# Hénon maps: a list of open problems

Julia Xénelkis de Hénon

###### Abstract

We propose a set of questions on the dynamics of Hénon maps from the real,
complex, algebraic and arithmetic points of view.

###### Contents

  1. 1 Introduction (C. Favre, T. Firsova, L. Palmisano, J. Raissy, and G. Vigny (Eds.))
  2. 2 Real Hénon maps (P. Berger)
    1. 2.1 Strange attractors
    2. 2.2 Non-statistical behavior
    3. 2.3 Conservative dynamics
  3. 3 Dissipative real Hénon maps (S. Crovisier and E. Pujals)
    1. 3.1 Mild dissipation
    2. 3.2 Maps with zero entropy
    3. 3.3 Set of periodic points
  4. 4 Symbolic dynamics for real Hénon and Lozi maps (S. Štimac)
    1. 4.0.1 Example
  5. 5 Hénon maps tangent to the identity (X. Buff)
  6. 6 Quasi-hyperbolicity and uniform hyperbolicity (E. Bedford)
    1. 6.1 Complex Hénon maps.
    2. 6.2 Quasi-expanding Hénon maps
    3. 6.3 Surface automorphisms
    4. 6.4 Real maps
  7. 7 Parameter loci for the Hénon family (Y. Ishii)
    1. 7.1 Connectedness locus
    2. 7.2 Horseshoe locus
      1. 7.2.1 Isentropes
  8. 8 Topology and rigidity of Hénon maps (R. Dujardin)
  9. 9 Statistical properties of complex Hénon maps (F. Bianchi and T.-C. Dinh)
    1. 9.1 Thermodynamics for Hénon maps
    2. 9.2 Statistical properties of equilibrium states and spectral gap for the transfer operators
    3. 9.3 Higher dimension and other generalizations
  10. 10 Towards higher dimensions and complex differential geometry (C. Favre)
    1. 10.1 Hénon maps and the group of polynomial automorphisms of $\mathbb{C}^{2}$.
    2. 10.2 Hénon maps and compact complex manifolds.
  11. 11 Hénon maps over number fields (P. Ingram)

##  1 Introduction (C. Favre, T. Firsova, L. Palmisano, J. Raissy, and G.
Vigny (Eds.))

A workshop ‘Dynamics of Hénon maps: Real, Complex and Beyond” took place at
BIRS, Banff in April 2023. The purpose of this meeting was to bring together
international experts working on various aspects of Hénon maps. Recall that
these maps are two-dimensional discrete dynamical systems which are ubiquitous
in low dimensional dynamics, and among the most studied examples exhibiting
chaotic behavior. Quadratic Hénon maps

$H_{a,c}(x,y):=(ay+x^{2}+c,ax)$

are arguably the simplest examples. Here $a$ and $c$ are fixed parameters and
$x,y$ are affine coordinates111The exact definition of (quadratic) Hénon maps
may differ from section to section in these notes, as one might want to
conjugate them by affine transformations to exploit various aspects of the
original equations.. These maps can be analyzed over the real numbers using
techniques from smooth dynamical systems, or over the complex numbers and then
complex analysis and geometry play crucial roles. They are amenable to
generalizations, by replacing $x^{2}+c$ by higher degree polynomials, or even
transcendental maps, and we may consider finite composition of such maps. We
may also consider them with coefficients in number fields, and look at them
from the perspective of arithmetic dynamics. Many of the most recent
breakthroughs were actually made by combining several techniques coming from
these different fields. It was delightful to attend series of talks blending
so many different ideas. Many interesting questions were raised during the
conference, a fact which encouraged us to collect them in a single text.

Hénon introduced his family of maps in the real domain as a simplified model
of the Poincaré section of the first return map of the Lorenz flow [85]. In
1976, Hénon made numerical experiments for the map
$H_{\sqrt{0.3},-1.4}$222Hénon actually considered the map $h(x,y)=(1-1.4\cdot
x^{2}+y\,,\,0.3\cdot x)$ which is affinely conjugate to it. and observed that
an initial point of the plane either approaches a set of points known since
then as the Hénon strange attractor, or diverges to infinity under iterations.
The Hénon attractor has a fractal nature: it is smooth in the unstable
direction and has a Cantor-like structure in the transversal direction. This
led Hénon to conjecture the existence of an ergodic measure which restricts to
the Lebesgue measure in the smooth direction (a.k.a. an SRB measure). In 1981,
Jakobson [93] proved the existence of a set of positive Lebesgue measure of
parameters $c$ for which $x^{2}+c$ displays an SRB measure. In the 90’s,
Benedicks and Carleson [24] reworked Jakobson’s theorem and further
generalized it to describe the dynamics of Hénon maps. They proved $H_{a,c}$
display a strange attractor for $a$ small and for a set of parameters $c$ of
positive Lebesgue measure. Benedicks-Carleson’s breakthrough has been further
developed by Mora-Viana [112], Wang-Young [141], and Takahasi [134].

In 1996, during his inaugural lecture at Collège de France, Yoccoz proposed an
alternative approach to prove the Hénon conjecture, with Sinai’s positive
entropy conjecture lying in the horizon. To this end, he introduced a
combinatorial and topological approach, based on the notion of strong
regularity, that he used to give yet another proof of Jakobson’s theorem
[145]. The second step of Yoccoz’ program was completed more recently by
Berger in [25] who generalized this notion of strong regularity, leading him
in particular to an alternative proof of Benedicks-Carleson theorem.

The theory of Hénon maps in the complex domain started with the seminal work
of Friedland and Milnor [79], who used Jung’s theorem to show that every
polynomial automorphism of the complex affine plane is affinely conjugate to
either an affine map, or a map preserving the pencil $x=\mathrm{cst}$, or to a
finite compositions of generalized Hénon maps (the latter class being usually
called complex Hénon maps nowadays). In the early 90’s Hubbard and his
collaborators developed a topological approach to the study of Henon maps
giving description of the Fatou sets [88, 90]. Hubbard pointed out that Hénon
maps appeared as natural generalizations of quadratic polynomials (by taking
$a\to 0$ in $H_{a,c}$), which he used with Oberste-Vorth to give topological
description of Julia sets [89].

An important breakthrough was the introduction of pluripotential techniques to
construct invariant currents by Fornæss and Sibony [77] and by Bedford and
Smillie [17]. The latter authors, partly with Lyubich, further developed in a
series of influential papers (e.g., [18, 16, 15]) a thorough study of the
ergodic properties of Hénon mappings and of their stability properties linked
to hyperbolicity. These works were pursued and generalized to other invertible
maps by many others including Diller, Cantat, Dujardin, Dinh and Sibony [54,
42, 68, 67, 57, 61]. Very recently, Bianchi and Dinh [30] made significant
progress in the study of the fine statistical properties of the maximal
entropy measure.

It is intriguing to see in retrospect how these seemingly simple maps have
produced such an elaborate and successful theory. Note however that the
results obtained so far are most complete in the case of dissipative maps
(that is $|a|<1$), while the understanding of the conservative case (that is
$|a|=1$) remains less developed.

In the past decade, the algebraic and arithmetic aspects of dynamical systems
defined by rational maps have also been developed extensively. We refer to the
survey [23] in which one can find a large set of open problems in this
emerging field. For Hénon maps defined over a number field or over a function
field, Silverman [133], and later Kawaguchi [97], constructed a suitable
height function that lead to interesting analogs of the Northcott property
(see also [91]). Hénon maps have also been the testing ground of some
important conjectures in arithmetic dynamics like the Kawaguchi-Silverman
conjecture on arithmetic degrees [98], or the dynamical Manin-Mumford problem
which was partially solved by Dujardin and Favre using both height and Pesin
theories [72]. Deep connections exist between the arithmetic of these systems
and pluripotential theoretic techniques: it is for instance possible to
retrieve the equidistribution of repelling periodic orbits using a theorem on
the equidistribution of points of small height by Yuan [147].

The study of Hénon maps is still very active, as shown by the recent
breakthroughs in the study of wandering domains. On the one hand, Ou [118] has
proved the absence of wandering domains for strongly dissipative doubly
infinitely period-doubling renormalizable real Hénon maps. On the other hand,
Berger and Biebler [27] exhibited wandering domains for complex Hénon maps in
2023 by mixing deep techniques coming from both real and complex dynamics. We
also witness exciting new developments extending the already rich theory of
Hénon maps to more general systems such as transcendental diffeomorphisms of
the complex plane [4, 5, 6], or higher dimensional invertible rational maps,
[53, 59, 62, 63, 83], where questions arising from complex dynamics led to
profound developments in complex geometry such as PB currents, density
currents and superpotential theory. Several developments have been made also
in the context of higher dimensional unfoldings of homoclinic tangencies with
a rank one saddle point, [120, 139, 142]. We hope that gathering these
questions and open problems at one place will reinforce the community and
attract new generations of researchers to work on these beautiful and rich
objects.

##### Acknowledgments

The idea to collect this list of problem arose after holding a problem session
at a BIRS conference which was lively chaired by M. Abate. We thank all
participants of this session and the other contributors to this list to have
generously shared their ideas and problems on Hénon maps. We are also grateful
to Zin Arai, André de Carvalho, Sébastien Gouëzel, Stéphane Lamy, Joseph
Silverman, and Liz Vivas for their careful reading and their comments on
previous versions of this manuscript.

##  2 Real Hénon maps (P. Berger)

We propose a set of questions on the dynamics of Hénon maps in the real
domain, or more generally on entire diffeomorphisms of $\mathbb{R}^{2}$.

###  2.1 Strange attractors

Attractors play an important role in the study of dynamical systems since the
60’s (Lorenz attractor, Hénon strange attractor, etc.). This notion is quite
flexible and can cover many different situations in which a substantial set of
points (either in the topological or measurable sense) is converging to some
invariant compact subset. We refer to Milnor [106] for a discussion of various
possible definitions of attractors. For instance, a measure-theoretical
attractor is an invariant compact subset which attracts a set of Lebesgue
positive measure and which is minimal with this property.

In the case of unimodal interval dynamics, measure-theoretical attractors can
be classified into four types: cyclic, solenoidal, interval and wild (see,
e.g., [36]). The latter two classes are arguably the most interesting.
Jakobson [93] proved the abundance of quadratic maps displaying a stochastic
interval of attractors (induced by an SRB measure). On the other hand, Bruin-
Keller-Nowicki-van Strien [39] showed the existence of a polynomial unimodal
map displaying a wild attractor: an invariant Cantor set attracting Lebesgue
almost every point and included in a transitive interval.

Van Strien [137, Question 1.9] asked whether a suitable analog of wild
attractors could exist for Hénon maps (of some degree). More precisely, we
ask:

######  Question 1.

Can we find a Hénon map (of some degree) which admits a wild attractor, i.e.,
a Cantor set which attracts a set of Lebesgue positive measure and which is
strictly included in a transitive set ?

Returning to stochastic attractors, observe that the existence of a parameter
$c\in\mathbb{R}$ for which the quadratic map $x\mapsto x^{2}+c$ displays an
absolutely continuous measure is easy to ensure. It suffices to pick a
parameter $c$ such that the post-critical orbit is finite but not periodic. In
dimension $2$, the existence of a positive measure set of parameters of Hénon
maps displaying an attractor supporting an invariant SRB measure 333i.e., a
measure whose conditional measures along unstable curves are absolutely
continuous. is a fundamental result, whose proof still remain difficult and
lengthy ([24, 25, 134, 141]).

######  Question 2.

Is there a quick proof for the existence of SRB for some parameters of the
Hénon maps?

A positive answer to this question might help finding new examples of
stochastic attractors in the Hénon maps.

###  2.2 Non-statistical behavior

Let $f$ be any smooth dynamical system. We say that a point $x$ has a non-
statistical behavior (or simply is non-statistical) if its sequence of
empirical measures $e_{n}(x):=\tfrac{1}{n}\sum_{k=0}^{n-1}\delta_{x_{n}}$ is
not converging. We say that $f$ is non-statistical if there is a positive
Lebesgue measure set of points with non convergent empirical measures. Ruelle
[127] asked whether non-statistical dynamics could exist persistently444Ruelle
used another terminology and talked about historical behavior..

In polynomial dynamics, two phenomena give rise to non-statistical dynamics.
The first one was discovered by Hofbauer and Keller:

######  Theorem 1 ([86]).

There exist uncountably many $c\in\mathbb{R}$ such that the quadratic
polynomial $P_{c}(x):=x^{2}+c$ has non-statistical dynamics. More precisely,
Lebesgue almost every non-escaping point $x$ has non-statistical behavior.

In [135], Talebi gave a counterpart of this result for rational functions on
the Riemann sphere. In these two results, the set of non-statistical points is
of full measure, but of empty interior.

The second occurrence of non-statistical dynamics is related to the notion of
wandering stable component that we now introduce.

######  Definition 1.

A stable domain of $f$ is a connected open subset $U$ such that

$\lim_{n\to\infty}d(f^{n}(x),f^{n}(y))=0$

for all $x,y\in U$. A stable component is a maximal stable domain. A stable
component is wandering if it is not preperiodic.

Colli and Vargas [48] gave the first example of a smooth dynamical system
having a wandering stable component formed by points with non-statistical
behavior. In [99], Kiriki and Soma constructed a locally dense set of such
dynamics in the $C^{r}$-category with $r<\infty$. This also occurs for
polynomial maps:

######  Theorem 2 ([27]).

There is a locally dense set of real sextic polynomials
$P(x)=x^{6}+a_{4}x^{4}+\cdots+a_{0}$, for which the map
$(x,y)\mapsto(P(x)-by,x)$ displays a wandering stable component containing
only points with non-statistical behavior.

The proof of this theorem actually implies the existence of a wandering Fatou
component at the same parameters for its complex counterpart. This is in sharp
contrast with the one-dimensional situation for which no wandering Fatou
components exist by Sullivan’s theorem.

Conversely, we can ask whether a counterpart of Hofbauer-Keller phenomenon
appears within the Hénon family. We can formulate this question in more
precise terms.

######  Question 3.

Do there exist $\varepsilon>0$ and a locally dense set555i.e., whose closure
has non-empty interior. $E$ of quadratic Hénon maps for which every point
starting in some set of Lebesgue measure at least $\varepsilon$ has non-
statistical behavior?

By [135, Theorem 1.14], this would imply the existence of a generic set of the
closure of $E$ of non-statistical dynamics.

###  2.3 Conservative dynamics

Let $f\colon S\to S$ be any homeomorphism of a closed surface $S$. An annular
rotation domain for $f$ is by definition an $f$-invariant open annulus that
does not contain any periodic points. Such domains play an important role in
conservative dynamics. The next result can be deduced from the works of [78,
100, 101, 144] (Le Calvez, private communication).

######  Theorem 3.

Let $S$ be a closed surface endowed with a symplectic form. Let $f\colon S\to
S$ be any symplectomorphism of class $C^{r}$ with $r\geq 1$ that contains at
least one periodic point and satisfying the following conditions.

  1. 1.

Any eigenvalue of any periodic point does not belong to $\\{e^{2\pi
ip/q}:p/q\in\mathbb{Q}\\}$.

  2. 2.

For every hyperbolic periodic points $P,Q\in\mathrm{Per}(f)$, $W^{s}(P)$ is
transverse to $W^{u}(Q)$.

  3. 3.

Every elliptic point $P\in\mathrm{Per}(f)$ is surrounded by arbitrarily close
KAM circles.

  4. 4.

There are no annular rotation domain.

Then $\bigcup_{P\in\mathrm{Per}(f)}W^{s}(P)$ is dense in $S$.

It is natural to ask whether the third condition is superfluous. The recent
result [117] suggests that this may be the case.

######  Question 4.

Is there a real, conservative, polynomial Hénon map with an annular rotation
domain? Is there an open set of such real, conservative, polynomial Hénon
maps?

An annular rotation domain is said to be trivial when the whole dynamics is
conjugate to a rotation.

######  Question 5.

Is there an entire666i.e., an analytic map which extends to a holomorphic map
of $\mathbb{C}^{2}$. symplectomorphism of $\mathbb{R}^{2}$ with a non trivial
annular rotation domain?

Recently, an entire map of the cylinder
$\mathbb{R}\times\mathbb{R}/\mathbb{Z}$ having a bounded rotation domain on
which the dynamics is not conjugate to a rotation has been exhibited in [26],
thereby disproving a conjecture by Birkhoff [32] . The construction in [26]
also gives an example of a symplectic entire automorphism of
$\mathbb{C}\times\mathbb{C}/\mathbb{Z}$ without periodic point and with a non-
empty set of (Lyapunov) unstable points777an orbit $(x_{n})_{n\geq 0}$ is
Lyapunov stable if for any $y_{0}$ close enough to $x_{0}$, then $y_{n}$ stays
close to $x_{n}$ for all $n\geq 0$.. Hence Question 5 may shed light on the
following intriguing problem.

######  Problem 1.

Construct an entire symplectic automorphism of $\mathbb{C}^{2}$ without
periodic point and with non-empty set of (Lyapunov) unstable points with
bounded orbit.

##  3 Dissipative real Hénon maps (S. Crovisier and E. Pujals)

###  3.1 Mild dissipation

Quadratic real Hénon maps

$f_{c,b}(x,y):=(x^{2}+c-by,x),$

with Jacobian $b$ close to zero share some properties of the quadratic family
on the interval: some results are obtained by perturbative methods (for
instance [24, 52]) and are known when $|b|$ is extremely tiny. In [49] another
approach has been introduced which allows to reduce the dynamics to a one-
dimensional system.

######  Definition 2.

The map $f_{c,b}$ is _mildly dissipative_ if it is dissipative (i.e., $|b|<1$)
and if for any ergodic invariant measure $\mu$ which is not supported on a
sink, and for $\mu$-almost every point $x$, both components of the stable
curve $W^{s}(x)\setminus\\{x\\}$ are unbounded.

Under this assumption, and in restriction to any open topological disc
$D\subset\mathbb{R}^{2}$ that is compactly mapped inside itself, the dynamics
is semi-conjugated to a (non-trivial) continuous map of a real tree. Other
strong consequences can be derived (e.g., a closing lemma, or a description of
zero entropy dynamics, see below). Using Wiman’s theorem (in the same spirit
like in [74, 104]), one can prove that Hénon maps are mildly dissipative once
$|b|<1/4$. In this way we obtain dynamical informations for _all_ Hénon maps
having their Jacobian in $(-1/4,1/4)$ and not only for those satisfying
$|b|\ll 1$. One expects that this property extends beyond the bound obtained
through Wiman’s theorem.

######  Question 6.

Which real Hénon maps $f_{c,b}$ are mildly dissipative? Is this property
satisfied by all Hénon maps with $|b|<1$?

In some cases [49] proves that the mild dissipation is an open property, but
we don’t know if this holds in general.

###  3.2 Maps with zero entropy

It is well-known that the quadratic maps $f_{c}(x):=x^{2}+c$ have their
topological entropy equal to zero exactly when $c$ belongs to some interval
$(-\infty,c_{0}]$. At the critical parameter $c_{0}$, the dynamics exhibits an
odometer, which is the limit set of an infinite sequence of successive
renormalizations of period $2$. This result persists inside any line $c\mapsto
f_{c,b}$, provided $|b|$ is smaller than some number $\varepsilon>0$ small, as
it has been shown in [52]. Let us consider the locus in the parameter space
where the topological entropy vanishes:

$\mathcal{E}_{0}:=\\{(b,c)\in\mathbb{R}^{2},h_{\text{top}}(f_{c,b})=0\\}.$

Inside the strip $(-\varepsilon,\varepsilon)\times\mathbb{R}$ this set is
bounded by an analytic arc $\\{(b,c_{0}(b)),|b|<\varepsilon\\}$. Moreover any
map $f_{c,b}$ with $c<c_{0}(b)$ can be renormalized at most finitely many
times; and for an open and dense subset of these parameters, the dynamics is
Morse Smale (i.e., is supported by finitely many hyperbolic periodic orbits).
When $c=c_{0}(b)$, the sequence of renormalizations converges towards a
particular unimodal map of the interval.

When $|b|$ is larger but smaller than $1/4$ (so that it is mildly
dissipative), [50] describes the dynamics of $f_{c,b}\in\mathcal{E}_{0}$. In
particular, all maps $f_{c,b}\in\partial\mathcal{E}_{0}$ are infinitely
renormalizable (with renormalization periods eventually equal to $2$), solving
a conjecture by Tresser (which is still open when we don’t assume the mild
dissipation). One may wonder if the converse holds.

######  Question 7.

Let us consider any infinitely renormalizable mildly dissipative map
$f_{c,b}\in\mathcal{E}_{0}$. Is it the limit of maps with positive entropy? Is
it the limit of Morse-Smale maps?

It is also natural to try to describe the boundary $\partial\mathcal{E}_{0}$:
is it a (piecewise smooth) arc? One would like to implement the strategy
developed in [52] for $b$ close to $0$:

######  Question 8.

Let us consider any infinitely renormalizable mildly dissipative map in
$\mathcal{E}_{0}$. Does the sequence of renormalizations converge?

We also don’t know if different combinatorics of the renormalizations may
occur.

######  Question 9.

Does there exist a mildly dissipative map $f_{c,b}\in\mathcal{E}_{0}$ which
admits arbitrarily deep renormalizations with odd periods?

###  3.3 Set of periodic points

One would like to describe the dynamics through its periodic orbits. To any
periodic point $p$, one associates two Lyapunov exponents
$\lambda^{-}(p)\leq\lambda^{+}(p)$. When they do not vanish and have different
sign, we say that $p$ is a saddle.

As mentioned previously, for mildly dissipative real Hénon maps the set of
periodic points is dense in the union of the supports of the invariant
probability measures [49]. The same property holds for any complex Hénon maps
[70].

######  Question 10.

For any real Hénon map, does the closure of the set of periodic orbits support
all the invariant probability measures?

The next step is to describe how periodic saddles are organized. We say that
two saddles $p,q$ are homoclinically related if there exists $k\geq 0$ such
that the invariant curves $W^{u}(p),W^{s}(f^{k}(q))$ (and
$W^{s}(p),W^{u}(f^{k}(q))$ as well) intersect transversally. This defines an
equivalence relation which decomposes the set of periodic saddles into its
homoclinic classes. There may exist infinitely many periodic saddles which are
not homoclinically related, but we conjecture that their hyperbolicity should
drop.

######  Question 11.

For any map $f_{c,b}$ and any infinite set of periodic saddles $(p_{n})$ which
are pairwise not homoclinically related, do we have
$\min(|\lambda^{-}(p_{n})|,|\lambda^{+}(p_{n})|)\underset{n\to\infty}{\longrightarrow}0$?

This questions goes beyond Hénon maps, but [50] implies a positive answer in
the particular case of mildly dissipative $f_{c,b}\in\mathcal{E}_{0}$.

##  4 Symbolic dynamics for real Hénon and Lozi maps (S. Štimac)

Kneading theory is a combinatorial tool to understand the dynamics of a
piecewise monotone map from the interval to itself and was developed by Milnor
and Thurston [107]. Applications extend from the topological classification to
the computation of the entropy, to the counting of periodic orbits, and the
construction of measures of maximal entropy. We propose several problems
connected to the extension of this theory to real Hénon maps and the Lozi maps
$\widetilde{H}_{a,b},L_{a,b}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$,

$\widetilde{H}_{a,b}(x,y)=(1+y-ax^{2},bx),\quad L_{a,b}(x,y)=(1+y-a|x|,bx),$

respectively888observe that $\widetilde{H}_{a,b}$ is affinely conjugated to
$H_{\sqrt{b},-a}$ from the introduction. The Lozi maps are piecewise affine
map that display the same fold and bend behavior as the Hénon maps, but are
usually easier to analyze technically [75, 109].

In [110], the authors developed a kneading theory for the Lozi maps $L_{a,b}$
with $(a,b)\in\mathcal{M}$, where $\mathcal{M}=\\{(a,b)\in\mathbb{R}^{2}:b>0,\
a\sqrt{2}-b>2,\ 2a+b<4\\}$ is a set of parameters for which Misiurewicz proved
the existence of a strange attractor (for details see [110] and [111]). A
kneading sequence $\bar{k}$ is defined as the itinerary of a turning point
$T$, where turning points are points of transversal intersections of the
$x$-axis and the unstable manifold $W^{u}$ of the fixed point $X$ of the
attractor. Any kneading sequence is a bi-infinite sequence of $+$ and $-$.

The kneading set $\mathfrak{K}=\\{\bar{k}^{n}:n\in\mathbb{Z}\\}$ is the set of
all kneading sequences $\bar{k}^{n}$, $n\in\mathbb{Z}$, and every kneading
sequence $\bar{k}=\bar{k}^{n}$, for some $n\in\mathbb{Z}$, has the following
form:

$\bar{k}=\hskip 5.69046pt+^{\infty}\hskip
5.69046ptw\pm\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}\overrightarrow{k}_{0},$

where $\hskip 5.69046pt+^{\infty}\hskip 5.69046pt=\dots+++$, $w=w_{0}\dots
w_{m}$, for some $m\in\mathbb{N}_{0}$,
$\overrightarrow{k}_{0}=k_{0}k_{1}k_{2}\dots$, $w_{0}=-$, $k_{0}=+$,
$w_{i},k_{j}\in\\{-,+\\}$ for $i=1,\dots,m$ and $j\in\mathbb{N}$, and the
little black square $\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}$
indicates where the 0th coordinate is located. Here for $\pm$ one can
substitute any of $+$ and $-$.

In [110], the authors prove that $\mathfrak{K}$ characterizes all itineraries
of all points of the attractor of $L_{a,b}$. The proof is given in two steps.
We say that an itinerary is $W^{u}$-admissible if it is realized by a point on
the unstable manifold $W^{u}$. We first have:

######  Theorem 4.

A sequence $\hskip 5.69046pt+^{\infty}\hskip 5.69046pt\overrightarrow{p}_{n}$,
where $\overrightarrow{p}_{n}=p_{n}p_{n+1}\dots$ such that $p_{n}=-$ for some
$n\in\mathbb{Z}$, is $W^{u}$-admissible if and only if for every kneading
sequence $\hskip 5.69046pt+^{\infty}\hskip
5.69046ptw\pm\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}\overrightarrow{k}_{0}$,
such that $w=p_{n}p_{n+1}\dots p_{n+m}$ for some $m\in\mathbb{N}_{0}$, we have
$\sigma^{m+2}(\overrightarrow{p}_{n})\preceq\overrightarrow{k}_{0}$, where
$\preceq$ is the parity-lexicographical ordering.

Next, we equip the symbolic space with its natural product topology. Using
topological arguments, one may prove:

######  Theorem 5.

A sequence $\bar{p}=\dots
p_{-2}p_{-1}\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}p_{0}p_{1}\dots$
is admissible if and only if for every positive integer $n$ there is a
$W^{u}$-admissible sequence $\bar{q}=\dots
q_{-2}q_{-1}\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}q_{0}q_{1}\dots$
such that $p_{-n}\dots p_{n}=q_{-n}\dots q_{n}$.

######  Problem 2.

Describe the set of kneading sequences $\mathfrak{K}$.

In [92], Ishii developed formulas that can be used to obtain a relation
between parameters $a,b$, a turning point $T=(x_{T},0)$ of the Lozi map
$L_{a,b}$, and its itinerary $\bar{k}$ (that is a kneading sequence of
$L_{a,b}$). This relation is $p(a,b,\bar{k})=x_{T}=q(a,b,\bar{k})$, where
$p=p(a,b,\bar{k})$ is given in formula [91, (4.2)] and $q=q(a,b,\bar{k})$ is
given in formula [91, (4.3)]. Therefore, every kneading sequence $\bar{k}$
gives an equation

$p(a,b,\bar{k})=q(a,b,\bar{k}).$ (1)

Numerical experiments show that if one has two kneading sequences,
$\bar{k}^{0}$ of the rightmost turning point $T_{0}$ and $\bar{k}^{-1}$ of the
leftmost turning point $T_{-1}$, and if these two turning points lie in the
stable manifolds of some periodic points with small periods, then it is
possible to calculate $a$ and $b$ from the corresponding two equations,
implying that these two kneading sequences govern all other kneading
sequences.

####  4.0.1 Example

Let $\bar{k}^{0}=\hskip 5.69046pt+^{\infty}\hskip
5.69046pt\pm\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}+--+^{\infty}$
and $\bar{k}^{-1}=\hskip 5.69046pt+^{\infty}\hskip
5.69046pt-\pm\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}(+-)^{\infty}$.
The equation $p(a,b,\bar{k}^{0})=q(a,b,\bar{k}^{0})$ reads

$a^{4}-6a^{2}-4a+4b^{2}+a^{2}b+(a^{3}+2a-ab)\sqrt{4b+a^{2}}=0,$ (2)

and the equation $p(a,b,\bar{k}^{-1})=q(a,b,\bar{k}^{-1})$ reads

$\frac{4(-a^{2}-2b^{2}+2b+a\sqrt{a^{2}-4b})}{a-2b-\sqrt{a^{2}-4b}}-\left(2+a-\sqrt{a^{2}+4b}\right)\left(3a-\sqrt{a^{2}+4b}\right)=0.$
(3)

Using the “NSolve” command of Wolfram Mathematica produces a unique solution
to this system of equations in the region $a\in[1,2]$, $b\in[0,1]$. This
solution is approximately
$a=1.655319602968851744592,b=0.2765071079677260998121$, see Figure 1.

Figure 1: Graph of (2) is in orange and of (3) in green. Graph of (1) for
$\bar{k}=\hskip 5.69046pt+^{\infty}\hskip
5.69046pt\pm\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}(+--)^{\infty}$
is in magenta and for $\bar{k}=\hskip 5.69046pt+^{\infty}\hskip
5.69046pt-\pm\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}(+--)^{\infty}$
in brown. Graph of the line $2a+b=4$, that is a boundary line of the
Misiurewicz set, is in red.

######  Question 12.

Is it true that any two distinct kneading sequences determine a unique pair of
parameters $(a,b)$, and in that way govern all the other kneading sequences of
$\mathfrak{K}$?

Very recently, in [38], the authors developed a kneading theory for the Hénon
maps $H_{a,b}$ within a set of parameters $\mathcal{WY}$ for which Wang and
Young proved the existence of a strange attractor. This set has positive
measure and consist of maps which are strongly dissipative. We refer to [38]
and [141] for details.

######  Problem 3.

Describe the set of kneading sequences $\mathfrak{K}$ of the Hénon map
$H_{a,b}$, with $(a,b)\in\mathcal{WY}$.

######  Question 13.

Is it true that any two distinct kneading sequences of the Hénon map
$H_{a,b}$, with $(a,b)\in\mathcal{WY}$, determine a unique pair of parameters
$(a,b)$, and in that way govern all the other kneading sequences in
$\mathfrak{K}$ of $H_{a,b}$?

##  5 Hénon maps tangent to the identity (X. Buff)

We propose to investigate the local dynamics of some specific Hénon maps.
Consider the quadratic complex Hénon map
$H_{2}\colon\mathbb{C}^{2}\to\mathbb{C}^{2}$ defined by

$H_{2}\left(x,y\right)=\left(y,x+y^{2}\right).$

The origin is a fixed point and $H_{2}^{\circ 2}$ is tangent to the identity
at the origin.

Note that $H_{2}$ restricts to an orientation reversing diffeomorphism
$H_{2}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$. The dynamics in $\mathbb{R}^{2}$
is well understood. There is an analytic map
$\phi_{2}\colon\mathbb{R}\to\mathbb{R}^{2}$ such that

$H_{2}\circ\phi_{2}(t)=\phi_{2}(t+1)\ \mathrm{and}\
\phi_{2}(t)\sim\left(\frac{-2}{t},\frac{-2}{t}\right)\text{ as }t\to+\infty.$

The curve $\phi_{2}(\mathbb{R})$ is invariant by $H_{2}$ and within
$\phi_{2}(\mathbb{R})$, every orbit converges to the origin in
$\mathbb{R}^{2}$. Outside the origin and $\phi_{2}(\mathbb{R})$, every orbit
diverges to infinity (see Figure 2).

Figure 2: Left: an orbit converging to the origin. Right: an orbit diverging
to infinity.

######  Question 14.

Can we describe the dynamics of $H_{2}$ near the origin in $\mathbb{C}^{2}$ ?

Before specifying this question, let us consider the Hénon map
$H_{3}\colon\mathbb{C}^{2}\to\mathbb{C}^{2}$ defined by

$H_{3}\left(x,y\right)=\left(y,x+y^{3}\right).$

This Hénon map also preserves $\mathbb{R}^{2}$ and the dynamics in
$\mathbb{R}^{2}$ is also completely understood. There is an analytic map
$\phi_{3}\colon\mathbb{R}\to\mathbb{R}^{2}$ such that

$H_{3}\circ\phi_{3}(t)=-\phi_{3}(t+1)\ \mathrm{and}\
\phi_{3}(t)\sim\left(\frac{1}{\sqrt{t}},\frac{-1}{\sqrt{t}}\right)\text{ as
}t\to+\infty.$

The curves $\phi_{3}(\mathbb{R})$ and $-\phi_{3}(\mathbb{R})$ are exchanged by
$H_{3}$. Within those curves, every orbit converges to the origin. Outside
those curves and the origin, every orbit diverges to infinity (see Figure 3
left).

Figure 3: Left: The points are colored according to whether $x+y$ tends to
$+\infty$ (dark grey) or to $-\infty$ (light grey). Right: The dynamics of
$H_{3}^{\circ 2}\colon\Pi_{1}\to\Pi_{1}$ exhibits KAM phenomena.

Set $\omega=e^{{\rm i}\frac{\pi}{8}}$ so that $\omega^{9}=-\omega$ and
consider the real planes $\Pi_{1}\subset\mathbb{C}^{2}$ and
$\Pi_{2}\subset\mathbb{C}^{2}$ defined by

$\Pi_{1}=\left\\{\left(\omega
x,\omega^{3}y\right)~{}:~{}x\in\mathbb{R},~{}y\in\mathbb{R}\right\\}\
\mathrm{and}\ \Pi_{2}=\left\\{\left(\omega^{3}x,\omega
y\right)~{}:~{}x\in\mathbb{R},~{}y\in\mathbb{R}\right\\}.$

Observe that $H_{3}$ exchanges the planes $\Pi_{1}$ and $\Pi_{2}$:

$H_{3}\left(\omega
x,\omega^{3}y\right)=\left(\omega^{3}y,\omega(x-y^{3})\right)\ \mathrm{and}\
H_{3}\left(\omega^{3}x,\omega y\right)=\left(\omega
y,\omega^{3}(x+y^{3})\right).$

The dynamics of $H_{3}^{\circ 2}\colon\Pi_{1}\to\Pi_{1}$ is much more complex
than that of $H_{3}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ (see Figure 3
right).

The second iterate of $H_{3}$ is tangent to the identity at the origin. More
precisely

$H_{3}^{\circ
2}\left(x,y\right)=\left(x,y\right)+\left(y^{3},x^{3}\right)+{\mathcal{O}}\bigl{(}\|x,y\|^{4}\bigr{)}.$

It follows that near the origin, the orbits of $H_{3}^{\circ 2}$ shadow the
orbits of the vector field

$\vec{v}_{3}=y^{3}\partial_{x}+x^{3}\partial_{y}.$

The vector field is a Hamiltonian vector field. It is tangent to the level
curves of the function

$\Phi_{3}=x^{4}-y^{4}.$

Note that

$\Phi_{3}\left(\begin{array}[]{c}\omega x\\\
\omega^{3}y\end{array}\right)={\rm i}(x^{4}+y^{4})$

so that the intersection of the level curves of $\Phi_{3}$ with the real plane
$\Pi_{1}$ are topological circles. Those topological circles are invariant by
the flow of the vector field $\vec{v}_{3}$. It follows from the theory of
Kolmogorov-Arnold-Moser that in any neighborhood of the origin, there is a set
of positive Lebesgue measure of topological circles which are invariant by
$H_{3}$ and on which $H_{3}$ is analytically conjugate to a rotation
$\mathbb{R}/\mathbb{Z}\ni t\mapsto t+\theta\in\mathbb{R}/\mathbb{Z}$ with
bounded type rotation number
$\theta\in(\mathbb{R}\smallsetminus\mathbb{Q})/\mathbb{Z}$. Those invariant
circles are separated by small saddle cycles and small elliptic cycles. The
analytic conjugacies extend to complex neighborhoods of
$\mathbb{R}/\mathbb{Z}$ in $\mathbb{C}/\mathbb{Z}$. This proves that $H_{3}$
has lots of Herman rings.

Coming back to our initial problem, observe that the second iterate of $H_{2}$
is also tangent to the identity at the origin with

$H_{2}^{\circ
2}(x,y)=(x,y)+\left(y^{2},x^{2}\right)+{\mathcal{O}}\bigl{(}\|x,y\|^{3}\bigr{)}.$

It follows that near the origin, the orbits of $H_{2}^{\circ 2}$ shadow the
orbits of the vector field

$\vec{v}_{2}=y^{2}\partial_{x}+x^{2}\partial_{y}.$

The vector field is also a Hamiltonian vector field. The vector field
$\vec{v}_{2}$ is tangent to the level curves of the function

$\Phi_{2}=x^{3}-y^{3}.$

We can no longer apply the theory of Kolmogorov-Arnold-Moser since there is no
invariant real-plane on which the level curves of $\Phi_{2}$ are topological
circles. However, we may wonder whether the complex dynamics of $H_{2}$
exhibits KAM phenomena.

We say that $H_{2}$ has small cycles if for any neighborhood $U$ of the origin
$\boldsymbol{0}$ in $\mathbb{C}^{2}$, there exists a cycle of $H_{2}$ which is
entirely contained in $U\smallsetminus\\{\boldsymbol{0}\\}$.

######  Question 15.

Does $H_{2}$ have small cycles?

######  Question 16.

Does $H_{2}$ have both small saddle cycles and small elliptic cycles?

We say that $H_{2}$ has a Herman ring with rotation number
$\theta\in(\mathbb{R}\smallsetminus\mathbb{Q})/\mathbb{Z}$ if there exists an
annulus $V=\bigl{\\{}z\in\mathbb{C}/\mathbb{Z}~{}:~{}{\rm Im}(z)<h\bigr{\\}}$
with $h>0$, a holomorphic map $\phi\colon V\to\mathbb{C}^{2}$, and an integer
$n\geq 2$ such that

$\forall z\in V,\quad H_{2}^{\circ n}\circ\phi(z)=\phi(z+\theta).$

######  Question 17.

Does $H_{2}$ have a Herman ring?

If the answer is yes, we may consider the set
$\Theta\subset(\mathbb{R}-\mathbb{Q})/\mathbb{Z}$ of rotation numbers $\theta$
such that $H_{2}$ has a Herman ring with rotation number $\theta$.

######  Question 18.

Does $\Theta$ have positive Lebesgue measure? More precisely, is $0$ a
Lebesgue density point of $\Theta$?

We believe that the answers to the previous questions are all affirmative.
Regarding the following question, we do not have an opinion.

######  Question 19.

Assume $H_{2}$ has a Herman ring with bounded type rotation number $\theta$.
Is it possible to find parameters $a\in\mathbb{D}\smallsetminus\\{0\\}$
arbitrarily close to $1$ such that the dissipative Hénon map
$H\colon\mathbb{C}^{2}\to\mathbb{C}^{2}$ defined by

$H(x,y)=\left(ay,x+y^{2}\right).$

has a Herman ring with rotation number $\theta$?

##  6 Quasi-hyperbolicity and uniform hyperbolicity (E. Bedford)

###  6.1 Complex Hénon maps.

Any Hénon map $H_{a,P}(x,y):=(ay+P(x),x)$ where $a\in\mathbb{C}^{*}$ and
$P\in\mathbb{C}[x]$ is a polynomial of degree $d\geq 2$ induces a polynomial
automorphism of the affine plane
$H_{a,P}\colon\mathbb{C}^{2}\to\mathbb{C}^{2}$.

For a general polynomial automorphism
$f\colon\mathbb{C}^{2}\to\mathbb{C}^{2}$, write $(f(x,y)=(P(x,y),Q(x,y))$ and
define its degree $\deg(f):=\max\\{\deg(P),\deg(Q)\\}$. In a celebrated
article [79], Friedland and Milnor have proved the following remarkable result
(see §10.1 below for more details). If the sequence of degrees $\deg(f^{n})$
is unbounded, then $f$ is actually conjugated to a composition of Hénon maps
$H_{a_{1},P_{1}}\circ\cdots\circ H_{a_{k},P_{k}}$, and
$\deg(f)=\deg(P_{1})\cdots\deg(P_{k})$.

Any such composition will be called a generalized Hénon map.

###  6.2 Quasi-expanding Hénon maps

Suppose $f$ is a generalized Hénon map of degree $d\geq 2$, and let
${\mathcal{S}}$ denote its set of (periodic) saddle points. It is known to be
infinite, and its distribution represents the unique measure of maximal
entropy, see [16].

Given a saddle point $p$, denote by
$E^{u}_{p}=\\{v\in\mathbb{C}^{2},\|Df^{n}_{p}v\|\to\infty\\}$ the associated
unstable direction, by $W^{u}_{\mathrm{loc}}(p)$ its local unstable manifold,
and by $W^{u}(p)$ its global unstable manifold.

We now introduce the following three conditions measuring the expansion of
$f$.

Condition 1  For each $p\in{\mathcal{S}}$, there is a metric on $E^{u}_{p}$
which is expanded by $Df_{p}$ with a uniform bound independent on $p$. More
specifically, this means that there exists $\kappa>1$ such that for each $p$,
there is a metric $\|\cdot\|_{j}$ on $E^{u}_{f^{j}(p)}$ so that for nonzero
$v\in E^{u}_{f^{j}(p)}$, we have

$\|Df_{f^{j}(p)}v\|_{j+1}\geq\kappa\|v\|_{j}$

for each $j$.

If $f$ is a Hénon map, we can consider its Green function (see [17])

$G^{+}(x,y)=\lim_{n\to\infty}\frac{1}{d^{n}}\log\max\\{1,|f^{n}(x,y)|\\}.$

We let $W_{r}^{u}(p)$ denote the connected component of $B(p,r)\cap W^{u}(p)$
containing $p$, where $W^{u}(p)$ is the unstable manifold at $p$ and $B(p,r)$
the euclidean ball in $\mathbb{C}^{2}$.

Condition 2  The unstable manifolds $W^{u}(p)$ satisfy the proper, locally
bounded area condition: there exist $\varepsilon>0$ and $A<\infty$ such that
for each $\delta>0$ there is an $\eta>0$ such that for each saddle point $p$
we have: $W_{\varepsilon}^{u}(p)$ is closed in $B(p,\varepsilon)$,
$\mathrm{Area}(W_{\varepsilon}^{u}(p))\leq A$, and
$\sup_{W_{\delta}^{u}(p)}G^{+}\geq\eta$ (see [20, Corollary 3.5]).

Recall that each unstable manifold is uniformized by an entire map
$\xi_{p}\colon\mathbb{C}\to W^{u}(p)\subset\mathbb{C}^{2}$ with
$\xi_{p}(0)=p$. Using the Green function, we may normalize it by putting
$\hat{\xi}_{p}(\zeta):=\xi_{p}(\alpha\zeta)$ so that

$\max_{|\zeta|\leq 1}G^{+}(\hat{\xi}_{p}(\zeta))=1.$ (4)

The last condition we want to mention is:

Condition 3  The normalized maps $\\{\hat{\xi}_{p}:p\in{\mathcal{S}}\\}$ form
a normal family of entire mappings.

A consequence of Condition 3 is that for all $x\in\overline{\mathcal{S}}$ we
define a family of unstable manifolds by setting
$W^{u}(x):=\xi_{x}(\mathbb{C})$.

For generalized Hénon maps, Conditions 1, 2, and 3 are equivalent (see [20]),
and in case one/all of them hold, we say that $f$ is quasi-expanding. A map
$f$ is said to be quasi-hyperbolic if both $f$ and $f^{-1}$ are quasi-
expanding.

Furthermore, a theorem from [12] asserts a quasi-hyperbolic map is uniformly
hyperbolic if and only if there is no tangency between $W^{u}(x_{1})$ and
$W^{s}(x_{2})$ for any $x_{1},x_{2}$ in the closure of ${\mathcal{S}}$.

Recall the following standard definitions (see [17]). Let $K^{\pm}$ denote the
set of points with bounded forward orbits for $f^{\pm}$. We denote
$J^{\pm}:=\partial K^{\pm}$. We also define $J:=J^{+}\cap J^{-}$ and $J^{*}$
to be the closure of ${\mathcal{S}}$ (hence $J^{*}\subset J$).

######  Question 20.

If $f$ is quasi-hyperbolic, then is ${\rm int}(K^{+})$ the union of a finite
number of basins of sink orbits?

######  Question 21.

If $f$ is quasi-hyperbolic, is $J=J^{*}$?

######  Question 22.

If $f$ is quasi-hyperbolic, then is there no wandering Fatou component?

######  Question 23.

If $f$ is quasi-hyperbolic, do the unstable slices satisfy a John-type
condition (as in [19])?

######  Question 24.

If $f$ is quasi-hyperbolic and dissipative, and if $J$ is connected, do the
external rays land at $J$? Is $J$ a finite quotient of the real solenoid?

The answers to all these questions are “yes” in the uniformly hyperbolic case
by [19]. We thus ask whether these properties remain true in the quasi-
hyperbolic case (in which case the dynamics is expected to be close to being
hyperbolic). Observe that Questions 20 and 23 have negative answer for a
general Hénon map, and that Question 22 was disproved recently by Berger and
Biebler for some Hénon map of degree $6$ (which are known not to be quasi-
hyperbolic). Question 21 is a well-known problem. Beside the hyperbolic case
[17], it has been solved in a few other cases [84, 71], but the case of a
generalized Hénon map remains elusive.

###  6.3 Surface automorphisms

Let $X$ be any compact complex Kähler surface, and let $f\colon X\to X$ be any
holomorphic automorphism having positive topological entropy $\log\lambda$,
$\lambda>1$. By Cantat [42], $X$ is isomorphic to either the blow-up of
$\mathbb{P}^{2}$ at at least $10$ points, or a K$3$ surface, or an Enriques
surface, or an Abelian surface.

In this context, Conditions 1 and 2 are still meaningful, but we do not have a
Green function $G^{+}$. However, since the dynamical degree of $f$ is
$\lambda>1$, there exists an expanded positive closed $(1,1)$ current $T^{+}$
with $f^{*}T^{+}=\lambda T^{+}$ (see [42]). In this case, we can replace the
normalization (4) with a condition involving the mass of a slice of the
current $T^{+}$:

${\rm Mass}(T^{+}|_{\xi_{p}(|\zeta|<1)})=1$ (5)

We can thus formulate a Condition $3^{\prime}$, which is Condition 3 with the
normalization (5).

######  Question 25.

Is Condition $3^{\prime}$ equivalent to 1 and 2? And do Questions 20, 21, 22
above hold for quasi-hyperbolic surface automorphisms?

###  6.4 Real maps

Let us now suppose that $f$ is a real surface automorphism. In other words, we
suppose $X$ to be projective and defined by real polynomial equations, and $f$
to be also defined over the real numbers. We may thus consider the restriction
map $f_{\mathbb{R}}$ to the set of real points $X_{\mathbb{R}}$. Observe that
$h_{\mathrm{t}op}(f_{\mathbb{R}})\leq\log\lambda$. It was proved in [20] that
for generalized Hénon maps $h_{\mathrm{t}op}(f_{\mathbb{R}})=\log\lambda$
implies $f$ to be quasi-hyperbolic.

######  Question 26.

If $f$ is a real surface automorphism such that the entropy of
$f_{\mathbb{R}}$ is the same as the entropy of $f$, does it follow that $f$ is
quasi-hyperbolic?

We refer to [56] for a discussion of real surface automorphisms satisfying
this condition on the entropy.

##  7 Parameter loci for the Hénon family (Y. Ishii)

###  7.1 Connectedness locus

Consider the complex Hénon family:

$f_{c,b}(x,y):=(x^{2}+c-by,x),$

where $(c,b)\in\mathbb{C}^{2}$ is a parameter.999We include the case $b=0$ to
simplify the presentation. Let $J_{c,b}$ be the Julia set of $f_{c,b}$: by
definition this is the intersection between the boundaries of the sets of
points having bounded forward (resp. backward) orbits. By extension, we let
$J_{c,0}$ be the Julia set of $p(z)=z^{2}+c$. The _connectedness locus_ of
$f_{c,b}$ is defined as

$\mathcal{M}=\big{\\{}(c,b)\in\mathbb{C}^{2}:J_{c,b}\mbox{ is
connected}\big{\\}}.$

######  Conjecture 1.

$\mathcal{M}$ is disconnected.

It has been shown that $\mathcal{M}\cap\mathbb{R}^{2}$ is disconnected [3],
which partially supports the conjecture above.

###  7.2 Horseshoe locus

We say that $f_{c,b}$ is a _complex hyperbolic horseshoe_ if $J_{c,b}$ is a
hyperbolic set for $f_{c,b}$ and the restriction $f_{c,b}:J_{c,b}\to J_{c,b}$
is topologically conjugate to the full $2$-shift. The _complex hyperbolic
horseshoe locus_ is defined as

$\mathcal{H}_{\mathbb{C}}=\big{\\{}(c,b)\in\mathbb{C}^{2}:f_{c,b}\mbox{ is a
complex hyperbolic horseshoe}\big{\\}}.$

One can see that $\mathcal{H}_{\mathbb{C}}$ is not simply connected since the
monodromy representation:

$\rho:\pi_{1}(\mathcal{H}_{\mathbb{C}})\longrightarrow\mathrm{Aut}(\\{0,1\\}^{\mathbb{Z}})$

of the fundamental group of $\mathcal{H}_{\mathbb{C}}$ (with the base-point at
$(c,b)=(-4,0)$) to the group of shift-commuting automorphisms of
$\\{0,1\\}^{\mathbb{Z}}$ is surjective (see, e.g., [1, 21]).

######  Question 27.

Is the locus $\mathcal{H}_{\mathbb{C}}$ connected?

For $(c,b)\in\mathbb{R}^{2}$, we can consider the restriction of $f_{c,b}$ to
$\mathbb{R}^{2}$ and we can analogously define the _real hyperbolic horseshoe
locus_ $\mathcal{H}_{\mathbb{R}}\subset\mathbb{R}^{2}$. One of the main result
of [2] states that $\mathcal{H}_{\mathbb{R}}$ is connected and simply
connected (see also [22]).

####  7.2.1 Isentropes

Take again $(c,b)\in\mathbb{R}^{2}$ and consider the restriction
$f_{c,b}|_{\mathbb{R}^{2}}:\mathbb{R}^{2}\to\mathbb{R}^{2}$. Let
$h_{\mathrm{top}}(f_{c,b}|_{\mathbb{R}^{2}})$ be the topological entropy of
the real Hénon map $f_{c,b}|_{\mathbb{R}^{2}}$. For every $0\leq\alpha\leq\log
2$, the _isentrope_ is defined as

$\mathcal{E}_{\alpha}=\big{\\{}(c,b)\in\mathbb{R}^{2}:h_{\mathrm{top}}(f_{c,b}|_{\mathbb{R}^{2}})=\alpha\big{\\}}.$

In a topological term, monotonicity of the topological entropy of the real
Hénon map $f_{c,b}|_{\mathbb{R}^{2}}$ can be formulated as

######  Question 28 (van Strien [138]).

Is the isentrope $\mathcal{E}_{\alpha}$ connected for any $0\leq\alpha\leq\log
2$ ?

Milnor and Tresser [108] showed it is true for cubic polynomials. The main
result of [2] implies that the locus $\mathcal{E}_{\log 2}$ is connected and
simply connected (see also [22]).

Several articles attempt at giving lower bounds for topological entropy of
real Hénon maps, e.g., [114, 115]. Among others, the paper [114] rigorously
showed that $h_{\mathrm{top}}(f_{c,b}|_{\mathbb{R}^{2}})>0.46469$ for the
classical Hénon’s parameter, and this bound is believed to be close to
optimal. For upper bounds, the paper [16] has shown that
$h_{\mathrm{top}}(f_{c,b}|_{\mathbb{R}^{2}})<\log 2$ if and only if the Julia
set of $f_{c,b}$ (as a complex dynamical system) is not contained in
$\mathbb{R}^{2}$. However, there is no algorithm which provides rigorous (non-
trivial) upper bounds.

We thus propose the following problem.

######  Question 29.

Construct an algorithm to compute a rigorous upper bound for the topological
entropy of a real Hénon map $f_{c,b}|_{\mathbb{R}^{2}}$.

Probably the only existing formula for (non-trivial) upper bound is given in
[146]. However, according to Yomdin himself, the bound in the current form is
far from sharp and would not give non-trivial ones.

##  8 Topology and rigidity of Hénon maps (R. Dujardin)

For polynomials and rational maps in dimension 1, there is a well-known list
of exceptional examples whose Julia sets and dynamical properties are
unexpectedly regular: Chebychev polynomials, monomial mappings and Lattès
examples. They can characterized in many different ways, see e.g., [148, 46].

For generalized Hénon maps (as defined in §6.1) it is expected that no such
exceptional example exists, but not so many actual results in this direction
are known:

  * •

Brunella proved in [40] that a generalized Hénon map cannot preserve an
algebraic foliation of $\mathbb{C}^{2}$, i.e., a singular algebraic foliation
by holomorphic curves. Here by preserving we mean that $f$ maps leaves into
leaves.

  * •

Bedford and Kim proved in [13, 14] that neither $J^{+}$ nor $J^{-}$ (see §6.2
for a definition) can be a smooth $C^{1}$ submanifold, nor a semi-analytic
set.

Here we propose a few rigidity questions related to these results.

The first question is about a quantitative reinforcement of the Bedford-Kim
theorem. Recall from the introduction, the definition of the standard
quadratic Hénon map $H_{a,c}(x,y):=(ay+x^{2}+c,ax)$, and the definition of
$K^{+}$ and $J^{+}$ from §6.2.

For $(a,c)$ close to $(0,0)$, $H_{a,c}$ is a small perturbation of the
monomial map $(x,0)\mapsto(x^{2},0)$, whose Julia set is smooth, and in this
case $J^{+}_{a,c}$ is a topological 3-manifold.

######  Question 30.

Give an asymptotic expansion of the Hausdorff dimension of $J^{+}_{a,c}$ as
$(a,c)$ tends to $(0,0)$. In particular is there a uniform lower bound of
$\dim(J^{+}_{a,c})$ of the form $\dim(J^{+}_{a,c})\geq 3+h(a)$ with $h(a)>0$
in the neighborhood of $c=0$?

Note that $(a,c)\mapsto\dim(J^{+}_{a,c})$ is real analytic in the domain where
$H_{a,c}$ is hyperbolic (this was proved for one-dimensional maps by Ruelle in
[125], and by Wolf [143] for polynomial automorphisms). It is not clear
whether the dimension of the Julia set remains real-analytic when $H_{a,c}$
degenerates to a unimodal map, for instance in a full neighborhood of
$(a,c)=(0,0)$.

Can we make Brunella’s theorem local? More precisely:

######  Question 31.

Is it possible to find a generalized complex Hénon map $f$, an open set
$U\subset\mathbb{C}^{2}$ intersecting $J^{+}$ and a holomorphic foliation of
$U$ such that $J^{+}\cap U$ is a union of leaves of this foliation?

We conjecture that the answer to this question is “no”. The answer is
presumably easier if we assume that $U\cap J^{*}\neq\emptyset$. It is also
possible that if $f$ is dissipative (i.e., $|\det(Df)|<1$), the assumption
that $J^{-}$ is foliated is stronger than the assumption that $J^{+}$ is
foliated (see [11, §2]).

This would imply in particular that a generalized Hénon map cannot preserve a
(transcendental) holomorphic foliation $\mathcal{F}$ of $\mathbb{C}^{2}$.
Indeed in such a case, consider the leaf $L$ through a saddle periodic point
$p$: $L$ must be mapped into itself by $f^{n}$, hence coincide with the stable
$W^{s}(p)$ or unstable $W^{u}(p)$ manifolds (see again §6.2 for a discussion
of these objects); changing $f$ to $f^{-1}$ if necessary, we may assume that
$L=W^{s}(p)$, and since $W^{s}(p)$ is dense in $J^{+}$ it follows that $J^{+}$
is a union of leaves of $\mathcal{F}$.

######  Remark 1.

Note that the basin of attraction of the super-attracting point at infinity
$\Omega(f):=\\{(x,y),|f^{n}(x,y)|\to\infty\\}$ carries a natural
(transcendental) holomorphic foliation which is $f$-invariant defined by the
holomorphic $1$-form $\partial G^{+}$, see [88, §7]. However this foliation
does not extend to $\mathbb{C}^{2}$ (otherwise it would extend holomorphically
to $\mathbb{P}^{2}$ which is absurd, see [69, §3]).

Related results include:

  * •

the classification of holomorphic Anosov diffeomorphisms on surfaces by Ghys
[80], in which a basic step is to prove that stable and unstable laminations
are actually holomorphic foliations;

  * •

the classification of birational maps preserving algebraic foliations by
Cantat and Favre [43];

  * •

the work of Pinto, Rand and others on the smooth rigidity of hyperbolic
diffeomorphisms on surfaces (see e.g., [122]).

If the stable lamination is holomorphic, then by holonomy the unstable slices
are holomorphically equivalent. We can now forget the foliation and ask about
holomorphic equivalence of stable/unstable slices.

######  Question 32.

Under which circumstances is it possible that there exist saddle points $p$
and $q$, and relative open subsets $U\subset W^{u}(p)$ and $V\subset W^{u}(q)$
such that $U\cap K^{+}$ is biholomorphic to $V\cap K^{+}$?

One obvious possibility is that $p$ and $q$ belong to the same cycle, and that
the biholomorphism is induced by the action of $f$. We suspect that this is
the only possibility.

A variant of this problem is when $p$ and $q$ are associated to different
mappings.

######  Question 33.

Let $f_{1}$, $f_{2}$ be two generalized Hénon maps. Under which circumstances
is it possible that some local unstable slice of $f_{1}$ (i.e., a set of the
form $K^{+}\cap U$, where $U$ is a relatively open subset of an unstable
manifold) is biholomorphically equivalent to an unstable slice of $f_{2}$?

Since a local unstable slice of a generalized Hénon map contains essentially
complete information, we expect that this can happen only if $f_{1}$ and
$f_{2}$ are related by some algebraic correspondence. This question was raised
in [72, Remark 4.4] for $f_{2}=f_{1}^{-1}$, and a complete understanding would
imply the main conjecture of [72]. The analogous question of existence of
local biholomorphisms between Julia sets for 1-dimensional rational maps was
addressed in [73, 95].

Since a local unstable slice of a generalized Hénon maps contains essentially
a complete information about unstable multipliers, the previous question is
reminiscent of the classical “spectral rigidity” problem:

######  Question 34.

To which extent is a generalized Hénon map determined by the list of its
unstable multipliers (resp. by the list of moduli of its unstable
multipliers)?

We refer to [94] for a proof that the list of moduli of all multipliers
determine a finite set of conjugacy classes of rational map of the Riemann
sphere.

##  9 Statistical properties of complex Hénon maps (F. Bianchi and T.-C. Dinh)

We denote in this section by $f$ a complex Hénon map and by $\mu$ its unique
measure of maximal entropy [17, 16, 132]. We are interested in the statistical
properties of $\mu$ and of other natural invariant measures associated to such
systems.

###  9.1 Thermodynamics for Hénon maps

Consider a continuous function $\phi\colon\mathbb{C}^{2}\to\mathbb{R}$, that
will be called a _weight_. Following [124] one can define the _pressure_
$P(\phi)$ as

$P(\phi):=\sup\big{(}h_{\nu}+\langle\nu,\phi\rangle\big{)},$

where the supremum is taken over all invariant probability measures $\nu$ for
$f$ and $h_{\nu}$ denotes the measure-theoretic entropy of $\nu$. A measure
$\nu_{0}$ maximising the above supremum is called an _equilibrium state_
associated to $\phi$ and is necessarily ergodic when it is unique. The
equilibrium state associated to $\phi\equiv 0$ is the measure of maximal
entropy $\mu$. We refer to [123] for an account on the properties of
equilibrium states in one-dimensional complex dynamics and to [7, 45, 47] and
references therein for the case of real Hénon maps and diffeomorphisms of
compact manifolds satisfying some hyperbolicity assumptions.

######  Problem 4.

Prove the existence and the uniqueness of the equilibrium state $\mu_{\phi}$
associated to any sufficiently regular weight $\phi$ (for instance, every
Hölder continuous $\phi$, and possibly with some bound on
$\max\phi-\min\phi$).

Recall that saddle points are equidistributed with respect to the measure of
maximal entropy [15]. Namely, we have

$\frac{1}{d^{n}}\sum_{x\in SP_{n}}\delta_{x}\to\mu,$ (6)

where $d$ is the algebraic degree of $f$ (or, equivalently, $\log d$ is the
topological entropy of $f$, and the measure-theoretic entropy of $\mu$) and
$SP_{n}$ is the set of the saddle $n$-periodic points of $f$.

######  Question 35.

Suppose $\phi$ is sufficiently regular, so that the equilibrium state
$\mu_{\phi}$ exists and is unique. Is it true that

$\frac{1}{e^{nP(\phi)}}\sum_{x\in
SP_{n}}e^{S_{n}(\phi)}\delta_{x}\to\mu_{\phi}\quad?$ (7)

A version of the previous question has been established in [29] in the
(expanding) setting of endomorphisms of $\mathbb{P}^{k}_{\mathbb{C}}$, and in
particular for polynomials maps on $\mathbb{C}$.

Of a somehow different flavour, we recall that an explicit speed of
convergence in (6) is unknown. We believe it is a very natural and challenging
question to quantify such convergence when testing against sufficiently
regular functions.

######  Question 36.

Is the convergence (6) exponentially fast against Hölder continuous
observables? Is that also the case for (7)?

###  9.2 Statistical properties of equilibrium states and spectral gap for the
transfer operators

Suppose the existence and the uniqueness of an equilibrium state $\mu_{\phi}$
have been established. The (deterministic) problem of describing all orbits in
the support of $\mu_{\phi}$ is essentially impossible as this support should
be contained in the set of points with chaotic behaviour in both forward and
backward time. It is natural to adopt a probabilistic (or statistical)
approach to this problem, to consider an _observable_
$g\colon\mathbb{C}^{2}\to\mathbb{R}$, and to view the sequence $\\{g\circ
f^{j}\\}_{j\in\mathbb{N}}$ as a sequence of random variables on the
probability space $(\mathbb{C}^{2},\mu_{\phi})$. Since $\mu_{\phi}$ is
invariant, these random variables have the same distribution. They are however
not independent, since they arise from a deterministic setting. The first goal
is thus to show that the correlations $\langle\mu_{\phi},g\circ f^{j_{1}}\cdot
g\circ f^{j_{2}}\rangle-\langle\mu_{\phi},g\rangle^{2}$ go to zero in a
quantifiable way, as $|j_{2}-j_{1}|\to\infty$, see for instance [140, Problem
2]. When this happens and the convergence is fast enough, the sequence
$\\{g\circ f^{j}\\}_{j\in\mathbb{N}}$ is then expected to satisfy a list of
properties which are typical of independent identically distributed (i.i.d.)
random variables.

As a first step, since $\mu_{\phi}$ is ergodic, Birkhoff theorem asserts that

$\frac{1}{n}S_{n}(g)(x):=\frac{1}{n}(g(x)+g\circ f(x)+\dots+g\circ
f^{n-1}(x))\to\langle\mu_{\phi},g\rangle:=\int_{\mathbb{C}^{2}}g\,{\rm
d}\mu_{\phi}$ (8)

for $\mu_{\phi}$-almost every $x$ and every $g\in L^{1}(\mu_{\phi})$. This can
be seen as a version of the law of large numbers in this setting. The next
step is to show the Central Limit Theorem (CLT) for sufficiently regular
observables. As in the case of i.i.d. random variables, this CLT gives the
rate of the above convergence (8).

######  Problem 5.

Show that $\mu_{\phi}$ satisfies the CLT for Hölder continuous observables.
Namely, prove that, for any Hölder continuous observable $g$, there exists
$\sigma\geq 0$ such that for any interval $I\subset\mathbb{R}$, we have

$\lim_{n\to\infty}\mu_{\phi}\left(\left\\{\frac{S_{n}(g)-n\langle\mu_{\phi},g\rangle}{\sqrt{n}}\in
I\right\\}\right)=\begin{cases}1\mbox{ when }I\mbox{ is of the form
}I=(-\delta,\delta)&\mbox{ if }\sigma^{2}=0,\\\
\frac{1}{\sqrt{2\pi\sigma^{2}}}\displaystyle\int_{I}e^{-t^{2}/(2\sigma^{2})}\,{\rm
d}t&\mbox{ if }\sigma^{2}>0.\end{cases}$

In the case of the maximal entropy measure $\mu$, the CLT was established in
[30]. A natural question is also to characterize the observables for which
$\sigma=0$.

Sequences of (almost) independent random variables are also expected to
satisfy _large deviations_ properties. Recall that a coboundary $g$ is an
observable of the form $\varphi\circ f-\varphi$.

######  Problem 6.

Show that $\mu_{\phi}$ satisfies the Large Deviation Principle (LDP) for
Hölder continuous observables. Namely, prove that, for any Hölder continuous
observable $g$ with $\langle\mu_{\phi},g\rangle=0$ and which is not a
coboundary, there exists a non-negative, strictly convex function $c$ which is
defined on a neighborhood of $0\in\mathbb{R}$, vanishes only at $0$, and such
that, for all $\epsilon>0$ sufficiently small,

$\lim_{n\to\infty}\frac{1}{n}\log\mu_{\phi}\left(\left\\{x\in
X\colon\frac{S_{n}(g)(x)}{n}>\epsilon\right\\}\right)=-c(\epsilon).$

Note that this question is still open even for the measure of maximal entropy.

A possible unified approach to the above statistical properties would be to
find Banach spaces (containing all Hölder continuous functions) where a
suitable _Ruelle-Perron-Frobenius (transfer) operator_ associated to $f$ would
turn out to be a strict contraction on the complement of an invariant line,
see for instance [7, 81, 124]. In the case of endomorphisms of
$\mathbb{P}^{k}_{\mathbb{C}}$ in any dimension (in particular for any
$1$-dimensional complex polynomial), this is the main result of [28] (see also
[105, 126]).

######  Question 37.

Do there exist norms for functions on the Julia set which are bounded on
Hölder continuous functions, contract (on the complement of an invariant line)
under the action of $f_{*}$ (or, more generally, of
$f_{*}(e^{\phi-P(\phi)}\cdot)$), and such that the contraction is stable by
perturbation of $\phi$?

In the case of hyperbolic maps, such a good Banach space has been introduced
by Blank-Keller-Liverani [35, 103]. The norm is obtained by combining a
regularity condition on the unstable manifolds together with a dual condition
on the stable manifolds. Note that this was the starting point of a long story
(see, e.g., [82, 8]). As Hénon maps are only non-uniformly hyperbolic (so that
stable and unstable manifolds do not behave nicely in general) the above
result does not apply here.

A positive answer to Question 37 would also give a unified proof for many
statistical properties of independent interest (including the Large Deviations
as in Problem 6), without the need of an _ad hoc_ proof for each of them. For
instance, the Local Central Limit Theorem (LCLT) and the Almost Sure
Invariance Principle (ASIP) are both satisfied by sequences of i.i.d., and
provide stronger results than the CLT, see, e.g., [121, 81] for definitions
and criteria.

######  Problem 7.

Let $\mu_{\phi}$ as in Problem 4. Show that all Hölder continuous observables
which are not coboundaries satisfy the LCLT and the ASIP with respect to
$\mu_{\phi}$.

Recall that the ASIP implies the Almost Sure Central Limit Theorem and the Law
of the Iterated Logarithm.

###  9.3 Higher dimension and other generalizations

Until now, we restricted our attention to Hénon maps, i.e., polynomial
automorphisms of $\mathbb{C}^{2}$. On the other hand, Problems 6 and 7, and
Question 37 make perfect sense for the equilibrium measures of general
rational maps once this measure has been successfully defined. We review below
some partial results that have been obtained in more general (invertible)
settings than Hénon maps.

A polynomial automorphism of $\mathbb{C}^{k}$ is said to be regular if the
indeterminacy sets of the extensions to $\mathbb{P}^{k}_{\mathbb{C}}$ of $f$
and $f^{-1}$ are non-empty and disjoint (observe that every Hénon map in
dimension 2 satisfies this assumption, as these two sets are two distinct
points). The construction of the measure of maximal entropy is given in [132],
and the equidistribution of saddle points with respect to this measure is
proved in [63] (see [30, 57] for the exponential mixing and the CLT in this
case). More generally, one can also consider birational meromorphic maps of
$\mathbb{P}^{k}_{\mathbb{C}}$, see [10, 55, 53, 68] for the construction of
the measure of maximal entropy and its properties.

Given integers $1\leq p<k$ and open bounded convex domains
$M\Subset\mathbb{C}^{p}$ and $N\Subset\mathbb{C}^{k-p}$, a _horizontal-like
map_ is a proper holomorphic map from a vertical subset to a horizontal subset
of $M\times N$ which geometrically expands in $p$ directions and contracts in
$k-p$ directions, see [61] for the precise definition. In this setting, the
unique measure of maximal entropy has been constructed and studied in [58, 61,
67]. In the invertible case, the CLT for this measure can be deduced from
[30].

One can also consider automorphisms of compact Kähler manifolds, see for
instance [42, 60, 68] for the construction of the measure of maximal entropy
and its properties. This setting shares a number of features with that of
Hénon maps (in dimension 2) and regular automorphisms (in any dimensions). On
the other hand, the compactness of the manifold makes it more difficult to
apply pluripotential techniques as in the case of Hénon maps. For instance,
the proof of the CLT for the measure of maximal entropy, given in [31],
requires the use of the theory of _superpotentials_ on such manifolds [62].

##  10 Towards higher dimensions and complex differential geometry (C. Favre)

###  10.1 Hénon maps and the group of polynomial automorphisms of
$\mathbb{C}^{2}$.

Let $\mathrm{Aut}[\mathbb{C}^{2}]$ be the group of polynomial automorphisms of
$\mathbb{C}^{2}$. Recall the definition of degrees of a polynomial
automorphism of the affine plane from §6.1. Jung [96] proved that the group
$\mathrm{Aut}[\mathbb{A}^{2}_{\mathbb{C}}]$ is generated by affine
transformations and triangular maps of the form $E_{P}(x,y):=(x,y+P(y))$. And
the more precise version of Friedland-Milnor’s main theorem ([79]) states that
either $f\in\mathrm{Aut}[\mathbb{A}^{2}_{\mathbb{C}}]$ is conjugated to a
generalized Hénon maps $H_{a_{1},P_{1}}\circ\cdots\circ H_{a_{k},P_{k}}$ and
$\deg(f^{n})\asymp(d_{1}\cdots d_{k})^{n}$ for all $n$; or $\deg(f^{n})$
remains bounded and $f$ is conjugated to an affine map or to a triangular map.

Let us state the following general problem in vague terms.

######  Problem 8.

Describe the growth type of the sequence $\\{\deg(f^{n})\\}$ for any
polynomial automorphism $f$ of $\mathbb{C}^{d}$, $d\geq 3$.

Very few results are known. Recall that Russakovski and Shiffman [129]
observed that

$\deg(f^{n+m})\leq\deg(f^{n})\deg(f^{m})$

for all $n,m\geq 0$ so that the following limit
$\lambda(f):=\lim_{n}\deg(f^{n})^{1/n}$ exists. We refer to it as the
dynamical degree of $f$.

The case of cubic automorphisms on $\mathbb{C}^{3}$, and the case of
automorphisms obtained as a composition of an affine transformation and a
triangular one were considered by Blanc and Van Santen [34, 33]. Their
computations lead them to formulate the following intriguing problem. A weak
Perron number is an algebraic integer $\lambda\geq 1$ such that all its Galois
conjugates satisfy $|\mu|\leq\lambda$.

######  Question 38.

Is the dynamical degree of any polynomial automorphism of $\mathbb{C}^{d}$
equal to a weak Perron number of degree $\leq d-1$?

It has been proven in [51], that $\lambda(f)$ is an algebraic number of degree
$\leq 6$ for any polynomial automorphism $f$ of $\mathbb{C}^{3}$.

The case $\lambda(f)=1$ is also particularly interesting.

######  Question 39.

Suppose $f$ is a polynomial automorphism of $\mathbb{C}^{d}$ satisfying
$\lambda(f)=1$. Is it true that $\deg(f^{n})\asymp n^{k}$ for some
$k\in\mathbb{N}$? Moreover, if $k\geq 1$, does $f$ preserve a rational
fibration?

Urech proved that $\deg(f^{n})$ tends to infinity whenever it is unbounded,
see [136]. His result was made stronger by Cantat and Xie in [44]: there
exists a universal function $\sigma\colon\mathbb{N}\to\mathbb{N}$ such that
$\limsup\sigma=\infty$ and $\deg(f^{n})\geq\sigma(n)$. They raised the
following weaker form of the previous problem.

######  Question 40.

Suppose that $\lambda(f)=1$, and $\deg(f^{n})$ is unbounded for some
polynomial automorphism $f$ of $\mathbb{C}^{d}$. Does there exist $C>0$ such
that $\deg(f^{n})\geq Cn$?

Let $\mathrm{Tame}(3)$ be the subgroup of polynomial automorphisms of
$\mathbb{C}^{3}$ which is generated by affine and triangular transformations.
A theorem of Shestakov and Umirbaev [130] states that $\mathrm{Tame}(3)$ is a
strict subgroup of the full group of polynomial automorphisms of
$\mathbb{C}^{3}$ (as opposed to the $2$-dimensional situation). A decisive
progress on the structure of $\mathrm{Tame}(3)$ was recently made by Lamy and
Przytycky [102], who constructed a $\mathrm{CAT}(0)$-complex $\mathcal{C}$
over which $\mathrm{Tame}(3)$ acts by isometries.

######  Question 41.

Is it possible to characterize those $f\in\mathrm{Tame}(3)$ for which
$\lambda(f)=1$ in terms of their action on $\mathcal{C}$?

###  10.2 Hénon maps and compact complex manifolds.

Consider any generalized Hénon map $f=H_{a_{1},P_{1}}\circ\cdots\circ
H_{a_{k},P_{k}}$ as in the previous section. Recall that $f$ extends to the
projective plane $\mathbb{P}^{2}_{\mathbb{C}}$ as a birational map contracting
the line at infinity to the super-attracting fixed point $p=[1:0:0]$. The
topology of the basin of attraction of this point
$\Omega(f):=\\{q\in\mathbb{C}^{2},\,f^{n}(q)\to p\\}$ has been explored by
Hubbard and Oberste-Vorth [88]. They also observed that the map $f$ acts
properly discontinuously on $\Omega(f)$ so that the space of orbits
$S(f):=\Omega(f)/\langle f\rangle$ is naturally a complex surface. One can
then construct a compact complex surface $\check{S}(f)$ having an isolated
normal singularity at a point $0\in\check{S}(f)$ such that
$\check{S}(f)\setminus\\{0\\}$ is biholomorphic to $S(f)$. The minimal
resolution of $\check{S}(f)$ is a compact complex surface $\bar{S}(f)$ that is
non-Kähler, contains no smooth rational curve of self-intersection $-1$, and
satisfies $b_{1}(\bar{S}(f))=1$. In Kodaira’s classification of surfaces [9],
it belongs to the class VII0 which is arguably the most mysterious class of
compact complex surfaces. Dloussky and Oeljeklaus [65] have investigated when
these surfaces carry global holomorphic vector fields.

######  Question 42.

Let $f$ be any generalized Hénon map. Describe the set of all generalized
Hénon maps $g$ such that $\bar{S}(f)$ is biholomorphic to $\bar{S}(g)$.

It follows from [76, Proposition 2.1] that under the preceding assumptions,
$f$ and $g$ have the same degree and the same jacobian. Some partial results
have been obtained by R. Pal for maps of the form $H_{a,P}$ [119] generalizing
former works on quadratic Hénon maps by Bonnot-Radu-Tanase [37].

Surfaces $\bar{S}(f)$ carry only finitely many rational curves that are all
contracted to the singular point $0\in\check{S}(f)$. One can also prove that
it carries a unique holomorphic foliation which is induced by the Levi flats
of the Green function $G^{+}$ on $\Omega(f)$, see Remark 1.

An interesting feature of the complex surface $\bar{S}(f)$ is that it admits a
family of charts $(U_{i},\phi_{i})$ where $U_{i}$ is an open cover of
$\bar{S}(f)$, and $\phi_{i}\colon U_{i}\to\mathbb{C}^{2}$ is an open immersion
such that $\phi_{ij}$ is the restriction to an open domain of a birational
self-map of $\mathbb{P}^{2}_{\mathbb{C}}$.

A complex manifold which admits a holomorphic atlas whose transition maps are
restriction of birational maps of $\mathbb{P}^{d}_{\mathbb{C}}$ is said to
carry a birational structure.

The following problem is extracted from [64].

######  Question 43.

Does any non-Kähler compact complex surface admit a birational structure?

This question is extremely challenging, and reduces to the case of VII0
surfaces. One can ask whether any deformation $S$ of a surface $\bar{S}(f)$
associated to a polynomial automorphism $f$ as above admits a birational
structure. This is true when the surface satisfies $b_{2}(S)\leq 3$, see [64].

Analogs of the construction of $\bar{S}(f)$ have been explored by Oeljeklaus
and Renaud in [116] for some quadratic polynomial automorphisms of
$\mathbb{C}^{3}$, and further expanded by Ruggiero [128, Chapter 4]. A
polynomial automorphism $f\in\mathrm{Aut}[\mathbb{C}^{3}]$ is said to be
regular if the indeterminacy locus $I(f)$ of its extension to
$\mathbb{P}^{3}_{\mathbb{C}}$ is disjoint from $I(f^{-1})$. This notion was
introduced by Sibony in [131]. Let $\Omega(f)$ be the basin of attraction of
$I(f^{-1})$: this is an open $f$-invariant set over which $f$ acts properly
discontinuously. As above, denote by $S(f)$ the quotient space
$\Omega(f)/\langle f\rangle$.

######  Problem 9.

Let $f\in\mathrm{Aut}[\mathbb{C}^{d}]$ be any regular polynomial automorphism.

  1. 1.

Prove that one can find a compact complex manifold $\bar{S}(f)$ and an open
immersion $S(f)\subset\bar{S}(f)$ such that the complement
$\bar{S}(f)\setminus S(f)$ is a divisor.

  2. 2.

Prove that $\bar{S}(f)$ is unique up to bimeromorphism.

  3. 3.

Describe complex objects on $\bar{S}(f)$ (analytic subvarieties, vector
fields, holomorphic foliations, positive closed currents,…). Compute its
deformation space.

It is unclear how to extend this construction to a larger class of polynomial
automorphisms of $\mathbb{C}^{3}$. However when
$\lambda(f)^{2}>\lambda(f^{-1})$ an invariant valuation on the ring of
polynomial functions in three variables101010One way to interpret
geometrically such an object is to say that it picks an irreducible subvariety
in _any_ algebraic compactification of $\mathbb{C}^{3}$ in a compatible way.
is known to exist by [51], which suggests the following question.

######  Question 44.

Let $f\in\mathrm{Aut}[\mathbb{C}^{3}]$ be any polynomial automorphism such
that $\lambda(f)^{2}>\lambda(f^{-1})$. Prove the existence of a projective
compactification $X$ of $\mathbb{C}^{3}$ such that the induced birational map
$f\colon X\dashrightarrow X$ admits a super-attracting fixed point $p$ on the
divisor at infinity.

Once such a compactification has been found, one can consider the basin of
attraction $\Omega$ of the point $p$ and try to construct a compactification
of the space of $f$-orbits in $\Omega$ as above.

##  11 Hénon maps over number fields (P. Ingram)

Consider a sufficiently large field $k$, for example a number field. In
general, one should expect to be able to construct Hénon maps of the form

$f(x,y)=(y,F(y)-\delta x)$

with cycles of length $\deg(F)+2$. Write $F(y)=a_{0}+\cdots+a_{d}y^{d}$, and
let $y_{0},...,y_{d+1}$ be variables ranging over $k$. Then, insisting that
$f$ sends

$(y_{0},y_{1})\to(y_{1},y_{2})\to\cdots\to(y_{d},y_{d+1})\to(y_{d+1},y_{0})\to(y_{0},y_{1})$
(9)

is the same as insisting that

$a_{0}+a_{1}y_{n}+\cdots+a_{d}y_{n}^{d}-\delta y_{n-1}=y_{n+1},$

for all $n\;(\mathrm{mod}\ d+2)$. One checks that the determinant of the
associated Vandermonde-like matrix

$\begin{pmatrix}1&y_{0}&\cdots&y_{0}^{d}&-y_{d+1}\\\
1&y_{1}&\cdots&y_{1}^{d}&-y_{0}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
1&y_{d+1}&\cdots&y_{d+1}^{d}&-y_{d}\end{pmatrix}$

is not identically zero (e.g., substituting $y_{d+1}=y_{d}$ into this
determinant gives $\pm(y_{d}-y_{d-1})\prod_{0\leq i<j\leq d}(y_{j}-y_{i})$,
which is itself not identically zero), and so this matrix is invertible over
some affine-open subset of $k^{d+1}$. Here, one can find coefficients
$a_{0},...,a_{d}$, and $\delta$ of $f$ which enact (9).

Recently, Hyde and Doyle [66] exhibited single-variable polynomials over
number fields with more preperiodic points than this sort of naive
interpolation construction gives. One might ask if similar phenomena could be
exploited for generalized Hénon maps.

######  Question 45.

Over a number field $K$, is it possible to construct infinite families of
generalized Hénon maps of algebraic degree $d$, and $K$-rational cycles of
length at least $d+3$? Can one construct maps with $N_{d}$ periodic points,
where $N_{d}-d\to\infty$, or even $N_{d}/d\to\infty$ as $d\to\infty$?

Next, it is natural to ask about bounds in the other direction. In analogy to
the Uniform Boundedness Conjecture of Morton and Silverman [113], it is
natural to posit the following, in which $\operatorname{Per}(f)$ is the set of
periodic points of $f$ over the algebraic closure of $K$.

######  Conjecture 2.

Let $K$ be a number field, let $B\geq 1$, and let $d\geq 2$. Then as $f$
varies over generalized Hénon maps of degree $d$ over $K$, the quantity

$\\#\\{P\in\operatorname{Per}(f):[K(P):K]\leq B\\}$

is bounded just in terms of $[K:\mathbb{Q}]$, $d$, and $B$.

We have already seen why this bound must depend on $d$, and adjoining periodic
points of $f$ to the base field shows why the bound must depend on
$[K:\mathbb{Q}]$. As a starting point for further exploration, we mention two
more readily falsifiable conjectures.

######  Conjecture 3 (see [91]).

Over $\mathbb{Q}$, $(x,y)\mapsto(y,y^{2}+c+x)$ has no point of period $N$,
other than $N\in\\{1,2,3,4,6,8\\}$.

######  Conjecture 4.

For all but finitely many $\delta\in\mathbb{Q}$, the $\mathbb{Q}$-rational
periodic points of any $f(x,y)=(y,y^{2}+c-\delta x)$ with $c\in\mathbb{Q}$
have period dividing 2.

It should be noted that some infinite families of examples show that both of
these conjectures, if true, would be sharp.

It seems reasonable to posit something even stronger than Conjecture 2. Write
$\log^{+}x=\log\max\\{1,x\\}$ for $x\in\mathbb{R}^{+}$, and for an absolute
value $|\cdot|_{v}$, set

$\|x_{1},...,x_{m}\|_{v}=\max\\{|x_{1}|_{v},...,|x_{m}|_{v}\\}.$

Recall that a number field $K$ comes equipped with a standard set $M_{K}$ of
absolute values, and we define the _naive Weil height_ of
$P\in\mathbb{A}^{N}(K)$ to be

$h(P)=\sum_{v\in
M_{K}}\frac{[K_{v}:\mathbb{Q}_{v}]}{[K:\mathbb{Q}]}\log^{+}\|P\|_{v}.$

Kawaguchi [97] constructed a _canonical height_ $\hat{h}_{f}$ associated to a
generalized Hénon map, which differs by a controllable amount from the naive
height, and interacts favorably with the dynamics of $f$, satisfying for
example

$\hat{h}_{f}\circ f+\hat{h}_{f}\circ
f^{-1}=\left(d+\frac{1}{d}\right)\hat{h}_{f},$

and $\hat{h}_{f}(P)=0$ if and only if $P$ is periodic. In light of partial
results in this direction [91], it seems reasonable to conjecture the
following strengthening of Conjecture 2.

######  Conjecture 5.

Let $K$ be a number field, let $B\geq 1$, and let $d\geq 2$. Then, there exist
an $\epsilon>0$ and constant $C$ (depending on these data) such that, as $f$
varies over generalized Hénon maps of degree $d$ over $K$, the quantity

$\\#\\{P\in(\overline{K})^{2}:[K(P):K]\leq B\text{ and
}\hat{h}_{f}(P)<\epsilon h(f)+C\\}$

is bounded uniformly, where $h(f)$ is the height of the tuple of coefficients
of $f$.

Finally, let $f$ be a generalized Hénon map defined over a number field $K$
with good reduction away from some finite set $S$ of primes (so, the
coefficients are $S$-integers, and $a_{d}$ and $\delta$ are $S$-units), and
let $P_{0}\in(\mathcal{O}_{K,S})^{2}$ be some non-periodic point. Set
$P_{n+1}=f(P_{n})$, for $n\geq 0$, and

$\mathfrak{a}_{n}=\gcd(x(P_{n})-x(P_{0}),y(P_{n})-y(P_{0}))\subseteq\mathcal{O}_{K,S}.$

That is, $\mathfrak{a}_{n}$ is the largest ideal such that $f^{n}(P)\equiv P$
modulo ${\mathfrak{a}_{n}}$. Then $\mathfrak{a}_{n}$ is a divisibility
sequence, i.e., $m\mid n\Rightarrow\mathfrak{a}_{m}\mid\mathfrak{a}_{n}$,
which we will call a _Hénon divisibility sequence_. In the case
$\mathcal{O}_{K,S}=\mathbb{Z}$, we may identify the ideals with their unique
positive generators, and think of this as a sequence of positive integers.

######  Example 1.

$f(x,y)=(y,y^{2}+x-2)$, $P=(2,3)$.

$\mathfrak{a}_{n}:1,1,8,3,1,8,1,3,8,1,5,48,11,1,8,51,1,8,1,3,8,5,7,288,13,11,8,3,1,...$

In analogy to other divisibility sequences, we probably expect this sequence
to grow slowly. By comparing to the height of $P_{n}$, one easily obtains an
upper bound of size $C^{d^{n}}$, for some $C$, on the norm of each of the
terms in the gcd, but the gcd itself should usually be much smaller.

######  Theorem 6 (Bugeaud, Corvaja, and Zannier [41]).

If $a,b\geq 2$ are multiplicatively independent integers and $\varepsilon>0$,
then

$\gcd(a^{n}-1,b^{n}-1)\ll_{\varepsilon}e^{\varepsilon n}.$

######  Theorem 7 (Huang [87]).

If $f(x),g(x)\in\mathbb{Z}[x]$ of degree $d\geq 2$, then “generically” and
under Vojta’s Conjecture

$\gcd(f^{n}(a)-\alpha,g^{n}(b)-\beta)\ll_{\varepsilon}e^{\varepsilon d^{n}}.$

On the other hand, it is certainly true that every prime
$\mathfrak{p}\subseteq\mathcal{O}_{K,S}$ divides some term in the sequence,
since the image of $P$ in the residue field $\mathcal{O}_{K,S}/\mathfrak{p}$
must be periodic of period at most $\operatorname{Norm}(\mathfrak{p})^{2}$.

######  Question 46.

Under what circumstance is it true that

$\log\operatorname{Norm}(\mathfrak{a}_{n})=o(d^{n})$

as $n\to\infty$? Under what circumstance does there exist an ideal
$\mathfrak{b}\subset\mathcal{O}_{K,S}$ such that
$\mathfrak{a}_{n}|\mathfrak{b}$ for infinitely many $n$? Note that, since
$\mathfrak{a}_{1}|\mathfrak{a}_{n}$ for all $n$,
$\mathfrak{b}=\mathfrak{a}_{1}$ is a reasonable candidate.

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