# Monolithically Integrated Wavelength-meter in InP with measurement bandwidth
of 100nm centered on the C band.

Andrea Volpini  University of Vigo, El Telecomunication - Campus Universitario
As Lagoas, 36310 Vigo, Spain  Corresponding author<EMAIL_ADDRESS>Damiano Massella  University of Vigo, El Telecomunication - Campus
Universitario As Lagoas, 36310 Vigo, Spain  David Alvarez-Outerelo  University
of Vigo, El Telecomunication - Campus Universitario As Lagoas, 36310 Vigo,
Spain  Francisco Soares  Soares Photonics, Lisbon, Portugal.  Francisco
J.Diaz-otero  University of Vigo, El Telecomunication - Campus Universitario
As Lagoas, 36310 Vigo, Spain  AtlanTTic research center, El Telecomunication -
Campus Universitario As Lagoas, 36310 Vigo, Spain

###### Abstract

In this paper we will explore the creation of a monolithically integrated
wavelength meter in InP. This type of devices are a key requirement for many
applications and it is especially important to have them integrated with
active components like lasers and gain sections. We present a wavelength meter
based on multiple ring resonators that has been realized in a commercial MPW
run and tested using a tunable laser.
The designed circuit is theoretically capable of resolution down to 1.6pm and
a measurement speed down to 500ps within a wavelength range of 100nm.

††journal: ol

##  1 Introduction

The determination of light wavelength is not limited to telecommunication
application, but it has also various applications in Wavelength Division
Multiplexing, spectroscopy, tunable lasers control and metrology [1, 2, 3].
Among all integrated photonics platforms, Indium Phosphide (InP) is the only
platform that allows monolithic integration of lasers and optical gain
sections[4]. It is important to have wavelength meters that can be co-
integrated with lasers, since this allows better control of integrated tunable
laser with all the benefits of integration: lower footprint, better
performance and easier fabrication.
InP platforms have a second advantage over the others as they enable higher
operation speeds, thanks to high speed modulators, and give future prospects
of integration into a more advanced compact photonic system.

A variety of different designs for integrated wavelength meters has been
reported in the literature [2, 5, 6]. All these works are focused on Silicon
or Silicon Nitride platform employing different techniques between ring
resonators or differential transmission thought multimode interferometers. The
most remarkable result so far have has been able to achieve good performance
both in range ($100nm$) and accuracy ($15pm$) [7].
Respect to Silicon and Silicon Nitride platform, InP suffers lower
performances since passive components have higher losses and higher bend
radius are used in the platforms. This makes particularly hard to design ring
resonators with high Free Spectral Range, that are at the hearth of wavelength
meters design.
So far the wider bandwidth reported in an InP platform is of 3nm [8].

The design that we present in this paper is specifically thought to mitigate
the limitation of the InP platform by use of different ring resonators.
In our design, in order to achieve a high operation bandwidth we employ four
microring resonators, each of them contains a phase modulator that can achieve
$2\pi$ phase shift and the through port of the ring is connected to an
independent photodiode (PD). The different length of the rings leads to
different Free Spectral Range (FSR) that can be used to increase the
operational bandwidth of the system. The setup work with the modulation of
spectral position of ring resonances while monitoring the PDs output. The
subsequent wavelength measurement is obtained by checking at what modulator
bias we observe a minimum in ring transmission. This information creates a set
of 4 values that will then be compared with a lookup table, obtained during
calibration, to estimate the light wavelength.

The presented design has been fabricated on the generic Indium Phosphide (InP)
platform of the Franhofer Heinrich Hertz Institute(HHI), the experimental
results in this letter have been obtained with a custom photonic setup in the
University of Vigo.

Figure 1: Schematic of the circuit, an unknown monochromatic source is routed
to four rings of different length using a combination of MMI splitters. Phase
modulators (yellow box) are used to tune the rings on resonance with the
source. Transmission spectra are recorded with photodetectors (PD).

##  2 Methods

In Figure 1 the schematic of the circuit is shown. The device works by routing
a monochromatic source to four micro-ring resonators. Each ring contains a
phase shifter (PHS) that allows to change the ring resonance when electrically
controlled. Please notice that no "drop" port is present along the rings. In
such way unnecessary losses are avoided allowing for higher cavity Q factor.
Calling $\theta(I)$ the phase added by the PHS, the subset of wavelength on
resonance with the ring are defined by the expression:

$\lambda(k)=\dfrac{L\cdot n_{eff}}{k+\dfrac{\theta(V)}{2\pi}}$ (1)

where $k\in\mathbb{N}$, $L$ is the ring length, $n_{eff}$ the effective
refractive index and $V$ the voltage applied to the phase shifter.

As mentioned before, the phase $\theta(V)$ can be changed by controlling the
value of the applied voltage $V$. If the phase shifter is long enough,
$\theta(V)$ can be made greater than $2\pi$ and the ring can be tuned on
resonance with any input wavelength. The resonance condition can be easily
identified since, when reached, minimum power is detected at the photo
detector.
Equation 1 does not define a bijective relation between the voltage $V$ and a
wavelength $\lambda$. In other words, any ring taken individually can operate
as a wavelength meter only if its free spectral range ($FSR$) is wider than
the band of the unknown source. This problem can be overcome by using
carefully designed rings of different length. This is because rings of
different lengths have different possible sets of resonances associated with
phase shifter voltage $V$. The intersection of such sets determines the input
wavelength. To obtain such sets of wavelengths the system is operated as
following.

The source to be measured is coupled in the device, and transmitted intensity
is measured at the detectors. PHSs are driven to sweep the phase $\theta(V)$
in the interval $[0,2\pi]$, in this way all PDs will have at least a minimum
in transmission (Figure 3) that can be associated to a driving voltage $V$
characteristic of each ring. From each individual resonant voltage the subset
of possible resonance wavelengths is deduced.

It is worth to point out that to predict exactly such sets of wavelength a
detailed model of: group index dispersion, phase control, temperature
dependence and fabrication defects would be necessary. Given the challenge of
the task, is much easier to proceed with an external calibration of the system
in order to create a look up table where wavelengths are associated to
specific voltages .

##  3 Ring design

Rings must be designed of different length in order to allow for bandwidth
expansion. It is advantageous to start with the design of ring $L_{1}$, that
has to be short as possible. Infact, the shorter the ring, the smaller the
propagation loss, thus leading to an high $Q$ factor and consequently a small
full width half maximum (FWHM) of the resonance. This is of critical
importance since FWHM poses a lower limit on the resolution achievable with
the device. Shorter rings also have a wider FSR, that makes the task of
expanding the detectable bandwidth easier.

In general, the minimum achievable length for a ring is fixed by a number of
design constrains. Every ring must have: a directional coupler, with a proper
length to guarantee coupling, a PHS, long enough to guarantee at least a
$2\pi$ phase shift, and two 180 degrees bends with the smallest radius of
curvature available on the platform.

The shortest ring we have designed in the HHI platform has a length of
$l=3318\mu m$. We call this ring $L_{1}$. When designing the ring, we were
afraid of not being able to reach phase shifts up to $2\pi$. We therefore used
a PHS of $800\mu m$ length. As it can be seen from figure 3, we could have
been less conservative. A PHS of $300\mu m$ would have worked as well and
would have required a voltage of about $5V$ to get a $2\pi$ phase shift.

$FSR=\frac{c}{n_{g}L}$ (2)

Figure 2: Top figure: Ring L1 and Ring L2 have similar FSR and consequently it
is not possible to resolve the resonance unambiguously in the interval.
Bottom figure:Ring L1 and Ring L3 have different spectral range resulting in
the possibility of determine the wavelength without ambiguity in the selected
interval.

For ring $L1$, assuming an effective index of $n_{e}ff=3.5$, using equation 2
we expect FSR close to $25$ GHz ($0.2nm$) at a central frequency of $193THz$.
With a process that resembles the vernier effect is possible to expand the
operational bandwidth of the device.

Equation 3 [9] predicts the combined FSR of two rings of length $l_{1}$,
$l_{2}$. We can verify that a length $L_{2}=3324\mu m$ corresponds to
$FSR_{1-2}=14.3THz$ ($100nm$).

$FSR_{1-2}=FSR_{1}\frac{L_{1}}{L_{1}-L_{2}}$ (3)

However, this is not sufficient to determine the unknown wavelength over such
broad spectrum. With the help of figure 2, we can understand better the
problem. In the upper plot the the spectra of ring $L_{1}$ and $L_{2}$ are
simulated.

Because of the finite $FWHM$ of the resonances we cannot determine if the
resonance is at $1550.00nm$, $1550.02nm$ or $1449.98nm$. In the lower plot
instead we see what happens when we compare compare the spectra of $L_{1}$ and
$L_{4}$. $L_{4}$ as a length of $l_{4}=3578\mu m$ such that
$FSR_{1}-FSR_{4}>FWHM$. The degeneracy has been removed and we can state the
resonance common to both rings is at $1550.00nm$. But why don’t we use only
the rings $L_{1}$ and $L_{4}$ then? Because in such case the new combined
$FSR$ would be of only $3.318nm$. According to our simulations, using only
three rings is not sufficient to cover accurately a $100nm$ wide band since
there are regions in the spectra where a single wavelength cannot be
identified. To finally solve the problem we have added ring $L_{3}$ with
$l_{3}=3578\mu m$.

##  4 Calibration procedure

At the moment, we have tested the concept of bandwidth expansion over the span
of $1nm$ at $1550nm$. For such limited bandwidth rings $L_{1}$ and $L_{4}$ are
sufficient to determine the wavelength without ambiguity. For that we have
used a laser source of known wavelength, and acquired the PD response in steps
of $20nm$. In Figure 3 we see an acquisition of the shortest ring at
$1549.96nm$. We see numerous peaks, that implies that we overestimated the
necessary PHS minimum length and we could have used a shorter PHS. The peaks
are clearly of two types: in particular we notice the odd peaks are much
deeper than the even one. This is likely caused by the propagation of two
different optical modes, likely fundamental $TE$ and $TM$. For our
calibration, we have decided to consider only the first three odd peaks, that
are clearly distinguishable in all our acquisitions. For any wavelength step
we have stored the voltages corresponding to such peaks. We did this both for
peaks of ring $L_{1}$ and $L_{4}$.

In Fig. 4 we have plotted the square of this voltage versus the corresponding
wavelength. We plot the square because PHs are thermally controlled, and the
electric power dissipated is proportional to the voltage squared. Every single
group of resonances is fitted with a linear function. Using this fit is
possible to create the look up table that associates the two voltages
($V_{1}$,$V_{2}$) to the corresponding wavelength. In the phase diagram of
Fig. 5 we have all the data displayed in the look up table. The horizontal
axis shows the squared voltage applicable to the ring $L_{1}$, the vertical
ones refers to the squared voltage applicable to $L_{2}$. Points in the phase
space are identified by couples of values ($V_{1}^{2}$,$V_{2}^{2}$). The color
of the point codifies the wavelength according to the color bar on the right.
Error bars are also displayed. They have been obtained by computing the
standard variation of experimental points from the predicting fitting model.
We see that such error-bars correspond to an uncertainty of $\pm 1.5pm$ on the
estimated wavelength. To operate the wavelength meter, a script where
experimental ($V_{1},V_{2}$) are associated to the corresponding wavelength is
necessary. Despite the extension of the error bars, the identification of an
unique wavelength as been always possible in the tested range of $1nm$.

Figure 3: Measured response of ring L1 at a wavelength of 1549.96. The figure
is obtained sweeping voltages between $0V$ and $5V$ and registering the power
at the integrated photodiode. Using a peak detection alghorithms we can
identify the voltages associated with minima, noticing two series of peak at
different depth we have decided to consider only odd peaks in our subsequent
analysis. Figure 4: Peak voltages of Ring L1 and Ring L4 respect to the
injection wavelength, each point in the graph correspond to a minima: red
point are minima of Ring L1, blue points are minima of Ring L4. The data are
fitted with a quadratic function in order to create a lookup table. Figure 5:
A visualization of the look up table, On the axes the voltages obtained by
fitting represented in Fig 4 for the two different ring. The resulting
wavelength is color coded, point are separate by $4.4pm$.

##  5 Reading speed limit

We have realized that there is a minimum speed limit necessary to estimate the
wavelength. Such limit is linked to light propagation into ta ring. Indeed,a
ring resonator has a typical cavity lifetime that is determined by the $Q$
factor: $Q=\omega\tau_{s}$ . A time $T=3\tau_{s}$ is necessary before
considering the ring to be stabilized[10]. To cover the full FSR of any of the
rings we need a number of steps in the voltage applied to the PHS. We chose
this number of steps ($m$), in order not to lose resolution but also not to
oversample the PD response. Given that the limit resolution is equal to the
$FWHM$ of the ring:

$m=\frac{2FSR}{FWHM}=\frac{2FSR\times Q}{\omega}$ (4)

We can express this equation in therms of $Q$ to estimate the time $T$
necessary for a complete measurement as:

$T=m\tau_{s}=\frac{6FSR\times Q^{2}}{\omega^{2}}$ (5)

We can notice from this equation that the measurement speed scales with the
square of the $Q$ factor. In the following table we estimate the speed of our
device for different Q factors at constant single ring $FSR=30$ GHz and
$\omega=193$ THz.

Table 1: Table representing the correlation between the Quality factor of the cavity, minimum measurement Time and measurement resolution. The results are obtained applying equation 5. Q factor | T | Resolution
---|---|---
$10^{4}$ | $500ps$ | $160pm$
$10^{5}$ | $50ns$ | $20pm$
$10^{6}$ | $5\mu s$ | $1.6pm$

An important remark is necessary, the measurement speed does not depend on the
number of rings since all the rings are measured simultaneously.

##  6 Conclusions

A photonic-integrated wavelength-meter based on multiple ring resonators has
been realized in InP. The correct operation of the device has been verified
for a limited bandwidth of $1nm$. New measures are expected to be carried out
with a much broader band in order to verify the stability and reliability over
a span of $100nm$. Moreover, upon realization of wire bonding connections, it
will be possible to test the maximum readout speed of the device.

##  7 Acknowledgements

Project developed in the framework of the European Doctorate in Indium
Phosphide PIC Fabrication Technology (EDIFY) project. This work has received
financial support from MSCA H2020-ITN-2018-EDIFY (Contract number 813467)

##  8 Disclosures

The authors declare no conflicts of interest.

## References

  * [1] C. Xiang, M. A. Tran, T. Komljenovic, J. Hulme, M. Davenport, D. Baney, B. Szafraniec, and J. E. Bowers, Opt. Lett. 41, 3309 (2016).
  * [2] R. M. Oldenbeuving, H. Song, G. Schitter, M. Verhaegen, E. J. Klein, C. J. Lee, H. L. Offerhaus, and K.-J. Boller, Opt. Express 21, 17042 (2013).
  * [3] T. Komljenovic, L. Liang, R.-L. Chao, J. Hulme, S. Srinivasan, M. Davenport, and J. E. Bowers, Applied Sciences 7 (2017).
  * [4] M. Smit, X. Leijtens, E. Bente, J. van der Tol, H. Ambrosius, D. Robbins, M. Wale, N. Grote, and M. Schell, “A generic foundry model for inp-based photonic ics,” in _Optical Fiber Communication Conference,_ (Optical Society of America, 2012), pp. OM3E–3.
  * [5] R. Dey, J. Doylend, J. Ackert, A. Evans, P. Jessop, and A. Knights, Opt. Express 21, 23450 (2013).
  * [6] C. Taballione, T. Agbana, G. Vdovin, M. Hoekman, L. Wevers, J. Kalkman, M. Verhaegen, P. J. van der Slot, and K.-J. Boller, “Temperature-drift-immune wavelength meter based on an integrated micro-ring resonator,” in _Integrated Optics: Physics and Simulations III,_ , vol. 10242 P. Cheben, J. Čtyroký, and I. Molina-Fernández, eds., International Society for Optics and Photonics (SPIE, 2017), pp. 39 – 48.
  * [7] P. Wang, A. M. Hatta, H. Zhao, W. Yang, J. Ren, Y. Fan, G. Farrell, and G. Brambilla, Opt. Express 25, 2939 (2017).
  * [8] R. Broeke, D. Melati, F. Morichetti _et al._ , “Design and performance of a packaged inp wavelength meter,” in _18th Annual Symposium of the IEEE Photonics Benelux Chapter,_ (2013), pp. 287–290.
  * [9] T. Komljenovic, L. Liang, R.-L. Chao, J. Hulme, S. Srinivasan, M. Davenport, and J. E. Bowers, Applied Sciences 7 (2017).
  * [10] S. Biasi, P. Guilleme, A. Volpini, G. Fontana, and L. Pavesi, Journal of Lightwave Technology 37, 5091 (2019).

30_references/references_new


$\eqref{ie-1}\Rightarrow\eqref{ie-4}$. Since $E$ is the fixed point set of a
Shapley operator, $E$ is a closed ambitropical cone. If this Shapley operator
is finitely generated, then by 8.3, $E$ can be written as the union of a
finite family of alcoved polyhedra $(X_{\tau})_{\tau\in\mathscr{T}}$. Then,
$\bar{E}^{\max}=\operatorname{clo}^{\downarrow}E^{\max}$ coincides with the
union of the lower closures $\operatorname{clo}^{\downarrow}X_{\tau}$, for
$\tau\in\mathscr{T}$. By 2.3, every $\operatorname{clo}^{\downarrow}X_{\tau}$
is a finitely generated $\mathbb{R}_{\max}$-semimodule. It follows that
$\bar{E}^{\max}$ is a finitely generated $\mathbb{R}_{\max}$-semimodule.

$\eqref{ie-4}\Rightarrow\eqref{ie-2}$. By 8.7, $E$ is the fixed point set of
the operator $\bar{Q}^{-}_{E}$ given in (29). By 8.5, every operator
$Q_{E_{k}}^{\pm}$ is finitely generated. Hence, $\bar{Q}^{-}_{E}$ is finitely
generated. Then, by 8.3, $E$ is a finite union of alcoved polyhedra. ∎

The description of an ambitropical polyhedron as the fixed point set of a
Shapley operator is analogous to the “external” description of a polyhedron.
We next show that ambitropical polyhedra admits an alternative description, by
generators.

######  Definition 8.2.

A description by generators of a closed ambitropical cone $E$ consists of a
pair $(U^{\max},U^{\min})$ such that $U^{\max}$ is a tropical generating set
of $\bar{E}^{\max}$ and $U^{\min}$ is a dual tropical generating set of
$\bar{E}^{\min}$. We say that the description is finite if the sets $U^{\max}$
and $U^{\min}$ are finite.

The generating sets $U^{\max}$ and $U^{\min}$ uniquely determine $E^{\max}$
and $E^{\min}$, and so they uniquely determine the ambitropical cone $E$,
which coincides with $\operatorname{Im}P^{\max}_{E}\circ P^{\min}_{E}$.

######  Corollary 8.9.

A closed ambitropical cone is an ambitropical polyhedron if and only if it has
a finite description by generators. Moreover, it is an ambitropical polytope
if and only if these generators belong to $\mathbb{R}^{n}$.

###### Proof.

If $E$ is an ambitropical polyhedron, it follows from 8.8, (4) that $E$ admits
a finite description by generators.

Conversely, if $E$ is a closed ambitropical cone that admits a finite
description by generators, $(U^{\max},U^{\min})$, by (7) and its dual,
$P^{\max}_{E}$ and $P^{\min}_{E}$ are finitely generated Shapley operators,
and so, $E$ is the fixed point set of the finitely generated Shapley operator
$P^{\max}_{E}\circ P^{\min}_{E}$. So, $E$ is an ambitropical polyhedron by
Theorem 8.8 (2).

Now, if $E$ is bounded in Hilbert’s seminorm, we have
$\bar{E}^{\max}=E^{\max}\cup\\{(-\infty,\dots,-\infty)\\}$, and dually,
$\bar{E}^{\min}=E^{\min}\cup\\{(+\infty,\dots,+\infty)\\}$, we get
$U^{\max}\subset E^{\max}$ and $E^{\min}\subset U^{\min}$, showing that the
generators belong to $\mathbb{R}^{n}$. Conversely, suppose that the elements
of $U^{\max}$ and $U^{\min}$ belong to $\mathbb{R}^{n}$, and let $R$ denote
the maximal Hilbert seminorm of these elements. We observe that balls in
Hilbert’s seminorm are invariant by the operations of suprema and infima, and
so, considering the formula (7), we deduce that
$E\subset\operatorname{Im}P_{E}^{\max}$ is included in the ball of radius $R$
in Hilbert’s seminorm. ∎

When $E$ is an ambitropical polytope, for any description by generators
$(U^{\max},U^{\min})$, $U^{\max}$ and $U^{\min}$ are necessarily subsets of
$E$, and so we actually get a proper notion of internal representation of $E$
by generators.

In particular, we have the following characterization of ambitropical
polytopes. inline]SG: I revised this part of the ms and added the following
cor, check

######  Corollary 8.10.

Every finite subset of $\mathbb{R}^{n}$ admits an ambitropical hull that is an
ambitropical polytope, and all the ambitropical polytopes arise in this way.

###### Proof.

By 4.17, an ambitropical hull of a finite subset $E=\\{u^{1},\dots,u^{k}\\}$
of $\mathbb{R}^{n}$ is given by the range of the finitely generator operator
$\bar{Q}^{+}_{E}=P^{\max}_{E}\circ P^{\min}_{E}$, which, by 8.9, is bounded in
Hilbert’s seminorm, and so, it is an ambitropical polytope.

Conversely, suppose that $F$ is an ambitropical polytope. Then, $F$ is a
finite union of alcoved polyhedra $F_{k}$ that are bounded in Hilbert
seminorm. Each of these alcoved polyhedra $F_{k}$ has a finite set of tropical
generators. By taking the union of these finite sets we obtain a (possibly
redundant) finite set $F^{+}$ of tropical generators of $E^{\max}$. The dual
constructions yields a finite set $F^{-}$ of dual tropical generators of
$E^{\min}$. Observe that the explicit construction of the retraction
$\bar{Q}^{+}_{E}$ given in 8.7 involves elementary operators
${Q}^{\pm}_{E_{k}}$ which only depend on the primal and dual tropical
generators of $E_{k}$. Hence, by taking for $E$ the union of the two sets
$F^{\pm}$, we get that $\bar{Q}^{+}_{E}=\bar{Q}^{+}_{F}$, showing that $F$ is
an ambitropical hull of the finite set $E$. ∎

######  Example 8.11.

8.10 is illustrated in Figure 3, showing an ambitropical hull of the points
$a_{1},\dots,a_{9}$ given by the columns of the matrix

$\bordermatrix{\leavevmode\nobreak\
&a_{1}&a_{2}&a_{3}&a_{4}&a_{5}&a_{6}&a_{7}&a_{8}&a_{9}\cr\leavevmode\nobreak\
&4&5&3&1&0&0&0&0&4\cr\leavevmode\nobreak\
&0&2&4&3&4&2&2&-1&0\cr\leavevmode\nobreak\ &0&0&0&0&2&4&2&0&3}$

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$a_{8}$$a_{9}$$x_{1}$$x_{2}$$x_{3}$0

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$a_{8}$$a_{9}$$x_{1}$$x_{2}$$x_{3}$0

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$a_{8}$$a_{9}$$x_{1}$$x_{2}$$x_{3}$0

Figure 3. A finite collection of points $E=\\{a_{1},\dots,a_{9}\\}$; the
tropical cone $E^{\max}$ that it generates (left); the dual tropical cone
$E^{\min}$ (right); and the ambitropical hull
$\operatorname{Im}P_{E}^{\max}\circ P_{E}^{\min}$ (middle).

##  9\. Homogeneous ambitropical polyhedra

We next study the class of ambitropical polyhedra that are homogeneous in the
sense of 6.1. We shall see that such polyhedra arise when studying “locally”
ambitropical polyhedra, or when considering their behavior at infinity.
Moreover, they admit a combinatorial characterization, in terms of posets.

######  Definition 9.1.

Let $C$ be an ambitropical polyhedron in $\mathbb{R}^{n}$ and $u\in C$. The
tangent cone of $C$ at point $u$, denoted by $\mathcal{T}_{u}(C)$, is the set
of vectors $v$ such that $u+sv\in C$ holds for all $s\geqslant 0$ small
enough.

Let us recall the following definition from variational analysis.

######  Definition 9.2.

A function $T:\mathbb{R}^{n}\to\mathbb{R}^{p}$ is semidifferentiable at point
$u\in\mathbb{R}^{n}$ if there exists a continuous map $T^{\prime}_{u}$,
(positively) homogeneous (so $T^{\prime}_{u}(\alpha x)=\alpha
T^{\prime}_{u}(x)$ holds for all $\alpha>0$ and $x\in\mathbb{R}^{n}$) such
that

(30) $\displaystyle T(u+h)=T(u)+T^{\prime}_{u}(h)+o(\|h\|)\enspace.$

Then, $T^{\prime}_{u}(h)$ must coincide with the one sided directional
derivative:

$T^{\prime}_{u}(h)=\lim_{s\to 0^{+}}s^{-1}(T(u+sh)-T(u))\enspace.$

Conversely, if $T$ is Lipschitz continuous, an application of Ascoli’s theorem
shows that if this directional derivate exists for all $h\in\mathbb{R}^{n}$,
then, $T$ is semidifferentiable at point $u$, see e.g. [AGN16, Lemma 3.2]. We
shall be consider especially the situation in which $T$ is continuous and
piecewise linear, meaning that $\mathbb{R}^{n}$ can be covered by finitely
many polyhedra on each of which the restriction of $T$ is an affine map. Then,
$T$ is automatically Lipschitz continuous, and the directional derivative
always exists, showing that $T$ is semidifferentiable. In this case, the
semiderivative $h\mapsto T^{\prime}_{u}(h)$ is also piecewise linear, and this
entails that the local expansion (30) is exact for $h$ small enough:

######  Proposition 9.1.

Let $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be piecewise linear, and let
$u\in\mathbb{R}^{n}$. Then, there exists a neighborhood $V$ of $0$ such that,
for all $h\in V$,

(31) $\displaystyle T(u+h)=T(u)+T^{\prime}_{u}(h)\enspace.$

∎

The chain rule extends to semidifferentiable maps: if $f,g$ are Lipschitz
continuous, if $f$ is semidifferentiable at point $u$, and if $g$ is
semidifferentiable at point $f(u)$, then

(32) $\displaystyle(g\circ f)^{\prime}_{u}=g^{\prime}_{f(u)}\circ
f^{\prime}_{u}\enspace,$

see Lemma 3.4 of [AGN16].

Let us also recall the rule of computation of directional derivatives of
suprema and infima. If $f$ is a function $\mathbb{R}^{n}\to\mathbb{R}$ that
can be written as a supremum of a finite family of functions $f=\max_{i\in
I}f_{i}$ and if each $f_{i}$ has one sided directional derivatives at point
$u$, then

(33) $\displaystyle f^{\prime}_{u}(h)=\max_{i\in
I^{*}(u)}(f^{\prime}_{i})_{u}(h),\qquad\text{where}\;I^{*}(u)=\\{i\in I\mid
f(u)=f_{i}(u)\\}\enspace,$

see [RW98, Exercise 10.27]. A dual rule applies to a function $f=\min_{i\in
I}f_{i}$.

We call homogeneous ambitropical polyhedron an ambitropical polyhedron that is
a homogeneous ambitropical cone.

######  Proposition 9.2.

Suppose $C$ is an ambitropical polyhedron, and let $u\in C$. Then, the tangent
cone $\mathcal{T}_{u}(C)$ is a homogeneous ambitropical polyhedron.

###### Proof.

It follows from the definition of $\mathcal{T}_{u}(C)$ that if
$v\in\mathcal{T}_{u}(C)$, then $sv\in\mathcal{T}_{u}(C)$ for all $s>0$. Since
$C$ is an ambitropical polyhedron, we have $C=\\{x\in\mathbb{R}^{n}\mid
T(x)=x\\}$ where $T$ is a finitely generated Shapley operator. We claim that

$\mathcal{T}_{u}(C)=\\{h\in\mathbb{R}^{n}\mid T^{\prime}_{u}(h)=h\\}\enspace.$

Let $h\in\mathbb{R}^{n}$ such that $T^{\prime}_{u}(h)=h$. For $s>0$ small
enough such that $sh$ belongs to the neighborhood $V$ of Proposition 9.1, we
have that $T(u+sh)=T(u)+T^{\prime}_{u}(sh)=u+sT^{\prime}_{u}(h)=u+sh$. It
follows that $h\in\mathcal{T}_{u}(C)$. Conversely if $h\in\mathcal{T}_{u}(C)$,
$u+sh\in C$ holds for all $s$ small enough, hence, $T(u+sh)=u+sh$ holds for
all such $s$, and using (31), we deduce that $T^{\prime}_{u}(h)=h$. Therefore,
$\mathcal{T}_{u}(C)$ is the fixed point set of the homogeneous Shapley
operator $T^{\prime}_{u}$. Using the chain rule (32), the rule of
semidifferentiation of suprema (33), and the dual rule of semidifferentiation
of infima, we deduce that $T^{\prime}_{u}$ is finitely generated.

∎

######  Definition 9.3.

Suppose $C$ is a finite union of (ordinary) polyhedra. Then, the recession
cone of $C$, $\hat{C}$, is the set of vectors $v$ such that there is a vector
$x\in C$ such that $x+sv$ belongs to $C$ for all $s\geqslant 0$.

If $T$ is piecewise linear $\mathbb{R}^{n}\to\mathbb{R}^{p}$, an in particular
if $T$ is finitely generated, then the recession function

$\hat{T}(x):=\lim_{s\to\infty}s^{-1}T(sx)$

is well defined. Observe that $\hat{T}$ is finitely generated as soon as $T$
is finitely generated (this follows from the proof of 7.2).

######  Proposition 9.3.

Suppose $C$ is an ambitropical polyhedron. Then, the recession cone $\hat{C}$
is a homogeneous ambitropical polyhedron.

###### Proof.

If $v\in\hat{C}$, then $\alpha v\in\hat{C}$ holds for all $\alpha>0$, showing
that $C$ is homogeneous. Suppose that $C=\\{x\in\mathbb{R}^{n}\mid T(x)=x\\}$,
where $T$ is a finitely generated Shapley operator. We claim that
$\hat{C}=\\{v\in\mathbb{R}^{n}\mid\hat{T}(v)=v\\}$. Let $v\in\hat{C}$. Then,
the exists $y\in C$ such that $y+sv\in C$ holds for all $s\geqslant 0$. So
$T(y+sv)=y+sv$. Dividing by $s$, using the nonexpansive character of $T$, and
letting $s$ tend to infinity, we deduce that $\hat{T}(v)=v$. Conversely,
suppose that $\hat{T}(v)=v$. Then, we can find a vector $y$ such that the ray
$[0,\infty)\ni s\mapsto y+sv$ is included in a region in which $T$ is affine.
Then, $T(y+sv)=w+sCv$ for some matrix $C$ and for some vector $w$.
Specializing at $s=0$, we deduce that $w=T(y)$. We also have $T(y+sv)/s\to Cv$
as $s\to\infty$, and so $\hat{T}(v)=Cv$. It follows that
$T(y+sv)=T(y)+s\hat{T}(v)$. Hence, $T(y+sv)=y+sv$, showing that $y+sv\in C$,
for all $s\geqslant 0$, and so $v\in\hat{C}$. Since $\hat{T}$ is a finitely
generated homogeneous Shapley operator, this entails that $\hat{C}$ is a
homogeneous ambitropical polyhedron. ∎

######  Theorem 9.4.

Let $E$ be a subset of $\mathbb{R}^{n}$, then the following are equivalent:

  1. (1)

$E$ is a homogeneous ambitropical polyhedron;

  2. (2)

There exists a homogeneous finitely generated Shapley operator such that
$P=P^{2}$ and $E=\\{x\in\mathbb{R}^{n}|x=P(x)\\}$;

  3. (3)

There exists a homogeneous finitely generated Shapley operator $T$ such that
$E=\\{x\in\mathbb{R}^{n}|x=T(x)\\}$.

###### Proof.

(1)$\Rightarrow$(2). If $E$ is a homogeneous ambitropical polyhedron, we can
write $E$ as a finite union $\cup_{k}E_{k}$ where the $E_{k}$ are homogeneous
alcoved polyhedra. Then, the operators $Q_{E_{k}}^{\pm}$ are homogeneous and
finitely generated. We conclude as in the proof of the implication
(1)$\Rightarrow$(2) of 8.8.

(2)$\Rightarrow$(3): trivial.

(3)$\Rightarrow$(1): by 8.8, $E$ is an ambitropical polyhedron. Since
$E=\\{x\in\mathbb{R}^{n}|x=T(x)\\}$, and $T$ is homogeneous, $E$ is
homogeneous.

∎

Given a homogeneous ambitropical polyhedron $C$ of $\mathbb{R}^{n}$, we define
the skeleton of $C$, $\operatorname{Sk}{C}$, to be the intersection of $C$
with $\\{0,1\\}^{n}$. Given an (ordered) partition
$\mathcal{I}=(I_{1},\dots,I_{S})$ of $[n]$, we define the Weyl cell of $C$ to
be

$W^{\mathcal{I}}=\\{x\in\mathbb{R}^{n}\mid(i\in I_{r},j\in I_{s},r\leqslant
s)\implies x_{i}\leqslant x_{j}\\}\enspace.$

E.g., $\\{x\in\mathbb{R}^{4}\mid x_{1}\leqslant x_{2}=x_{3}\leqslant x_{4}\\}$
is the Weyl cell corresponding to the partition $(\\{1\\},\\{2,3\\},\\{4\\})$
of the set $\\{1,2,3,4\\}$. When each of the sets $I_{1},\dots,I_{S}$ has
exactly one element, $W^{\mathcal{I}}$ is a Weyl chamber of $A_{n}$ type,
i.e., a set of the form $\\{x\in\mathbb{R}^{n}\mid
x_{\sigma(1)}\leqslant\dots\leqslant x_{\sigma(n)}\\}$ for some permutation
$\sigma$.

We shall need the following observation, which is a variation on the
construction of the canonical triangulation of order polytopes by Stanley
[Sta86, § 5].

######  Lemma 9.5.

Any homogeneous ambitropical polyhedron is a union of Weyl cells.

###### Proof.

Any ambitropical polyhedron is a finite union of alcoved polyhedra, and if
this polyhedron is homogeneous, the alcoved polyhedra must be homogeneous. It
suffices to show that a homogeneous alcoved polyhedron is a finite union of
Weyl cells. A homogeneous alcoved polyhedron is of the form $E=\\{x\mid
x_{i}\leqslant x_{j},\forall(i,j)\in L\\}$ where $L$ is a subset of
$[n]\times[n]$. We shall think of $L$ as a relation on the set $[n]$, and,
since, $E$ is unchanged if $L$ is replaced by its reflexive and transitive
closure, we assume that $L$ is a preorder. Then, we define the equivalence
relation $\equiv_{L}$, on $[n]$, such that for any $i,j\in[n]$, $i\equiv_{L}j$
if and only if $(i,j)\in L$ and $(j,i)\in L$. The equivalence classes
determined by this relation are nonempty subsets of $[n]$, that constitute a
partition of $[n]$. The relation $L$ determines a partial order
$\leqslant_{L}$ on the set of these equivalence classes, the order being
defined by $I\leqslant_{L}J$ if $(i,j)\in L$ for all $i\in I$ and $j\in J$,
and for all equivalence classes $I,J$. We choose a linear extension of this
partial order, allowing us to write the equivalence classes as
$I_{1},\dots,I_{S}$, in such a way that $I_{k}\leqslant_{L}I_{l}\implies
k\leqslant l$. Such a linear extension determines then a Weyl cell
$W^{\mathcal{I}}$ with $\mathcal{I}=(I_{1},\dots,I_{S})$. By construction,
$W^{\mathcal{I}}\subset E$. Moreover, if $x\in E$, then, taking
$J_{1}:=\operatornamewithlimits{argmin}_{i\in[n]}x_{i}$, we see that $J_{1}$
must be a union of equivalence classes $I_{i_{1}},\dots,I_{i_{m_{1}}}$.
Similarly, $J_{2}=\operatornamewithlimits{argmin}_{i\in[n]\setminus
I_{1}}x_{i}$ must be a union of equivalence classes
$I_{i_{m_{1}+1}},\dots,I_{i_{m_{2}}}$. Continuing in this way, setting
$J_{k}:=\operatornamewithlimits{argmin}_{i\in[n]\setminus
I_{k-1}}x_{i}=I_{i_{m_{k-1}}}\cup\dots\cup I_{m_{k}}$until $J_{1}\cup\dots\cup
J_{k}=[n]$, we get that $x\in W^{\mathcal{I}}$ with
$\mathcal{I}=(I_{i_{1}},\dots,I_{i_{s}})$. This shows that $E$ is the union of
the Weyl cells $W^{\mathcal{I}}$ arising from all the linear extensions of the
order $\leqslant_{L}$. Note that different extensions give different Weyl
cells. ∎

The following theorem characterizes the polyhedral complexes associated with
homogeneous ambitropical polyhedra, showing that they are in bijection with
lattices included in $\\{0,1\\}^{n}$. These lattices have been studied by
Crapo [Cra82].inline]check that it works for cells not necessarily of maximal
dimension, them it describes the full polyhedral complex. Explain how a chain
yields a cell (this part of proof is missing)

######  Theorem 9.6 (Homogeneous ambitropical cones are equivalent to lattices
in $\\{0,1\\}^{n}$).

The map $C\mapsto\operatorname{Sk}C$ establishes a bijective correspondence
between homogeneous ambitropical polyhedra of $\mathbb{R}^{n}$ and subsets of
the partially ordered set $(\\{0,1\\}^{n},\leqslant)$ that contain the bottom
and the top element, and that are lattices in the induced order. Moreover,
there is a one-to-one correspondence between the chains in
$\operatorname{Sk}C$ and the Weyl cells included in $C$; the cardinality of
each of these chains coincides with the dimension of the corresponding Weyl
cell plus one unit; and the collection of these Weyl cells constitutes a
polyhedral fan with support $C$.

###### Proof.

If $C$ is a homogeneous ambitropical polyhedron, then, by 9.4, there is a
homogeneous finitely generated Shapley operator
$P=P^{2}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that
$C=\\{x\in\mathbb{R}^{n}\mid x=P(x)\\}$. The coordinates of a homogeneous
finitely generated Shapley operator can be written as min-max formula without
additive translations (i.e., as a monotone Boolean formula), and so, $P$
admits a restriction $\\{0,1\\}^{n}\to\\{0,1\\}^{n}$. Since
$\operatorname{Sk}(C)=C\cap\\{0,1\\}^{n}$, this entails that
$\operatorname{Sk}(C)=P(\\{0,1\\}^{n})$ is an order preserving retract of
$\\{0,1\\}^{n}$. Moreover, since $P$ is positive homogeneous, the identically
zero vector is fixed by $P$, and since $P$ commutes with the addition of a
constant vector, the unit vector is also fixed by $P$. This implies
$P(\\{0,1\\}^{n})$ is a lattice in the induced order of $\\{0,1\\}^{n}$,
containing the bottom and top elements.

We now claim that for all permutations $\sigma$ of $[n]$, the action of $P$ on
the chamber $W^{\sigma}:=\\{x\mid x_{\sigma(1)}\leqslant\dots\leqslant
x_{\sigma(n)}\\}$ is uniquely determined by its action on $\\{0,1\\}^{n}$. In
fact, $P$ is linear on the chamber $W^{\sigma}$, and since this chamber is a
cone with a generating family consisting of vectors in $\\{0,1\\}^{n}$, it
follows that $P$ is uniquely determined by its restriction to $\\{0,1\\}^{n}$.
In particular, $P$ fixes the full chamber $W^{\sigma}$ if and only if it fixes
each generator of each chamber belonging to $\\{0,1\\}^{n}$. So, the fixed
point set of $P$, which is $C$, is uniquely determined by the fixed point set
of $P$ restricted to $\\{0,1\\}^{n}$, which is $\operatorname{Sk}(C)$. This
shows that the correspondence between a homogeneous ambitropical polyhedron
and its skeleton is bijective.

We now show the following claim: every subset $S$ of $\\{0,1\\}^{n}$ that is a
lattice in the induced order and contains the bottom and top elements of
$\\{0,1\\}^{n}$, can be realized as a skeleton of a homogeneous ambitropical
polyhedron. The map $T(x):=\sup_{S}\\{u\in\\{0,1\\}^{n}\mid u\leqslant x\\}$
is an order preserving self-map of $\\{0,1\\}^{n}$ such that
$S=\\{x\in\\{0,1\\}^{n}\mid x=T(x)\\}$.Now, by a standard result, any order
preserving map $f$ from $\\{0,1\\}^{n}$ to $\\{0,1\\}$ such that
$f(0,\dots,0)=0$ and $f(1,\dots,1)=1$ can be represented by a monotone Boolean
function. Indeed, let $F:=\\{y\in\\{0,1\\}^{n}\mid f(y)=1\\}$, and consider
the monotone Boolean function

$g(x):=\bigvee_{y\in F}\bigwedge_{i\in[n],\,y_{i}=1}x_{i}\enspace.$

We have that $f(x)=g(x)$ holds for all $x\in\\{0,1\\}^{n}$. This establishes
the claim.

By 9.5, $C$ is a finite union of Weyl cells. We observe that the intersection
of the Weyl cell $W^{\mathcal{I}}$ with $\\{0,1\\}^{n}$ is a chain. Indeed,
the bottom element of $W^{\mathcal{I}}$ is the zero vector. The smallest non-
zero element of $W^{\mathcal{I}}$ is the vector $x$ gotten by setting
$x_{i}=0$ for all $i\in\cup_{s<S}I_{s}$ and $x_{i}=1$ for all $i\in I_{S}$.
The smallest element of $W^{\mathcal{I}}$ greater than $x$ is the vector $y$
gotten by setting $y_{i}=0$ for all $i\in\cup_{s<S-1}I_{s}$ and $y_{i}=1$ for
all $i\in\cup_{s\geqslant S-1}I_{s}$, etc. This yields a chain of length
$S+1$. Moreover, consider the linear space
$H^{\mathcal{I}}=\\{x\in\mathbb{R}^{n}\mid x_{i}=x_{j},\forall i,j\in
I_{s},\forall s\in[S]\\}$. Every set $I_{s}$ yields $|I_{s}|-1$ independent
linear relations, so, $H^{\mathcal{I}}$ is of dimension
$n-(\sum_{s=1}^{S}|I_{s}|-1)=n+S-\sum_{s=1}^{S}|I_{s}|=S$. We have
$W^{\mathcal{I}}\subset H^{\mathcal{I}}$, and since for all
$0<\alpha_{1}<\dots<\alpha_{S}$, the vector $x$ such that $x_{i}=\alpha_{s}$
for all $i\in I_{s}$, we deduce that $W^{\mathcal{I}}$ is of dimension at
least $S$. It follows that $W^{\mathcal{I}}$ is of dimension equal to $S$.

Finally, a face of the Weyl cell associated with an ordered partition
$I_{1},\dots,I_{S}$ is again a Weyl cell, associated with a new partition
obtained by merging several consecutive sets of the partition in a single
class (this corresponds to the operation of taking a subchain in the
skeleton). Moreover, taking the intersection of Weyl cells corresponds to
taking the intersection of the associated chains, which entails that the
collection of these Weyl cells constitutes a polyhedral fan. ∎

We deduce that the class of ambitropical sets is not closed under projection:

######  Example 9.7.

Consider the subset $L$ of $\\{0,1\\}^{5}$ given by bottom, top and the
following elements $(0,1,0,0,1),\\\ (0,0,1,0,1),(0,1,1,1,0),(1,1,1,0,1)$, with
induced order. The set $L$ is a lattice, so we know by the previous result
that it corresponds to an homogeneous ambitropical polyhedron $C$ of
$\mathbb{R}^{5}$, of which it is the skeleton $L=C\cap\\{0,1\\}^{5}$. Let us
consider now the projection $\operatorname{proj}(C)$ on $\mathbb{R}^{4}$ of
$C$ which is obtained by taking the first 4 coordinates, and observe that it
is a homogeneous polyedron. Assume that it is an ambitropical cone. Then, by
9.6 again, its skeleton would be a lattice. But the skeleton of
$\operatorname{proj}(C)$ is the projection of the skeleton $L$ of $C$, which
is not a lattice since the sup of the two elements $(0,1,0,0)$ and $(0,0,1,0)$
is not well defined, because the two elements $(0,1,1,1)$ and $(1,1,1,0)$ are
minimal upper bounds. This shows that $\operatorname{proj}(C)$ is not an
ambitropical cone.

######  Example 9.8.

Consider the finitely generated Shapley operator
$T:\mathbb{R}^{3}\to\mathbb{R}^{3}$,

$T(y)=\big{(}(x_{1}\wedge x_{2})\vee(x_{1}\wedge x_{3})\vee(x_{2}\wedge
x_{3}),x_{2},x_{3}\big{)}\enspace.$

The fixed point set $E$ of $T$ is the butterfly shaped polyhedral complex with
two full dimensional cells $E_{1}$ and $E_{2}$, shown in Figure 4. Explicitly,
$E_{1}=\\{x\in\mathbb{R}^{3}\mid x_{2}\geqslant x_{1}\geqslant x_{3}\\}$,
$E_{2}=\\{x\in\mathbb{R}^{3}\mid x_{3}\geqslant x_{1}\geqslant x_{2}\\}$.
Using 8.6, we get that the tropical projections are given by:

$\displaystyle P^{\max}_{E}\begin{pmatrix}x_{1}\\\ x_{2}\\\
x_{3}\end{pmatrix}=\begin{pmatrix}(x_{1}\wedge(x_{2}\vee x_{3}))\\\ x_{2}\\\
x_{3}\end{pmatrix}\text{ and }P^{\min}_{E}\begin{pmatrix}x_{1}\\\ x_{2}\\\
x_{3}\end{pmatrix}=\begin{pmatrix}(x_{1}\vee(x_{2}\wedge x_{3}))\\\ x_{2}\\\
x_{3}\end{pmatrix}$

######  Example 9.9.

A union of Weyl cells that is not ambitropical is shown at the left of Figure
7. This union is not ambitropical because it does not contain the unique
geodesic between two specific points of this union, contradicting the
conclusion of 4.18.

######  Example 9.10.

Consider the ambitropical polyhedron $E$ in Figure 6. We have that in this
case

$Q^{-}_{E}\begin{pmatrix}x_{1}\\\ x_{2}\\\
x_{3}\end{pmatrix}=\begin{pmatrix}(x_{1}\wedge
x_{2}\wedge(1+x_{3}))\vee(x_{1}\wedge(1+x_{2})\wedge x_{3})\vee(x_{2}\wedge
x_{3}\wedge(1+x_{1}))\\\ x_{2}\wedge(1+x_{1})\wedge(1+x_{3})\\\
x_{3}\wedge(1+x_{2})\wedge(1+x_{1})\end{pmatrix}$

The construction of the sets $E^{\max}$ and $E^{\min}$, as well as 4.12,
showing that $P^{\max}\circ P^{\min}$ and $P^{\min}\circ P^{\max}$ are
retractions on an ambitropical set $E$, are illustrated in the figure.

$x_{1}$$x_{2}$$x_{3}$$g_{1}$$g_{2\cdot}$$h$$f_{1}$$f_{2}$

$x_{1}$$x_{2}$$x_{3}$$g_{1}$$g_{2\cdot}$$h$$f_{1}$$f_{2}$

$x_{1}$$x_{2}$$x_{3}$$g_{1}$$g_{2\cdot}$$h$$f_{1}$$f_{2}$

Figure 4. A homogeneous ambitropical polyhedron $E$ consisting of two
unbounded alcoved polyhedra (left). The tropical polyhedral cones $E^{\max}$
(center) and $E^{\min}$ (right). See Example 9.8.
$(1,1,1)$$(0,1,1)$$(1,0,1)$$(1,1,0)$$(0,0,1)$$(0,1,0)$$(1,0,0)$$\mathbf{(}0,0,0)$
Figure 5. The skeleton (in bold) of the ambitropical polyhedron of Figure 4
(the two nodes $(1,0,0)$ and $(0,1,1)$ in gray do not belong to the skeleton).
The two maximal chains (of length $4$) yield the representation of $C$ as the
union of two Weyl cells (each being of dimension $3$).

$x_{1}$$x_{2}$$x_{3}$$g_{1}$$g_{2\cdot}$$h$$f_{1}$$f_{2}$

$x_{1}$$x_{2}$$x_{3}$$g_{1}$$g_{2\cdot}$$h$$f_{1}$$f_{2}$

$x_{1}$$x_{2}$$x_{3}$$g_{1}$$g_{2\cdot}$$h$$f_{1}$$f_{2}$

Figure 6. An ambitropical polyhedron $E$ consisting of two alcoved polyhedra
(left). The tropical polyhedral cones $E^{\max}$ (center) and $E^{\min}$
(right). The homogeneous ambitropical polyhedron of Figure 4 is precisely the
tangent cone $\mathcal{T}_{(0,0,0)}E$.

$x_{1}$$x_{2}$$x_{3}$$g_{1}$$g_{2\cdot}$$a$$b$$h$$f_{1}$$f_{2}$

$x_{1}$$x_{2}$$x_{3}$$a$$b$

$x_{1}$$x_{2}$$x_{3}$$a$$b$

Figure 7. Illustrating the metric convexity property of ambitropical cones. A
fan which is not ambitropical (left). The geodesic in Hilbert’s projective
metric connecting the two points $a$ and $b$ (dotted segment) is unique, but
it is not included in the fan. An ambitropical fan (middle, and right). An
example of geodesic connecting $a$ and $b$, included in the fan is shown (dark
blue broken line, middle). Another example of such a geodesic is shown at
right.

$x_{1}$$x_{2}$$x_{3}$$a$$b$$E^{\max}$$E^{\min}$

Figure 8. Three examples of ambitropical hulls of a set with two elements
$E=\\{a,b\\}$. The range of $\bar{Q}^{+}_{E}$ is the tropical cone $E^{\max}$
shown in blue, whereas the range of $\bar{Q}^{-}_{E}$ is the dual tropical
cone $E^{\min}$ shown in red. Another ambitropical hull is represented by the
zigzag line (in black). By 4.17, all these ambitropical hulls are isomorphic.

######  Example 9.11.

The construction of the ambitropical hulls by means of 4.17 is illustrated in
Figure 8. Here, $E=\\{a,b\\}$ where $a=(1,0,0)$ and $b=(0,1,0)$. In this
special case, the sets $E^{\max}$ and $E^{\min}$ are the ranges of
$\bar{Q}^{+}_{E}$ and $\bar{Q}^{-}_{E}$, respectively, and so, they provides
ambitropical hulls of $\\{a,b\\}$. There is an infinite family of ambitropical
hulls interpolating between $E^{\max}$ and $E^{\min}$. One element of this
family is shown in black. It constitutes a polyhedral complex whose cells are
cones, and whose rays are generated by the vectors $a,b$ and by the integer
vectors $(i,4-j,0)$ for $i=1,2,3$ and $(i-1,4-j,0)$ for $i=1,\dots,4$.

######  Example 9.12.

We now give an example in dimension $4$. Consider

$E^{I}=\\{x\in\mathbb{R}^{4}\mid x_{1}\leqslant x_{2}\leqslant x_{3}\leqslant
x_{4}\\}$

and for the circular permutation $\gamma$ with cycle $(1,2,3,4)$,

$E^{\gamma}=\\{x\in\mathbb{R}^{4}\mid x_{4}\leqslant x_{1}\leqslant
x_{2}\leqslant x_{3}\\}\enspace.$

The union of the chambers $E^{I}$ and $E^{\gamma}$ is shown in Figure 9, as
well as the range of ${Q}^{-}_{E}$, which is larger than this union, implying
that $E$ is not ambitropical. AN example of non-trivial ambitropical cone in
$\mathbb{R}^{4}$ is shown in Figure 10.

$x_{1}$$x_{2}$$x_{3}$$x_{4}$

$x_{1}$$x_{2}$$x_{3}$$x_{4}$

Figure 9. The union $E$ of the chambers $x_{4}\geqslant x_{3}\geqslant
x_{2}\geqslant x_{1}$ and $x_{3}\geqslant x_{2}\geqslant x_{1}\geqslant x_{4}$
(left) is not ambitropical. Indeed, the fixed point set of ${Q}^{-}_{E}$
(middle) is the union of these chambers with the chambers $x_{3}\geqslant
x_{4}\geqslant x_{2}\geqslant x_{1}$ and $x_{3}\geqslant x_{2}\geqslant
x_{4}\geqslant x_{1}$, which coincides with the alcoved polyhedron defined by
$x_{3}\geqslant x_{2}\geqslant x_{1}$. The latter coincides with the range of
$P^{\max}_{E}$ and with the range of $P^{\min}_{E}$.

$x_{1}$$x_{2}$$x_{3}$$x_{4}$

$x_{1}$$x_{2}$$x_{3}$$x_{4}$

$x_{1}$$x_{2}$$x_{3}$$x_{4}$

Figure 10. The union $E$ of the three chambers $\\{x_{4}\geqslant
x_{3}\geqslant x_{2}\geqslant x_{1}\\}$, $\\{x_{2}\geqslant x_{3}\geqslant
x_{4}\geqslant x_{1}\\}$ and $\\{x_{2}\geqslant x_{3}\geqslant x_{1}\geqslant
x_{4}\\}$ (left) is an ambitropical cone. Tropical cone $E^{\max}$ (middle)
and $E^{\min}$ (right).

## Acknowledgments

The second author thanks Jean-Bernard Baillon, for having pointed out related
works concerning hyperconvexity, and he also thanks Gleb Koshevoy for helpful
comments and references. The first two authors thank Antoine Hochart for
helpful discussions.

## References

  * [AGG12] M. Akian, S. Gaubert, and A. Guterman.  Tropical polyhedra are equivalent to mean payoff games.  International Journal of Algebra and Computation, 22(1):125001 (43 pages), 2012.
  * [AGG13] X. Allamigeon, S. Gaubert, and E. Goubault.  Computing the vertices of tropical polyhedra using directed hypergraphs.  Discrete Comp. Geom., 49(2):247–279, 2013.
  * [AGGCG19] Marianne Akian, Stéphane Gaubert, Julien Grand-Clément, and Jérémie Guillaud.  The operator approach to entropy games.  Theor. Comp. Sys., 63(5):1089–1130, July 2019.
  * [AGH20] M. Akian, S. Gaubert, and A. Hochart.  A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors.  Discrete & Continuous Dynamical Systems - A, 40:207–231, 2020\.
  * [AGL08] M. Akian, S. Gaubert, and A. Lakhoua.  The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis.  SIAM J. Control Optim., 47(2):817–848, 2008.
  * [AGN16] M. Akian, S. Gaubert, and R. Nussbaum.  Uniqueness of the fixed point of nonexpansive semidifferentiable maps.  Trans. of AMS, 368(2):1271–1320, February 2016.
  * [AGNS11] M. Akian, S. Gaubert, V. Nitica, and I. Singer.  Best approximation in max-plus semimodules.  Linear Algebra and its Applications, 435(12):3261–3296, 2011.
  * [AGW05] M. Akian, S. Gaubert, and C. Walsh.  Discrete max-plus spectral theory.  In G. L. Litvinov and V. P. Maslov, editors, Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics, pages 19–51. American Mathematical Society, 2005.  Also ESI Preprint 1485.
  * [AGW09] M. Akian, S. Gaubert, and C. Walsh.  The max-plus Martin boundary.  Doc. Math., 14:195–240, 2009.
  * [AP56] N. Aronszajn and P. Panitchpakdi.  Extension of uniformly continuous transformations and hyperconvex metric spaces.  Pacific J. Math., 6:405–439, 1956.
  * [Bai88] J.B. Baillon.  Nonexpansive mappings and hyperconvex spaces.  In R.F. Brown, editor, Fixed Point Theory and Its Applications, volume 72 of Contemporary Math., pages 11–19. Amer. Math. Soc., 1988.
  * [BCOQ92] F. Baccelli, G. Cohen, G. J. Olsder, and J. P. Quadrat.  Synchronization and linearity.  Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Ltd., Chichester, 1992.  An algebra for discrete event systems.
  * [BF68] D. Blackwell and T. S. Ferguson.  The big match.  The Annals of Mathematical Statistics, 39(1):159–163, 1968.
  * [BGV15] J. Bolte, S. Gaubert, and G. Vigeral.  Definable zero-sum stochastic games.  Math. Oper. Res., 40(1):171–191, 2015.
  * [BK76] T. Bewley and E. Kohlberg.  The asymptotic theory of stochastic games.  Math. Oper. Res., 1(3):197–208, 1976.
  * [Bla62] D. Blackwell.  Discrete dynamic programming.  Ann. Math. Stat., 33:719–726, 1962.
  * [BNS03] A. D. Burbanks, R. D. Nussbaum, and C. T. Sparrow.  Extension of order-preserving maps on a cone.  Proc. Roy. Soc. Edinburgh Sect. A, 133(1):35–59, 2003.
  * [Bru73] R. E. Bruck.  Properties of fixed-point sets of nonexpansive mappings in Banach spaces.  Trans. Amer. Math. Soc., 179:251–262, 1973.
  * [Bru74] R. E. Bruck, Jr.  A characterization of Hilbert space.  Proc. Amer. Math. Soc., 43:173–175, 1974.
  * [BSS07] P. Butkovič, H. Schneider, and S. Sergeev.  Generators, extremals and bases of max cones.  Linear Algebra and its Applications, 421(2):394 – 406, 2007.  Special Issue in honor of Miroslav Fiedler.
  * [But10] P. Butkovič.  Max-linear Systems: Theory and Algorithms.  Springer Monogr. Math. Springer, London, 2010.
  * [CC19] A. Connes and C. Consani.  Homological algebra in characteristic one.  Higher structures, 3(1):155–247, 2019.
  * [CGQ96] G. Cohen, S. Gaubert, and J.P Quadrat.  Kernels, images and projections in dioids.  In Proceedings of third international workshop on Discrete Event Systems (WODES’96), pages 151–158, Edinburgh, August 1996. IEE.
  * [CGQ04] G. Cohen, S. Gaubert, and J.-P. Quadrat.  Duality and separation theorems in idempotent semimodules.  Linear Algebra and Appl., 379:395–422, 2004.
  * [Cra82] H. Crapo.  Ordered sets: retracts and connections.  J. Pure Applied Algebra, 23:13–28, 1982.
  * [CT80] M. G. Crandall and L. Tartar.  Some relations between nonexpansive and order preserving mappings.  Proc. Amer. Math. Soc., 78(3):385–390, 1980.
  * [CTGG99] J. Cochet-Terrasson, S. Gaubert, and J. Gunawardena.  A constructive fixed point theorem for min-max functions.  Dynamics and Stability of Systems, 14(4):407–433, 1999.
  * [Dre84] Andreas W.M Dress.  Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces.  Advances in Mathematics, 53(3):321–402, September 1984.
  * [DS04] M. Develin and B. Sturmfels.  Tropical convexity.  Doc. Math., 9:1–27 (electronic), 2004.
  * [Dyn69] E.B. Dynkin.  Boundary theory of Markov processes (the discrete case).  Russian Math. Surveys, 24(7):1–42, 1969.
  * [EK01] R. Espínola and M. A. Khamsi.  Introduction to hyperconvex spaces.  In Handbook of Metric Fixed Point Theory, pages 391–435. Springer Netherlands, 2001.
  * [EM79] A. Ehrenfeucht and J. Mycielski.  Positional strategies for mean payoff games.  Internat. J. Game Theory, 8(2):109–113, 1979.
  * [Eve57] H. Everett.  Recursive games.  In Contributions to the theory of games, vol. 3, Annals of Mathematics Studies, no. 39, pages 47–78. Princeton University Press, Princeton, N. J., 1957.
  * [Fat08] A. Fathi.  Weak KAM theorem in Lagrangian dynamics.  Tenth preliminary version. Available on line., 2008.
  * [FS04] A. Fathi and A. Siconolfi.  Existence of $c^{1}$ critical subsolutions of the Hamilton-Jacobi equation.  Inventiones Mathematicae, 155(2):363–388, February 2004.
  * [FS05] A. Fathi and A. Siconolfi.  PDE aspects of aubry-mather theory for quasiconvex hamiltonians.  Calculus of Variations, 22(2):185–228, February 2005.
  * [GG04] S. Gaubert and J. Gunawardena.  The Perron-Frobenius theorem for homogeneous, monotone functions.  Trans. of AMS, 356(12):4931–4950, 2004.
  * [GK95] J. Gunawardena and M. Keane.  On the existence of cycle times for some nonexpansive maps.  Technical Report HPL-BRIMS-95-003, Hewlett-Packard Labs, 1995.
  * [GK07] S. Gaubert and R. Katz.  The Minkowski theorem for max-plus convex sets.  Linear Algebra and Appl., 421:356–369, 2007.
  * [GR84] K. Goebel and S. Reich.  Uniform convexity, Hyperbolic geometry, and nonexpansive mappings.  Marcel Dekker, 1984.
  * [GV12] S. Gaubert and G. Vigeral.  A maximin characterization of the escape rate of nonexpansive mappings in metrically convex spaces.  Math. Proc. of Cambridge Phil. Soc., 152:341–363, 2012.
  * [IM07] H. Ishii and H. Mitake.  Representation formulas for solutions of hamilton-jacobi equations with convex hamiltonians.  Indiana University Mathematics Journal, 56(5):2159–2183, 2007.
  * [Isb64] J. R. Isbell.  Six theorems about injective metric spaces.  Commentarii Mathematici Helvetici, 39(1):65–76, December 1964.
  * [Kak39] S. Kakutani.  Some characterizations of Euclidean space.  Jpn. J. Math., 16:93–97, 1939.
  * [KcK16] Mehmet Kılıç and Şahin Koçak.  Tight span of subsets of the plane with the maximum metric.  Advances in Mathematics, 301:693–710, 2016.
  * [Kle60] V. Klee.  Circumspheres and inner products.  Math. Scan., 9:363–370, 1960.
  * [KM97a] V. N. Kolokoltsov and V. P. Maslov.  Idempotent analysis and its applications, volume 401 of Mathematics and its Applications.  Kluwer Academic Publishers Group, Dordrecht, 1997.
  * [KM97b] V. N. Kolokoltsov and V. P. Maslov.  Idempotent analysis and its applications, volume 401 of Mathematics and its Applications.  Kluwer Academic Publishers Group, Dordrecht, 1997.  Translation of Idempotent analysis and its application in optimal control (Russian), “Nauka” Moscow, 1994 [ MR1375021 (97d:49031)], Translated by V. E. Nazaikinskii, With an appendix by Pierre Del Moral.
  * [Koh74] E. Kohlberg.  Repeated games with absorbing states.  Ann. Statist., 2:724–738, 1974.
  * [Koh80] E. Kohlberg.  Invariant half-lines of nonexpansive piecewise-linear transformations.  Math. Oper. Res., 5(3):366–372, 1980.
  * [KR07] E. Kopecká and S. Reich.  Nonexpansive retracts in Banach spaces.  In Fixed point theory and its applications, volume 77 of Banach Center Publ., pages 161–174. Polish Acad. Sci. Inst. Math., Warsaw, 2007\.
  * [KY92] L. A. Kontorer and S. Yu. Yakovenko.  Nonlinear semigroups and infinite horizon optimization.  In Idempotent analysis, volume 13 of Adv. Soviet Math., pages 167–210. Amer. Math. Soc., Providence, RI, 1992.
  * [LL69] T. M. Liggett and S. A. Lippman.  Stochastic games with perfect information and time average payoff.  SIAM Rev., 11:604–607, 1969.
  * [LMS01] G. L. Litvinov, V. P. Maslov, and G. B. Shpiz.  Idempotent functional analysis. An algebraic approach.  Mat. Zametki, 69(5):758–797, 2001.
  * [LN12] B. Lemmens and R. Nussbaum.  Nonlinear Perron-Frobenius Theory.  Cambridge University Press, 2012.
  * [LP07] T. Lam and A. Postnikov.  Alcoved polytopes. I.  Discrete Comput. Geom., 38(3):453–478, 2007.
  * [Mou76] H. Moulin.  Prolongement des jeux à deux joueurs de somme nulle. une théorie abstraite des duels.  Bulletin Soc. Math. Fr, 45, 1976.
  * [MS15] D. Maclagan and B. Sturmfels.  Introduction to Tropical Geometry, volume 161 of Grad. Stud. Math. AMS, Providence, RI, 2015.
  * [MSZ15a] J.-F. Mertens, S. Sorin, and S. Zamir.  Repeated games.  Econometric Society Monographs. Cambridge University Press, New York, 2015\.  With a foreword by Robert J. Aumann.
  * [MSZ15b] Jean-François Mertens, Sylvain Sorin, and Shmuel Zamir.  Repeated games.  Cambridge University Press, 2015.
  * [Ney03] A. Neyman.  Stochastic games and nonexpansive maps.  In Stochastic games and applications (Stony Brook, NY, 1999), volume 570 of NATO Sci. Ser. C Math. Phys. Sci., pages 397–415. Kluwer Acad. Publ., Dordrecht, 2003.
  * [PS79] P. L. Papini and I. Singer.  Best coapproximation in normed linear spaces.  Monatshefte Math., 88:27–44, 1979.
  * [PT09] A. Papadopoulos and M. Troyanov.  Weak Finsler structures and the Funk weak metric.  Math. Proc. Cambridge Philos. Soc., 147(2):419–437, 2009.
  * [Rei77] S. Reich.  Extension problems for accretive sets in Banach spaces.  J. Funct. Anal., 26:378–395, 1977.
  * [Ren11] J. Renault.  Uniform value in dynamic programming.  J. Eur. Math. Soc. (JEMS), 13(2):309–330, 2011.
  * [RS01a] D. Rosenberg and S. Sorin.  An operator approach to zero-sum repeated games.  Israel J. Math., 121:221–246, 2001.
  * [RS01b] A. M. Rubinov and I. Singer.  Topical and sub-topical functions, downward sets and abstract convexity.  Optimization, 50(5-6):307–351, 2001.
  * [RW98] R. T. Rockafellar and R. J-B. Wets.  Variational Analysis.  Springer, 1998.
  * [Sha53] L.S. Shapley.  Stochastic games.  Proc. Natl. Acad. Sci. U. S. A., 39:1095–1100, 1953.
  * [Sta86] R. P. Stanley.  Two poset polytopes.  Discrete Comput. Geom., 1:9–23, 1986.
  * [Wal18] C. Walsh.  Hilbert and Thompson geometries isometric to infinite-dimensional Banach spaces.  Ann. Inst. Fourier (Grenoble), 68(5):1831–1877, 2018.
  * [Wes92] U. Westphal.  Contractive projections on $\ell^{1}$.  Constructive approximation, 8:223–231, 1992.
  * [Zil16] B. Ziliotto.  A tauberian theorem for nonexpansive operators and applications to zero-sum stochastic games.  Mathematics of Operations Research, 41(4):1522–1534, 2016.