, see Theorem
1.4(iv). The boundedly compact hypothesis is necessary to obtain such a
uniform convergence on compact sets, see the example in Remark 3.24.

###  3.1. Continuous norms along a segment

In this section we give a preliminary and technical discussion about norms
depending on a parameter that we will later use.

Let $N:[0,1]\times\mathbb{E}\to[0,\infty)$ be a continuous function so that,
for every $t\in[0,1]$, the restriction $N_{t}:=N(t,\cdot)$ is a norm on the
finite-dimensional vector space $\mathbb{E}$. We define, for a measurable
$u:[0,1]\to\mathbb{E}$,

$\displaystyle\|u\|_{N,1}$
$\displaystyle:=\int_{0}^{1}N_{t}(u(t))\,\mathrm{d}t,$
$\displaystyle\|u\|_{N,\infty}$
$\displaystyle:=\operatorname*{{esssup}}_{t\in[0,1]}N_{t}(u(t)).$

Notice that, since $N$ is continuous, $\|u\|_{N,1}$ is a Banach norm on
$L^{1}([0,1];\mathbb{E})$ and $\|u\|_{N,\infty}$ is a Banach norm on
$L^{\infty}([0,1];\mathbb{E})$, and both are bi-Lipschitz equivalent to the
standard Banach norms on those spaces.

If $\|\cdot\|$ is a Banach norm on a vector space, we denote by
$\|\cdot\|^{*}$ its Banach dual norm on the dual space. With this convention
in mind, we denote by $N^{*}$ the continuous function
$[0,1]\times\mathbb{E}^{*}\to[0,\infty)$ that, for each $t\in[0,1]$, gives the
norm $N_{t}^{*}$ on $\mathbb{E}^{*}$ dual to $N_{t}$, i.e.,

$N_{t}^{*}(w)=\sup\left\\{\langle w|u\rangle:u\in\mathbb{E},\ N_{t}(u)\leq
1\right\\}.$

######  Remark 3.1.

Recall the following consequence of Hahn–Banach Theorem, cfr. [5, Corollary
1.4]: if $(V,\|\cdot\|)$ is a Banach space, then, for every $u\in V$,

$\|u\|=\max\\{\langle w|u\rangle:w\in V^{*},\ \|w\|^{*}\leq 1\\}.$

######  Lemma 3.2.

Let $N:[0,1]\times\mathbb{E}\to[0,\infty)$ be a continuous function so that,
for every $t\in[0,1]$, the restriction $N_{t}:=N(t,\cdot)$ is a norm. Then

(14) $\begin{array}[]{ll}\|\cdot\|_{N^{*},\infty}=\|\cdot\|_{N,1}^{*}&\text{
on }L^{\infty}([0,1];\mathbb{E}^{*})\text{ and}\\\
\|\cdot\|_{N,\infty}=\|\cdot\|_{N^{*},1}^{*}&\text{ on
}L^{\infty}([0,1];\mathbb{E}),\end{array}$

and

(15) $\begin{array}[]{cl}\|\cdot\|_{N^{*},1}=\|\cdot\|_{N,\infty}^{*}&\text{
on }L^{1}([0,1];\mathbb{E}^{*})\text{ and}\\\
\|\cdot\|_{N,1}=\|\cdot\|_{N^{*},\infty}^{*}&\text{ on
}L^{1}([0,1];\mathbb{E}).\end{array}$

In particular, if $u:[0,1]\to\mathbb{E}$, then

(16) $\displaystyle\|u\|_{N,1}$ $\displaystyle=\max\left\\{\langle
w|u\rangle:w\in L^{\infty}([0,1];\mathbb{E}^{*}),\ \|w\|_{N^{*},\infty}\leq
1\right\\},$ (17) $\displaystyle\|u\|_{N,\infty}$
$\displaystyle=\sup\left\\{\langle w|u\rangle:w\in
L^{1}([0,1];\mathbb{E}^{*}),\ \|w\|_{N^{*},1}\leq 1\right\\}.$

Moreover, if $w\in L^{\infty}([0,1];\mathbb{E}^{*})$ is an argument of the
maximum of (16), then, for almost every $t\in[0,1]$, $N_{t}^{*}(w(t))=1$ and
$\langle w(t)|u(t)\rangle=N_{t}(u(t))$.

Notice that (14) is a pair of genuine identities of norms, because
$L^{\infty}([0,1];\mathbb{E}^{*})$ is the dual space of
$L^{1}([0,1];\mathbb{E})$ and $L^{\infty}([0,1];\mathbb{E})$ is the dual space
of $L^{1}([0,1];\mathbb{E}^{*})$. The second pair of identities (15) is
understood via the standard embedding of a Banach space into its bi-dual,
because $L^{1}$ is not reflexive, i.e., the dual of
$L^{\infty}([0,1];\mathbb{E}^{*})$ (resp. $L^{\infty}([0,1];\mathbb{E})$) is
not $L^{1}([0,1];\mathbb{E})$ (resp. $L^{1}([0,1];\mathbb{E}^{*})$).

The fact that $\|u\|_{N,1}$ is a maximum is an application of Remark 3.1. On
the contrary, $\|u\|_{N,\infty}$ is only a supremum and we can see this
phenomenon with $\mathbb{E}=\mathbb{R}$ and $N_{1}\equiv|\cdot|$: indeed, on
the one hand, $u(x):=x$ is an element of $L^{\infty}([0,1];\mathbb{R})$ and
$\operatorname*{{esssup}}_{x}|u(x)|=1$; on the other hand, there is no
function $w\in L^{1}([0,1];\mathbb{R})$ so that
$\int_{0}^{1}|w(x)|\,\mathrm{d}x\leq 1$ and
$\int_{0}^{1}xw(x)\,\mathrm{d}x=1$. One can show the latter claim from the
fact that it would hold

$1=\int_{0}^{1}xw(x)\,\mathrm{d}x\leq\int_{0}^{1}|w(x)|\,\mathrm{d}x\leq 1,$

from which it would follow that
$\int_{0}^{1}xw(x)\,\mathrm{d}x=\int_{0}^{1}|w(x)|\,\mathrm{d}x$. Writing $w$
as the difference between its positive and negative part, one gets a
cotradiction.

###### Proof of Lemma 3.2.

Notice that in both (14) and (15), the two identities are equivalent because
$\mathbb{E}$ has finite dimension.

Let us now prove the second identity in (14). Fix $u\in
L^{\infty}([0,1];\mathbb{E})$. Up to choosing a representative, we can assume
$u$ is defined on the whole $[0,1]$. Notice that, by definition of dual norm,

$\|u\|_{N^{*},1}^{*}=\sup\\{\langle w|u\rangle:w\in
L^{1}([0,1];\mathbb{E}^{*}),\ \|w\|_{N^{*},1}\leq 1\\},$

where $\langle w|u\rangle:=\int_{0}^{1}\langle w(t)|u(t)\rangle\,\mathrm{d}t$.
Moreover, for every $w\in L^{1}([0,1];\mathbb{E}^{*})$, we have that

$\int_{0}^{1}\langle
w(t)|u(t)\rangle\,\mathrm{d}t\leq\int_{0}^{1}N_{t}(u(t))N_{t}^{*}(w(t))\,\mathrm{d}t\leq\operatorname*{{esssup}}_{t\in[0,1]}N_{t}(u(t))\int_{0}^{1}N_{t}^{*}(w(t))\,\mathrm{d}t$

Hence we get $\|u\|_{N^{*},1}^{*}\leq\|u\|_{N,\infty}$.

In order to show that $\|u\|_{N^{*},1}^{*}\geq\|u\|_{N,\infty}$, it is
sufficient to prove that, for every $\varepsilon>0$, there exists
$w_{\varepsilon}\in L^{1}([0,1];\mathbb{E}^{*})$ such that
$\|w_{\varepsilon}\|_{N^{*},1}\leq 1+\varepsilon$ and $\langle
w_{\varepsilon}|u\rangle\geq\|u\|_{N,\infty}-2\varepsilon$, and taking
$\varepsilon\to 0$. So, for $0<\varepsilon<1/2$, define

$B_{\varepsilon}:=\left\\{s\in[0,1]:N_{s}(u(s))\geq\|u\|_{N,\infty}-\varepsilon\right\\},$

and notice that $\mathscr{L}^{1}(B_{\varepsilon})>0$. Let
$\\{w_{j}\\}_{j\in\mathbb{N}}$ be a dense subset of $\mathbb{E}^{*}$ and
define

$A_{\varepsilon,j}:=\left\\{t\in B_{\varepsilon}:N_{t}^{*}(w_{j})\leq
1+\varepsilon\text{ and }\langle w_{j}|u(t)\rangle\geq
N_{t}(u(t))-\varepsilon\right\\},$

and notice that $B_{\varepsilon}=\bigcup_{j\in\mathbb{N}}A_{\varepsilon,j}$.
Finally, define $w_{\varepsilon}:[0,1]\to\mathbb{E}^{*}$ by

$w_{\varepsilon}(t):=\frac{1}{\mathscr{L}^{1}(B_{\varepsilon})}\sum_{j=1}^{\infty}\chi_{A_{\varepsilon,j}\setminus\bigcup_{i=1}^{j-1}A_{\varepsilon,i}}(t)\cdot
w_{j}.$

From how $w_{\varepsilon}$ is defined, we obtain that
$N_{t}^{*}(w_{\varepsilon}(t))\leq(1+\varepsilon)/(\mathscr{L}^{1}(B_{\varepsilon}))$
for every $t\in B_{\varepsilon}$ and $w_{\varepsilon}(t)=0$ for every
$t\in[0,1]\setminus B_{\varepsilon}$. Hence
$\|w_{\varepsilon}\|_{N^{*},1}=\int_{0}^{1}N_{t}^{*}(w_{\varepsilon}(t))\,\mathrm{d}t\leq
1+\varepsilon$. We conclude that

$\int_{0}^{1}\langle
w_{\varepsilon}(t)|u(t)\rangle\,\mathrm{d}t\geq\int_{B_{\varepsilon}}\frac{N_{t}(u(t))-\varepsilon}{\mathscr{L}^{1}(B_{\varepsilon})}\,\mathrm{d}t\geq\|u\|_{N,\infty}-2\varepsilon.$

Next, we prove the second identity in (15): Fix $u\in L^{1}([0,1];\mathbb{E})$
and notice that

$\|u\|_{N^{*},\infty}^{*}=\sup\left\\{\langle w|u\rangle:w\in
L^{\infty}([0,1];\mathbb{E}^{*}),\ \|w\|_{N^{*},\infty}\leq 1\right\\}.$

Since

(18) $\langle w|u\rangle\leq\|w\|_{N^{*},\infty}\cdot\|u\|_{N,1},$

then $\|u\|_{N,1}\geq\|u\|_{N^{*},\infty}^{*}$. For the opposite inequality,
it is sufficient to prove that, for every $\varepsilon>0$ there exists
$w_{\varepsilon}\in L^{\infty}([0,1];\mathbb{E}^{*})$ with
$\|w_{\varepsilon}\|_{N^{*},\infty}\leq 1+\varepsilon$ and $\langle
w_{\varepsilon}|u\rangle\geq\|u\|_{N,1}-\varepsilon$, and then taking
$\varepsilon\to 0$. Indeed, let $\\{w_{j}\\}_{j\in\mathbb{N}}$ be a dense
subset of $\mathbb{E}^{*}$ and, for $\varepsilon>0$ and each $j\in\mathbb{N}$,
define the sets

$A_{\varepsilon,j}:=\left\\{t\in[0,1]:N_{t}^{*}(w_{j})\leq 1+\varepsilon\text{
and }\langle w_{j}|u(t)\rangle\geq N_{t}(u(t))-\varepsilon\right\\},$

and the function $w_{\varepsilon}\in L^{\infty}([0,1];\mathbb{E}^{*})$,

$w_{\varepsilon}(t)=\sum_{j=1}^{\infty}\chi_{A_{\varepsilon,j}\setminus\bigcup_{i=1}^{j-1}A_{\varepsilon,i}}(t)\cdot
w_{j}.$

Then we have $[0,1]=\bigcup_{j\in\mathbb{N}}A_{\varepsilon,j}$,
$\|w_{\varepsilon}\|_{N^{*},\infty}\leq 1+\varepsilon$ and

$\int_{0}^{1}\langle
w_{\varepsilon}(t)|u(t)\rangle\,\mathrm{d}t\geq\int_{0}^{1}N_{t}(u(t))\,\mathrm{d}t-\varepsilon.$

Both suprema in (16) and (17) are a direct consequence of the previous
identities. The fact that the supremum in (16) is attained, is an application
of Remark 3.1, since we know that the dual space of
$(L^{1}([0,1];\mathbb{E}),\|\cdot\|_{N,1})$ is
$(L^{\infty}([0,1];\mathbb{E}^{*}),\|\cdot\|_{N^{*},1})$, thanks to the first
identity in (14).

To prove the last statement of the lemma, let $w$ be an argument of the
maximum in (16). First, we claim that $N_{t}^{*}(w(t))=1$ for almost every
$t\in[0,1]$. If this were not the case, then there would be a set
$A\subset[0,1]$ with positive measure such that
$N_{t}^{*}(w(t))\leq\lambda<1$, for some $\lambda>0$ and all $t\in A$. Define
$\tilde{w}(t):=\chi_{[0,1]\setminus A}(t)w(t)+\chi_{A}(t)w(t)/\lambda$. Then
$\|\tilde{w}\|_{N^{*},\infty}\leq 1$ and
$\langle\tilde{w}(t)|u(t)\rangle>\langle w(t)|u(t)\rangle$ for $t\in A$, in
contradiction to the maximality of $w$.

Finally on the one hand from (18) we consequently have $\langle
w(t)|u(t)\rangle\leq N_{t}(u(t))$ for almost every $t\in[0,1]$. On the other
hand, by maximality of $w$ we have

$\int_{0}^{1}\langle
w(t)|u(t)\rangle\,\mathrm{d}t=\int_{0}^{1}N_{t}(u(t))\,\mathrm{d}t.$

Consequently $\langle w(t)|u(t)\rangle=N_{t}(u(t))$ for almost every
$t\in[0,1]$. ∎

###  3.2. Energy and length

As in Section 2, we are in the setting where $M$ is a smooth manifold and
$\mathbb{E}$ is a finite-dimensional Banach space. Let $\mathcal{N}$ be the
space of all continuously varying norms on $M\times\mathbb{E}$; that is
$N\in\mathcal{N}$ if $N$ is a continuous function
$N:M\times\mathbb{E}\to[0,+\infty)$ that is a norm on fibers. We endow
$\cal{N}$ with the topology of uniform convergence on compact sets. We now
define the energy and the lengths associated to controls, i.e., elements of
$L^{\infty}([0,1];\mathbb{E})$, and we study some of their basic properties.

######  Definition 3.3 (Energy and length).

Let $(o,\mathtt{f},u)\in\mathscr{D}_{\mathrm{End}}$ and
$\gamma(t):=\mathrm{End}^{\mathtt{f}}_{o}(tu)$ for every $t\in[0,1]$. We
define the _energy_

$\mathtt{J}(o,\mathtt{f},u,N):=\operatorname*{{esssup}}_{t\in[0,1]}N(\gamma(t),u(t)),$

and the _length_

$\ell(o,\mathtt{f},u,N):=\int_{0}^{1}N(\gamma(t),u(t))\,\mathrm{d}t.$

For the next result, we recall that $\mathscr{D}_{\mathrm{End}}$ is the domain
of the End-point map.

######  Proposition 3.4 (Semi-continuity of energy and length).

Both functions $\mathtt{J}$ and $\ell$ from
$\mathscr{D}_{\mathrm{End}}\times\mathcal{N}$ to $\mathbb{R}$ are lower-
semicontinuous. In other words, if
$(o_{n},\mathtt{f}_{n},u_{n},N_{n})\to(o_{\infty},\mathtt{f}_{\infty},u_{\infty},N_{\infty})$
in $\mathscr{D}_{\mathrm{End}}\times\cal{N}$, then

(19)
$\displaystyle\mathtt{J}(o_{\infty},\mathtt{f}_{\infty},u_{\infty},N_{\infty})$
$\displaystyle\leq\liminf_{n\to\infty}\mathtt{J}(o_{n},\mathtt{f}_{n},u_{n},N_{n}),$
$\displaystyle\ell(o_{\infty},\mathtt{f}_{\infty},u_{\infty},N_{\infty})$
$\displaystyle\leq\liminf_{n\to\infty}\ell(o_{n},\mathtt{f}_{n},u_{n},N_{n}).$

###### Proof.

Let $(o_{n},\mathtt{f}_{n},u_{n},N_{n})\in\mathscr{D}_{\mathrm{End}}\times\cal
N$ be a sequence converging to

$(o_{\infty},\mathtt{f}_{\infty},u_{\infty},N_{\infty})\in\mathscr{D}_{\mathrm{End}}\times\cal
N.$

Let us show that
$u\mapsto\operatorname*{{esssup}}_{t\in[0,1]}N_{\infty}(\gamma_{\infty}(t),u(t))$
is lower-semicontinuous with respect to the weak* convergence. Notice that the
previous one is a Banach norm on $L^{\infty}([0,1];\mathbb{E})$. In
particular, it is the dual norm of the norm defined on
$L^{1}([0,1];\mathbb{E}^{*})$ via

$v\to\int_{0}^{1}N_{\infty}^{*}(\gamma_{\infty}(t),v(t))\,\mathrm{d}t,$

see Lemma 3.2. Since the weak* convergence on $L^{\infty}([0,1];\mathbb{E})$
does not depend on the choice of biLipschitz equivalent Banach norms on the
predual $L^{1}([0,1];\mathbb{E}^{*})$, and since every dual norm is weakly*
lower-semicontinuous, we get the sought claim.

Let us now prove the first inequality in (19). Notice that, for every
$n\in\mathbb{N}$ and $t\in[0,1]$ we have

(20) $N_{n}(\gamma_{n}(t),u_{n}(t))\geq
N_{\infty}(\gamma_{\infty}(t),u_{n}(t))-\left|N_{n}(\gamma_{n}(t),u_{n}(t))-N_{\infty}(\gamma_{\infty}(t),u_{n}(t))\right|,$

where, for $j\in\mathbb{N}\cup\\{\infty\\}$, and $t\in[0,1]$,
$\gamma_{j}(t):=\mathrm{End}_{o_{j}}^{\mathtt{f}_{j}}[tu_{j}]$. As a
consequence of Proposition 2.9 we have that
$\gamma_{n}(t)\to\gamma_{\infty}(t)$ uniformly on $[0,1]$. Hence, the
functions $(t,v)\mapsto N_{n}(\gamma_{n}(t),v)$ converge uniformly on compact
subsets of $[0,1]\times\mathbb{E}$ to $(t,v)\mapsto
N_{\infty}(\gamma_{\infty}(t),v)$, and since the $u_{n}$ are uniformly
bounded, the second term in the lower bound (20) goes to 0 as $n\to\infty$,
uniformly in $t$.

The first inequality in (19) thus follows from the following computation

$\displaystyle\liminf_{n\to\infty}$
$\displaystyle\mathtt{J}(o_{n},\mathtt{f}_{n},u_{n},N_{n})$
$\displaystyle\geq\liminf_{n\to\infty}\operatorname*{{esssup}}_{t\in[0,1]}(N_{\infty}(\gamma_{\infty}(t),u_{n}(t))+$
$\displaystyle\phantom{+}-\left|N_{n}(\gamma_{n}(t),u_{n}(t))-N_{\infty}(\gamma_{\infty}(t),u_{n}(t))\right|)$
$\displaystyle=\liminf_{n\to\infty}\operatorname*{{esssup}}_{t\in[0,1]}N_{\infty}(\gamma_{\infty}(t),u_{n}(t))$
$\displaystyle\geq\mathtt{J}(o_{\infty},\mathtt{f}_{\infty},u_{\infty},N_{\infty}).$

Similarly, to prove that $\ell$ is lower-semicontinuous, we need, in addition
to (20), to prove that
$u\mapsto\int_{0}^{1}N_{\infty}(\gamma_{\infty}(s),u(s))\,\mathrm{d}s$ is
lower-semicontinuous on $L^{\infty}([0,1];\mathbb{E})$ (endowed with the weak*
topology).

Let
$\mathbb{S}^{*}(t)=\\{v\in\mathbb{E}^{*}:N_{\infty}^{*}(\gamma_{\infty}(t),v)=1\\}$,
so that, for almost every $t\in[0,1]$, we have

$N_{\infty}(\gamma_{\infty}(t),u_{\infty}(t))=\sup_{v\in\mathbb{S}^{*}(t)}\langle
v|u_{\infty}(t)\rangle.$

Moreover, if we set $\mathbb{S}^{*}=\\{v\in
L^{\infty}([0,1];\mathbb{E}^{*}):v(t)\in\mathbb{S}^{*}(t)\,\text{for almost
every $t\in[0,1]$}\\}$, then, by Lemma 3.2, for every $u\in
L^{1}([0,1];\mathbb{E})$ there is $v\in\mathbb{S}^{*}$ such that $\langle
v(t)|u(t)\rangle=N_{\infty}(\gamma_{\infty}(t),u(t))$ for a.e. $t\in[0,1]$.
Therefore, for every $u\in L^{1}([0,1];\mathbb{E})$,

$\int_{0}^{1}N_{\infty}(\gamma_{\infty}(t),u(t))\,\mathrm{d}t=\max_{v\in\mathbb{S}^{*}}\int_{0}^{1}\langle
v(t)|u(t)\rangle\,\mathrm{d}t.$

We can finally conclude, by using the fact that since $u_{n},u_{\infty}\in
L^{\infty}([0,1];\mathbb{E})$ we have a fortiori that $u_{n},u_{\infty}\in
L^{1}([0,1];\mathbb{E})$, with the following chain of inequalities

$\displaystyle\int_{0}^{1}N_{\infty}(\gamma_{\infty}(t),u_{\infty}(t))\,\mathrm{d}t$
$\displaystyle=\max_{v\in\mathbb{S}^{*}}\int_{0}^{1}\langle
v(t)|u_{\infty}(t)\rangle\,\mathrm{d}t$
$\displaystyle=\max_{v\in\mathbb{S}^{*}}\liminf_{n\to\infty}\int_{0}^{1}\langle
v(t)|u_{n}(t)\rangle\,\mathrm{d}t$
$\displaystyle\leq\max_{v\in\mathbb{S}^{*}}\liminf_{n\to\infty}\max_{w\in\mathbb{S}^{*}}\int_{0}^{1}\langle
w(t)|u_{n}(t)\rangle\,\mathrm{d}t$
$\displaystyle=\liminf_{n\to\infty}\max_{w\in\mathbb{S}^{*}}\int_{0}^{1}\langle
w(t)|u_{n}(t)\rangle\,\mathrm{d}t$
$\displaystyle=\liminf_{n\to\infty}\int_{0}^{1}N_{\infty}(\gamma_{\infty}(t),u_{n}(t))\,\mathrm{d}t.$

∎

With an abuse of language, we say that a curve $\gamma$ has constant $N$-speed
or simply constant speed if $\gamma=\gamma_{(o,\mathtt{f},u)}$ with $t\mapsto
N(\gamma(t),u(t))$ almost everywhere constant for $t\in[0,1]$. That a curve
$\gamma$ has constant speed, really depends on all the data
$(o,\mathtt{f},u,N)$.

######  Lemma 3.5 (Constant speed reparametrization).

If $(o,\mathtt{f},u,N)\in\mathscr{D}_{\mathrm{End}}\times\cal N$, then there
exists $v\in L^{\infty}([0,1],\mathbb{E})$ such that
$(o,\mathtt{f},v)\in\mathscr{D}_{\mathrm{End}}$, $\gamma_{(o,\mathtt{f},u)}$
is a reparametrization of $\gamma_{(o,\mathtt{f},v)}=:\gamma$, and

(21)
$N(\gamma(t),v(t))=\mathtt{J}(o,\mathtt{f},v,N)=\ell(o,\mathtt{f},v,N)=\ell(o,\mathtt{f},u,N),$

for almost every $t\in[0,1]$. In particular, $\gamma$ has constant speed and
$\mathtt{J}(o,\mathtt{f},v,N)\leq\mathtt{J}(o,\mathtt{f},u,N)$.

###### Proof.

Let us assume that $\ell:=\ell(o,\mathtt{f},u,N)\neq 0$, otherwise the result
is trivial. Define

$\psi(t):=\frac{1}{\ell}\int_{0}^{t}N(\eta(s),u(s))\,\mathrm{d}s,$

where $\eta:=\gamma_{(o,\mathtt{f},u)}$. Notice that $\psi:[0,1]\to[0,1]$ is a
Lipschitz function with $\psi(0)=0$, $\psi(1)=1$ and $\psi^{\prime}\geq 0$.
Since $\psi$ is Lipschitz, the image set
$E:=\psi(\\{t:\psi^{\prime}(t)=0\text{ or $\psi^{\prime}(t)$ does not
exist}\\})$ has measure zero by the area formula. Notice that for every
$s\in[0,1]\setminus E$ there exists a unique $t\in[0,1]$ such that
$\psi(t)=s$, and thus the formula

(22) $v(s)=\frac{u(t)}{\psi^{\prime}(t)},$

defines a measurable function $v:[0,1]\to\mathbb{E}$. Moreover

(23) $N(\eta(t),v(s))=\frac{N(\eta(t),u(t))}{\psi^{\prime}(t)}=\ell,$

and then, since $N(\eta(\cdot),\cdot)$ is a continuous family of norms, we get
that $v\in L^{\infty}([0,1];\mathbb{E})$. Furthermore, notice that from (22)
it follows $\gamma_{(o,\mathtt{f},u)}(t)=\gamma_{(o,\mathtt{f},v)}(\psi(t))$.
Therefore, for every $s\in[0,1]\setminus E$, we have $N(\gamma(s),v(s))=\ell$.
Finally, since $N(\gamma(\cdot),v(\cdot))$ is constant, then it is equal to
both the energy and the length of the curve $\gamma$. ∎

###  3.3. Carnot-Carathéodory distances

In this section we define the Carnot-Carathéodory distance, CC distance for
brevity, associated to $(\mathtt{f},N)$ and we investigate some of its
properties. We use the notation $\mathrm{Lip}\Gamma$ from Section 2.1, and
$\mathcal{N}$ from Section 3.2.

######  Definition 3.6 (CC distance).

Given $\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ and
$N\in\cal N$, we define _the Carnot-Carathéodory distance_ , or _CC distance_
,

(24) $d_{(\mathtt{f},N)}(p,q):=\inf\\{\mathtt{J}(p,\mathtt{f},u,N):u\in
L^{\infty}([0,1];\mathbb{E}),\mathrm{End}^{\mathtt{f}}_{p}(u)=q\\}.$

It is immediate from Lemma 3.5 that

(25) $d_{(\mathtt{f},N)}(p,q)=\inf\\{\ell(p,\mathtt{f},u,N):u\in
L^{\infty}([0,1];\mathbb{E}),\mathrm{End}^{\mathtt{f}}_{p}(u)=q\\}.$

We will denote the open $d_{(\mathtt{f},N)}$-ball of radius $r$ and center $p$
by $B_{(\mathtt{f},N)}(p,r)$. Moreover, if $M^{\prime}\subset M$ is open, we
denote by $d_{(\mathtt{f},N)}|_{M^{\prime}}$ the distance defined by
restricting $(\mathtt{f},N)$ to $M^{\prime}$. In particular, the infimum
defining $d_{(\mathtt{f},N)}|_{M^{\prime}}$ is taken over the family of
controls whose integral curves lie in $M^{\prime}$. Thus,
$d_{(\mathtt{f},N)}|_{M^{\prime}}(p,q)\geq d_{(\mathtt{f},N)}(p,q)$ for every
$p,q\in M^{\prime}$.

######  Lemma 3.7.

Let
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
be a norm on $\mathbb{E}$, and define on $L^{\infty}([0,1];\mathbb{E})$ the
Banach norm

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}=\operatorname*{{esssup}}_{t\in[0,1]}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u(t)\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}.$

Let $\rho$ be a complete Riemannian metric on $M$, which induces a norm
$|\cdot|_{\rho}$ on $TM$.

Let $\mathscr{K}\subset\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$,
$K\subset M$, and $\cal K\subset\cal N$ be compact sets, and $R>0$. Then there
exists $L\geq 1$ such that

(26) $\begin{aligned}
1/L{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|v\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\leq
N(p,v)&\leq
L{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|v\right|\kern-1.07639pt\right|\kern-1.07639pt\right|},\\\
|\mathtt{f}(p)[v]|_{\rho}&\leq
L{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|v\right|\kern-1.07639pt\right|\kern-1.07639pt\right|},\end{aligned}\qquad\forall
p\in\bar{B}_{\rho}(K,R),\ \mathtt{f}\in\mathscr{K},\ N\in\cal
K,v\in\mathbb{E}.$

Moreover, the following hold for every $\mathtt{f}\in\mathscr{K}$, $p\in K$,
$u\in L^{\infty}([0,1];\mathbb{E})$ and $N\in\cal K$:

  1. (a)

if
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}<\frac{R}{L}$,
then $(p,\mathtt{f},u)\in\mathscr{D}_{\mathrm{End}}$ and
$\gamma_{(p,\mathtt{f},u)}(t)\in B_{\rho}(K,R)$ for every $t\in[0,1]$;

  2. (b)

if $(p,\mathtt{f},u)\in\mathscr{D}_{\mathrm{End}}$ and
$\ell(p,\mathtt{f},u,N)<R/L^{2}$, then $\gamma_{(p,\mathtt{f},u)}(t)\in
B_{\rho}(K,R)$ for every $t\in[0,1]$;

  3. (c)

if $(p,\mathtt{f},u)\in\mathscr{D}_{\mathrm{End}}$ and
$\gamma_{(p,\mathtt{f},u)}(t)\in\bar{B}_{\rho}(K,R)$ for every $t\in[0,1]$,
then

(27) $\rho(p,\mathrm{End}^{\mathtt{f}}_{p}(u))\leq
L{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}\leq
L^{2}\mathtt{J}(p,\mathtt{f},u,N)\leq
L^{3}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty};$

  4. (d)

if $q\in M$ is such that $d_{(\mathtt{f},N)}(p,q)<\frac{R}{L^{2}}$, then

(28) $\frac{1}{L^{2}}\rho(p,q)\leq
d_{(\mathtt{f},N)}(p,q)=d_{(\mathtt{f},N)}|_{B_{\rho}(K,R)}(p,q).$

###### Proof.

The existence of $L$ that satisfies (26) is a consequence of the compactness
of $\bar{B}_{\rho}(K,R)\times\mathscr{K}\times\cal K$, and the continuity of
$\mathtt{f}$ and $N$.

Let $p\in K$, $\mathtt{f}\in\mathscr{K}$, $u\in L^{\infty}([0,1];\mathbb{E})$
and $\gamma:[0,t_{0}]\to M$ be a solution of

(29) $\begin{cases}\gamma^{\prime}(t)=\mathtt{f}(\gamma(t))[u(t)],\\\
\gamma(0)=p,\end{cases}$

for some $t_{0}>0$. Then, for every $a,b\in[0,t_{0}]$ with
$\gamma([a,b])\subset B_{\rho}(K,R)$, we have

(30) $\int_{a}^{b}|\gamma^{\prime}(s)|_{\rho}\,\mathrm{d}s\leq
L\int_{a}^{b}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u(s)\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\,\mathrm{d}s\leq
L{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}|b-a|.$

It follows that, if
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}<\frac{R}{L}$,
then $\gamma$ can be extended, as solution to (29), to a curve
$\gamma:[0,1]\to B_{\rho}(K,R)$, i.e., part (a) holds.

Similarly, we infer part (b) from the estimate

(31) $\int_{a}^{b}|\gamma^{\prime}(s)|_{\rho}\,\mathrm{d}s\leq
L\int_{a}^{b}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u(s)\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\,\mathrm{d}s\leq
L^{2}\int_{a}^{b}N(\gamma(s),u(s))\,\mathrm{d}s\leq
L^{2}\ell(p,\mathtt{f},u,N),$

that holds whenever $\gamma([a,b])\subseteq B_{\rho}(K,R)$.

The first inequality in (27) follows from (30). The other two estimates in
(27) are obtained from the first line of (26), and also part (c) is proven.

For part (d), we see from (25) and item (b) that, if
$d_{(\mathtt{f},N)}(p,q)<R/L^{2}$, then the infimum in (25) can be taken on
curves laying in $\bar{B}_{\rho}(K,R)$. This shows the equality in (28); the
first inequality in (28) is then a direct consequence of (24) and item (c). ∎

######  Lemma 3.8.

Let $\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ an $N\in\cal
N$. Denote by $\tau_{M}$ the manifold topology of $M$ and by
$\tau_{(\mathtt{f},N)}$ the topology of $(M,d_{(\mathtt{f},N)})$. Then
$\tau_{M}\subset\tau_{(\mathtt{f},N)}$.

###### Proof.

Let $U\in\tau_{M}$ and $\hat{p}\in U$. We need to show that there is $r>0$
such that $B_{(\mathtt{f},N)}(\hat{p},r)\subset U$. Fix a complete Riemannian
metric $\rho$ on $M$ and let $L\geq 1$ be a constant that satisfies (26) for
$R=1$, $K=\\{\hat{p}\\}$. By part (d) of Lemma 3.7, if
$d_{(\mathtt{f},N)}(\hat{p},q)<1/L^{2}$, then $\rho(\hat{p},q)\leq
L^{2}d_{(\mathtt{f},N)}(\hat{p},q)$. Let $\varepsilon>0$ be such that
$B_{\rho}(\hat{p},\varepsilon)\subset U$ and fix
$0<r<\min\\{1/L^{2},\varepsilon/L^{2}\\}$. If $q\in
B_{(\mathtt{f},N)}(\hat{p},r)$, then $d_{(\mathtt{f},N)}(\hat{p},q)<1/L^{2}$
and thus $\rho(\hat{p},q)\leq L^{2}d_{(\mathtt{f},N)}(\hat{p},q)<\varepsilon$.
Therefore, $B_{(\mathtt{f},N)}(\hat{p},r)\subset
B_{\rho}(\hat{p},\varepsilon)\subset U$. ∎

In the setting of the above Lemma 3.8, the two topologies may not be equal. As
an example, consider a structure $\mathtt{f}$ defined by a constant line field
on $\mathbb{R}^{2}$: the integral lines of such a structure are open sets in
$\tau_{(\mathtt{f},N)}$ but not in the standard topology of $\mathbb{R}^{2}$.
We will later show that the two topologies do agree under an essentially non-
holomic condition on $\mathtt{f}$, see Theorem 1.4.

######  Proposition 3.9 (Local existence of geodesics).

Let $\mathscr{K}\subset\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$,
$K\subset M$ and $\cal K\subset\cal N$ be compact sets. Then there is a
constant $C>0$ so that, for every $\mathtt{f}\in\mathscr{K}$, every $N\in\cal
K$, every $p\in K$, and every $q\in M$ with $d_{(\mathtt{f},N)}(p,q)\leq C$,
there exists $u\in L^{\infty}([0,1],\mathbb{E})$ of constant $N$-speed such
that $\mathrm{End}^{\mathtt{f}}_{p}(u)=q$ and

$\mathtt{J}(p,\mathtt{f},u,N)=\ell(p,\mathtt{f},u,N)=d_{(\mathtt{f},N)}(p,q).$

In particular, the curve
$\gamma_{(p,\mathtt{f},u)}:[0,1]\to(M,d_{(\mathtt{f},N)})$ is a homothetic
embedding, i.e., a $d_{(\mathtt{f},N)}$-length minimizing curve.

In the setting of Lemma 3.7, we can take $C=\frac{R}{L^{2}}$, for $R,L>0$
satisfying (26).

###### Proof.

In the setting of Lemma 3.7, fix $R>0$ and the corresponding $L>0$ and set
$C:=\frac{R}{L^{2}}$. Let $\mathtt{f}\in\mathscr{K}$, $N\in\cal K$, $p\in K$
and $q\in M$ with $d_{\mathtt{f}}(p,q)\leq C$. By Lemma 3.7.(d), there exists
a sequence $u_{n}\in L^{\infty}([0,1],\mathbb{E})$ such that
$\mathrm{End}^{\mathtt{f}}_{p}(u_{n})=q$,
$\lim_{n\to\infty}\mathtt{J}(p,\mathtt{f},u_{n},N)=d_{(\mathtt{f},N)}(p,q)$,
and $\gamma_{(p,\mathtt{f},u_{n})}(t)\in B_{\rho}(K,R)$ for all $t\in[0,1]$
and all $n$.

From Lemma 3.7.(c),
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u_{n}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}$
is uniformly bounded in $n$. Therefore, up to passing to a subsequence,
$u_{n}$ weakly* converge to some $u_{\infty}\in L^{\infty}([0,1],\mathbb{E})$,
with

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u_{\infty}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}\leq\liminf_{n\to\infty}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u_{n}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}\leq\liminf_{n\to\infty}L\mathtt{J}(p,\mathtt{f},u_{n},N)=Ld_{(\mathtt{f},N)}(p,q)<\frac{R}{L}.$

By Lemma 3.7.(a), we thus have
$(p,\mathtt{f},u_{\infty})\in\mathscr{D}_{\mathrm{End}}$. Finally, From
Theorem 2.5 we get $\mathrm{End}^{\mathtt{f}}_{p}(u_{\infty})=q$, while from
Proposition 3.4 we get

$\mathtt{J}(p,\mathtt{f},u_{\infty},N)\leq\lim_{n\to\infty}\mathtt{J}(p,\mathtt{f},u_{n},N)=d_{(\mathtt{f},N)}(p,q),$

and thus $d_{(\mathtt{f},N)}(p,q)=\mathtt{J}(p,\mathtt{f},u_{\infty},N)$.

We claim that $N(\gamma(t),u_{\infty}(t))$ is constant for almost every
$t\in[0,1]$. Indeed, if not, we have that
$\ell(p,\mathtt{f},u_{\infty},N)<J(p,\mathtt{f},u_{\infty},N)$. Moreover,
since we have (24), (25), and the trivial inequality $\ell\leq\mathtt{J}$, we
get that on controls that realize the distance one has $\ell=\mathtt{J}$,
which gives the sought contradiction. Therefore, from the minimality, we get
$N(\gamma(t),u_{\infty}(t))=d_{(\mathtt{f},N)}(p,q)$ for almost every
$t\in[0,1]$. Notice that, for curves of constant speed, energy and length are
equal.

Finally, the claim that
$\gamma=\gamma_{(p,\mathtt{f},u)}:[0,1]\to(M,d_{(\mathtt{f},N)})$ is a
homothetic embedding, i.e., that for every $s,t\in[0,1]$

$d_{(\mathtt{f},N)}\left(\gamma(s),\gamma(t)\right)=d_{(\mathtt{f},N)}(\gamma(0),\gamma(1))\cdot|t-s|,$

is a direct consequence of the minimality of $u$ and the fact that $t\mapsto
N(\gamma(t),u(t))=d_{(\mathtt{f},N)}(\gamma(0),\gamma(1))$ for a.e.
$t\in[0,1]$. ∎

######  Proposition 3.10.

Let $\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ and $N\in\cal
N$. Suppose that there exist a complete Riemannian metric $\rho$, a norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
on $\mathbb{E}$ and $L>0$ so that (26) holds for every $R>0$ and every compact
set $K\subseteq M$. Then $(M,d_{(\mathtt{f},N)})$ is a complete and geodesic
metric space, whose closed bounded sets are compact. However,
$d_{(\mathtt{f},N)}$ may take value $\infty$ and be not continuous with
respect to the manifold topology.

###### Proof.

By Lemma 3.7.(d), the bound (28) holds for every $p,q\in M$. Therefore, a
$d_{(\mathtt{f},N)}$-closed set is also $\rho$-closed, and a
$d_{(\mathtt{f},N)}$-bounded set is also $\rho$-bounded. In particular, a
$d_{(\mathtt{f},N)}$-closed and $d_{(\mathtt{f},N)}$-bounded set is compact.
Hence closed bounded sets in $(M,d_{(\mathtt{f},N)})$ are compact, and thus
$d_{(\mathtt{f},N)}$-Cauchy sequences converge. Proposition 3.9 readily
implies that $(M,d_{(\mathtt{f},N)})$ is a geodesic space. ∎

######  Remark 3.11.

We know that for every $R>0$ there are $L_{R}^{(1)}>0$ and $L_{R}^{(2)}>0$ so
that the two conditions in (26) hold, with two independent constants. One can
modify Proposition 3.10 by requiring that the growth of $L_{R}^{(1)}$ and
$L_{R}^{(2)}$ as $R\to\infty$ are slow enough, although not bounded. However,
we don’t need such a finer analysis.

###  3.4. CC distances and sub-Finsler lengths of curves

We use the notation $\mathrm{Lip}\Gamma$ from Section 2.1, and $\mathcal{N}$
from Section 3.2. In this section we prove that the distance
$d_{(\mathtt{f},N)}$ is obtained as the infimum of the length of curves, where
the length element is the natural sub-Finsler structure on $TM$ associated to
$(\mathtt{f},N)$, see (32).

For $\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ and $N\in\cal
N$, define

(32) $|v|_{(\mathtt{f},N)}:=\inf\\{N(p,u):u\in\mathbb{E},\
\mathtt{f}(p,u)=v\\},$

for $p\in M$ and $v\in T_{p}M$. Notice that $|v|_{(\mathtt{f},N)}<\infty$ if
and only if $v\in\mathtt{f}(p,\mathbb{E})$.

######  Lemma 3.12.

Let $\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ and $N\in\cal
N$. If $\gamma:[0,1]\to(M,d_{(\mathtt{f},N)})$ is Lipschitz, then
$\gamma:[0,1]\to M$ is absolutely continuous and there exists $u\in
L^{\infty}([0,1];\mathbb{E})$ such that
$\gamma^{\prime}(t)=\mathtt{f}(\gamma(t),u(t))$ for a.e. $t\in[0,1]$.

###### Proof.

By Lemma 3.8, $\gamma:[0,1]\to M$ is continuous, hence $\gamma([0,1])$ is
compact. We fix a complete Riemannian metric $\rho$ on $M$ and apply (28) with
$K=\gamma([0,1])$, and $R=1$, obtaining that the curve
$\gamma:[0,1]\to(M,\rho)$ is locally Lipschitz and thus an absolutely
continuous curve in the manifold $M$.

Without losing in generality, we can assume that $\gamma$ is 1-Lipschitz, that
is, for every $t,s\in[0,1]$,

(33) $d_{(\mathtt{f},N)}(\gamma(s),\gamma(t))\leq|t-s|.$

By Proposition 3.9 (with $\mathscr{K}=\\{\mathtt{f}\\}$, $K=\gamma([0,1])$ and
$\cal K=\\{N\\}$), for every $n\in\mathbb{N}$ large enough and $0\leq j\leq
2^{n}-1$ integer, there is a control $u^{(n)}_{j}\in
L^{\infty}([0,1];\mathbb{E})$ such that the corresponding integral curve
$\gamma^{(n)}_{j}:[0,1]\to M$ starting from
$\gamma^{(n)}_{j}(0)=\gamma(\frac{j}{2^{n}})$ is a geodesic parametrized with
constant speed, with end point
$\gamma^{(n)}_{j}(1)=\gamma(\frac{j+1}{2^{n}})$, and, by also exploiting (33),

$\operatorname*{{esssup}}_{t\in[0,1]}N(\gamma^{(n)}_{j}(t),u^{(n)}_{j}(t))=d_{(\mathtt{f},N)}\left(\gamma\left(\frac{j}{2^{n}}\right),\gamma\left(\frac{j+1}{2^{n}}\right)\right)\leq
1/2^{n}.$

Thus, from Lemma 3.7.(b) and (c), we obtain that there exists $L$ such that,
for $n$ large enough,
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u^{(n)}_{j}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}\leq
L/2^{n}$ (here
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}$
is a reference norm as in Lemma 3.7). Define

$u^{(n)}(t):=2^{n}\sum_{j=0}^{2^{n}-1}u^{(n)}_{j}(2^{n}(t-j/2^{n})),$

where we mean that $u^{(n)}_{j}\equiv 0$ outside $[0,1]$ for every
$n\in\mathbb{N}$ and $0\leq j\leq 2^{n}-1$. Hence $u^{(n)}\in
L^{\infty}([0,1];\mathbb{E})$,
$(\gamma(0),\mathtt{f},u^{(n)})\in\mathscr{D}_{\mathrm{End}}$, and
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u^{(n)}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}\leq
L$. Since the controls $u^{(n)}$ are uniformly bounded in $n$, there is a
subsequence $\\{u^{(n_{k})}\\}_{k}$ that weakly* converges to some
$u^{(\infty)}$.

We claim that $u^{(\infty)}$ is a control for $\gamma$. Let $\gamma_{n}$ be
the integral curve of $u^{(n)}$ with starting point $\gamma_{n}(0)=\gamma(0)$.
Notice that $\gamma_{n}$ is the reparametrization on $[0,1]$ of the
concatenation of the $\gamma^{(n)}_{j}$ for $j$ from $0$ to $2^{n}-1$. In
particular, $\gamma_{n}(\frac{j}{2^{n}})=\gamma(\frac{j}{2^{n}})$ for all $n$
and $j$.

For every $\hat{n}$ large enough and every $\hat{j}<2^{\hat{n}}$, there exist
$\varepsilon_{\hat{n},\hat{j}}>0$ for which the control $u^{(\infty)}$ can be
integrated on the interval
$\left[\frac{\hat{j}}{2^{\hat{n}}},\frac{\hat{j}}{2^{\hat{n}}}+\varepsilon_{\hat{n},\hat{j}}\right]$.
Namely, there exists a curve

$\eta_{\hat{n},\hat{j}}:\left[\frac{\hat{j}}{2^{\hat{n}}},\frac{\hat{j}}{2^{\hat{n}}}+\varepsilon_{\hat{n},\hat{j}}\right]\to
M$

that has control $u^{(\infty)}$ and starting point
$\eta_{\hat{n},\hat{j}}(\frac{\hat{j}}{2^{\hat{n}}})=\gamma(\frac{\hat{j}}{2^{\hat{n}}})$.
By Proposition 2.9 (and by taking an affine reparametrization), the
restrictions

$\gamma_{n_{k}}|_{\left[\frac{\hat{j}}{2^{\hat{n}}},\frac{\hat{j}}{2^{\hat{n}}}+\epsilon_{\hat{n},\hat{j}}\right]}$

uniformly converge to $\eta_{\hat{n},\hat{j}}$ on compact subsets. By
continuity of $\eta_{\hat{n},\hat{j}}$, and by the previous convergence, it
follows that $\eta_{\hat{n},\hat{j}}(t)=\gamma(t)$ for all $t$ in the
respective domains, and thus $\gamma$ is an integral curve of $u^{(\infty)}$.
∎

In the rest of this section we will not need the following Lemma 3.13, since
it will be enough to use Lemma 3.12. We decided to keep this result here for
an independent interest because the proof is different and because it says
something more precise, that is, that $\gamma$ is tangent to the image of
$\mathtt{f}$ exactly at all points of differentiability of $\gamma$.

######  Lemma 3.13.

Let $\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ and $N\in\cal
N$. If $\gamma:[0,1]\to(M,d_{(\mathtt{f},N)})$ is Lipschitz, then $\gamma$ is
absolutely continuous and
$\gamma^{\prime}(t)\in\mathtt{f}(\gamma(t),\mathbb{E})$ for every $t\in[0,1]$
where $\gamma$ is differentiable.

###### Proof.

Let $L$ be the Lipschitz constant of $\gamma$. By Lemma 3.8, $\gamma:[0,1]\to
M$ is continuous, hence $\gamma([0,1])$ is compact. Hence, by (28) applied
with $K=\gamma([0,1])$, and $R=1$, for a complete Riemannian metric $\rho$,
the curve $\gamma:[0,1]\to(M,\rho)$ is locally Lipschitz and thus an
absolutely continuous curve in the manifold $M$. Let $t_{0}$ be a point of
differentiability for $\gamma$, define $Z:=\gamma^{\prime}(t_{0})$ and,
arguing by contradiction, suppose that
$Z\notin\mathtt{f}(\gamma(t_{0}),\mathbb{E})$.

We choose coordinates $(x_{1},\dots,x_{n})$ so that $\gamma(t_{0})=0$,
$Z=\partial_{n}$, and

$\mathtt{f}(\gamma(t_{0}),\mathbb{E})\subset\operatorname{\mathrm{span}}\\{\partial_{1},\dots,\partial_{n-1}\\}.$

For $v\in\mathbb{R}^{n}$, we write $v_{n}$ for the last coordinate of $v$.

Using the fact that $\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes
TM)$, that $N$ is continuous, and that $\mathtt{f}(0,v)_{n}=0$ for every
$v\in\mathbb{E}$, we deduce that there are $r>0$ and $C>0$ such that, for
every $p\in\mathbb{R}^{n}$ with $|p|\leq r$ and every $v\in\mathbb{E}$,

$|\mathtt{f}(p,v)_{n}|\leq C|p|N(p,v).$

By definition of $d_{(\mathtt{f},N)}$, for every $t$ there is $u^{t}\in
L^{\infty}([0,1];\mathbb{E})$ such that
$\mathrm{End}_{0}^{\mathtt{f}}(u^{t})=\gamma(t)$ and
$\mathtt{J}(0,\mathtt{f},u^{t},N)\leq 2d_{(\mathtt{f},N)}(0,\gamma(t))$. Let
$\sigma^{t}:[0,1]\to M$ be the curves with control $u^{t}$ and starting point
$0$.

We claim that $\sigma^{t}$ uniformly converge to the constant curve
$\sigma^{t_{0}}\equiv 0$ as $t\to t_{0}$. Indeed, since $\gamma$ is Lipschitz,
then, as $t\to t_{0}$, $d_{(\mathtt{f},N)}(0,\gamma(t))\to 0$ and thus
$\mathtt{J}(0,\mathtt{f},u^{t},N)\to 0$. By Lemma 3.7.(d) (with $K=\\{0\\}$ in
there) for $|t-t_{0}|$ small enough, we have $\sigma^{t}([0,1])\subset
B_{\rho}(0,r)$. Therefore, $u^{t}\to 0$ in $L^{\infty}([0,1];\mathbb{E})$ as
$t\to t_{0}$. By Proposition 2.9, the curves $\sigma^{t}$ uniformly converge
to the constant curve $\sigma^{t_{0}}\equiv 0$ as $t\to t_{0}$.

Since $\gamma^{\prime}(t_{0})=\partial_{n}$ then
$\gamma_{n}(t)\geq\frac{|t-t_{0}|}{2}$ for $|t-t_{0}|$ small enough. So, we
can make the following estimates for $|t-t_{0}|$ small enough

$\displaystyle\frac{|t-t_{0}|}{2}$
$\displaystyle\leq\gamma_{n}(t)=\sigma^{t}_{n}(1)$
$\displaystyle\leq\int_{0}^{1}|\dot{\sigma}^{t}_{n}(s)|\,\mathrm{d}s=\int_{0}^{1}|\mathtt{f}(\sigma^{t}(s),u^{t}(s))_{n}|\,\mathrm{d}s$
$\displaystyle\leq
C\int_{0}^{1}|\sigma^{t}(s)|N(\sigma^{t}(s),u^{t}(s))\,\mathrm{d}s$
$\displaystyle\leq
C\mathtt{J}(0,\mathtt{f},u^{t},N)\int_{0}^{1}|\sigma^{t}(s)|\,\mathrm{d}s$
$\displaystyle\leq
2Cd_{(\mathtt{f},N)}(0,\gamma(t))\int_{0}^{1}|\sigma^{t}(s)|\,\mathrm{d}s$
$\displaystyle\leq 2CL|t-t_{0}|\int_{0}^{1}|\sigma^{t}(s)|\,\mathrm{d}s$

Hence, $\int_{0}^{1}|\sigma^{t}(s)|\,\mathrm{d}s\geq\frac{1}{4CL}>0$ for $t$
close to $t_{0}$, which is in contradiction with the uniform convergence of
$\sigma^{t}$ to $0$. ∎

######  Lemma 3.14 (Borel right inverse).

Let $(X,d)$ be a locally compact, complete, separable metric space, $Y$ a
topological space and $f:X\to Y$ a surjective continuous function. Then there
is a Borel function $g:Y\to X$ such that $f(g(y))=y$ for all $y\in Y$.

###### Proof.

Let $\zeta:\mathbb{N}\to X$ be a function such that $\zeta(\mathbb{N})$ is
dense in $X$ and define $Z_{n}=\zeta(\\{0,1,\dots,n\\})$. For every
$n\in\mathbb{N}$, we define a function $g_{n}:Y\to Z_{n}\subset X$ iteratively
as follows: $g_{0}(y)=\zeta(0)$ for every $y\in Y$; if $g_{n}$ is given,
define $g_{n+1}(y):=g_{n}(y)$, unless there exists $x\in Z_{n+1}\cap
B(g_{n}(y),2d(g_{n}(y),f^{-1}(y)))$ such that
$d(x,f^{-1}(y))<\frac{1}{4}d(g_{n}(y),f^{-1}(y))$, in which case we set
$g_{n+1}(y):=x$. Notice that, since $Z_{n}$ is finite, $g_{n}$ is well defined
and Borel.

By construction, we have that for every $y\in Y$ the following holds

(34) $g_{k}(y)\in B(g_{n}(y),4d(g_{n}(y),f^{-1}(y)))\qquad\forall k>n,$

and, since $\bigcup_{n\in\mathbb{N}}Z_{n}$ is dense in $X$, we have
$\lim_{n\to\infty}d(g_{n}(y),f^{-1}(y))=0$. Therefore, taking into account
(34), every $y\in Y$ gives a Cauchy sequence $\\{g_{n}(y)\\}_{n}$ with
$\lim_{n\to\infty}d(g_{n}(y),f^{-1}(y))=0$. Since $X$ is complete, we can
define

$g(y):=\lim_{n\to\infty}g_{n}(y).$

As $g$ is the pointwise limit of a sequence of Borel functions, $g$ is also
Borel. Moreover, $d(g(y),f^{-1}(y))=0$, i.e., there is a sequence $x_{j}\in
f^{-1}(y)$ converging to $g(y)$. Hence, by the continuity of $f$, we have
$f(g(y))=\lim_{j\to\infty}f(x_{i})=y$. ∎

######  Lemma 3.15.

Let $\mathbb{E}_{1}$ and $\mathbb{E}_{2}$ be finite-dimensional Banach spaces,
$N:[0,1]\times\mathbb{E}_{1}\to[0,+\infty)$ a continuous function that is a
norm on $\mathbb{E}_{1}$ for each $t\in[0,1]$ and
$\pi:[0,1]\times\mathbb{E}_{1}\to\mathbb{E}_{2}$ a continuous function that is
a linear map $\mathbb{E}_{1}\to\mathbb{E}_{2}$ for each $t\in[0,1]$. If
$v:[0,1]\to\mathbb{E}_{2}$ is a measurable (resp., Borel) function such that
$v(t)\in\pi(t,\mathbb{E}_{1})$ for almost every $t\in[0,1]$, then there exists
a measurable (resp., Borel) function $u:[0,1]\to\mathbb{E}_{1}$ such that, for
almost every $t\in[0,1]$, $\pi(t,u(t))=v(t)$ and
$N(t,u(t))=\inf\\{N(t,w):\pi(t,w)=v(t)\\}$.

###### Proof.

Let

$Z:=\\{(t,\pi(t,w)):t\in[0,1],\
w\in\mathbb{E}_{1}\\}\subset[0,1]\times\mathbb{E}_{2},$

and $S:Z\to[0,\infty)$ defined as $S(t,v):=\min\\{N(t,w):\pi(t,w)=v\\}$.
Notice that $S$ is semi-continuous, in particular Borel. Indeed, if
$(t_{k},v_{k})\in Z$ converge to $(t_{\infty},v_{\infty})$, then there are
$w_{k}\in\mathbb{E}_{1}$ with $\pi(t_{k},w_{k})=v_{k}$ and
$N(t_{k},w_{k})=S(t_{k},v_{k})$. If $S(t_{k},v_{k})$ is uniformly bounded in
$k$, then, up to passing to a subsequence, $w_{k}\to w_{\infty}$ with
$\pi(t_{\infty},w_{\infty})=\lim_{k\to\infty}\pi(t_{k},w_{k})=v_{\infty}$ and
$S(t_{\infty},v_{\infty})\leq
N(t_{\infty},w_{\infty})=\lim_{k\to\infty}N(t_{k},w_{k})=\lim_{k\to\infty}S(t_{k},v_{k})$.
So,

$S(t_{\infty},v_{\infty})\leq\liminf_{(t,v)\to(t_{\infty},v_{\infty})}S(t,v).$

Define
$f:[0,1]\times\mathbb{E}_{1}\to[0,1]\times\mathbb{E}_{2}\times[0,+\infty)$ as
$f(t,w)=(t,\pi(t,w),N(t,w))$. We apply Lemma 3.14 with
$X=[0,1]\times\mathbb{E}_{1}$ and $Y=f(X)$, obtaining a Borel function $g:Y\to
X$ with $f(g(y))=y$ for all $y\in Y$. Notice that $g(t,v,\lambda)=(t,w)$ with
$\pi(t,w)=v$ and $N(t,w)=\lambda$.

Now, let $v:[0,1]\to\mathbb{E}_{2}$ be a Borel function with
$v(t)\in\pi(t,\mathbb{E}_{1})$ for almost every $t\in[0,1]$. Define
$\bar{v}(t):=(t,v(t),S(t,v(t)))$, which is a Borel function $[0,1]\to Y$. Then
$\bar{u}(t):=g(\bar{v}(t))$ is also a Borel function of the form
$\bar{u}(t)=(t,u(t))$, with $u:[0,1]\to\mathbb{E}_{1}$ Borel such that
$\pi(t,u(t))=v(t)$ and $N(t,u(t))=S(t,v(t))$.

If $v$ is only measurable, then there is a Borel function $v^{\prime}$ that is
equal to $v$ almost everywhere, and so we can apply the proposition from the
Borel setting. ∎

######  Lemma 3.16.

Let $\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$, $N\in\cal N$
and $\gamma:[0,1]\to M$. The following statements are equivalent

  1. (i)

$\gamma:[0,1]\to(M,d_{(\mathtt{f},N)})$ is Lipschitz;

  2. (ii)

$\gamma$ is absolutely continuous and
$\operatorname*{{esssup}}_{t\in[0,1]}|\gamma^{\prime}(t)|_{(\mathtt{f},N)}<\infty$;

  3. (iii)

there is $u\in L^{\infty}([0,1];\mathbb{E})$ such that
$\gamma=\gamma_{(\gamma(0),\mathtt{f},u)}$, and
$|\gamma^{\prime}(t)|_{(\mathtt{f},N)}=N(\gamma(t),u(t))$ for almost every
$t\in[0,1]$;

  4. (iv)

there is $u\in L^{\infty}([0,1];\mathbb{E})$ such that
$\gamma=\gamma_{(\gamma(0),\mathtt{f},u)}$.

###### Proof.

The implication $(i)\Rightarrow(iv)$ is proven in Lemma 3.12. The implication
$(iv)\Rightarrow(i)$ is a direct consequence of the definition of
$d_{(\mathtt{f},N)}$.

The implication $(iv)\Rightarrow(ii)$ comes from the definition of
$|\cdot|_{(\mathtt{f},N)}$. The implication $(ii)\Rightarrow(iii)$ is a
consequence of Lemma 3.15. The implication $(iii)\Rightarrow(iv)$ is trivial.
∎

Notice that the statement (iv) in Lemma 3.16 does not depend on $N$: that a
curve is Lipschitz does not depend on the particular norm we choose. We can
thus say that a curve $\gamma:[a,b]\to M$ is _$\mathtt{f}$ -Lipschitz_ if, up
to an affine reparametrization, statement (iv) in Lemma 3.16 holds.

We can define the length of $\mathtt{f}$-Lipschitz curves in three ways, which
we will show being equivalent.

######  Proposition 3.17.

Let $\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ and $N\in\cal
N$. For every $\mathtt{f}$-Lipschitz curve $\gamma:[a,b]\to M$ the following
three quantities are equal:

$\displaystyle L_{1}(\gamma)$
$\displaystyle=\sup\left\\{\sum_{j=1}^{k}d_{(\mathtt{f},N)}(\gamma(t_{j}),\gamma(t_{j+1})):a=t_{1}\leq
t_{2}\leq\dots\leq t_{k}\leq t_{k+1}=b\right\\},$ $\displaystyle
L_{2}(\gamma)$
$\displaystyle=\int_{a}^{b}|\gamma^{\prime}(t)|_{(\mathtt{f},N)}\,\mathrm{d}t,$
$\displaystyle L_{3}(\gamma)$
$\displaystyle=\inf\left\\{\ell(\mathtt{f},\gamma(0),u,N):u\in
L^{\infty}([0,1];\mathbb{E})\text{ s.t.\leavevmode\nobreak\
}[t\mapsto\gamma(a+t(b-a))]=\gamma_{(\gamma(0),\mathtt{f},u)}\right\\}.$

Moreover, the infimum in the definition of $L_{3}(\gamma)$ is a minimum.

###### Proof.

Notice that all three quantities $L_{j}(\gamma)$ are invariant under affine
reparametrizations of $\gamma$, so we may assume $a=0$, and $b=1$.

The quantities $L_{2}(\gamma)$ and $L_{3}(\gamma)$ are equal by Lemma 3.16,
which also shows that the infimum in the definition of $L_{3}(\gamma)$ is a
minimum.

Let us now show that $L_{1}(\gamma)=L_{2}(\gamma)$. Let us first show that
$L_{1}(\gamma)\leq L_{2}(\gamma)$.

Since $\gamma:[0,1]\to(M,d_{(\mathtt{f},N)})$ is Lipschitz, we have that its
metric derivate

$|\gamma^{\prime}(t)|:=\lim_{\varepsilon\to
0}\frac{d_{(\mathtt{f},N)}(\gamma(t+\varepsilon),\gamma(t))}{|\varepsilon|},$

exists for almost every $t\in(0,1)$, and
$L_{1}(\gamma)=\int_{0}^{1}|\gamma^{\prime}(t)|\,\mathrm{d}t$, see [3, Theorem
4.1.1]. By Lemma 3.16(iii) we have that there is $u\in
L^{\infty}([0,1];\mathbb{E})$ such that
$\gamma=\gamma_{(\gamma(0),\mathtt{f},u)}$, and
$|\gamma^{\prime}(t)|_{(\mathtt{f},N)}=N(\gamma(t),u(t))$ for almost every
$t\in[0,1]$. Let us now fix $t_{0}\in(0,1)$ a point at which the metric
derivative $|\gamma^{\prime}(t_{0})|$ exists and
$t\mapsto|\gamma^{\prime}(t)|_{(\mathtt{f},N)}$ has a Lebesgue point at
$t_{0}$. Hence, by definition of $d_{(\mathtt{f},N)}$, we have that for every
$\varepsilon>0$ small enough

$\frac{d_{(\mathtt{f},N)}(\gamma(t_{0}+\varepsilon),\gamma(t_{0}))}{\varepsilon}\leq\frac{1}{\varepsilon}\int_{t_{0}}^{t_{0}+\varepsilon}N(\gamma(s),u(s))\,\mathrm{d}s=\frac{1}{\varepsilon}\int_{t_{0}}^{t_{0}+\varepsilon}|\gamma^{\prime}(s)|_{(\mathtt{f},N)}\,\mathrm{d}s.$

Hence, taking $\varepsilon\to 0$ in the previous inequality, we get that

$|\gamma^{\prime}(t_{0})|\leq|\gamma^{\prime}(t_{0})|_{(\mathtt{f},N)}.$

Thus we conclude that
$|\gamma^{\prime}(t)|\leq|\gamma^{\prime}(t)|_{(\mathtt{f},N)}$ for almost
every $t\in[0,1]$. Integrating the previous inequality between $0$ and $1$ we
thus get $L_{1}(\gamma)\leq L_{2}(\gamma)$, which is the sought claim.

Let us now show $L_{1}(\gamma)\geq L_{2}(\gamma)$. Let us argue similarly as
in the last part of the proof of Lemma 3.12. Let us assume without loss of
generality that $\gamma:[0,1]\to(M,d_{(\mathtt{f},N)})$ is $1$-Lipschitz. By
Proposition 3.9, for every $n\in\mathbb{N}$ large enough and $0\leq j\leq
2^{n}-1$ integer, there is a control $u^{(n)}_{j}\in
L^{\infty}([0,1];\mathbb{E})$ such that the corresponding integral curve
$\gamma^{(n)}_{j}:[0,1]\to M$ starting from
$\gamma^{(n)}_{j}(0)=\gamma(\frac{j}{2^{n}})$ is a geodesic parametrized with
constant speed (see Lemma 3.5), with end point
$\gamma^{(n)}_{j}(1)=\gamma(\frac{j+1}{2^{n}})$, and, by the fact that
$\gamma$ is $1$-Lipschitz, the following holds for almost every $t\in[0,1]$,

$N(\gamma^{(n)}_{j}(t),u^{(n)}_{j}(t))=d_{(\mathtt{f},N)}\left(\gamma\left(\frac{j}{2^{n}}\right),\gamma\left(\frac{j+1}{2^{n}}\right)\right)\leq
1/2^{n}.$

Thus, from Lemma 3.7.(b) and (c), we obtain that there exists $L$ such that,
for $n$ large enough,
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u^{(n)}_{j}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}\leq
L/2^{n}$ (here
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}$
is a reference norm as in Lemma 3.7). Define

$u^{(n)}(t):=2^{n}\sum_{j=0}^{2^{n}-1}u^{(n)}_{j}(2^{n}(t-j/2^{n})),$

where we mean that $u^{(n)}_{j}\equiv 0$ outside $[0,1]$ for every
$n\in\mathbb{N}$ and $0\leq j\leq 2^{n}-1$. Hence $u^{(n)}\in
L^{\infty}([0,1];\mathbb{E})$,
$(\gamma(0),\mathtt{f},u^{(n)})\in\mathscr{D}_{\mathrm{End}}$, and
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u^{(n)}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}\leq
L$. Notice that $u^{(n)}$ is a control associated to the curve $\gamma^{(n)}$
which is the reparametrization on $[0,1]$ of the concatenation of all the
curves $\gamma^{(n)}_{j}$ for $0\leq j\leq 2^{n}-1$. Since the controls
$u^{(n)}$ are uniformly bounded in $n$, there is a subsequence
$\\{u^{(n_{k})}\\}_{k}$ that weakly* converges to some $u^{(\infty)}$. In the
proof of Lemma 3.12 we showed that $u^{(\infty)}$ is a control for $\gamma$.
Let us rename $n_{k}=n$.

Let us now fix $\varepsilon>0$. Since $u^{(n)}$ weakly* converges to
$u^{(\infty)}$, by Proposition 3.4 we get that

$\ell(\gamma(0),\mathtt{f},u^{(\infty)},N)\leq\liminf_{n\to+\infty}\ell(\gamma^{(n)}(0),\mathtt{f},u^{(n)},N).$

Hence, from the previous inequality, we have, up to subsequences and for $n$
large enough,

$\int_{0}^{1}N(\gamma(t),u^{(\infty)}(t))\,\mathrm{d}t\leq\int_{0}^{1}N(\gamma^{(n)}(t),u^{(n)}(t))\,\mathrm{d}t+\varepsilon.$

Hence, using that $u^{(\infty)}$ is a control for $\gamma$, the previous
inequality, the fact that $\gamma^{(n)}_{j}$ are geodesics parametrized with
constant speed, we have, for $n$ large enough,

$\begin{split}\int_{0}^{1}|\gamma^{\prime}(t)|_{(\mathtt{f},N)}\,\mathrm{d}t&\leq\int_{0}^{1}N(\gamma(t),u^{(\infty)}(t))\,\mathrm{d}\leq\int_{0}^{1}N(\gamma^{(n)}(t),u^{(n)}(t))\,\mathrm{d}t+\varepsilon\\\
&=\sum_{j=0}^{2^{n}-1}\int_{0}^{1}N(\gamma^{(n)}_{j}(t),u^{(n)}_{j}(t))\,\mathrm{d}t+\varepsilon\\\
&=\sum_{j=0}^{2^{n}-1}d_{(\mathtt{f},N)}\left(\gamma\left(\frac{j}{2^{n}}\right),\gamma\left(\frac{j+1}{2^{n}}\right)\right)+\varepsilon\\\
&\leq L_{1}(\gamma)+\varepsilon,\end{split}$

where in the last inequality we have used the definition of the length
$L_{1}(\gamma)$. Taking $\varepsilon\to 0$ in the previous inequality we get
$L_{1}(\gamma)\geq L_{2}(\gamma)$, which is the sought inequality.

Thus we finally get that $L_{1}(\gamma)=L_{2}(\gamma)$, and the proof is
concluded. ∎

######  Corollary 3.18.

Let $\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ and $N\in\cal
N$. The metric derivative of a Lipschitz curve
$\gamma:[a,b]\to(M,d_{(\mathtt{f},N)})$ (cfr. [3]) is
$t\mapsto|\gamma^{\prime}(t)|_{(\mathtt{f},N)}$.

###### Proof.

It is a direct consequence of Proposition 3.17, according to which
$L_{1}=L_{2}$, and [3, Theorem 4.1.1]. ∎

######  Corollary 3.19.

If $\mathtt{f}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ and $N\in\cal
N$, then, for every $p,q\in M$,

$d_{(\mathtt{f},N)}(p,q)=\inf\left\\{\int_{a}^{b}|\gamma^{\prime}(t)|_{(\mathtt{f},N)}\,\mathrm{d}t:\begin{array}[]{l}\gamma:[a,b]\to
M\text{ absolutely continuous,}\\\ \text{with }\gamma(a)=p,\
\gamma(b)=q\end{array}\right\\}$

###### Proof.

It is a direct consequence of Proposition 3.17, according to which
$L_{1}=L_{2}$, and [6, Proposition 2.4.1.]. ∎

###  3.5. Limits of CC distances

We use the notation $\mathrm{Lip}\Gamma$ from Section 2.1, and $\mathcal{N}$
from Section 3.2. In this section we investigate what happens when one takes
the limit of Carnot-Carathéodory distances associated to $(\mathtt{f},N)$. We
prove a relaxation property of the limit distance, and finally the main
theorem of this section, i.e., Theorem 1.4.

######  Proposition 3.20.

If $\mathtt{f}_{n}\to\mathtt{f}_{\infty}$ in
$\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$, $N_{n}\to N_{\infty}$ in $\cal
N$, and $p,q\in M$, then

(35)
$d_{(\mathtt{f}_{\infty},N_{\infty})}(p,q)\geq\inf\left\\{\liminf_{n\to\infty}d_{(\mathtt{f}_{n},N_{n})}(p_{n},q_{n}):p_{n}\to
p,\ q_{n}\to q\right\\}.$

Moreover, if $K\subset M$ is a compact set, there exists $C>0$ such that
whenever $p\in K$ and $q\in M$ with
$d_{(\mathtt{f}_{\infty},N_{\infty})}(p,q)\leq C$, one has that

(36)
$d_{(\mathtt{f}_{\infty},N_{\infty})}(p,q)=\inf\left\\{\liminf_{n\to\infty}d_{(\mathtt{f}_{n},N_{n})}(p_{n},q_{n}):p_{n}\to
p,\ q_{n}\to q\right\\}.$

In the setting of Lemma 3.7, we can take $C=\frac{R}{L^{2}}$, for $R,L>0$
satisfying (26) on the compact set $\bar{B}_{\rho}(K,1)$ in place of $K$ and
with $\mathscr{K}=\\{\mathtt{f}_{n}\\}_{n}\cup\\{\mathtt{f}_{\infty}\\}$,
$\cal K=\\{N_{n}\\}_{n}\cup\\{N_{\infty}\\}$.

###### Proof.

Let us first prove (35). If
$d_{(\mathtt{f}_{\infty},N_{\infty})}(p,q)=\infty$, there is nothing to prove,
so we assume $d_{(\mathtt{f}_{\infty},N_{\infty})}(p,q)<\infty$. Fix
$\varepsilon>0$. Then there is $u\in L^{\infty}([0,1];\mathbb{E})$ such that
$\mathrm{End}^{\mathtt{f}_{\infty}}_{p}(u)=q$ and
$\mathtt{J}(p,\mathtt{f}_{\infty},u,N_{\infty})\leq
d_{(\mathtt{f}_{\infty},N_{\infty})}(p,q)+\varepsilon$. Let $p_{n}\equiv p$
and $q_{n}=\mathrm{End}^{\mathtt{f}_{n}}_{p}(u)$ that exists for $n$ big
enough thanks to Proposition 2.9(1). By Theorem 2.5, $q_{n}\to q$, and
moreover, by the definition of the distance,
$d_{(\mathtt{f}_{n},N_{n})}(p,q_{n})\leq\mathtt{J}(p,\mathtt{f}_{n},u,N_{n})$.

Let $\gamma_{n}$ be the curve
$\gamma_{n}(t):=\mathrm{End}^{\mathtt{f}_{n}}_{p}(tu)$. By Proposition 2.9,
$\gamma_{n}\to\gamma_{\infty}$ uniformly. Hence, by possibly passing to
subsequences,

$\begin{split}\lim_{n\to\infty}\mathtt{J}(p,\mathtt{f}_{n},u,N_{n})&=\lim_{n\to\infty}\operatorname*{{esssup}}\\{N_{n}(\gamma_{n}(t),u(t)):t\in[0,1]\\}\\\
&=\operatorname*{{esssup}}\\{N_{\infty}(\gamma_{\infty}(t),u(t)):t\in[0,1]\\}=\mathtt{J}(p,\mathtt{f}_{\infty},u,N_{\infty}).\end{split}$

We conclude that
$\liminf_{n\to\infty}d_{(\mathtt{f}_{n},N_{n})}(p,q_{n})\leq\mathtt{J}(p,\mathtt{f}_{\infty},u,N_{\infty})\leq
d_{(\mathtt{f}_{\infty},N_{\infty})}(p,q)+\varepsilon$. Since $\varepsilon$
can be taken arbitrarily small, we get the sought claim.

Let us now prove (36). In the notation of Lemma 3.7, take
$\mathscr{K}:=\\{\mathtt{f}_{n}\\}_{n\in\mathbb{N}}\cup\\{\mathtt{f}_{\infty}\\}$,
$\cal K:=\\{N_{n}\\}_{n\in\mathbb{N}}\cup\\{N_{\infty}\\}$, and $R>0$. Let $L$
be the constant of Lemma 3.7 associated to these choices on the compact set
$\bar{B}_{\rho}(K,1)$ in place of $K$, and set $C:=R/L^{2}$.

Suppose by contradiction that (36) is not true. Hence, since (35) always
holds, there exist $p\in K$, $q\in M$ with
$d_{(\mathtt{f}_{\infty},N_{\infty})}(p,q)\leq R/L^{2}$, $p_{n}\to p$,
$q_{n}\to q$ such that, up to passing to subsequences,

(37)
$d_{(\mathtt{f}_{\infty},N_{\infty})}(p,q)>\lim_{n\to\infty}d_{(\mathtt{f}_{n},N_{n})}(p_{n},q_{n}).$

Hence, for $n$ large enough, we have
$d_{(\mathtt{f}_{n},N_{n})}(p_{n},q_{n})<R/L^{2}$, and
$p_{n}\in\bar{B}_{\rho}(K,1)$, since $p_{n}\to p\in K$.

Therefore, by applying Lemma 3.7(d), thanks to the choice of the compact
$\bar{B}_{\rho}(K,1)$, we know that we can take, for $n$ large enough,
$u_{n}\in L^{\infty}([0,1];\mathbb{E})$ such that
$\gamma_{(p_{n},\mathtt{f}_{n},u_{n})}\subseteq\bar{B}_{\rho}(K,R+1)$ and

$J(p_{n},\mathtt{f}_{n},u_{n},N_{n})\leq
d_{(\mathtt{f}_{n},N_{n})}(p_{n},q_{n})+1/n.$

Therefore, by using (27), we have that

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u_{n}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}\leq
L(R/L^{2}+1/n),$

for $n$ large enough. Henceforth, up to subsequences, $u_{n}\to u_{\infty}$
weakly*. From Theorem 2.5 we get
$\mathrm{End}^{\mathtt{f}_{\infty}}_{p}(u_{\infty})=q$, while from Proposition
3.4 we get

$d_{(\mathtt{f}_{\infty},N_{\infty})}(p,q)\leq\mathtt{J}(p,\mathtt{f}_{\infty},u_{\infty},N_{\infty})\leq\liminf_{n\to\infty}\mathtt{J}(p_{n},\mathtt{f}_{n},u_{n},N_{n})\leq\liminf_{n\to\infty}d_{(\mathtt{f}_{n},N_{n})}(p_{n},q_{n}).$

TWe obtained a contradiction with (37). ∎

######  Lemma 3.21.

Let $\mathtt{f}_{n}\to\mathtt{f}_{\infty}$ in
$\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$, $N_{n}\to N_{\infty}$ in
$\mathcal{N}$, and $K\subset M$ be a compact set. Assume that
$\mathtt{f}_{\infty}$ is essentially non-holonomic and fix a complete
Riemannian metric $\rho$ on $M$.

Then, for every $\varepsilon>0$ there exist $\delta>0$ and
$n_{0}\in\mathbb{N}$ such that, for all $n\geq n_{0}$ (also $n=\infty$) and
all $p\in K$,

$B_{\rho}(p,\delta)\subset B_{(\mathtt{f}_{n},N_{n})}(p,\varepsilon).$

###### Proof.

First, we claim that for every $\hat{p}\in K$ there are $n_{1}\in\mathbb{N}$
and a neighborhood $U$ of $\hat{p}$ such that, for all $n\geq n_{1}$,

(38) $U\subset B_{(\mathtt{f}_{n},N_{n})}(\hat{p},\varepsilon/2).$

To prove the claim, we use Lemma 3.7 with $R=1$ and the compact sets $K$,
$\\{\mathtt{f}_{n}\\}_{n\in\mathbb{N}}\cup\\{\mathtt{f}_{\infty}\\}$, and
$\\{N_{n}\\}_{n\in\mathbb{N}}\cup\\{N_{\infty}\\}$. Let $L$ be as in Lemma
3.7. Define the set

$\mathscr{U}=\left\\{u\in
L^{\infty}([0,1];\mathbb{E}):{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}\leq\min\left(\frac{\varepsilon}{2L},\frac{1}{L}\right)\right\\},$

where
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}$
is as in Lemma 3.7. By Theorem 2.13, there are $n_{1}\in\mathbb{N}$ and a
neighborhood $U$ of $\hat{p}$ so that
$U\subset\mathrm{End}^{\mathtt{f}_{n}}_{\hat{p}}(\mathscr{U})$, for all
$n>n_{1}$. By Lemma 3.7(a) and Lemma 3.7(c), for every $n>n_{1}$ and every
$u\in\mathscr{U}$, we have
$d_{(\mathtt{f}_{n},N_{n})}(\hat{p},\mathrm{End}^{\mathtt{f}_{n}}_{\hat{p}}(u))\leq\mathtt{J}(\hat{p},\mathtt{f}_{n},u,N_{n})\leq
L{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\infty}\leq\varepsilon/2$,
that is,

$U\subset\mathrm{End}^{\mathtt{f}_{n}}_{\hat{p}}(\mathscr{U})\subset
B_{(\mathtt{f}_{n},N_{n})}(\hat{p},\varepsilon/2).$

Second, by the compactness of $K$, we obtain that there are
$\hat{p}_{1},\dots,\hat{p}_{k}\in K$, open neighborhoods $U_{j}$ of
$\hat{p}_{j}$ with $K\subset\bigcup_{j}U_{j}$, and $n_{0}\in\mathbb{N}$, such
that, for all $n>n_{0}$ and all $j$,

$U_{j}\subset B_{(\mathtt{f}_{n},N_{n})}(\hat{p}_{j},\varepsilon/2).$

Next, on the one hand, there is $\delta>0$ such that for every $p\in K$ there
is $j$ with $B_{\rho}(p,\delta)\subset U_{j}$. On the other hand, if $p\in
B_{(\mathtt{f},N)}(\hat{p}_{j},\varepsilon/2)$, then
$B_{(\mathtt{f},N)}(\hat{p}_{j},\varepsilon/2)\subset
B_{(\mathtt{f},N)}(p,\varepsilon)$.

We conclude that, if $n>n_{0}$, then for every $p\in K$ there is $j$ such that

$B_{\rho}(p,\delta)\subset U_{j}\subset
B_{(\mathtt{f}_{n},N_{n})}(\hat{p}_{j},\varepsilon/2)\subset
B_{(\mathtt{f}_{n},N_{n})}(p,\varepsilon),$

which concludes the proof. ∎

######  Proposition 3.22.

Let $\mathtt{f}_{n}\to\mathtt{f}_{\infty}$ in
$\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$, $N_{n}\to N_{\infty}$ in
$\mathcal{N}$, and $K\subset M$ be a compact set. Assume that
$\mathtt{f}_{\infty}$ is essentially non-holonomic and fix a complete
Riemannian metric $\rho$ on $M$.

Then for every $\varepsilon>0$ there are $n_{0}\in\mathbb{N}$ and $\delta>0$
so that for every $n>n_{0}$ and $p,p^{\prime},q,q^{\prime}\in K$

(39)
$\rho(p,p^{\prime})+\rho(q,q^{\prime})<\delta\quad\Rightarrow\quad\left|d_{(\mathtt{f}_{n},N_{n})}(p,q)-d_{(\mathtt{f}_{n},N_{n})}(p^{\prime},q^{\prime})\right|\leq\varepsilon$

###### Proof.

We define

$\beta_{k}(s):=\sup\\{d_{(\mathtt{f}_{n},N_{n})}(p,q):p,q\in K,\ n\geq k,\
\rho(p,q)\leq s\\}.$

Lemma 3.21 immediately implies that, for every $\varepsilon$, there are
$\delta>0$ and $n_{0}$ such that, for every $s\in[0,\delta]$, we have
$\beta_{n_{0}}(s)<\varepsilon/2$. Now, if $p,p^{\prime},q,q^{\prime}\in K$,
$\rho(p,p^{\prime})+\rho(q,q^{\prime})<\delta$ and $n\geq n_{0}$, then

$\displaystyle\left|d_{(\mathtt{f}_{n},N_{n})}(p,q)-d_{(\mathtt{f}_{n},N_{n})}(p^{\prime},q^{\prime})\right|$
$\displaystyle\leq
d_{(\mathtt{f}_{n},N_{n})}(p,p^{\prime})+d_{(\mathtt{f}_{n},N_{n})}(q,q^{\prime})$
$\displaystyle\leq 2\beta_{n_{0}}(\delta)\leq\varepsilon.$

∎

######  Corollary 3.23.

If $\hat{\mathtt{f}}$ is essentially non-holonomic and $\hat{N}\in\cal N$,
then, for every $p,q\in M$,

$\limsup_{(\mathtt{f},N)\to(\hat{\mathtt{f}},\hat{N})}d_{(\mathtt{f},N)}(p,q)\leq
d_{(\hat{\mathtt{f}},\hat{N})}(p,q).$

###### Proof.

The proof of this corollary is a slight modification of the proof of (35),
where we now have that $d_{(\mathtt{f}_{n},N_{n})}(q_{n},q)\to 0$ by
Proposition 3.22. ∎

We are now ready to prove the main result of the paper, namely Theorem 1.4.

###### Proof of Theorem 1.4.

In order to prove item (i), we define for $p\in M$

$\mathscr{U}(p)=\\{q\in M:d_{(\hat{\mathtt{f}},\hat{N})}(p,q)<\infty\\}.$

Notice that, if $\mathscr{U}(p)\cap\mathscr{U}(q)\neq\emptyset$ then
$\mathscr{U}(p)=\mathscr{U}(q)$. Therefore, $\\{\mathscr{U}(p)\\}_{p\in M}$ is
a partition of $M$. Moreover, by Lemma 3.21 applied to the constant sequence
$(\hat{\mathtt{f}},\hat{N})$ and the compact set $\\{p\\}$, $p$ is always in
the interior of $\mathscr{U}(p)$. Now, if $p^{\prime}\in\mathscr{U}(p)$ then
$p^{\prime}$ is in the interior of $\mathscr{U}(p^{\prime})=\mathscr{U}(p)$
and thus we proved that $\mathscr{U}(p)$ is open for every $p\in M$. Since $M$
is connected, we conclude that $\mathscr{U}(p)=M$.

The proof of item (ii) has two parts and uses an auxiliary complete Riemannian
distance $\rho$ on $M$. First, if $U\subset M$ is
$d_{(\hat{\mathtt{f}},\hat{N})}$-open and $p\in U$, then there is
$\varepsilon>0$ so that $B_{(\hat{\mathtt{f}},\hat{N})}(p,\varepsilon)\subset
U$. By Lemma 3.21 applied to the constant sequence
$(\hat{\mathtt{f}},\hat{N})$ and the compact set $\\{p\\}$, there is
$\delta>0$ such that $B_{\rho}(p,\delta)\subset
B_{(\hat{\mathtt{f}},\hat{N})}(p,\varepsilon)$. Since $p$ is an arbitrary
point in $U$, we have proven that $U$ is $\rho$-open. The fact that a
$d_{(\hat{\mathtt{f}},\hat{N})}$-open set is also a $\rho$-open set is a
direct consequence of Lemma 3.8.

Item (iii) is proven as follows. We show that every $o\in M$ has a compact
neighborhood $U$ so that $d_{(\mathtt{f}_{n},N_{n})}\to
d_{(\hat{\mathtt{f}},\hat{N})}$ pointwise on $U\times U$. By Proposition 3.22,
$\\{d_{(\mathtt{f}_{n},N_{n})}\\}_{n\in\mathbb{N}}\cup\\{d_{(\hat{\mathtt{f}},\hat{N})}\\}$
is an equicontinuous family of functions $U\times U\to\mathbb{R}$, hence
pointwise convergence would imply the uniform convergence on $U\times U$.

Let us show the pointwise convergence. Let $r>0$ so that
$\bar{B}_{(\hat{\mathtt{f}},\hat{N})}(o,r)$ is compact. This can be done thank
to the item (ii) that we previously proved. By Proposition 3.20 there is $C>0$
so that for every $p\in\bar{B}_{(\hat{\mathtt{f}},\hat{N})}(o,r)$ and $q\in M$
with $d_{(\hat{\mathtt{f}},\hat{N})}(p,q)\leq C$, one has (36). We may assume
$r<C/2$, that is, that (36) holds for every
$p,q\in\bar{B}_{(\hat{\mathtt{f}},\hat{N})}(o,r)$. So, let
$p,q\in\bar{B}_{(\hat{\mathtt{f}},\hat{N})}(o,r)$ and $p_{n}\to p$ and
$q_{n}\to q$ so that

$d_{(\hat{\mathtt{f}},\hat{N})}(p,q)=\lim_{n\to\infty}d_{(\mathtt{f}_{n},N_{n})}(p_{n},q_{n}).$

Then

$\limsup_{n\to\infty}\left|d_{(\hat{\mathtt{f}},\hat{N})}(p,q)-d_{(\mathtt{f}_{n},N_{n})}(p,q)\right|\\\
\leq\limsup_{n\to\infty}\left|d_{(\hat{\mathtt{f}},\hat{N})}(p,q)-d_{(\mathtt{f}_{n},N_{n})}(p_{n},q_{n})\right|+\left|d_{(\mathtt{f}_{n},N_{n})}(p_{n},q_{n})-d_{(\mathtt{f}_{n},N_{n})}(p,q)\right|\\\
=\limsup_{n\to\infty}\left|d_{(\mathtt{f}_{n},N_{n})}(p_{n},q_{n})-d_{(\mathtt{f}_{n},N_{n})}(p,q)\right|.$

We need to show that the previous limit is zero. Let $\varepsilon>0$. By
Proposition 3.22, there are $n_{0}>0$ and $\delta>0$ so that, if $n>n_{0}$, we
have $\rho(p_{n},p)+\rho(q_{n},q)<\delta$, and then
$\left|d_{(\mathtt{f}_{n},N_{n})}(p_{n},q_{n})-d_{(\mathtt{f}_{n},N_{n})}(p,q)\right|<\varepsilon$.
This shows that the limit is zero.

Item (iv) is a consequence of item (iii) together with the forthcoming metric
Lemma 3.25. ∎

######  Remark 3.24 (About the completeness assumption in Theorem 1.4(iv)).

In this remark we show that the assumption of the completeness of
$d_{(\hat{\mathtt{f}},\hat{N})}$ in Theorem 1.4(d) is necessary in order to
have the uniform convergence
$\lim_{(\mathtt{f},N)\to(\hat{\mathtt{f}},\hat{N})}d_{(\mathtt{f},N)}=d_{(\hat{\mathtt{f}},\hat{N})}$
on compact subsets. In the following example we show that one may not even
have pointwise convergence. Thus Corollary 3.23 cannot be improved in general.

Let us fix $M:=\mathbb{R}\times(-1,1)\subseteq\mathbb{R}^{2}$ and $p:=(-2;0)$,
$q:=(2;0)$. Let us take, for every $n\in\mathbb{N}$, a smooth function
$g_{n}:\mathbb{R}^{2}\to[1,+\infty)$ such that

$\begin{cases}g_{n}=1&\text{outside $[-2,2]\times(-1+1/(4n),1-1/(4n))$}\\\
g_{n}=10&\text{inside $[-1,1]\times(-1+1/(2n),1-1/(2n))$}.\end{cases}$

Let $d_{n}$ be the Riemannian distance associated to the Riemannian tensor
$g_{n}(\,\mathrm{d}x\otimes\,\mathrm{d}x+\,\mathrm{d}y\otimes\,\mathrm{d}y)$
on $M$. We can take $g_{n}$ such that we have $g_{n}\to g_{\infty}$ uniformly
on compact subsets of $M$, where $g_{\infty}$ is a smooth function with
$g_{\infty}=10$ inside $[-1,1]\times(-1,1)$. Let $d_{\infty}$ the Riemannian
distance associated to the Riemannian tensor
$g_{\infty}(\,\mathrm{d}x\otimes\,\mathrm{d}x+\,\mathrm{d}y\otimes\,\mathrm{d}y)$
on $M$. We have that $d_{n}\to d_{\infty}$ locally uniformly on $M$, i.e.,
every $p\in M$ has a neighborhoof $U$ such that $d_{n}\to d_{\infty}$
uniformly on $U\times U$. Nevertheless, $d_{n}$ does not converge uniformly to
$d_{\infty}$ on compact subsets of $M$. Indeed, we have that $d_{n}(p,q)\leq
6$, while $d_{\infty}(p,q)\geq 10$.

######  Lemma 3.25.

Let $\Lambda$ be endowed with a sequential topology. Let $X$ be a set. For
$t\in\Lambda$, let $d_{t}$ be a length metric on $X$. Assume that for some
$t_{0}\in\Lambda$ we have that $(X,d_{t_{0}})$ is boundedly compact. Assume
that for every point $x\in X$ and every sequence $t_{n}\to t_{0}$ there exists
a $d_{t_{0}}$-neighborhood $U$ of $x$ such that

(40)
$\sup_{(p,q)\in\overline{U}\times\overline{U}}|d_{t_{n}}(p,q)-d_{t_{0}}(p,q)|\to
0,$

as $n\to+\infty$. Hence for every $d_{t_{0}}$-compact set $K$ we have

(41) $\sup_{(p,q)\in K\times K}|d_{t_{n}}(p,q)-d_{t_{0}}(p,q)|\to
0,\qquad\text{for every sequence ${t_{n}}\to{t_{0}}$}.$

Moreover,we have that, for every $x\in X$, and for every sequence ${t_{n}}\to
t_{0}$,

(42) $(X,d_{t_{n}},x)\to(X,d_{t_{0}},x),$

in the pointed Gromov–Hausdorff sense.

###### Proof.

Let us first prove (41). Fix $K$ a $d_{t_{0}}$-compact set. Let
$D:=\operatorname{\mathrm{diam}}_{d_{t_{0}}}K$ and let
$K^{\prime}:=B_{d_{t_{0}}}(K,D+3)$ be the closed $(D+3)$-tubular neighborhood
of $K$. Since $(X,d_{t_{0}})$ is boundedly compact, $K^{\prime}$ is compact.

Suppose by contradiction that (41) does not hold for some sequence $t_{n}\to
t_{0}$. Hence, up to passing to subsequences, we have that for some
$0<\varepsilon<1$ and every $n\in\mathbb{N}$ the following holds

(43) $\sup_{(p,q)\in K\times K}|d_{t_{n}}(p,q)-d_{t_{0}}(p,q)|\geq
2\varepsilon.$

For every $s\in K^{\prime}$ there exists $U_{s}$ a $d_{t_{0}}$-neighborhood of
$s$ such that (40) holds for the sequence $\\{t_{n}\\}_{n\in\mathbb{N}}$.
Since $K^{\prime}$ is compact, we can extract a finite covering of
$K^{\prime}$ from $\\{U_{s}\\}_{s\in K^{\prime}}$. Hence there exists
$m\in\mathbb{N}$ and $s_{1},\dots,s_{m}\in K^{\prime}$ such that

$K^{\prime}\subseteq U_{s_{1}}\cup\dots\cup U_{s_{m}}.$

For the sake of simplicity we rename $U_{s_{i}}=:U_{i}$ for every
$i=1,\dots,m$. Let us take $N$ big enough such that for every $n\geq N$ and
every $i=1,\dots,m$ we have

(44)
$\sup_{(p,q)\in\overline{U}_{i}\times\overline{U}_{i}}|d_{t_{n}}(p,q)-d_{t_{0}}(p,q)|\leq\frac{\varepsilon}{2m}.$

We now aim at showing that for every $p,q\in K$ and every $n\geq N$ we have

(45) $d_{t_{n}}(p,q)\leq d_{t_{0}}(p,q)+\varepsilon.$

Since $d_{t_{0}}$ is a length distance, given $p,q\in K$, there exists a curve
$\gamma:[0,1]\to X$ such that

(46) $L_{d_{t_{0}}}(\gamma)\leq d_{t_{0}}(p,q)+\varepsilon/2.$

For every $\alpha\in[0,1]$ we have that

$d_{t_{0}}(p,\gamma(\alpha))\leq L_{d_{t_{0}}}(\gamma)\leq
d_{t_{0}}(p,q)+\varepsilon/2\leq D+1/2,$

and hence $\gamma\subseteq\mathrm{int}(K^{\prime})$. We now aim at finding on
$\gamma$ a finite number $i$, with $i\leq m$, of points
$p=p_{1},p_{2},\dots,p_{i}=q$ such that for every $j=1,\dots,i-1$ we have that
that $p_{j},p_{j+1}$ are in the same $\overline{U}_{{k_{j}}}$, for some
$k_{j}\in\\{1,\dots,m\\}$. We define such a sequence inductively.

First, since $p\in K$, there exists a $k_{1}\in\\{1,\dots,m\\}$ such that
$p\in\overline{U}_{{k_{1}}}$. Let us suppose that the sequence
$p=p_{1},\dots,p_{\ell}$ has been defined for some $\ell\in\mathbb{N}$, in
such a way that

  * (i)

for every $j=1,\dots,\ell$, there exist $k_{j}\in\\{1,\dots,m\\}$ that are
pairwise distinct such that

  * (ii)

$p_{j},p_{j+1}\in\overline{U}_{k_{j}}$ for every $j=1,\dots,\ell-1$, and

  * (iii)

$p_{\ell}\in\overline{U}_{k_{\ell}}$.

Hence define

(47)
$\alpha_{\ell+1}:=\max\\{\alpha:\gamma(\alpha)\in\overline{U}_{k_{\ell}}\\},\quad
p_{\ell+1}:=\gamma(\alpha_{\ell+1}).$

Obviously we have $p_{\ell+1}\in\overline{U}_{k_{\ell}}$. If
$\alpha_{\ell+1}=1$ the process ends and
$q=p_{\ell+1}\in\overline{U}_{k_{\ell+1}}$ with $k_{\ell+1}$ distinct from
every $k_{1},\dots,k_{\ell}$ by the inductive definition of the $\alpha$’s. If
not, we now show that $p_{\ell+1}\in\overline{U}_{k_{\ell+1}}$ for some
$k_{\ell+1}\in\\{1,\dots,m\\}$ different from every $k_{1},\dots,k_{\ell}$.
This is true since for every $\eta>0$ small enough we have that
$\gamma(\alpha+\eta)\in K^{\prime}$ and hence $\gamma(\alpha+\eta)\in
U_{k_{\eta}}$, where $k_{\eta}\in\\{1,\dots,m\\}$. Since $k_{\eta}$ ranges in
a finite set, there exists $k_{\ell+1}\in\\{1,\dots,m\\}$ such that
$\gamma(\alpha+\eta_{j})\in U_{k_{\ell+1}}$ for a sequence $\eta_{j}\to 0$.
Moreover $k_{\ell+1}$ has to be different from every $k_{1},\dots,k_{\ell}$,
since it is inductively defined by means of (47). This eventually proves that,
after at most $m$ steps, we end the process at $q$, since also
$q\in\overline{U}_{k_{i}}$ for some $k_{i}\in\\{1,\dots,m\\}$. Hence the claim
is shown.

Hence we now want to obtain (45). Fix $p,q\in K$, $n\geq N$, and take the
chain of points $p=p_{1},\dots,p_{i}=q$ previously defined. Hence

(48)
$\begin{split}d_{t_{n}}(p,q)&\leq\sum_{\ell=1}^{i-1}d_{t_{n}}(p_{\ell},p_{\ell+1})\leq\sum_{\ell=1}^{i-1}d_{t_{0}}(p_{\ell},p_{\ell+1})+\varepsilon/2\\\
&\leq L_{d_{t_{0}}}(\gamma)+\varepsilon/2\leq
d_{t_{0}}(p,q)+\varepsilon.\end{split}$

where the first inequality is an application of the triangle inequality; the
second inequality comes from (44), the fact that
$p_{\ell},p_{\ell+1}\in\overline{U}_{k_{\ell}}$ for some
$k_{\ell}\in\\{1,\dots,m\\}$, and the fact that $i\leq m$; the third
inequality comes from the definition of length; and the fourth is a
consequence of (46).

With a slight variation of the previous argument, we now aim at showing that
for every $p,q\in K$ and every $n\geq N$ we have

(49) $d_{t_{0}}(p,q)\leq d_{t_{n}}(p,q)+\varepsilon.$

Given $p,q\in K$ and $n\geq N$, since $d_{t_{n}}$ is a length distance, there
exists a curve $\gamma_{t_{n}}:[0,1]\to X$ such that

(50) $L_{d_{t_{n}}}(\gamma_{t_{n}})\leq d_{t_{n}}(p,q)+\varepsilon/2.$

We do not know a priori if $\gamma_{t_{n}}\subseteq K^{\prime}$, but
nevertheless we may argue as before, paying attention to one more detail.
Again, we aim at finding on $\gamma_{t_{n}}$ a finite number $i$, with $i\leq
m$, of points $p=p_{1},p_{2},\dots,p_{i}=q$ such that for every
$j=1,\dots,i-1$ we have that that $p_{j},p_{j+1}$ are in the same
$\overline{U}_{{k_{j}}}$, for some $k_{j}\in\\{1,\dots,m\\}$. We proceed by
induction.

Since $p\in K$, there exists a $k_{1}\in\\{1,\dots,m\\}$ such that
$p\in\overline{U}_{{k_{1}}}$. Let us suppose that the sequence
$p=p_{1},\dots,p_{\ell}$ has been defined for some $\ell\in\mathbb{N}$, in
such a way that items (i), (ii), and (iii) above hold. Hence define

(51)
$\alpha_{\ell+1}:=\max\\{\alpha:\gamma_{t_{n}}(\alpha)\in\overline{U}_{k_{\ell}}\\},\quad
p_{\ell+1}:=\gamma_{t_{n}}(\alpha_{\ell+1}).$

Clearly $p_{\ell+1}\in\overline{U}_{k_{\ell}}$. We now first show that
$p_{\ell+1}\in\mathrm{int}(K^{\prime})$. Indeed

(52)