#  Optimized measurements of chaotic dynamical systems via the information
bottleneck

Kieran A. Murphy  Dept. of Bioengineering, School of Engineering & Applied
Science,  Dani S. Bassett  Dept. of Bioengineering, School of Engineering &
Applied Science,  Dept. of Electrical & Systems Engineering, School of
Engineering & Applied Science,  Dept. of Neurology, Perelman School of
Medicine,  Dept. of Psychiatry, Perelman School of Medicine,  Dept. of Physics
& Astronomy, College of Arts & Sciences, University of Pennsylvania,
Philadelphia, PA 19104, USA  The Santa Fe Institute, Santa Fe, NM 87501, USA

###### Abstract

Deterministic chaos permits a precise notion of a “perfect measurement” as one
that, when obtained repeatedly, captures all of the information created by the
system’s evolution with minimal redundancy. Finding an optimal measurement is
challenging, and has generally required intimate knowledge of the dynamics in
the few cases where it has been done. We establish an equivalence between a
perfect measurement and a variant of the information bottleneck. As a
consequence, we can employ machine learning to optimize measurement processes
that efficiently extract information from trajectory data. We obtain
approximately optimal measurements for multiple chaotic maps and lay the
necessary groundwork for efficient information extraction from general time
series.

Encapsulated in deterministic chaos is the fundamental obstruction to
predictability that can result from nonlinearity in a system’s evolution, even
in the absence of randomness [shaw1981, eckmannruelle1985]. Signatures of
chaos are found broadly, from weather [tsonis1989chaosweather,
slingo2011weatheruncertainty] to the brain [kargarnovin2023evidence,
skinner1992application], and tools developed in the study of chaos have been
applied more broadly still [nicolis2012foundations, bradley2015nonlinear].
Advancing capabilities to forecast chaotic dynamics thus has marked potential
for impact. The challenge may be glimpsed through the relation between
precision and predictability: for any predictive model utilizing less than
infinite precision, the error of prediction grows exponentially
[wales1991calculating]. Given the inherent difficulties and the potential for
impact, the field has recently turned to machine learning [pathak2018model,
amil2019machine, tang2020introduction, gilpin2023forecast].

Machine learning approaches to forecasting chaotic dynamics generically
utilize full-precision states as input to the predictive model. Yet, the
elusive determinism of chaos gives rise to a curious fact: beyond a certain
precision per state, the ability to forecast given a partial trajectory
saturates [shaw1981]. There is thus a measurement capacity beyond which
resources—whether to acquire the measurement, or to record the trajectory—are
wasted. Instead, it is sufficient to discretize the continuous-valued states,
yielding an analogue system with symbolic dynamics that simplifies the
statistical analysis of the system [williams2004introduction,
nicolis2012foundations, beal2013symbolic, lind2021symbolicdyn] and can be used
for various applications [hirata2023review] including anomaly detection
[ray2004symbolic] and communication [hayes1994experimental]. Here we employ
machine learning to optimize the measurement of, and equivalently the
extraction of information from, a chaotic system.

The requisite precision per measurement, in the form of a number of bits per
state, is a fundamental quantity of the system: the metric entropy
[eckmannruelle1985]. Also known as the Kolmogorov-Sinai (KS) entropy, it
corresponds to the rate of information creation, generated from infinitesimal
scales by the expansion of nearby points under the dynamics and commonly
referred to as sensitive dependence on initial conditions [shaw1981,
james2014chaosforgets]. For many systems of interest, the metric entropy is
equal to the sum of the system’s positive Lyapunov exponents
[eckmannruelle1985].

A finite metric entropy—which can be used to define chaos [gaspard1993noise,
beck1995thermodynamics, schurmanngrassberger1996] and quantify the extent of
chaos in a system [cohenprocaccia1985, wales1991calculating]—implies that a
system’s continuous-valued trajectory through state space has the same
information content, in the asymptotic limit, as a corresponding sequence of
discrete-valued measurements, although only if the measurement process is
optimal. A discretization of the state colors state space according to a
partition, and a measurement that is optimal in the above sense is termed a
generating partition [badii1997complexity]. Generating partitions may be
remarkably coarse [kennelbuhl2003gp]; one approximate generating partition for
the Ikeda map [ikeda1979] divides the attractor into two parts (Fig. 1a).
Despite the intricate structure of the attractor, a measurement of the state
with the capacity of only one bit captures all of the information created
under the dynamics.

Finding a coarse generating partition is challenging, and in the cases where
it has been done, has required intimate knowledge of the dynamics
[grassberger1985henon, davidchack2000UPOs, plumecoq2000templateanalysisII,
mitchell2012partitioning, chai2021symbolic, zhang2022koopman]. Here we
establish an equivalence between the definition of a maximally coarse
generating partition and a particular optimization objective from rate-
distortion theory known as the distributed information bottleneck
[aguerri2018DIB]. As a consequence of the equivalence, we are able to use
machine learning and trajectory data to optimize a partition of the state
space and recover approximate generating partitions.

The distributed information bottleneck extracts information from a composite
random variable by lossily compressing each of its components independently of
one another [aguerri2018DIB, dib_ml], thereby distributing an information
bottleneck (IB) [tishbyIB2000] to each one. The extracted information is
prioritized by relevance to an auxiliary “relevance” random variable. The
identification of meaningful information has been used for the study of
complex systems, serving to decompose the information contained in many
microscale measurements in terms of relevance to some macroscale behavior
[dib2]. Here we use a finite trajectory as both the composite random variable
to be compressed, and as the relevance variable. The product of optimization
is a lossy compression of the attractor in state space that conveys maximal
information about a trajectory and minimal information about a single state.

Figure 1: Measurement schemes for the Ikeda map. (a) The attractor of the
Ikeda map and several points of a trajectory. An optimal measurement of the
continuous-valued state to a discrete-valued outcome is called a generating
partition. States colored blue (orange) are measured as outcome A (B). The
proposed method optimizes the measurement of a chaotic system using artificial
neural networks that apply the information bottleneck and vector quantization
to each continuous-valued state. For an optimal measurement and in the
asymptotic limit, the sequence of discrete measurements contains the same
information as the sequence of continuous-valued states from which it was
derived. (b) The space of possible discrete measurements visualized in terms
of the entropy of a single measurement, $H(U)$, and the rate of entropy
production in the infinite duration limit, $h_{\infty}(U)$. The largest
attainable entropy rate is the metric entropy $h_{\textnormal{KS}}$, and is
achievable with a generating partition. The blue point corresponds to the
approximate generating partition in panel a as identified by the proposed
method. The black points correspond to discrete measurements parameterized by
neural networks with randomly sampled weights (details in Supp.).

Let a state $x\in\mathcal{M}$ exist in $\mathbb{R}^{d}$, where $d$ is the
dimension of the state space. A map
$\mathcal{F}(x):\mathcal{M}\rightarrow\mathcal{M}$ propagates a state forward
in time by one iteration, i.e. $\mathcal{F}(x_{n})=x_{n+1}$. We consider only
discrete time maps in this work, though the analysis is as applicable to
straightforward discretizations of continuous-time flows [eckmannruelle1985].
The dynamics are fully described by the map above, but a probabilistic view is
often more natural [nicolis2012foundations] and allows us to utilize
information theory. For ergodic dynamical systems, which will be our concern
in this work, the natural probability distribution over states $p(x)$ is an
invariant measure, $\mathcal{F}(p(x))=p(x)$, and can be obtained by iterating
forward a long trajectory [eckmannruelle1985].

Given a probability distribution over states, we can define a random variable
$X_{n}$ for the state at timestep $n$, and a random process $\bm{X}_{n:n+L}$
for a sequence of random variables $X_{n},X_{n+1},...,X_{n+L-1}$. For a
stationary process, the start of the trajectory is unimportant, and instead we
consider the statistics of subsequences of length $L$, which we denote
$\bm{X}_{L}$.

A continuous-valued state can be measured, and specifically, discretized,
through the use of a partition that divides the support of $p(x)$ into
disjoint subsets $\mathcal{A}_{i}$ [kennelbuhl2003gp]. The outcome of a
measurement, a random variable $U=f(X)$, converts the continuous-valued state
to the index of the subset to which it belongs. Thus a sequence $\bm{X}_{L}$
is replaced by a sequence of discrete measurements $\bm{U}_{L}$.

How much information does a measurement $U$ convey about the original state
$X$? Intuitively, information gained serves to reduce uncertainty. The amount
of uncertainty about the outcome of a random variable $Z$ may be quantified
via the Shannon entropy, $H(Z)=\mathbb{E}_{z\sim p(z)}[-\textnormal{log}\
p(z)]$ [shannon1948mathematical]. The mutual information contained in two
random variables is given by the reduction of entropy in one variable after
finding the value of the other, $I(Z_{1};Z_{2})=H(Z_{1})-H(Z_{1}|Z_{2})$
[cover1999elements]. The act of measuring a state—by recording in which subset
of a partition it resides—conveys $I(U;X)=H(U)$ bits of information because
the mapping from $X$ to $U$ is deterministic (i.e., $H(U|X)=0$).

As a random process plays out, entropy is generated from the uncertainty about
each subsequent outcome. For a repeated discrete measurement $\bm{U}_{L}$ the
amount of generated entropy is equivalent to the accumulated information about
the underlying trajectory $\bm{X}_{L}$. The entropy rate is defined as the
average entropy generated per step in the limit where the sequence length
becomes infinite, $h_{\infty}(U)=\lim_{L\rightarrow\infty}H(\bm{U}_{L})/L$
[cover1999elements]. The largest achievable entropy rate of any partition $U$
is the KS entropy [eckmannruelle1985],

$h_{\textnormal{KS}}=\sup_{U}h_{\infty}(U).$ (1)

A partition $U$ whose entropy rate is equal to $h_{\textnormal{KS}}$ is a
generating partition [cohenprocaccia1985]. While a “trivial” generating
partition can be approximated by a fine discretization of state space
[cohenprocaccia1985], there often exist generating partitions which are
coarse, with minimal entropy [kennelbuhl2003gp]. Fig. 1a shows an approximate
generating partition of the Ikeda map, whose corresponding infinite sequence
of observations $...BBABA...$ contains the same information as the continuous-
valued trajectory.

A coarse generating partition is desirable because it leverages the specific
transformation $\mathcal{F}$ in order to convey minimal redundant information
that would be gathered at multiple points in the trajectory. A maximally
coarse generating partition $U^{*}$ can be defined as

$U^{*}=\operatorname*{arg\,min}_{U:h_{\infty}(U)=h_{\text{KS}}}H(U).$ (2)

Given the restriction to discrete measurements, we can rewrite Eqn. 2 in terms
of mutual information rather than entropy, to more closely resemble a rate-
distortion function [cover1999elements],

$U^{*}=\operatorname*{arg\,min}_{U:\lim_{L\rightarrow\infty}I(\bm{U}_{L};\bm{X}_{L})/L=h_{\text{KS}}}I(U;X).$
(3)

The constrained optimization of Eqn. 3 is more manageable as a Lagrangian cost
function $\mathcal{L}$, with $U^{*}=\operatorname*{arg\,min}\mathcal{L}$. We
relax the constraint of a specific entropy rate to instead define with the
Lagrangian a Pareto front of optimal partitions, and the optimum with maximal
entropy rate corresponds to the coarsest generating partition $U^{*}$. By
replacing the entropy rate’s infinite limit with a large but finite $L$, we
recover the Lagrangian for the distributed IB [aguerri2018DIB, dib_ml],

$\mathcal{L}=-I(\bm{U}_{L};\bm{X}_{L})+\beta\sum_{i}^{L}I(U_{i};X_{i}),$ (4)

where $\beta$ is the parameter that determines the relative weighting of the
two terms in the Lagrangian. Note: we have made use of the fact that $X$ is
stationary, so that $I(U;X)=\sum^{L}_{i}I(U_{i};X_{i})/L$. In contrast to
previous applications of the distributed IB [aguerriDVIB2021,
steiner2021distributedcompression, dib_ml], the trajectory $\bm{X}_{L}$ serves
both as the input composite random variable and as the relevance random
variable.

Figure 2: Entropy rate of optimized partitions and the trajectory length
during training. The fractional deviation of the entropy rate $h_{\infty}(U)$
from the metric entropy $h_{\text{KS}}$ as a function of the length of the
trajectory $L$ used for training for (a) Ikeda and (b) Hénon maps with
standard parameters, and (c) the logistic map with $r=3.7115$. The
distribution of values of the entropy rate over 20 trials for each $L$ are
shown as violin plots, with the extrema and median indicated by the black
horizontal marks; the dotted line indicates $h_{\infty}(U)=0.99h_{\text{KS}}$.
For each system, the partition found with the largest entropy rate for $L=12$
is shown on the right. For the logistic map, the probability density of states
$p(x)$ is colored according to the optimized partition.

The space of possible partitions is vast, filled with colorings of the
attractor that encapsulate suboptimal information [bollt2001misplaced,
cafaro2015causation]. Fig. 1b displays example partitions of the Ikeda map by
their single measurement entropy $H(U)$ and by their entropy rate,
$h_{\infty}(U)$, which is upper bounded by $H(U)$ and $h_{\textnormal{KS}}$.
By replacing the mutual information terms in Eqn. 4 with appropriate
bounds—namely InfoNCE [oord2018InfoNCE] as a lower bound on
$I(\bm{U}_{L};\bm{X}_{L})$ and the variational upper bound central to
variational autoencoders on $I(U_{i};X_{i})$ [betavae, alemiVIB2016]—we can
optimize the distributed IB with machine learning and leverage the
expressivity of deep learning to search the space of partitions [alemiVIB2016,
aguerriDVIB2021, dib_ml].

To map a state $x$ to a measurement $u$, we followed a two-step process, where
each step was performed by a distinct multilayer perceptron (MLP) (Fig. 1a).
First $x$ was mapped to an intermediate latent space where the information
penalty was enforced by a variational information bottleneck [alemiVIB2016],
and then the outcome was passed through a vector quantization MLP. To perform
the vector quantization, we followed a method that parameterizes the spectrum
of soft-to-hard compression schemes (a soft compression has $H(U|X){>}0$) and
gradually hardens over the course of training [agustsson2017softtohard].
Successful partitions were obtained without requiring a hardening schedule;
instead a simple discontinuous hardening was applied after training. For a
sequence of length $L$, all states $\bm{x}_{:L}$ were ‘measured’ and then the
corresponding measurements $\bm{u}_{:L}$ were processed together by another
MLP to predict a reference state from the original sequence, e.g. $x_{1}$. For
a chaotic system, a single continuous-valued state contains the same
information as the infinite trajectory; we found optimization to fail when
matching with the full sequence of length $L$. The magnitude of the
information penalty $\beta$ was annealed to gradually allow more information
into the measurement process, $I(U;X)$. Training began with no information
passing through to the measurement $U$ and then stopped after $I(U;X)$
exceeded one bit. Architecture and training details are in the Supp.

Figure 3: Influence of training parameters on the optimized partitions. (a)
For the Ikeda map, the reference state (i.e., the one predicted from the
sequence of measurements) strongly influences the optimized partition. Each
partition displayed had the largest entropy rate out of ten trials. Coordinate
axes have been suppressed. (b) The same as in panel a, but with half the rate
of information annealing $\dot{\beta}$ during training. (c) Two of the
partitions of the Hénon map, identified with the same annealing rate as in
panel a. (d) Same as in panel c for the logistic map with $r=3.7115$. (e)
Colorings of the Ikeda attractor where, given a sequence of three measurements
$\bm{u}_{:3}$, the final state $x_{3}$ is predicted, utilizing for the
measurement $U$ the partition labelled (i) (left) and (ii) (right) in a and b,
respectively. Partition (i) yields a simpler coloring than partition (ii) when
predicting the final state of a sequence of three iterations.

To quantify the performance of an optimized measurement, we used the
difference between the known metric entropy and the measurement’s entropy
rate, quantifying the amount of information per step that the measurement
fails to capture. Estimating the entropy rate from finite sequences of symbols
is challenging, with several methods proposed over the years
[schurmanngrassberger1996, kennel2002ctw, nemenman2004entropy,
gao2008estimating, martiniani2019quantifying]. We used a form of data
compression known as context tree weighting [willems1995ctw], and estimated
$h_{\infty}(U)$ from finite size scaling with the ansatz proposed in
schurmanngrassberger1996. While strict equivalence between the distributed IB
and a maximally coarse generating partition is valid only in the limit
$L\rightarrow\infty$, we found that $L\approx 10$ was sufficient to
consistently find partitions with an entropy rate at least $0.99h_{\text{KS}}$
for the chaotic maps studied (Fig. 2). Partitions with the largest
$h_{\infty}(U)$ found for $L=12$ are shown on the right.

The generating partition for a map is not unique [jaegerkantz1997]. Any of the
infinite forward or reverse iterates of a generating partition is also one;
additionally there can be nontrivial variants that cross through certain
homoclinic tangencies, where the stable and unstable manifolds run parallel to
one another [jaegerkantz1997]. We found that certain details of the training
process steered optimization to qualitatively different partitions (Fig. 3).
Varying the reference state drove the optimization to different iterates of
the same base partition (Fig. 3a-d; iterates in Supp.), and a slower annealing
of the information $\dot{\beta}$ reached deeper into the forward and backward
iterates of the base partition (Fig. 3b). The different iterates require
simpler or more complicated functions to integrate multiple measurements (Fig.
3e); the iterates found through optimization allow for simpler integration
functions. The base partition found for the Ikeda map (Fig. 3a, reference
state 5/12) closely resembles what has been found in prior work
[davidchack2000UPOs, kennelbuhl2003gp, hirata2004shadowing,
ghalyan2018locally]. For the Hénon and logistic maps, the base partitions
(shown in Fig 2) are slight variations of reported generating partitions
[grassberger1985henon, schurmanngrassberger1996].

In this Letter, we established an equivalence between the definition of a
maximally coarse generating partition and the distributed information
bottleneck, and leveraged the equivalence to optimize measurement of a chaotic
dynamical system with machine learning. Operating without knowledge of the
dynamics and without reliance on properties specific to generating partitions
such as the unique description of periodic orbits [davidchack2000UPOs],
homoclinic tangencies [grassberger1985henon, jaegerkantz1997,
mitchell2012partitioning, chai2021symbolic], or the Koopman operator
[zhang2022koopman], the method is not restricted to chaotic systems. Instead,
the deterministic chaos serves as a testbed where the notion of an optimal
measurement scheme has been made precise and can be quantitatively evaluated.
Optimizing a measurement to extract maximal information about an underlying
state is a broad goal in understanding inference in biological and
computational systems. Prior research has predominantly focused on a singular
compression that extracts maximal information from the present and/or past
about the future [creutzig2009PIB, still2010optimal, palmer2015predictive,
marzen2016predictive]. A notable exception is the recursive information
bottleneck [still2014information], in which a sequence of measurements are
recursively aggregated and each step’s measurement scheme can vary based on
what has been previously observed. By contrast, the distributed information
bottleneck setup of this work optimizes a fixed measurement scheme that is
repeatedly applied for maximal aggregate information, a plausible scenario for
sensing organisms or sensory devices.

The current study focused exclusively on relatively simple chaotic maps that
have been well characterized so as to establish the capabilities of the
proposed method. The variegated faces of chaos present exciting opportunities
for the optimization of measurement processes and can serve as a rich testbed
for machine learning methods of compression [oord2017vqvae,
jang2016categorical, gilpin2023forecast]. In the large majority of systems
where there is no precedent generating partition with which to compare, the
metric entropy can be readily estimated from data [cohenprocaccia1985,
wales1991calculating] and other means of quantitatively evaluating partitions
can be used [plumecoq2000templateanalysisII] to ground the measurement schemes
learned by the proposed method.

To our knowledge, the connection between metric entropy and a rate-distortion
objective was previously unconsidered in the literature. The transformation
$\mathcal{F}$ whose repeated application creates the strange attractor evolves
points in state space such that a fixed measurement continually acquires new
information about the continuous-valued trajectory ad infinitum. The maximally
coarse generating partition is the minimally redundant lossy compression of
the attractor that can uniquely identify every trajectory given an infinite
time horizon. By optimizing the lossy compression, the spawned information
that serves to limit predictability for any finite model is manifest as a
specific coloring of the attractor that divides state space with a remarkably
crude, yet highly specific cut.

##  I Acknowledgements

We gratefully acknowledge Sam Dillavou, Sarah Marzen and Kevin A. Mitchell for
helpful discussions, and Jason Z. Kim, and Suman Kulkarni for comments on the
manuscript.

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Supplemental Material

##  S1 Code availability

The full code base has been released on Github at the following location:
distributed-information-bottleneck.github.io. Every analysis included in this
work can be repeated from scratch in Google Colab with the iPython notebook in
this directory.

##  S2 Appendix A: The chaotic maps studied

We studied three chaotic maps in this work. The logistic map
[beck1995thermodynamics],

$x_{n+1}=rx_{n}(1-x_{n}),$ (5)

is a one-dimensional map defined over the unit interval. We used the value
$r=3.7115$ for the main text, and additionally used $r=4$ to benchmark the
entropy rate estimation in App. B. The Hénon map [grassberger1985henon],

$\begin{split}x_{n+1}&=1-ax_{n}^{2}+by_{n}\\\ y_{n+1}&=x_{n},\end{split}$ (6)

is a two-dimensional map, and we used the extensively studied parameters
$a=1.4$, $b=0.3$. Finally, the Ikeda map [ikeda1979],

$\begin{split}\phi&=\kappa-\eta/(1+x_{n}^{2}+y_{n}^{2})\\\
x_{n+1}&=a+b(x_{n}\cos(\phi)-y_{n}\sin(\phi))\\\
y_{n+1}&=b(x_{n}\sin(\phi)-y_{n}\cos(\phi)),\end{split}$ (7)

is another two-dimensional map, and we again followed precedent with the
parameters $a=1$, $b=0.9$, $\kappa=0.4$, $\eta=6$ [davidchack2000UPOs].

##  S3 Appendix B: Estimation of entropy rate

Given a sequence of symbols that can take any of a discrete set of values, we
wish to estimate the asymptotic rate of entropy produced per symbol,

$h_{\infty}(U)=\lim_{L\rightarrow\infty}\frac{H(\bm{U}_{L})}{L}=\lim_{L\rightarrow\infty}H(U_{L}|\bm{U}_{L-1}).$
(8)

The second equality holds in the case of stationary processes
[gao2008estimating] and makes clearer the interpretation of the entropy rate
as the rate of information creation.

The entropy rate can be at most $H(U)$, when every symbol is independent from
the rest, as is the case for a sequence of coin flips. Regularity in the
process reduces the entropy rate, and the lowest entropy rate of zero
corresponds to determinism in the symbolic dynamics (e.g., a periodic signal).
Note the distinction between determinism in the symbolic dynamics, where every
symbol is known with certainty given the past, and determinism in the
continuous-valued state space. The latter can yield uncertainty given only the
results of discrete measurements, which have discarded information.

Supplemental Material, Figure S1: Estimating entropy rate. (a) The logistic
map ($r=4$) generating partition with two symbols; $h_{\textnormal{KS}}=1$ bit
per iteration of the map. (b) The logistic map ($r=4$) generating partition
with four symbols; $h_{\textnormal{KS}}=1$ bit per iteration of the map. (c)
The logistic map ($r=3.7115$) generating partition with two symbols;
$h_{\textnormal{KS}}=0.5203$ bit per iteration of the map. (d) The Hénon map
($a=1.4,b=0.3$); $h_{\textnormal{KS}}=0.6048$ bits per iteration of the map.
Error bars indicate the standard error from five repeats.

Methods of file compression leverage regularities in the bit stream of a file,
and commonly form the basis for estimates of entropy rate. Lempel-Ziv (LZ)
compression, of which there are several variants, constructs a codebook of
previously unobserved sequences during a single pass through the data. Context
tree weighting (CTW) is another method of file compression that constructs a
suffix tree storing every subsequence contained in the data. While LZ
compression is fast to compute, the meticulous record keeping of CTW converges
more quickly with file size to the entropy rate [kennel2002ctw]. We
implemented the infinite depth CTW following kennel2005bayesian; the C++ code
may be found in the linked repository.

We compare in Fig. S1 various estimates of the entropy rate for generating
partitions of the logistic and Hénon maps where $h_{\textnormal{KS}}$ and
generating partitions are known. We included a cross parsing variant of LZ
whereby a codebook is made with one sequence of length $L$ and evaluated on
another [ro2022xparse]. Additionally, we compared to a bias-corrected form of
block entropy [schurmann2015blockent], based on the entropy of subsequences of
length $B$. Even with the bias correction, estimating the entropy rate using
block entropy is problematic because there is often no indication about what
length $B$ yields the best estimate [kennel2005bayesian]. Here and for all
other entropy rate estimates in the manuscript, sequences of fifteen lengths
$L$ logarithmically spaced between two thousand and two million were randomly
selected from a dataset of 20 million points. Five repeats were evaluated for
each $L$.

We found that the convergence of CTW, while faster than LZ, was highly
variable for different partitions, and that computing the entropy rate from
CTW for a reasonably large file size would not suffice in general. Instead, we
followed schurmanngrassberger1996, who fit an ansatz of sequence length
dependence of the entropy rate measured via construction of a suffix tree
(though not CTW). We found the same ansatz to work well for the finite size
scaling of the CTW estimates,

$h_{N}=h_{\infty}+c\frac{\log N}{N^{\gamma}},$ (9)

and used nonlinear least squares (scipy.optimize.curve_fit) for the estimate
of $h_{\infty}$ and its standard error.

Supplemental Material, Figure S2: Iterates of optimized partitions and
established generating partitions. The partitions in the gray box, labelled
timestep $n$, were iterated forward and backward three steps while retaining
the coloring from timestep $n$. Coordinate axes have been suppressed. (a) The
Ikeda map, with the partition at timestep $n$ the result of a run with $L=12$
and reference timestep $6/12$. (b) The Hénon map with standard parameters,
optimized with the current method on the left and found by
grassberger1985henon by threading certain homoclinic tangencies. (c) The
logistic map with $r=3.7115$, optimized with the current method on the left
and the known GP, with a boundary at the critical point at $x=0.5$ (true for
any value of $r$). The invariant measure $p(x)$ is displayed vertically, and
the bins are shaded according to the proportion of partition assignments at
timestep $n$.

##  S4 Appendix C: Machine learning implementation details

All experiments were implemented in TensorFlow and run on a single computer
with a 12 GB GeForce RTX 3060 GPU. Each optimization and its entropy rate
estimation took several minutes.

For all maps, subsequences of length $L$ iterations were randomly sampled from
a sequence of length $10^{6}$ for training, and every $x_{i}$ was individually
compressed by the same two-step encoder. The first stage of the encoder was a
multilayer perceptron (MLP) with two layers of 128 units each and LeakyReLU
activations, followed by a final projection to 8-dimensional “information
bottleneck” space over $\tilde{U}$. The second stage was another MLP with two
layers of 128 LeakyReLU units, taking $\tilde{U}$ to the soft discretization
space over $U$: a two dimensional space followed by a softmax activation.

In the information bottleneck space, transmitted information was penalized
through the Kullback-Leibler (KL) divergence between the encoded conditional
distributions $p(\tilde{u}|x)$ and the prior $r(\tilde{u})$, a standard normal
distribution $\mathcal{N}(\bm{0},\bm{1})$. As is typical for variational
autoencoders [vae], the conditional distributions were parameterized as
Gaussians with diagonal covariance matrices. Values were sampled from the
conditional distributions following the reparameterization trick [vae], and
the sampled values $\tilde{u}$ were passed through the second stage. The
softmax activation mapped embeddings to a probability distribution over two
outcomes (representing assignment to a binary partition). The temperature
parameter in the softmax was taken to be one during training and changed to
zero discontinuously after training (i.e., making the partition assignment an
arg max operation on the probability vector).

The $L$ embeddings $u_{1},...,u_{L}$ were then concatenated and passed as
input to the predictive model, $g(u_{1},...,u_{L})$, an MLP consisting of two
layers of 256 units with LeakyReLU activation, followed by a linear projection
to 32 dimensions (i.e., a final dense layer with 32 units and no activation
function). The 32-dimensional space was shared with the output of a reference
state encoder $h(x_{\text{ref}})$, another MLP with two layers of 256 units
with LeakyReLU activation, followed by a linear projection to 32 dimensions.
In this 32-dimensional shared latent space the InfoNCE loss [oord2018InfoNCE]
was evaluated by pairing up each length $L$ trajectory with its reference
state, using the squared Euclidean distance as a measure of similarity between
the embedding vectors. The remainder of the batch size of 2048 provided
negative samples for the InfoNCE loss.

To traverse the low information Pareto front, we used nonlinear IB
[kolchinsky2019nonlinear], squaring the information bottleneck penalty
($I(U_{i};X_{i})$) in the Lagrangian (Eqn. 4 in the main text). The loss was

$\mathcal{L}_{\textnormal{VIB}}=\mathcal{L}_{\text{InfoNCE}}(g(u_{1},...,u_{L}),h(x_{\text{ref}}))+\beta\sum_{i}^{L}\left(D_{\textnormal{KL}}(p(\tilde{u}_{i}|x_{i})||r(\tilde{u}))\right)^{2}.$
(10)

We decreased $\beta$ in equally spaced logarithmic steps from $10$ to
$10^{-4}$ over 20,000 steps, and stopped once the information per measurement
$I(U;X)$ exceeded one bit (with $I(U;X)$ estimated as in Ref. [dib2] using
lower and upper bounds from poole2019variational). The Adam optimizer was used
with a learning rate of $3\times 10^{-4}$.

The low-dimensional continuous-valued states underwent positional encoding, a
term from natural language processing that has been used in certain computer
vision contexts to facilitate learning on low-dimensional features. Before
being passed to the MLP, inputs were mapped to
$x\leftarrow(x,\sin\omega_{1}x,\sin\omega_{2}x,...)$) with frequencies
$\omega_{k}=2^{k}$ where $k\in\\{1,2,...10\\}$ [dib_ml].

After training concluded, there remained a difficulty around how to use the
conditional distributions in the bottleneck space. The embeddings are the
distributions; we sampled from them during training thanks to the
reparameterization trick, but sampling during the conversion to a partition
would leave stochasticity in the assignment, i.e. $H(U|X){>}0$. We can instead
view the distribution in the information bottleneck space as an ensemble of
hard partitions, and the transmitted information is the degree to which the
downstream layers can “pinpoint” the member of the ensemble. From this
perspective it becomes sensible to sample a fixed number of noise vectors,
representing as many members of the ensemble, and use those noise vectors to
assign each $x$ to a $u$. The majority assignment of each point is then used
for the partition, sans stochasticity.

###  S4.1 Randomly sampled discrete measurements (Fig. 1 of main text)

To generate a random sample of discrete measurements, we initialized MLPs with
1, 2, and 3 layers of 64 units each, with ReLU or tanh activation. The weights
and biases were sampled from a normal distribution with a mean of 0.05 and a
standard deviation of 0.5. The dimension of the input was 2, corresponding to
the coordinates of the Ikeda map, and the output dimension was 2 or 4,
corresponding to the alphabet size for the measurement (i.e., the number of
colors). The continuous-valued outputs, one for each input point, were
discretized by selecting the dimension in the output with the largest absolute
value. 20 networks were sampled for each configuration (number of layers,
activation, alphabet size), for 240 in total.

##  S5 Appendix D: Iterates of partitions found in the paper

We display in Fig. S2 three forward and three reverse iterations of partitions
for the Ikeda, Hénon , and logistic maps studied in this work. The optimized
partitions for the Hénon and logistic maps differ from previously reported
generating partitions, with modifications that increase the entropy: 0.94 bits
up from 0.91 bits for Hénon, and 0.99 bits up from 0.82 bits for the logistic
map.

##  S6 Appendix E: Citation Diversity Statement

Science is a human endeavour and consequently vulnerable to many forms of
bias; the responsible scientist identifies and mitigates such bias wherever
possible. Meta-analyses of research in multiple fields have measured
significant bias in how research works are cited, to the detriment of scholars
in minority groups [maliniak2013gender, caplar2017quantitative,
chakravartty2018communicationsowhite, dion2018gendered, dworkin2020extent,
teich2022citation]. We use this space to amplify studies, perspectives, and
tools that we found influential during the execution of this research
[zurn2020citation, dworkin2020citing, zhou2020gender, budrikis2020growing]. We
sought to proactively consider choosing references that reflect the diversity
of the field in thought, form of contribution, gender, race, ethnicity, and
other factors. The gender balance of papers cited within this work was
quantified using a combination of automated gender-api.com estimation and
manual gender determination from authors’ publicly available pronouns. By this
measure (and excluding self-citations to the first and last authors of our
current paper), the references of the main text contain 3%
woman(first)/woman(last), 7% man/woman, 16% woman/man, and 74% man/man. This
method is limited in that a) names, pronouns, and social media profiles used
to construct the databases may not, in every case, be indicative of gender
identity and b) it cannot account for intersex, non-binary, or transgender
people. We look forward to future work that could help us to better understand
how to support equitable practices in science.

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#  Homogenization of the G–Equation: a metric approach

Antonio Siconolfi  Department of Mathematics
Università degli Studi di Roma “La Sapienza”
Piazzale Aldo Moro 5
00185 Roma
Italy<EMAIL_ADDRESS>
###### Abstract.

The aim of the paper is to recover some results of
Cardaliaguet–Nolen–Souganidis in [3] and Xin– Yu in [9] about the
homogenization of the G–equation, using different and simpler techniques. The
main mathematical issue is the lack of coercivity of the Hamiltonians. In our
approach we consider a multivalued dynamics without periodic invariants sets,
a family of intrinsic distances and perform an approximation by a sequence of
coercive Hamiltonians.

Key words. homogenization, Hamilton–Jacobi equations, effective Hamiltonian.
AMS subject classifications. 35B40, 35F21, 37J99, 49L25

##  1\. Introduction

True to the title, the aim of these notes is to study the homogenization of
time dependent Hamilton–Jacobi equations with Hamiltonian

$H(x,p)=|p|+p\cdot V(x)\qquad(x,p)\in\mathbb{R}^{N}\times\mathbb{R}^{N},$

where the advection field $V$ is $\mathbb{Z}^{N}$–periodic, Lipschitz
continuous and with suitably small divergence. This Hamiltonian appears in
combustion theory, where the homogenization result can be interestingly
interpreted, see [3], [9]. However, we do not treat this side of the issue,
and stay solely concerned with the theoretical aspects of the matter.

As it is well known, the relevant mathematical point is that $H$ looses
coercivity whenever $|V|\geq 1$ and consequently, as pointed out in [3], [9],
there could be no homogenization if $V$ is just assumed to be Lipschitz
continuous or even more regular, some additional requirements are needed,
specifically the condition on the divergence. This is in contrast to the
homogenization theory for coercive Hamiltonians, see [7].

It is opportune to make clear that our outputs are not new, in [3] it has been
already established, among other things, that homogenization takes place under
assumptions just slightly different from ours, and for vector fields $V$ also
depending on time, which adds further complications.

The interest of our contribution, if any, is that the proofs are different
and, we believe, more simple and intuitive. They rely on a metric approach to
the matter. Note that metric techniques have been already used in
homogenization of Hamilton–Jacobi equations, in particular in the stationary
ergodic case, see for instance [2], [5].

To begin with, we define the effective Hamiltonian $\overline{H}$, appearing
in the limit equation, as

$\overline{H}(P)=\inf\\{a\mid H(x,Du+P)=a\;\hbox{ admits an usc bdd periodic
subsolution}\\},$

without passing through ergodic approximation. From this formulation, we see
that the subsolutions of $H(x,Du+P)=\overline{H}(P)+\delta$, for $\delta>0$
small, can be used as approximate correctors to prove, via the perturbed test
function method, see [4], that the weak lower semilimit of the solutions to
the $\varepsilon$–oscillatory equations is supersolution of the limit equation
with the effective Hamiltonian. This depends on the very definition of
$\overline{H}$, and can be done without using the assumption on the divergence
of $V$.

The solution of the problem therefore boils down to show the existence of lsc
bounded periodic supersolutions to the critical equation
$H(x,Du+P)=\overline{H}(P)$. This should actually allow performing the other
half of the homogenization procedure.

Backtracking the issue, we see that such supersolutions do exist provided that
some periodic distances related to $H$ are bounded. In other terms we
discover, not surprisingly, that some controllability condition is needed in
order for homogenization to take place, compare for instance with [1], where
other noncoercive models are considered.

The boundedness of the distances is in turn equivalent, see Sections 5–6, to
the nonexistence of periodic invariant sets for a multivalued dynamics related
to $H$. This is the key point where the condition on the divergence of $V$
enters into play.

The key point in this respect is Theorem 4.2, where it is proved that if such
a set, say $\Omega$, does exist, then it possesses Lipschitz boundary, but
then the compression of $V$ toward the interior of $\Omega$ is not compatible
with its small divergence. This result is deduced through divergence theorem
and isoperimetric inequality, see Lemma 4.12, Theorem 4.9. More precisely, we
apply the isoperimetric inequality on the flat torus
$\mathbb{T}^{N}=\mathbb{R}^{N}/\mathbb{Z}^{N}$, so that our condition on $V$
reads

$\|\operatorname{div}V\|_{L^{N}(\mathbb{T}^{N})}\leq\frac{1}{\chi},$

where $\chi$ is the isoperimetric constant on $\mathbb{T}^{N}$, which
coincides with that of $Q_{\frac{1}{2}}=(0,1/2)^{N}$, see Section 2. This is
actually the same assumption of [3] except that they use the isoperimetric
constant of $Q_{1}=(0,1)^{N}$. We are not aware of comparison results between
these two constants, and we do not know if this difference is just a technical
detail, or some more deep fact is involved.

The paper is organized as follows: in Section 2 we give notations and
preliminary results. In Section 3 we write down the problem with the standing
assumptions. Section 4 is devoted to the definition of a relevant multivalued
dynamics and the proof of the nonexistence of periodic invariant sets. In
section 5 we define some distances intrinsically related to $H$ and study
their properties. In Section 5 we approximate $H$ by a sequence $H_{k}$ of
coercive Hamiltonians and prove that the effective Hamiltonians
$\overline{H}_{k}$ converge to $\overline{H}$. From this result we deduce the
existence of lsc periodic bounded supersolutions to the critical equation
related to $H$, and complete the proof of the homogenization result. Finally,
in the Appendix we present some proofs.

##  2\. Preliminaries

Given a subset $E$ of $\mathbb{R}^{N}$, we write $\overline{E}$,
$\operatorname{int}E$, $\partial E$, $E^{c}$ to indicate its closure,
interior, boundary, complement, respectively. For $x\in\mathbb{R}^{N}$, $r>0$,
we denote by $\mathcal{B}(x,r)$ the open ball centered at $x$ with radius $r$.
Given $a>0$, we denote by $Q_{a}$ the open hypercube $(0,a)^{N}$. We write
$\mathbb{T}^{N}=\mathbb{R}^{N}/\mathbb{Z}^{N}$ to indicate the flat torus.
Namely the torus with the metric induced by the Euclidean one on
$\mathbb{R}^{N}$. With this choice $\mathbb{R}^{N}$ and $\mathbb{T}^{N}$ are
locally isometric.

We identify $\mathbb{R}^{N}$ and its dual, and use the symbol $\cdot$ to
denote both the duality pairing and the scalar product.

Given a convex subset $C$ of $\mathbb{R}^{N}$, the negative polar cone $C^{-}$
is defined through

$C^{-}=\\{p\in\mathbb{R}^{N}\mid p\cdot q\leq 0\;\hbox{for any $q\in C$}\\}.$

All the curves we will consider are assumed to be Lipschitz continuous.

The acronyms usc /lsc stand for upper/lower semicontinuous. For any bounded
function $u:\mathbb{R}^{N}\to\mathbb{R}$, and $\delta>0$ we define
$\delta$–inf and sup convolutions, respectively, as follows:

$\displaystyle u_{\delta}(x)$ $\displaystyle=$
$\displaystyle\inf\left\\{u(y)+\frac{1}{2\delta}\,|x-y|^{2}\mid
y\in\mathbb{R}^{N}\right\\}$ $\displaystyle u^{\delta}(x)$ $\displaystyle=$
$\displaystyle\sup\left\\{u(y)-\frac{1}{2\delta}\,|x-y|^{2}\mid
y\in\mathbb{R}^{N}\right\\}$

Given $x_{0}\in\mathbb{R}^{N}$, we say that $y_{0}$ is $u_{\delta}$–optimal
with respect to $x_{0}$ if $y_{0}$ realizes the infimum in the formula of inf
convolution. The notion of $u^{\delta}$– optimal point is given similarly.

Given a sequence of locally bounded functions $u_{n}$, we define the
lower/upper weak semilimit as follows

$\displaystyle\operatorname{lim\,inf_{\\#}}u_{n}(x)$ $\displaystyle=$
$\displaystyle\inf\\{\liminf\\{u_{n}(x_{n})\mid x_{n}\to x\\}$
$\displaystyle\operatorname{lim\,sup^{\\#}}u_{n}(x)$ $\displaystyle=$
$\displaystyle\sup\\{\limsup\\{u_{n}(x_{n})\mid x_{n}\to x\\}$

The term super/sub solution for a given PDE must be understood in the
viscosity sense. We denote by $D^{+}u(x)$ (resp. $D^{-}u(x)$) the viscosity
superdifferential (resp. subdifferential).

######  Definition 2.1.

We say that a subset $\Omega\subset\mathbb{R}^{N}$ has Lipschitz boundary if
for any $x\in\partial\Omega$ there is a neighborhood $U$, an hyperplane of the
form $x+p^{\perp}$, for some nonvanishing vector $p$, and a Lipschitz function
$\psi$ defined in $(x+p^{\perp})\cap U$ with

  * (i)

$\partial\Omega\cap U=\\{y+\psi(y)\,p\mid y\in(x+p^{\perp})\cap U\\}$

  * (ii)

$\operatorname{int}\Omega\cap U=\\{y+\lambda\,p\mid y\in(x+p^{\perp})\cap
U,\,\lambda>\psi(y)\\}$

The isoperimetric constant $\chi$ on the torus $\mathbb{T}^{N}$ is the
smallest constant such that

$|\Theta|^{1-1/N}\leq\chi\,\mathcal{H}^{N-1}(\partial\Theta)$

for any subset $\Theta$ of $\mathbb{T}^{N}$ with Lipschitz boundary. Here
$\mathcal{H}^{N-1}$ stands for the $(N-1)$–dimensional Hausdorff measure and
$|\cdot|$ for the $N$–dimensional Lebesgue measure.

The (relative) isoperimetric constant $\chi_{a}$ on $Q_{a}$ is the smallest
constant such that

$|\Theta|^{1-1/N}\leq\chi_{a}\,\mathcal{H}^{N-1}(\partial\Theta\cap Q_{a})$

for all subset $\Theta\subset Q_{a}$ with Lipschitz boundary.

According to [8], the isoperimetric problems in $\mathbb{T}^{N}$ in
$Q_{\frac{1}{2}}$ are equivalent, and consequently the constants $\chi$ and
$\chi_{\frac{1}{2}}$ coincide.

A formulation of the divergence Theorem on the torus is:

Let $\Theta\subset\mathbb{T}^{N}$ be a set with Lipschitz boundary and
$V:\mathbb{T}^{N}\to\mathbb{R}^{N}$ a Lipschitz continuous vector field, then

$\int_{\Theta}\operatorname{div}V\,dx=\int_{\partial\Theta}V\cdot
n_{\Theta}\,d\,\mathcal{H}^{N-1},$

where $n_{\Theta}$ is the outward unit normal on $\partial\Theta$.

######  Remark 2.2.

We recall that if $\Theta$ has Lipschitz boundary then the outer normal
$n_{\Theta}(x)$ exists for $\mathcal{H}^{N-1}$ a.e. $x\in\partial\Theta$ and
satisfies the following property:

$\operatorname{proj}_{\Theta}(x+t\,n_{\Theta}(x))=x\qquad\hbox{for $t>0$
suitably small,}$

where $\operatorname{proj}_{\Theta}$ indicates the projection on
$\overline{\Theta}$.

##  3\. The problem

We consider the Hamiltonian

$H(x,p)=|p|+p\cdot V(x)\qquad\hbox{in $\mathbb{R}^{N}\times\mathbb{R}^{N}$,}$

where $V:\mathbb{R}^{N}\to\mathbb{R}^{N}$, is a vector field that we assume

  * (A1)

$\mathbb{Z}^{N}$–periodic and Lipschitz continuous;

  * (A2)

satisfying
$\|\operatorname{div}V\|_{L^{N}(\mathbb{T}^{N})}\leq\frac{1}{\chi}$, where
$\chi$ is the isoperimetric constant in $Q_{\frac{1}{2}}$ and in
$\mathbb{T}^{N}$ as well, see Section 2.

Throughout the paper we will denote by $L_{V}$, $M_{V}$, the Lipschitz
constant of $V$ and the maximum of $|V(x)|$, respectively. We consider, for
$\varepsilon>0$, the family of time–dependent Hamilton–Jacobi equations

(HJε)
$\left\\{\begin{array}[]{cc}(u_{\varepsilon})_{t}(x/\varepsilon,t)+H(x/\varepsilon,Du_{\varepsilon})&=0\qquad\hbox{in
$\mathbb{R}^{N}\times(0,+\infty)$}\\\
u_{\varepsilon}(\cdot,0)&=u_{0}\qquad\hbox{in $\mathbb{R}^{N}$}\\\
\end{array}\right.$

where

  * (A3)

$u_{0}$ is bounded uniformly continuous.

Our goal is to study the asymptotic behavior of the (unique) solutions
$u^{\varepsilon}$ of (HJε), as $\varepsilon$ goes to $0$, and to prove that
they locally uniformly converges to a function $u^{0}$ which is (unique)
solution of a problem of the form

($\overline{\mathrm{HJ}}$)
$\left\\{\begin{array}[]{cc}u_{t}(x,t)+\overline{H}(Du)&=0\qquad\hbox{in
$\mathbb{R}^{N}\times(0,+\infty)$}\\\ u(\cdot,0)&=u_{0}\qquad\hbox{in
$\mathbb{R}^{N}$}\\\ \end{array}\right.$

where $\overline{H}$ is a suitable limit Hamiltonian convex and positively
homogeneous.

Conditions ${\bf(A_{1})}$, ${\bf(A_{2})}$, ${\bf(A_{3})}$ will be assumed
throughout the paper, without any further mentioning.

##  4\. Invariant sets

We introduce a set–valued vector field related to $V$. We set for any
$x\in\mathbb{R}^{N}$,

$F(x)=F_{1}(x)=\overline{\hbox{\rm
co\,}}\\{\mathcal{B}(V(x),1)\cup\\{0\\}\\},$

where $\overline{\hbox{\rm co\,}}$ stands for closed convex hull. We also
define for $\delta\in(0,1)$

$F_{\delta}(x)=\overline{\hbox{\rm
co\,}}\\{\mathcal{B}(V(x),\delta)\cup\\{0\\}\\}.$

The set–valued vector fields $F_{\delta}(x)$ are Hausdorff–continuous with
compact convex values for any $\delta\in(0,1]$. We will say that a curve
$\xi:[0,T]\to\mathbb{R}^{N}$ is an integral trajectory of $F$ if

$\dot{\xi}(t)\in F(\xi(t))\qquad\hbox{for a.e. $t\in[0,T]$.}$

######  Definition 4.1.

We say that a set $\Omega\subset\mathbb{R}^{N}$ is invariant for $F$ if
$\Omega$ is a proper subset of $\mathbb{R}^{N}$ and for any integral curve
$\xi$ of $F$ defined in some interval $[0,t]$ with $\xi(0)\in\Omega$ (resp.
$\xi(t)\in\Omega$), we have

$\xi(s)\in\Omega\qquad\hbox{for any $s\in[0,t]$.}$

Our first result is:

######  Theorem 4.2.

Any invariant set for the set–valued dynamic $\dot{\xi}\in F(\xi)$ in
$\mathbb{R}^{N}$ has Lipschitz boundary.

######  Remark 4.3.

The same statement holds clearly true for the dynamics given by
$\dot{\xi}\in\mathcal{B}(V(\xi),1)$, since the integral curves of the two
problems are the same, up to change of parameter. The choice of $F$ fits our
analysis better.

######  Remark 4.4.

The above statement is nontrivial only if $H$ is noncoercive, or in other
terms if $|V(x)|\geq 1$ for some $x$. If on the contrary $|V(x)|<1$ for any
$x$ then $0\in\operatorname{int}F(x)$ and any curve of $\mathbb{R}^{N}$ is an
integral trajectory of $F$, up to change of parameter, so that invariant
subsets cannot exist. Actually, we reach in our setting the conclusion that no
periodic invariant set exist, but some mathematical effort is needed.

We need some preliminary material. The next assertion is easy to check.

######  Lemma 4.5.

Given $\delta_{2}>\delta_{1}$ in $(0,1]$

$F_{\delta_{1}}(x)\subset F_{\delta_{2}}(y)\qquad\hbox{for any
$x\in\mathbb{R}^{N}$, $y\in\mathcal{B}(x,r_{V}(\delta_{1},\delta_{2}))$,}$

with $r_{V}(\delta_{1},\delta_{2})=\frac{\delta_{2}-\delta_{1}}{L_{V}}$.
CORRECTION

######  Lemma 4.6.

Let $\Omega$ be an invariant set for $F$, then

$|V(x)|\geq 1\qquad\hbox{for any $x_{0}\in\partial\Omega$.}$

###### Proof.

Assume for purposes of contradiction that there is $x_{0}\in\partial\Omega$
with $|V(x_{0})|<1$, then we find $r>0$ such that the same strict inequality
holds true in $\mathcal{B}(x_{0},r)$. This implies that

$0\in\operatorname{int}F(x)\qquad\hbox{for any $x\in\mathcal{B}(x_{0},r)$.}$

Any curve lying in $\mathcal{B}(x_{0},r)$ is therefore an integral curve of
$F$, up to change of parameter. This is true in particular for a segment
connecting a point in $\mathcal{B}(x_{0},r)\cap\Omega$ to another point in
$\mathcal{B}(x_{0},r)\setminus\Omega$, which is in contrast with the
invariance of $\Omega$. ∎

######  Lemma 4.7.

Let $\Omega$, $x$, $\delta$ be an invariant set for $F$, a point in
$\mathbb{R}^{N}$ and a constant in $(0,1)$, then

$\displaystyle(y+\operatorname{int}F_{\delta}(x))\cap\mathcal{B}(x,r_{V}(\delta,1))$
$\displaystyle\subset$ $\displaystyle\operatorname{int}\Omega\;\qquad\hbox{
for any $y\in\overline{\Omega}\cap U$}$
$\displaystyle(y-\operatorname{int}F_{\delta}(x))\cap\mathcal{B}(x,r_{V}(\delta,1))$
$\displaystyle\subset$ $\displaystyle\operatorname{int}\Omega^{c}\qquad\hbox{
for any $y\in\overline{\Omega^{c}}\cap U$.}$

###### Proof.

To ease notations, we set $U=\mathcal{B}(x,r_{V}(\delta,1))$. First take
$y\in\Omega\cap U$ and $q\in F_{\delta}(x)$ with $y+q\in U$, then the curve
$y+t\,q$, $t\in[0,1]$, is an integral curve of $F$ in force of Lemma 4.5. This
shows

$(y+F_{\delta}(x))\cap U\subset\Omega$

and consequently

(1) $\operatorname{int}\big{(}(y+F_{\delta}(x))\cap
U)=(y+\operatorname{int}F_{\delta}(x))\cap
U\subset\operatorname{int}\Omega\qquad\hbox{for any $y\in\Omega\cap U$.}$

Now consider $y\in\overline{\Omega}\cap U$ and denote by $y_{n}$ a sequence in
$\Omega\cap U$ converging to $y$. If $q\in\operatorname{int}F_{\delta}(x)$
with $y+q\in U$, then

$q+y-y_{n}\in\operatorname{int}F_{\delta}(x)\qquad\hbox{for $n$ sufficiently
large}$

and consequently

$y+q=y_{n}+(q+y-y_{n})\in(y_{n}+\operatorname{int}F_{\delta}(x))\cap U.$

This proves

$(y+\operatorname{int}F_{\delta}(x))\cap
U\subset\bigcup_{n}\,(y_{n}+\operatorname{int}F_{\delta}(x))\cap U,$

since $y_{n}\in\Omega\cap U$ for any $n$, all the sets in the right hand–side
of the previous formula are contained in $\operatorname{int}\Omega$ by (1).
The first inclusion of the claim is then proved. The latter can be deduced by
slightly adapting the same argument. ∎

######  Proof of Theorem 4.2.

Let $\Omega$ be an invariant set. Given $x\in\partial\Omega$, we select
$\delta\in(0,1)$ and set $U=\mathcal{B}(x,r_{V}(\delta,1))$. We denote by
$\operatorname{proj}$ the orthogonal projection on the space $x+V(x)^{\perp}$
and consider the set

$I:=\operatorname{proj}((x+\operatorname{int}F_{\delta}(x))\cap
U)\cap\operatorname{proj}((x-\operatorname{int}F_{\delta}(x))\cap U),$

which is an open convex neighborhood of $x$.

Claim 1. For any $y\in I$ there is one and only one
$\lambda=\lambda(y)\in\mathbb{R}$ such that
$y+\lambda\,V(x)\in\partial\Omega\cap U$.

Let $y$ be in $I$, by the very definition of $I$ there are $\lambda_{1}$,
$\lambda_{2}$ with

$\displaystyle y+\lambda_{2}\,V(x)$ $\displaystyle\in$
$\displaystyle(x+\operatorname{int}F_{\delta}(x))\cap U$ $\displaystyle
y+\lambda_{1}\,V(x)$ $\displaystyle\in$
$\displaystyle(x-\operatorname{int}F_{\delta}(x))\cap U$

note that by Lemma 4.6 $|V(x)|\geq 1$, then $V(x)^{\perp}$ is a supporting
hyperplane for both $x+F_{\delta}(x)$ and $x-F_{\delta}(x)$, and

$\displaystyle x+\operatorname{int}F_{\delta}(x)$ $\displaystyle\subset$
$\displaystyle\\{z\mid(z-x)\cdot V(x)>0\\}$ $\displaystyle
x-\operatorname{int}F_{\delta}(x)$ $\displaystyle\subset$
$\displaystyle\\{z\mid(z-x)\cdot V(x)<0\\}.$

This implies that $\lambda_{2}>0$ and $\lambda_{1}<0$. In addition, owing to
the convexity of $U$ we have

(2) $y+\lambda\,V(x)\in U\qquad\hbox{for any
$\lambda\in[\lambda_{1},\lambda_{2}]$.}$

We deduce from Lemma 4.7

$\displaystyle y+\lambda_{2}\,V(x)$ $\displaystyle\in$
$\displaystyle\operatorname{int}\Omega$ $\displaystyle y+\lambda_{1}\,V(x)$
$\displaystyle\in$ $\displaystyle\operatorname{int}\Omega^{c}.$

We set

$\displaystyle\lambda^{\prime}$ $\displaystyle=$
$\displaystyle\inf\\{\lambda\in(\lambda_{1},\lambda_{2})\mid
y+\lambda\,V(x)\in\operatorname{int}\Omega\\}$
$\displaystyle\lambda^{\prime\prime}$ $\displaystyle=$
$\displaystyle\sup\\{\lambda\in(\lambda_{1},\lambda_{2})\mid
y+\lambda\,V(x)\in\operatorname{int}\Omega^{c}\\}.$

We proceed proving by contradiction that
$\lambda^{\prime}=\lambda^{\prime\prime}$. In fact, if
$\lambda^{\prime}>\lambda^{\prime\prime}$ then

$y+\lambda\,V(x)\in\partial\Omega\cap U\qquad\hbox{for any
$\lambda\in[\lambda^{\prime\prime},\lambda^{\prime}]$},$

on the other side, we choose $\lambda$ in
$[\lambda^{\prime\prime},\lambda^{\prime}]$ so close to
$\lambda^{\prime\prime}$ that

$(\lambda-\lambda^{\prime\prime})\,V(x)\in\operatorname{int}F_{\delta}(x),$

and we deduce from (2) and Lemma 4.7

$\displaystyle y+\lambda\,V(x)$ $\displaystyle=$ $\displaystyle
y+\lambda^{\prime\prime}\,V(x)+(\lambda-\lambda^{\prime\prime})\,V(x)$
$\displaystyle\in$
$\displaystyle((y+\lambda^{\prime\prime}\,V(x))+\operatorname{int}F_{\delta}(x))\cap
U\subset\operatorname{int}\Omega,$

which is impossible. If instead $\lambda^{\prime}<\lambda^{\prime\prime}$,
then we set

$\lambda_{0}=\sup\\{\lambda\in[\lambda^{\prime},\lambda^{\prime\prime}]\mid
y+\lambda\,V(x)\in\operatorname{int}\Omega\\}.$

We find $\lambda<\lambda_{0}$ close to $\lambda_{0}$ with

$\displaystyle y+\lambda\,V(x)$ $\displaystyle\in$
$\displaystyle\operatorname{int}\Omega$
$\displaystyle(\lambda_{0}-\lambda)\,V(x)$ $\displaystyle\in$
$\displaystyle\operatorname{int}F_{\delta}(x)$

and

$\displaystyle y+\lambda_{0}\,V(x)$ $\displaystyle=$ $\displaystyle
y+\lambda\,V(x)+(\lambda_{0}-\lambda)\,V(x)$ $\displaystyle\in$
$\displaystyle((y+\lambda\,V(x))+\operatorname{int}F_{\delta}(x))\cap
U\subset\operatorname{int}\Omega,$

contrast with the definition of $\lambda_{0}$. We have in conclusion
established by contradiction that $\lambda^{\prime}=\lambda^{\prime\prime}$.

Claim 2. The function

$y\mapsto\\{\lambda\mid y+\lambda\,V(x)\in\partial\Omega\\}$

from $I$ to $\mathbb{R}$ is Lipschitz–continuous.

The function, that we denote by $\psi$, is univocally defined by the previous
claim, and is continuous by its very definition and the fact that
$\partial\Omega$ is closed. By squeezing a bit $I$, we can in addition suppose
without loosing generality that $\psi$ is uniformly continuous so that we can
determine $\varepsilon$ with

$|y_{1}-y_{2}|<\varepsilon\,\Rightarrow\,|\psi(y_{1})-\psi(y_{2})|<1.$

We pick such a pair $y_{1}$, $y_{2}$, and assume that
$\psi(y_{1})>\psi(y_{2})$, therefore

$\mathcal{B}((\psi(y_{1})-\psi(y_{2}))\,V(x),(\psi(y_{1})-\psi(y_{2}))\,\delta)\
\subset\operatorname{int}F_{\delta}(x)$

and consequently by Lemma 4.7

$((y_{2}+\psi(y_{2})\,V(x))+\mathcal{B}((\psi(y_{1})-\psi(y_{2}))\,V(x),(\psi(y_{1})-\psi(y_{2}))\,\delta))\cap
U\subset\operatorname{int}\Omega.$

Since $y_{1}+\psi(y_{1})\,V(x)\in\partial\Omega\cap U$, we must therefore have

$\displaystyle|y_{2}-y_{1}|$ $\displaystyle=$
$\displaystyle|y_{1}+\psi(y_{1})\,V(x)-y_{2}-\psi(y_{2})\,V(x)-(\psi(y_{1})-\psi(y_{2}))\,V(x)|$
$\displaystyle\geq$ $\displaystyle(\psi(y_{1})-\psi(y_{2}))\delta$

This shows the claim. To conclude, it is enough to notice, see Definition 2.1,
that by Lemma 4.7 the set

$\\{y+\lambda\,\psi(y)\mid y\in I,\,\lambda>\psi(y)\\}$

is contained in $\operatorname{int}\Omega$. ∎

######  Definition 4.8.

We say that a subset $A\subset\mathbb{R}^{N}$ is periodic if

$x\in A\Rightarrow x+z\in A\qquad\hbox{for any $z\in\mathbb{Z}^{N}$.}$

The main consequence of the Theorem 4.2 is:

######  Theorem 4.9.

The dynamics given by $F$ does not have any invariant periodic subset.

######  Remark 4.10.

Note that invariant non periodic sets for $F$ could exist in our setting. It
is enough to consider for instance $V(x)$ constant.

We preliminarily give two lemmata. We define for $x\in\mathbb{R}^{N}$

$Z^{\infty}(x)=\\{p\mid H(x,p)\leq 0\\},$

It is apparent that $Z^{\infty}(x)$ is the recession cone of the sublevels
$\\{p\mid H(x,p)\leq a\\}$ for $a>0$.

######  Lemma 4.11.

$F(x)^{-}=Z^{\infty}(x)\qquad\hbox{for any $x\in\mathbb{R}^{N}$.}$

###### Proof.

An element of $F(x)$ can be written in the form

$\lambda\,(V(x)+q)\qquad\hbox{for $0\leq\lambda\leq 1$, $|q|\leq 1$.}$

Let $p\in Z^{\infty}(x)$, then

$p\cdot(V(x)+q)\leq-|p|+p\cdot q\leq 0,$

which shows that

$Z^{\infty}(x)\subset F(x)^{-}.$

Conversely, let $p\in F(x)^{-}$, then $V(x)+\frac{p}{|p|}\in F(x)$, and so

$0\geq p\cdot\left(V(x)+\frac{p}{|p|}\right)=H(x,p),$

which implies that $p\in Z^{\infty}(x)$. This concludes the proof. ∎

######  Lemma 4.12.

Let $\Omega\subset\mathbb{R}^{N}$ be an invariant set for $F$. Then

(3) $n_{\Omega}(x)\cdot V(x)\leq-1\qquad\hbox{for $\mathcal{H}^{N-1}$ a.e.
$x\in\partial\Omega$,}$

where $n_{\Omega}$ denotes the outer unit normal to $\Omega$ and
$\mathcal{H}^{N-1}$ is the $(N-1)$–dimensional Hausdorff measure.

###### Proof.

Recall that $\Omega$ has Lipschitz boundary by Theorem 4.2. Let $x$ be a point
of $\partial\Omega$ where there exists the outer normal $n_{\Omega}(x)$, see
Remark 2.2. We recall that by Lemma 4.6 $|V(x)|\geq 1$. By Lemma 4.7, we can
find for any $\delta\in(0,1)$ an open ball $U$ centered at $x$ with

$(x+\operatorname{int}F_{\delta}(x))\cap U\subset\operatorname{int}\Omega,$

which implies that $n_{\Omega}(x)$ is normal to $x+F_{\delta}(x)$ at $x$ and
consequently $n_{\Omega}(x)\in F_{\delta}(x)^{-}$. Therefore, by Lemma 4.11

$n_{\Omega}(x)\in\bigcap_{\delta\in(0,1)}\,F_{\delta}(x)^{-}=F(x)^{-}=Z^{\infty}(x),$

or, in other terms

$0\geq H(x,n_{\Omega}(x))\geq 1+n_{\Omega}(x)\cdot V(x).$

This proves (3).

∎

###### Proof of Theorem 4.9 .

Assume for purposes of contradiction that there exists a periodic invariant
set for $F$. We denote by $\Theta$ its projection on $\mathbb{T}^{N}$, by
Theorem 4.2 $\Theta$ possess Lipschitz boundary, so that we are in position to
apply divergence Theorem and, owing to Lemma 4.12, we get

$\mathcal{H}^{N-1}(\partial\Theta)\leq\int_{\partial\Theta}-n_{\Theta}\cdot
V\,d\,\mathcal{H}^{N-1}=\left|\int_{\Theta}\operatorname{div}V\,dx\right|\leq\|\operatorname{div}\,V\|_{L^{N}(\mathbb{T}^{N})}\,|\Theta|^{1-1/N}.$

Exploiting assumption (A2) on $\operatorname{div}V$ we deduce

$\mathcal{H}^{N-1}(\partial\Theta)<\frac{1}{\chi}\,|\Theta|^{1-1/N},$

which is contrast with the isoperimetric inequality, see Section 2. ∎

##  5\.  Intrinsic distances

We associate to the $1$–sublevels of $H$ an intrinsic distance. The first step
is to define for any $x$, $q$ in $\mathbb{R}^{N}$

$\displaystyle Z(x)$ $\displaystyle=$ $\displaystyle\\{p\mid H(x,p)\leq 1\\}$
$\displaystyle\sigma(x,q)$ $\displaystyle=$ $\displaystyle\sup\\{p\cdot q\mid
p\in Z(x)\\}\ $

The set–valued function $Z$ is closed convex valued and $0$ is an interior
point of $Z(x)$ for any $x$. This implies that $\sigma(x,q)$ is strictly
positive when $q\neq 0$.

For $|V(x)|$ increasing, the sublevel $Z(x)$ shrinks in the direction of
$V(x)$ and stretches in the opposite one, assuming more or less the shape of a
water drop, till $|V(x)|$ reaches the threshold value of $1$, where it breaks
apart and the unbounded recession cone $Z^{\infty}(x)$ pops up as a subset.
Accordingly, the support function $\sigma(x,\cdot)$ becomes infinite at some
$q$.

The next result is straightforward, we provide a proof in the Appendix for
completeness.

######  Proposition 5.1.

The function $(x,q)\mapsto\sigma(x,q)$ from
$\mathbb{R}^{n}\times\mathbb{R}^{N}$ to $\mathbb{R}^{+}\cup\\{+\infty\\}$ is
lower semicontinuous. In addition $q\mapsto\sigma(x,q)$ is convex positively
homogeneous, for any fixed $x$.

The next result puts in relation the present construction and $F$.

######  Lemma 5.2.

We have

$\sigma(x,q)\leq 1\qquad\hbox{for any $x\in\mathbb{R}^{N}$, $q\in F(x)$.}$

###### Proof.

Given

(4) $q=V(x)+\bar{q}\qquad\hbox{with $\bar{q}$ unit vector},$

$p\in Z(x)$, we have

$p\cdot q=p\cdot V(x)+p\cdot\bar{q}\leq p\cdot V(x)+|p|\leq 1,$

which shows that $\sigma(x,q)\leq 1$. Since $F(x)$ is the convex hull of
vectors of the form (4) plus $0$, we get the assertion.

∎

For any curve $\xi:[0,T]\to\mathbb{R}^{N}$ we define the intrinsic length
functional $\ell_{V}$ by

$\ell_{V}(\xi)=\int_{0}^{T}\sigma(\xi,\dot{\xi})\,dt.$

Notice that the above integral is invariant for orientation–preserving change
of parameter, as an intrinsic length should be. It is in addition positive,
and can be infinite. The corresponding path–metric $S_{1}$ is given by

(5) $S_{1}(x,y)=\inf\\{\ell_{V}(\xi)\mid\;\hbox{$\xi$ joining $x$ to $y$}\\},$

for any $x$, $y$ in $\mathbb{R}^{N}$. The (semi)distance $S_{1}$ is strictly
positive whenever the two arguments are distinct, and can be infinite. Note
that $S_{1}(x,x)=0$ for any $x$, in addition $S_{1}$ enjoys the triangle
property, but it is not in general symmetric.

We provide a periodic version of $S_{1}$ via the formula

(6) $\overline{S}_{1}(x,y)=\inf\\{S_{1}(x+z,y+z^{\prime})\mid
z,\,z^{\prime}\in\mathbb{Z}^{N}\\},$

Due to the periodic character of $H$, we also have

$\overline{S}_{1}(x,y)=\inf\\{S_{1}(x+z,y)\mid
z\in\mathbb{Z}^{N}\\}=\inf\\{S_{1}(x,y+z)\mid z\in\mathbb{Z}^{N}\\}.$

The following lemma will be used in Section 6.

######  Lemma 5.3.

Let $P$ be an element of $\mathbb{R}^{N}$, $\delta<1$. For any
$x\in\mathbb{R}^{N}$ there are two open sets $U_{x}$, $W_{x}$ with
$x\in\overline{U_{x}}\cap\overline{W_{x}}$ such that

(7) $\displaystyle S_{1}(x,y)+P\cdot(x-y)$
$\displaystyle<1+|P|\,r_{V}(\delta,1)$ for any $y\in U_{x}$ (8) $\displaystyle
S_{1}(y,x)+P\cdot(y-x)$ $\displaystyle<1+|P|\,r_{V}(\delta,1)$
$\displaystyle\qquad\hbox{for any $y\in W_{x}$}.$

In addition both families $\\{U_{x}\\}$, $\\{W_{x}\\}$, for
$x\in\mathbb{R}^{N}$, make up an open covering of the whole space.

###### Proof.

Given $x\in\mathbb{R}^{N}$, we have by Lemma 4.5 that

(9) $F_{\delta}(x)\subset F(y)\qquad\hbox{for any
$y\in\mathcal{B}(x,r_{V}(\delta,1))$}.$

We set

$\displaystyle U_{x}$ $\displaystyle=$
$\displaystyle(x+\operatorname{int}F_{\delta}(x))\cap\mathcal{B}(x,r_{V}(\delta,1))$
$\displaystyle W_{x}$ $\displaystyle=$
$\displaystyle(x-\operatorname{int}F_{\delta}(x))\cap\mathcal{B}(x,r_{V}(\delta,1)),$

if $y\in U_{x}$ then $y-x\in\operatorname{int}F_{\delta}(x)$ and
$x+t\,(y-x)\in\mathcal{B}(x,r_{V}(\delta,1))$, for $t\in[0,1]$, we therefore
derive from (9) and Lemma 5.2

$\sigma(x+t\,(y-x),y-x)<1\qquad\hbox{for $t\in[0,1]$.}$

This implies that the intrinsic length of the segment $x+t\,(y-x)$,
$t\in[0,1]$ is less than $1$ showing (7). Item (8) can be proven following the
same lines.

We proceed proving that for any $x\in\mathbb{R}^{N}$ there is $y$ with $x\in
U_{y}$. We select $\beta>0$ with

$\beta<\min\left\\{1,\,\frac{1-\delta}{M_{V}L_{V}}\right\\},$

and set

$y=x-\beta\,V(x).$

We get

$|y-x|=\beta\,|V(x)|\leq\beta\,M_{V}<\frac{1-\delta}{L_{V}}=r_{V}(\delta,1)$

and

$|x-y-\beta\,V(y)|=\beta\,|V(x)-V(y)|<\beta.$

The above inequalities show that $x\in U_{y}$. Adapting the above argument, we
also find $z$ with $x\in W_{z}$. This concludes the proof. ∎

######  Remark 5.4.

Slightly adapting the above construction, a distance, denoted by $S_{a}$,
$\overline{S}_{a}$ can be defined starting from the sublevels $\\{p\mid
H(x,p)\leq a\\}$, for any $a>0$. Due to the positive homogeneity of $H$ in
$p$, we have for any $x$, $y$ in $\mathbb{R}^{N}$

$\displaystyle S_{a}(x,y)$ $\displaystyle=$ $\displaystyle a\,S_{1}(x,y)$
$\displaystyle\overline{S}_{a}(x,y)$ $\displaystyle=$ $\displaystyle
a\,\overline{S}_{1}(x,y)$

Furthermore, it is easy to check that the intrinsic periodic distance
corresponding to the level $0$ is identically vanishing, thanks to Theorem
4.9, while $\overline{S}_{a}$ is identically equal to $-\infty$ for $a<0$.

##  6\. Homogenization

We consider the family of equations

(10) $H(x,Du+P)=a\quad a\in\mathbb{R},$

with $P\in\mathbb{R}^{N}$ fixed, and define the effective Hamiltonian
$\overline{H}$ as

(11) $\overline{H}(P)=\inf\\{a\mid\hbox{\eqref{hjp} admits an usc bounded
periodic subsolution}\\}.$

The effective Hamiltonian is the one appearing in the limit equation
($\overline{\mathrm{HJ}}$) of the homogenization procedure.

######  Lemma 6.1.

The effective Hamiltonian $\overline{H}$ is convex positively homogeneous and,
in addition

$\min_{x\in\mathbb{R}^{N}}H(x,P)\leq\overline{H}(P)\leq\max_{x\in\mathbb{R}^{N}}H(x,P)\qquad\hbox{for
any $P\in\mathbb{R}^{N}$}$

###### Proof.

The effective Hamiltonian $\overline{H}$ directly inherits convexity and
positive homogeneity from $H$. If $a\geq\max_{x\in\mathbb{R}^{N}}H(x,P)$ then
any constant function is a subsolution to (10), which gives the rightmost
inequality in the statement.

Next, let $u$ be a bounded periodic usc subsolution to (10). We denote by
$x_{0}$ a maximizer of it in $\mathbb{R}^{N}$. Then $H(x_{0},P)\leq a$ and so
$\min_{x\in\mathbb{R}^{N}}H(x,P)\leq a$. This completes the proof. ∎

We proceed fixing $P$. Following the same procedure as in Section 5, we see
that the intrinsic distance related to the sublevel $H(x,p+P)\leq a$ is

(12) $D_{a}(x,y)=S_{a}(x,y)+P\cdot(x-y),$

and the periodic version is

$\displaystyle\overline{D}_{a}(x,y)$ $\displaystyle=$
$\displaystyle\inf_{z,\,z^{\prime}\in\mathbb{Z}^{N}}\\{S_{a}(x+z,y+z^{\prime})+P\cdot(x+z-y-z^{\prime})\\}$
$\displaystyle=$
$\displaystyle\inf_{z\in\mathbb{Z}^{N}}\\{S_{a}(x,y+z)+P\cdot(x-y-z)\\}$

######  Remark 6.2.

Note that $\overline{D}_{a}$ could be finite even for $a<0$, despite the fact
that $\overline{S}_{a}$ is identically equal to $-\infty$. This is related to
the problem of the possible coercivity of $\overline{H}$. We do not treat this
issue here, see [3], [9].

The next result is a relevant consequence of Theorem 4.9.

######  Theorem 6.3.

The distance $\overline{D}_{1}$ is bounded from above in
$\mathbb{R}^{N}\times\mathbb{R}^{N}$.

###### Proof.

We first show that $\overline{D}_{1}$ is bounded from above in
$\mathbb{R}^{N}\times\mathbb{R}^{N}$, provided it is finite. Given $\delta<1$,
we consider the families of open sets $\\{U_{x}\\}$, $\\{W_{x}\\}$, for
$x\in\overline{Q}_{1}$, introduced in Lemma 5.3, and extract finite
subfamilies $\\{U_{i}\\}$, $i=1,\cdots,M_{1}$, $\\{W_{j}\\}$,
$j=1,\cdots,M_{2}$, respectively, corresponding to points $x_{i}$,
$\overline{x}_{j}$, covering $\overline{Q}_{1}$.

Due to Lemma 5.3, we have that for any $x\in\overline{Q}_{1}$ there are
indices $i_{0}$, $j_{0}$ with

(14) $\overline{D}_{1}(x_{i_{0}},x)\leq
1+|P|\,r_{V}(\delta,1)\qquad\hbox{and}\qquad\overline{D}_{1}(x,\overline{x}_{j_{0}})\leq
1+|P|\,r_{V}(\delta,1).$

Since we are supposing that $\overline{D}_{1}$ is finite, we have

$\overline{D}_{1}(\overline{x}_{j},x_{i})\leq\alpha\qquad\hbox{ for
$i=1,\cdots,M_{1}$, $j=1,\cdots,M_{2}$, some $\alpha>0$.}$

We take any pair of points $y$, $z$ in $\overline{Q}_{1}$ and assume that (14)
holds true with $z$ and $y$ in place of $x$, respectively. Then

$\overline{D}_{1}(y,z)\leq\overline{D}_{1}(y,\overline{x}_{j_{0}})+\overline{D}_{1}(\overline{x}_{j_{0}},x_{i_{0}})+\overline{D}_{1}(x_{i_{0}},z)\leq
2+2\,|P|\,r_{V}(\delta,1)+\alpha,$

showing the claim.

We proceed proving that since periodic invariant sets for $F$ do not exist in
force of Theorem 4.9, then $\overline{D}_{1}$ is actually finite in
$\mathbb{R}^{N}\times\mathbb{R}^{N}$. It is sufficient to show that, for any
given $y_{0}\in\mathbb{R}^{N}$, the set

$\Omega=\\{x\in\mathbb{R}^{N}\mid\overline{D}_{1}(y_{0},x)<+\infty\\}$

is periodic invariant for $F$. This in fact implies that
$\Omega=\mathbb{R}^{N}$, from which we in turn deduce, by the arbitrariness of
$y$, that $\overline{D}_{1}$ is finite.

It is clear by its very definition that $\Omega$ is periodic. To show the
invariance, we take $x_{0}\in\Omega$ and consider an integral curve $\xi$ of
$F$ defined in $[0,T)$ , for some $T\in\mathbb{R}^{+}\cup\\{+\infty\\}$,
issued from $x_{0}$. Then we have by Lemma 5.2

$\displaystyle\overline{D}_{1}(y_{0},\xi(t))$ $\displaystyle\leq$
$\displaystyle\overline{D}_{1}(y_{0},x_{0})+\ell_{V}\left(\xi\big{|}_{[0,t]}\right)+P\cdot(x_{0}-\xi(t))$
$\displaystyle\leq$
$\displaystyle\overline{D}_{1}(y_{0},x_{0})+t+P\cdot(x_{0}-\xi(t))\qquad\hbox{for
any $t\in[0,T)$.}$

Therefore the entire curve $\xi$ is contained in $\Omega$, as claimed. ∎

We proceed defining a sequence of coercive Hamiltonians approximating $H$. We
set for $k\in\mathbb{N}$

$H_{k}(x,p)=\max\\{H(x,p),-k\\}+f_{k}(p)$

where

$f_{k}(p)=\left\\{\begin{array}[]{cc}0&\qquad\hbox{if $|p|<k$}\\\
|p|-k&\qquad\hbox{if $|p|\geq k$}\\\ \end{array}\right.$

and define the related intrinsic distances $S^{k}_{a}$,
$\overline{S}^{k}_{a}$, $D_{a}^{k}$, $\overline{D}_{a}^{k}$ adapting formulae
(5), (6), (12), (6).

The Hamiltonians $H_{k}$ are continuous, periodic in the state and convex in
the momentum variable. They in addition satisfy:

  * –

the sequence $H_{k}(x,p)$ is nonincreasing for any $(x,p)$;

  * –

$H_{k}$ converges to $H$ locally uniformly in
$\mathbb{R}^{N}\times\mathbb{R}^{N}$ by Dini’s monotone convergence theorem.

The previous properties clearly imply that

$H(x,p)\leq H_{k}(x,p)\qquad\hbox{ for any $k$, $x$, $p$.}$

The effective Hamiltonians $\overline{H}_{k}$, corresponding to $H_{k}$, are
defined as in (11) with obvious adaptations. Notice that, being $H_{k}$
coercive, any periodic subsolution to (10), with $H_{k}$ in place of $H$, is
automatically Lipschitz–continuous. In addition the critical equation
$H_{k}(x,P+Du)=\overline{H}_{k}(P)$ possess solutions. We also have that the
$\overline{H}_{k}$ are convex coercive.

The main result of the section is:

######  Theorem 6.4.

  * –

For any $P\in\mathbb{R}^{N}$, $\lim_{k}\overline{H}_{k}(P)=\overline{H}(P)$;

  * –

there exist bounded periodic usc subsolutions and lsc supersolutions to the
equation $H(x,Du+P)=\overline{H}(P)$.

Note that, in absence of coerciveness of $H$, the convergence of
$\overline{H}_{k}$ to $\overline{H}$ is not a straightforward consequence of
the fact that $H_{k}\to H$ locally uniformly.

One half of the previous theorem is actually easy and is indeed a direct
consequence of the properties of $H_{k}$. In fact, if $u$ is a subsolution to
$H_{k}(x,Du+P)=a$, for some a, then it enjoys the same property with respect
to $H_{j}(x,Du+P)=a$ , whenever $j<k$, and with respect to (10). This actually
shows:

######  Lemma 6.5.

The sequence $\overline{H}_{k}(P)$ is nonincreasing and

(15) $\overline{H}_{k}(P)\geq\overline{H}(P)\qquad\hbox{ for any $k$.}$

######  Remark 6.6.

Since the matter is somehow intricate, let us summarize the different
monotonicity arising in the interplay between the $H_{k}$ and $H$.

  1. –

$H_{k}(x,p)$ is nonincreasing and $H_{k}(x,p)\to H(x,p)$, for any $(x,p)$;

  2. –

$S^{k}_{a}(x,y)$ is nondecreasing and $S^{k}_{a}(x,y)\leq S_{a}(x,y)$, for any
$x$, $y$, $a$;

  3. –

$\overline{H}_{k}(P)$ is nonincreasing and the possible convergence of
$\overline{H}_{k}(P)$ to $\overline{H}(P)$ is the subject we are presently
investigating.

We need some preliminary material for the proof of Theorem 6.4.

######  Lemma 6.7.

Let $a\geq\overline{H}_{1}(P)$, then $\overline{D}_{a}$ is bounded in
$\mathbb{R}^{N}\times\mathbb{R}^{N}$.

###### Proof.

We start proving that it is bounded from above. If $a\leq 0$ then
$\overline{D}_{a}\leq\overline{D}_{1}$, if $a>0$ then

$\overline{D}_{a}(x,y)=a\,\inf_{z\in\mathbb{Z}^{N}}\\{S_{1}(x,y)+P/a\cdot(x-y)\\}.$

It is then enough to prove that $\overline{D}_{1}$ is bounded from above. This
property actually comes from Theorem 6.3. To see that $\overline{D}_{a}$ is
also bounded from below, we recall that there is a periodic solution $u$ of

$H_{1}(x,Dv+P)=\overline{H}_{1}(P).$

Then $u$ is a subsolution to

$H(x,Dv+P)=a$

and, since $u$ is Lipschitz continuous for the coerciveness of $H_{1}$, we
have

$u(x)-u(y)\leq\overline{D}_{a}(y,x)\qquad\hbox{for any $x$, $y$}$

showing the claim. ∎

We derive:

######  Lemma 6.8.

The family of the distances
$\big{(}\overline{D}_{\overline{H}_{k}(P)}^{k}\big{)}_{k}$ are equibounded in
$\mathbb{R}^{N}\times\mathbb{R}^{N}$.

###### Proof.

We start proving the boundedness from above. Taking into account Remark 6.6,
we have

$\overline{D}_{\overline{H}_{k}(P)}^{k}(x,y)\leq\overline{D}^{k}_{\overline{H}_{1}(P)}(x,y)\leq\overline{D}_{\overline{H}_{1}(P)}(x,y).$

To get the converse estimate, we first notice that

(16)
$0=\overline{D}_{\bar{H}_{k}(P)}^{k}(x,x)=\inf_{z\in\mathbb{Z}^{N}}\\{S^{k}_{\overline{H}_{k}(P)}(x,x+z)-P\cdot
z\\}$

for any $x\in\mathbb{R}^{N}$. We consider a pair of points $x$, $y$ in
$\mathbb{R}^{N}$. We select $\bar{z}\in\mathbb{Z}^{N}$ in such a way that

$S_{\overline{H}_{1}(P)}(y,x+\bar{z})+P\cdot(y-x-\bar{z})\leq\overline{D}_{\overline{H}_{1}(P)}(y,x)+1.$

We compute taking into account (16), Remark 6.6 and Lemma 6.7

$\displaystyle\overline{D}_{\overline{H}(P)}^{k}(x,y)$ $\displaystyle=$
$\displaystyle\inf_{z\in\mathbb{Z}^{N}}\\{S^{k}_{\overline{H}_{k}(P)}(x+z,y))+P\cdot(x+z-y)\\}$
$\displaystyle\geq$
$\displaystyle\inf_{z\in\mathbb{Z}^{N}}\\{S^{k}_{\overline{H}_{k}(P)}(x+z,x+\bar{z})-S^{k}_{\overline{H}_{k}(P)}(y,x+\bar{z})$
$\displaystyle+$ $\displaystyle P\cdot(z-\bar{z})-P\cdot(y-x-\bar{z})\\}$
$\displaystyle\geq$
$\displaystyle\inf_{z\in\mathbb{Z}^{N}}\\{S^{k}_{\overline{H}_{k}(P)}(x+z,x+\bar{z})+P\cdot(z-\bar{z})\\}$
$\displaystyle+$ $\displaystyle
P\cdot(x+\bar{z}-y)-S^{k}_{\overline{H}_{k}(P)}(y,x+\bar{z})$
$\displaystyle\geq$ $\displaystyle
P\cdot(x+\bar{z}-y)-S_{\overline{H}_{k}(P)}(y,x+\bar{z})$ $\displaystyle\geq$
$\displaystyle
P\cdot(x+\bar{z}-y)-S_{\overline{H}_{1}(P)}(y,x+\bar{z})\geq-\overline{D}_{\overline{H}_{1}(P)}(y,x)-1.$

This concludes the proof. ∎

######  Lemma 6.9.

If for a given $a$ there is a bounded periodic usc subsolution and a bounded
periodic lsc supersolution of (10), then $a=\overline{H}(P)$.

The proof is basically the same as in coercive case, with some more
precaution. We present it in the Appendix.

We need the following result, see for instance [6].

######  Proposition 6.10.

For any $k$, any $y$ in the Aubry set related to $H_{k}$, the function
$x\mapsto\overline{D}_{\overline{H}_{k}(P)}^{k}(y,x)$ is a periodic Lipschitz
continuous solution of $H_{k}(x,P+Du)=\overline{H}_{k}(P)$.

######  Proof of Theorem 6.4.

We know from Proposition 6.10, that we can select, for any $k$, $y_{k}$ in
such a way that $v_{k}:=\overline{D}_{\overline{H}_{k}(P)}^{k}(y_{k},\cdot)$
is solution to

$H_{k}(x,Du+P)=\overline{H}_{k}(P).$

Since the $v_{k}$ are periodic, equibounded by Lemma 6.8, the functions

$v:=\operatorname{lim\,inf_{\\#}}v_{k}\qquad\hbox{and}\qquad
w:=\operatorname{lim\,sup^{\\#}}v_{k}$

are bounded periodic and, taking into account the local uniform convergence of
$H_{k}$ to $H$ plus standard stability properties of viscosity solutions
theory, are indeed lsc supersolution and usc subsolution, respectively, to

$H(x,Du+P)=\lim_{k}\overline{H}_{k}(P).$

This implies by Lemma 6.9 that $\overline{H}(P)=\lim_{k}\overline{H}_{k}(P)$.
The assertion then follows.

∎

Exploiting Theorem 6.4, we get the homogenization result in a quite standard
way. We state it below and provide the proof in the Appendix.

######  Theorem 6.11.

The solutions to (HJε) locally uniformly converge to the solution of
($\overline{\mathrm{HJ}}$), where $\overline{H}$ is the effective Hamiltonian
defined in (11).

##  Appendix A Some Proofs

###### Proof of Proposition 5.1.

Let us consider $(x_{0},q_{0})$ and an approximating sequence $(x_{n},q_{n})$.
First assume $\sigma(x_{0},q_{0})<+\infty$ and pick $p_{0}\in Z(x_{0})$ with
$\sigma(x_{0},q_{0})\leq p_{0}\cdot q_{0}+\delta$, for some positive $\delta$.
Taking into account that the interior of $Z(x_{n})$ is nonempty, there is a
sequence $p_{n}\in Z(x_{n})$ converging to $p_{0}$. We find

$\liminf_{n}\sigma(x_{n},q_{n})\geq\lim_{n}p_{n}\cdot q_{n}=p_{0}\cdot
q_{0}\geq\sigma(x_{0},q_{0})-\delta,$

and the claimed semicontinuity follows since $\delta$ has been arbitrarily
chosen .

If $\sigma(x_{0},q_{0})=+\infty$, then taking any $\beta>0$, we find $p_{0}\in
Z(x_{0})$ with $p_{0}\cdot q_{0}\geq\beta$, then

$\liminf_{n}\sigma(x_{n},q_{n})\geq\lim_{n}p_{0}\cdot q_{n}=p_{0}\cdot
q_{0}\geq\beta,$

which implies $\lim_{n}\sigma(x_{n},q_{n})=+\infty$.

The claimed positive homogeneity and subadditivity are immediate.

∎

###### Proof of Lemma 6.9.

The argument is by contradiction. Let $a$ be such that (10) has supersolution
and subsolution as specified in the statement, denoted by $v$, $u$,
respectively. If $a>\overline{H}(P)$, then we can select $b<a$ with

$H(x,Du+P)\leq b\qquad\hbox{ in the viscosity sense.}$

We denote, for any $\delta>0$, by $u^{\delta}$ the sup–convolutions of $u$. By
basic properties of it, we know that

$H\left(y,\frac{x-y}{\delta}\right)\leq b$

for any $\delta$, any $x\in\mathbb{R}^{N}$, $y$ $u^{\delta}$–optimal to $x$.
We proceed considering a sequence

(17) $x_{\delta}\in\arg\min_{\mathbb{R}^{N}}\\{v-u^{\delta}\\},$

note that such minimizers do exist because $v$ is lsc, $u^{\delta}$ usc, and
both $v$ and $u^{\delta}$ are periodic. We denote, for any $\delta$, by
$y_{\delta}$ an element $u^{\delta}$–optimal for $x_{\delta}$, we have

(18) $\lim_{\delta}\frac{|x_{\delta}-y_{\delta}|^{2}}{\delta}=0.$

Consequently, taking into account the inequality

$\left|H\left(x_{\delta},\frac{x_{\delta}-y_{\delta}}{\delta}\right)-H\left(y_{\delta},\frac{x_{\delta}-y_{\delta}}{\delta}\right)\right|\leq
L_{V}\,\frac{|x_{\delta}-y_{\delta}|^{2}}{\delta}$

and that $b<a$, we conclude that

(19) $H\left(x_{\delta},\frac{x_{\delta}-y_{\delta}}{\delta}\right)<a,$

for $\delta$ small enough. On the other side, keeping in mind that
$\frac{x_{\delta}-y_{\delta}}{\delta}\in D^{-}u^{\delta}(x_{\delta})$ and that
$u^{\delta}$ is subtangent to $v$ at $x_{\delta}$, we get

$H\left(x_{\delta},\frac{x_{\delta}-y_{\delta}}{\delta}\right)\geq
a\quad\qquad\hbox{for any $\delta$,}$

which contradicts (19).

∎

###### Proof of Theorem 6.11.

Let $u_{\varepsilon}$ the solutions to (HJε), they are equibounded by (A3). We
set

$v=\operatorname{lim\,inf_{\\#}}u_{\varepsilon}\qquad\hbox{and}\qquad
u=\operatorname{lim\,sup^{\\#}}u_{\varepsilon}.$

As already pointed out in the Introduction, it is standard to show that $v$ is
a supersolution of ($\overline{\mathrm{HJ}}$).

We focus on proving that $u$ is subsolution to ($\overline{\mathrm{HJ}}$). We
consider $(x_{0},t_{0})\in\mathbb{R}^{N}\times(0,+\infty)$ and a $C^{1}$
strict supertangent $\psi$ to $u$ at $(x_{0},t_{0})$. We denote by $w$ a
bounded periodic lsc supersolution to the problem

$H(x,Dv+D(\psi(x_{0},t_{0})))=\overline{H}(D(\psi(x_{0},t_{0}))).$

We have that $\varepsilon\,w(x/\varepsilon)$ is supersolution to

(20)
$H(x/\varepsilon,Dv+D(\psi(x_{0},t_{0})))=\overline{H}(D(\psi(x_{0},t_{0}))).$

The point $(x_{0},t_{0})$ is the unique maximizer of $u-\psi$ in
$\overline{U}$, for a suitable choice of an open neighborhood $U$ of
$(x_{0},t_{0})$ in $\mathbb{R}^{N}\times(0,+\infty)$. We proceed considering
the functions

(21) $u_{\varepsilon}(x,t)-\psi(x,t)-\varepsilon\,w_{\delta}(x/\varepsilon),$

where $w_{\delta}$ stands for the $\delta$ inf–convolution of $w$ for
$\delta>0$. We have that the quantity $\varepsilon\,w_{\delta}(x/\varepsilon)$
converges uniformly to $0$ as $\varepsilon\to 0$, uniformly with respect to
$\delta$. Owing to the stability properties of maximizers, we deduce that a
sequence of maximizers of (21) denoted by
$(x^{\delta}_{\varepsilon},t^{\delta}_{\varepsilon})$, belongs to $U$ for
$\varepsilon$ small enough and

(22)
$(x^{\delta}_{\varepsilon},t^{\delta}_{\varepsilon})\to(x_{0},t_{0})\qquad\hbox{as
$\varepsilon\to 0$, uniformly in $\delta$.}$

Since

$(\psi_{t}(x^{\delta}_{\varepsilon},t^{\delta}_{\varepsilon}),D\psi(x^{\delta}_{\varepsilon},t^{\delta}_{\varepsilon})+p^{\delta}_{\varepsilon})\in
D^{+}u_{\varepsilon}(x^{\delta}_{\varepsilon},t^{\delta}_{\varepsilon})$

for any $p^{\delta}_{\varepsilon}\in\partial
w_{\delta}(x^{\delta}_{\varepsilon})=D^{+}w_{\delta}(x^{\delta}_{\varepsilon})$,
and $u_{\varepsilon}$ is subsolution to (HJε), we derive

(23)
$\psi_{t}(x^{\delta}_{\varepsilon},t^{\delta}_{\varepsilon})+H(x^{\delta}_{\varepsilon}/\varepsilon,D\psi(x^{\delta}_{\varepsilon},t^{\delta}_{\varepsilon})+p^{\delta}_{\varepsilon})\leq
0\quad\hbox{for any $p^{\delta}_{\varepsilon}\in\partial
w_{\delta}(x^{\delta}_{\varepsilon})$.}$

Since $w_{\delta}$ is inf–convolution of $w$, and $w$ is supersolution to the
equation (20), we further have

$H(y^{\delta}/\varepsilon,Dw_{\delta}(x/\varepsilon)+D\psi(x_{0},t_{0}))\geq\overline{H}(D\psi(x_{0},t_{0}))$

at any $x$ where $w_{\delta}$ is differentiable, with $y^{\delta}$ denoting
the $w_{\delta}$–optimal point for $x$. Consequently

(24)
$H(y^{\delta}_{\varepsilon}/\varepsilon,q^{\delta}_{\varepsilon}+D\psi(x_{0},t_{0}))\geq\overline{H}(D\psi(x_{0},t_{0}))$

for any $\varepsilon$, some $q^{\delta}_{\varepsilon}\in\partial
w_{\delta}(x^{\delta}_{\varepsilon})$, with $y^{\delta}_{\varepsilon}$
denoting an $u_{\delta}$–optimal point for $x^{\delta}_{\varepsilon}$. In
order to combine (23) and (24), we need to estimate

(25)
$|H(x^{\delta}_{\varepsilon}/\varepsilon,D\psi(x^{\delta}_{\varepsilon},t_{\varepsilon})+q^{\delta}_{\varepsilon})-H(y^{\delta}_{\varepsilon}/\varepsilon,D\psi(x_{0},t_{0})+q^{\delta}_{\varepsilon})|=:I.$

After straightforward computations, we get

$\displaystyle I$ $\displaystyle\leq$
$\displaystyle|D\psi(x_{0},t_{0})-D\psi(x^{\delta}_{\varepsilon},t^{\delta}_{\varepsilon})|\,(1+|V(x^{\delta}_{\varepsilon}/\varepsilon)|)$
$\displaystyle+$
$\displaystyle|D\psi(x_{0},t_{0})+q_{\varepsilon}^{\delta}|\,L_{V}\,\frac{|x^{\delta}_{\varepsilon}-y^{\delta}_{\varepsilon}|}{\varepsilon}$
$\displaystyle\leq$
$\displaystyle\operatorname{O}(\varepsilon)+\frac{\operatorname{o}(\sqrt{\delta})}{\varepsilon}+\frac{\operatorname{o}(\delta)}{\delta}\,\frac{1}{\varepsilon}.$

We freeze $\varepsilon$ and move $\delta$ in order to obtain a
$\delta_{\varepsilon}$ for which

$I\leq\operatorname{O}(\varepsilon)+\frac{\varepsilon^{2}}{\varepsilon}+\frac{\varepsilon^{2}}{\varepsilon}=\operatorname{O}(\varepsilon).$

We use the above estimate plus (23), (24) to finally get

$\displaystyle 0$ $\displaystyle\geq$
$\displaystyle\psi_{t}(x^{\delta_{\varepsilon}}_{\varepsilon}/,t^{\delta_{\varepsilon}}_{\varepsilon})+H(x^{\delta_{\varepsilon}}_{\varepsilon}/\varepsilon,D\psi(x^{\delta_{\varepsilon}}_{\varepsilon},t^{\delta_{\varepsilon}}_{\varepsilon})+q^{\delta_{\varepsilon}}_{\varepsilon})$
$\displaystyle\geq$
$\displaystyle\psi_{t}(x^{\delta_{\varepsilon}}_{\varepsilon},t^{\delta_{\varepsilon}}_{\varepsilon})+H(y^{\delta_{\varepsilon}}_{\varepsilon}/\varepsilon,D\psi(x_{0},t_{0})+q^{\delta_{\varepsilon}}_{\varepsilon})-\operatorname{O}(\varepsilon)$
$\displaystyle\geq$
$\displaystyle\psi_{t}(x^{\delta_{\varepsilon}}_{\varepsilon},t^{\delta_{\varepsilon}}_{\varepsilon})+\overline{H}(D\psi(x_{0},t_{0}))-\operatorname{O}(\varepsilon).$

Sending $\varepsilon\to 0$, we see that $u$ satisfies the subsolution test at
$(x_{0},t_{0})$ with respect to the supertangent $\psi$, as was claimed.

Finally, taking into account that ($\overline{\mathrm{HJ}}$) satisfies a
comparison principle, we get that the $u_{\varepsilon}$ locally uniformly
converge to $v=u$.

∎

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