1 Introduction
Denote by
$k$
an algebraically closed field of characteristic
$0$
.
The aim of this paper is to prove the following result.
Theorem 1.1. Let
$f:\mathbb{A}_{k}^{2}\rightarrow \mathbb{A}_{k}^{2}$
be a dominant polynomial endomorphism. If there are no nonconstant rational functions
$g$
satisfying
$g\circ f=g$
, then there exists a point
$p\in \mathbb{A}^{2}(k)$
such that the orbit
$\{f^{n}(p)\mid n\geqslant 0\}$
of
$p$
is Zariski dense in
$\mathbb{A}_{k}^{2}$
.
We cannot ask
$g$
in Theorem 1.1 to be a polynomial. Indeed, let
$P(x,y)$
be a polynomial which is neither zero nor a root of unity. Let
$f:\mathbb{A}_{k}^{2}\rightarrow \mathbb{A}_{k}^{2}$
be the endomorphism defined by
$(x,y)\mapsto (P(x,y)x,P(x,y)y)$
. It is easy to see that
$g\circ f=g$
if
$g=y/x$
, but there does not exist any polynomial
$h$
satisfying
$h\circ f=h$
.
The following conjecture was proposed by Medvedev and Scanlon [Reference Medvedev and ScanlonMS09, Conjecture 5.10] and also by Amerik et al. [Reference Amerik, Bogomolov and RovinskyAB11].
Conjecture 1.2. Let
$X$
be a quasiprojective variety over
$k$
and
$f:X\rightarrow X$
be a dominant endomorphism for which there exists no nonconstant rational function
$g$
satisfying
$g\circ f=g$
. Then there exists a point
$p\in X(k)$
whose orbit is Zariski dense in
$X$
.
Conjecture 1.2 strengthens the following conjecture of Zhang [Reference ZhangZha06].
Conjecture 1.3. Let
$X$
be a projective variety and
$f:X\rightarrow X$
be an endomorphism defined over
$k$
. If there exists an ample line bundle
$L$
on
$X$
satisfying
$f^{\ast }L=L^{\otimes d}$
for some integer
$d>1$
, then there exists a point
$p\in X(k)$
whose orbit
$\{f^{n}(p)\mid n\geqslant 0\}$
is Zariski dense in
$X(k)$
.
Theorem 1.1 settles Conjecture 1.2 for polynomial endomorphisms of
$\mathbb{A}_{k}^{2}$
.
When
$k$
is uncountable, Conjecture 1.2 was proved by Amerik and Campana [Reference Amerik and CampanaAC08]. In [Reference FakhruddinFak14], Fakhruddin proved Conjecture 1.2 for
$generic$
Footnote
1
endomorphisms on projective spaces over arbitrary algebraically closed fields
$k$
of characteristic
$0$
. In [Reference XieXie15b], the author proved Conjecture 1.2 for birational surface endomorphisms with dynamical degree greater than
$1$
. Recently, in [Reference Medvedev and ScanlonMS14], Medvedev and Scanlon proved Conjecture 1.2 when
$f:=(f_{1}(x_{1}),\ldots ,f_{N}(x_{N}))$
is an endomorphism of
$\mathbb{A}_{k}^{N}$
, where the
$f_{i}$
are one-variable polynomials defined over
$k$
.
We mention that in [Reference AmerikAme11], Amerik proved that there exists a nonpreperiodic algebraic point when
$f$
is of infinite order. In [Reference Bell, Ghioca and TuckerBGT15], Bell et al. proved that if
$f$
is an automorphism, then there exists a subvariety of codimension
$2$
whose orbit under
$f$
is Zariski dense.
We note that Conjecture 1.2 is not true in the case when
$k$
is the algebraic closure of a finite field, since in this case all orbits of
$k$
-points are finite.
Our proof of Theorem 1.1 is based on the valuative techniques developed in [Reference Favre and JonssonFJ04, Reference Favre and JonssonFJ07, Reference Favre and JonssonFJ11, Reference XieXie15a]. Here is an outline of the proof.
For simplicity, suppose that
$f$
is a dominant polynomial map
$f:=(F(x,y),G(x,y))$
defined over
$\mathbb{Z}$
.
When
$f$
is birational, our Theorem 1.1 is essentially proved in [Reference XieXie15b]. So, we may suppose that
$f$
is not birational.
By [Reference Favre and JonssonFJ11], there exists a projective compactification
$X$
of
$\mathbb{A}^{2}$
for which the induced map by
$f$
at infinity is algebraically stable, i.e. it does not contract any curve to a point of indeterminacy. Moreover, we can construct an ‘attracting locus’ at infinity, in the sense that:
-
(i) either there exists a superattracting fixed point
$q\in X\setminus \mathbb{A}^{2}$
such that there is no branch of the curve at
$q$
which is periodic under
$f$
; -
(ii) or there exists an irreducible component
$E\in X\setminus \mathbb{A}^{2}$
such that
$f^{\ast }E=dE+F$
, where
$d\geqslant 2$
and
$F$
is an effective divisor supported by
$X\setminus \mathbb{A}^{2}$
.
In Case (i), we can find a point
$p\in \mathbb{A}^{2}(\bar{\mathbb{Q}})$
near
$q$
with respect to the Euclidean topology. Then we have
$\lim _{n\rightarrow \infty }f^{n}(p)=q$
. It is easy to show that the orbit of
$p$
is Zariski dense in
$\mathbb{A}^{2}$
.
In Case (ii),
$E$
is defined over
$\mathbb{Q}$
. There exists a prime number
$\mathfrak{p}\geqslant 3$
such that
$f_{\mathfrak{p}}|_{E_{\mathfrak{p}}}$
is dominant, where
$f_{\mathfrak{p}}:=f\,\text{mod}\,\mathfrak{p}$
and
$E_{\mathfrak{p}}:=E\,\text{mod}\,\mathfrak{p}$
.
We first treat the case
$f^{n}|_{E}\neq \operatorname{id}$
for all
$n\geqslant 1$
. After replacing
$f$
by a suitable iterate, we may find a fixed point
$x\in E_{\mathfrak{p}}$
such that
$df_{\mathfrak{p}}(x)=1$
in
$\bar{\mathbb{F}}_{\mathfrak{p}}$
. Denote by
$U$
the
$\mathfrak{p}$
-adic open set of
$X(\mathbb{Q}_{\mathfrak{p}})$
consisting of the points
$y$
such that
$y\,\text{mod}\,\mathfrak{p}=x$
. Then
$U$
is fixed by
$f$
. By [Reference PoonenPoo14, Theorem 1], all the preperiodic points in
$U\cap E$
are fixed. Moreover,
$\bigcap _{n\geqslant 0}f^{n}(U)=U\cap E$
. Denote by
$S$
the set of fixed points in
$U\cap E$
. Then
$S$
is finite. If
$S$
is empty, pick a point
$p\in \mathbb{A}^{2}(\bar{\mathbb{Q}}\cap \mathbb{Q}_{p})\cap U$
; it is easy to see that the orbit of
$p$
is Zariski dense in
$\mathbb{A}^{2}$
. If
$S$
is not empty, by [Reference AbateAba01, Theorem 3.1.4], at each point
$q_{i}\in S$
there exists at most one algebraic curve
$C_{i}$
passing through
$q_{i}$
which is preperiodic. Set
$C_{i}=\emptyset$
if no such curve does exist. We have that
$C_{i}$
is fixed. Pick a point
$p\in \mathbb{A}^{2}(\bar{\mathbb{Q}}\cap \mathbb{Q}_{p})\setminus C_{1}$
very close to
$q_{1}$
. We can show that the orbit of
$p$
is Zariski dense in
$\mathbb{A}^{2}$
.
Next, we treat the case
$f|_{E}=\operatorname{id}$
. By [Reference AbateAba01, Theorem 3.1.4], at each point
$q\in E$
there exists at most one algebraic curve
$C_{q}$
passing through
$q$
which is preperiodic. Set
$C_{q}=\emptyset$
if such curve does not exist. We have that
$C_{q}$
is fixed and transverse to
$E$
. If
$C_{q}=\emptyset$
for all but finitely many
$q\in E$
, then there exist
$q\in E$
and a
$\mathfrak{p}$
-adic neighborhood
$U$
of
$q$
such that for any point
$y\in U\cap E$
,
$C_{y}=\emptyset$
,
$f(U)\subseteq U$
and
$\bigcap _{\infty }f^{n}(U)=U\cap E$
. Then, for any point
$p\in \mathbb{A}^{2}(\bar{\mathbb{Q}}\cap \mathbb{Q}_{p})\cap U$
, the orbit of
$p$
is Zariski dense in
$\mathbb{A}^{2}$
. Otherwise there exists a sequence of points
$q_{i}\in E$
such that
$C_{i}:=C_{q_{i}}$
is an irreducible curve. Since
$f|_{C_{i}}$
is an endomorphism of
$C_{i}\cap \mathbb{A}^{2}$
of degree at least
$2$
,
$C_{i}$
has at most two branches at infinity. Since
$C_{i}$
is transverse to
$E$
at
$q_{i}$
, we can bound the intersection number
$(E\cdot C_{i})$
. By the technique developed in [Reference XieXie15a], we can also bound the intersection of
$C_{i}$
with the other irreducible components of
$X\setminus \mathbb{A}^{2}$
. Then we bound the degree of
$C_{i}$
, which allows us to construct a nonconstant invariant rational function.
The article is organized in two parts.
In Part I, we gather some results on the geometry and dynamics at infinity and metrics on projective varieties defined over a valued field. We first introduce the valuative tree at infinity in § 2, and then we recall the main properties of the action of a polynomial map on the valuation space in § 3. Next, we introduce the Green function on the valuative tree for a polynomial endomorphism in § 4. Finally, we give background information on metrics on projective varieties defined over a valued field in § 5.
In Part II, we prove Theorem 1.1. We first prove it in some special cases in § 6. In most of these cases, we find a Zariski dense orbit in some attracting locus. Then we study totally invariant curves in § 7 and prove Theorem 1.1 when there are infinitely many such curves. Finally, we finish the proof of Theorem 1.1 in § 8.
Part I. Preliminaries
In this part, we denote by
$k$
an algebraically closed field of characteristic
$0$
. We also fix affine coordinates on
$\mathbb{A}_{k}^{2}=\operatorname{Spec}k[x,y]$
.
2 The valuative tree at infinity
We refer to [Reference Jonsson, Nicaise, Ducros and FavreJon15] for details; see also [Reference Favre and JonssonFJ04, Reference Favre and JonssonFJ07, Reference Favre and JonssonFJ11].
2.1 The valuative tree at infinity
In this article by a valuation on a unitary
$k$
-algebra
$R$
we shall understand a function
$v:R\rightarrow \mathbb{R}\cup \{+\infty \}$
such that the restriction of
$v$
to
$k^{\ast }=k-\{0\}$
is constant equal to
$0$
, and
$v$
satisfies
$v(fg)=v(f)+v(g)$
and
$v(f+g)\geqslant \min \{v(f),v(g)\}$
. It is usually referred to as a semivaluation in the literature; see [Reference Favre and JonssonFJ04]. We will however make a slight abuse of notation and call it a valuation.
Denote by
$V_{\infty }$
the space of all normalized valuations centered at infinity, i.e. the set of valuations
$v:k[x,y]\rightarrow \mathbb{R}\cup \{+\infty \}$
satisfying
$\min \{v(x),v(y)\}=-1$
. The topology on
$V_{\infty }$
is defined to be the weakest topology making the map
$v\mapsto v(P)$
continuous for every
$P\in k[x,y]$
.
The set
$V_{\infty }$
is equipped with a partial ordering defined by
$v\leqslant w$
if and only if
$v(P)\leqslant w(P)$
for all
$P\in k[x,y]$
. Then
$-\text{deg}:P\mapsto -\text{deg}(P)$
is the unique minimal element.
Given any valuation
$v\in V_{\infty }\setminus \{-\text{deg}\}$
, the set
$\{w\in V_{\infty }\mid -\text{deg}\leqslant w\leqslant v\}$
is isomorphic as a poset to the real segment
$[0,1]$
endowed with the standard ordering. In other words,
$(V_{\infty },\leqslant )$
is a rooted tree in the sense of [Reference Favre and JonssonFJ04, Reference Jonsson, Nicaise, Ducros and FavreJon15].
Given any two valuations
$v_{1},v_{2}\in V_{\infty }$
, there is a unique valuation in
$V_{\infty }$
which is maximal in the set
$\{v\in V_{\infty }\mid v\leqslant v_{1}\text{ and }v\leqslant v_{2}\}$
. We denote it by
$v_{1}\wedge v_{2}$
.
The segment
$[v_{1},v_{2}]$
is by definition the union of
$\{w\,|\,v_{1}\wedge v_{2}\leqslant w\leqslant v_{1}\}$
and
$\{w\,|\,v_{1}\wedge v_{2}\leqslant w\leqslant v_{2}\}$
.
Pick any valuation
$v\in V_{\infty }$
. We say that two points
$v_{1},v_{2}$
lie in the same direction at
$v$
if the segment
$[v_{1},v_{2}]$
does not contain
$v$
. A direction (or a tangent vector) at
$v$
is an equivalence class for this relation. We write
$\operatorname{Tan}_{v}$
for the set of directions at
$v$
.
When
$\operatorname{Tan}_{v}$
is a singleton, then
$v$
is called an end point. In
$V_{\infty }$
, the set of end points is exactly the set of all maximal valuations. When
$\operatorname{Tan}_{v}$
contains exactly two directions, then
$v$
is said to be regular. When
$\operatorname{Tan}_{v}$
has more than three directions, then
$v$
is a branch point.
Pick any
$v\in V_{\infty }$
. For any tangent vector
$\vec{v}\in \operatorname{Tan}_{v}$
, denote by
$U(\vec{v})$
the subset of those elements in
$V_{\infty }$
that determine
$\vec{v}$
. This is an open set whose boundary is reduced to the singleton
$\{v\}$
. If
$v\neq -\text{deg}$
, the complement of
$\{w\in V_{\infty }\mid w\geqslant v\}$
is equal to
$U(\vec{v}_{0})$
, where
$\vec{v}_{0}$
is the tangent vector determined by
$-\text{deg}$
.
It is a fact that finite intersections of open sets of the form
$U(\vec{v})$
form a basis for the topology of
$V_{\infty }$
.
2.2 Compactifications of
$\mathbb{A}_{k}^{2}$
A compactification of
$\mathbb{A}_{k}^{2}$
is the data of a projective surface
$X$
together with an open immersion
$\mathbb{A}_{k}^{2}\rightarrow X$
with dense image.
A compactification
$X$
dominates another one
$X^{\prime }$
if the canonical birational map
$X{\dashrightarrow}X^{\prime }$
induced by the inclusion of
$\mathbb{A}_{k}^{2}$
in both surfaces is in fact a regular map.
The category
${\mathcal{C}}$
of all compactifications of
$\mathbb{A}_{k}^{2}$
forms an inductive system for the relation of domination.
Recall that we have fixed affine coordinates on
$\mathbb{A}_{k}^{2}=\operatorname{Spec}k[x,y]$
. We let
$\mathbb{P}_{k}^{2}$
be the standard compactification of
$\mathbb{A}_{k}^{2}$
and denote by
$l_{\infty }:=\mathbb{P}_{k}^{2}\setminus \mathbb{A}_{k}^{2}$
the line at infinity in the projective plane.
An admissible compactification of
$\mathbb{A}_{k}^{2}$
is by definition a smooth projective surface
$X$
endowed with a birational morphism
$\unicode[STIX]{x1D70B}_{X}:X\rightarrow \mathbb{P}_{k}^{2}$
such that
$\unicode[STIX]{x1D70B}_{X}$
is an isomorphism over
$\mathbb{A}_{k}^{2}$
with the embedding
$\unicode[STIX]{x1D70B}^{-1}|_{\mathbb{A}_{k}^{2}}:\mathbb{A}_{k}^{2}\rightarrow X$
. Recall that
$\unicode[STIX]{x1D70B}_{X}$
can then be decomposed as a finite sequence of point blow-ups.
We shall denote by
${\mathcal{C}}_{0}$
the category of all admissible compactifications. It is also an inductive system for the relation of domination. Moreover,
${\mathcal{C}}_{0}$
is a subcategory of
${\mathcal{C}}$
and, for any compactification
$X\in {\mathcal{C}}$
, there exists
$X^{\prime }\in {\mathcal{C}}_{0}$
that dominates
$X$
.
2.3 Divisorial valuations
Let
$X\in {\mathcal{C}}$
be a compactification of
$\mathbb{A}_{k}^{2}=\operatorname{Spec}k[x,y]$
and
$E$
be an irreducible component of
$X\setminus \mathbb{A}^{2}$
. Set
$b_{E}:=-\text{min}\{\operatorname{ord}_{E}(x),\operatorname{ord}_{E}(y)\}$
and
$v_{E}:=b_{E}^{-1}\operatorname{ord}_{E}$
. Then we have
$v_{E}\in V_{\infty }$
.
By Poincaré duality, there exists a unique dual divisor
${\check{E}}$
of
$E$
defined as the unique divisor supported on
$X\setminus \mathbb{A}^{2}$
such that
$({\check{E}}\cdot F)=\unicode[STIX]{x1D6FF}_{E,F}$
for all irreducible components
$F$
of
$X\setminus \mathbb{A}^{2}$
.
Remark 2.1. Recall that
$l_{\infty }$
is the line at infinity of
$\mathbb{P}^{2}$
. Let
$s$
be a formal curve centered at some point
$q\in l_{\infty }$
. Suppose that the strict transform of
$s$
in
$X$
intersects
$E$
transversally at a point in
$E$
which is smooth in
$X\setminus \mathbb{A}^{2}$
. Then we have
$(s\cdot l_{\infty })=b_{E}$
.
2.4 Classification of valuations
There are four kinds of valuations in
$V_{\infty }$
. The first one corresponds to the divisorial valuations which we have defined above. We now describe the three remaining types of valuations.
Irrational valuations
Consider any two irreducible components
$E$
and
$E^{\prime }$
of
$X\setminus \mathbb{A}_{k}^{2}$
for some compactification
$X\in {\mathcal{C}}$
of
$\mathbb{A}_{k}^{2}$
intersecting at a point
$p$
. There exists local coordinates
$(z,w)$
at
$p$
such that
$E=\{z=0\}$
and
$E^{\prime }=\{w=0\}$
. To any pair
$(s,t)\in (\mathbb{R}^{+})^{2}$
satisfying
$sb_{E}+tb_{E^{\prime }}=1$
, we attach the valuation
$v$
defined on the ring
$O_{p}$
of germs at
$p$
by
$v(\sum a_{ij}z^{i}w^{j})=\min \{si+tj\mid a_{ij}\neq 0\}$
. Observe that it does not depend on the choice of coordinates. By first extending
$v$
to the common fraction field
$k(x,y)$
of
$O_{p}$
and
$k[x,y]$
and then restricting it to
$k[x,y]$
, we obtain a valuation in
$V_{\infty }$
, called quasimonomial. It is divisorial if and only if either
$t=0$
or the ratio
$s/t$
is a rational number. Any divisorial valuation is quasimonomial. An irrational valuation is by definition a nondivisorial quasimonomial valuation.
Curve valuations
Recall that
$l_{\infty }$
is the line at infinity of
$\mathbb{P}_{k}^{2}$
. For any formal curve
$s$
centered at some point
$q\in l_{\infty }$
, denote by
$v_{s}$
the valuation defined by
$P\mapsto (s\cdot l_{\infty })^{-1}\operatorname{ord}_{\infty }(P|_{s})$
. Then we have
$v_{s}\in V_{\infty }$
and we call it a curve valuation.
Let
$C$
be an irreducible curve in
$\mathbb{P}_{k}^{2}$
. For any point
$q\in C\cap l_{\infty }$
, denote by
$O_{q}$
the local ring at
$q$
,
$m_{q}$
the maximal ideal of
$O_{q}$
and
$I_{C}$
the ideal of height 1 in
$O_{q}$
defined by
$C$
. Denote by
$\widehat{O}_{q}$
the completion of
$O_{q}$
with respect to
$m_{q}$
,
$\widehat{m}_{q}$
the completion of
$m_{q}$
and
$\widehat{I}_{C}$
the completion of
$I_{C}$
. For any prime ideal
$\widehat{p}$
of height 1 containing
$\widehat{I}_{C}$
, the morphism
$\operatorname{Spec}\widehat{O}_{q}/\widehat{p}\rightarrow \operatorname{Spec}\widehat{O}_{q}$
defines a formal curve centered at
$q$
. Such a formal curve is called a branch of
$C$
at infinity.
Infinitely singular valuations
Let
$h$
be a formal series of the form
$h(z)=\sum _{k=0}^{\infty }a_{k}z^{\unicode[STIX]{x1D6FD}_{k}}$
with
$a_{k}\in k^{\ast }$
and
$\{\unicode[STIX]{x1D6FD}\}_{k}$
an increasing sequence of rational numbers with unbounded denominators. Then
$P\mapsto -\text{min}\{\operatorname{ord}_{\infty }(x),\operatorname{ord}_{\infty }(h(x^{-1}))\}^{-1}\operatorname{ord}_{\infty }P(x,h(x^{-1}))$
defines a valuation in
$V_{\infty }$
called an infinitely singular valuation.
A valuation
$v\in V_{\infty }$
is a branch point in
$V_{\infty }$
if and only if it is divisorial, it is a regular point in
$V_{\infty }$
if and only if it is an irrational valuation and it is an end point in
$V_{\infty }$
if and only if it is a curve valuation or an infinitely singular valuation. Moreover, given any smooth projective compactification
$X$
in which
$v=v_{E}$
, one proves that the map sending an element
$V_{\infty }$
to its center in
$X$
induces a map
$\operatorname{Tan}_{v}\rightarrow E$
that is a bijection.
2.5 Parameterizations
The skewness function
$\unicode[STIX]{x1D6FC}:V_{\infty }\rightarrow [-\infty ,1]$
is the unique function on
$V_{\infty }$
that is continuous on segments and satisfies
$$\begin{eqnarray}\unicode[STIX]{x1D6FC}(v_{E})=\frac{1}{b_{E}^{2}}({\check{E}}\cdot {\check{E}}),\end{eqnarray}$$
where
$E$
is any irreducible component of
$X\setminus \mathbb{A}_{k}^{2}$
of any compactification
$X$
of
$\mathbb{A}_{k}^{2}$
and
${\check{E}}$
is the dual divisor of
$E$
as defined above.
The skewness function is strictly decreasing and upper semicontinuous. In an analogous way, one defines the thinness function
$A:V_{\infty }\rightarrow [-2,\infty ]$
as the unique, increasing, lower semicontinuous function on
$V_{\infty }$
such that for any irreducible exceptional divisor
$E$
in some compactification
$X\in {\mathcal{C}}$
, we have
$$\begin{eqnarray}A(v_{E})=\frac{1}{b_{E}}\,(1+\operatorname{ord}_{E}(dx\wedge dy)).\end{eqnarray}$$
Here we extend the differential form
$dx\wedge dy$
to a rational differential form on
$X$
.
2.6 Computation of local intersection numbers of curves at infinity
Let
$s_{1}$
,
$s_{2}$
be two different formal curves at infinity. We denote by
$(s_{1}\cdot s_{2})$
the intersection number of these two formal curves in
$\mathbb{P}^{2}$
. This intersection number is always nonnegative, and it is positive if and only if
$s_{1}$
and
$s_{2}$
are centered at the same point.
Denote by
$l_{\infty }$
the line at infinity in
$\mathbb{P}_{k}^{2}$
. Denote by
$v_{s_{1}},v_{s_{2}}$
the curve valuations associated to
$s_{1}$
and
$s_{2}$
.
By [Reference XieXie15a, Proposition 2.2], we have
$$\begin{eqnarray}(s_{1}\cdot s_{2})=(s_{1}\cdot l_{\infty })(s_{2}\cdot l_{\infty })(1-\unicode[STIX]{x1D6FC}(v_{s_{1}}\wedge v_{s_{2}})).\end{eqnarray}$$
3 Background on dynamics of polynomial maps
Recall that the affine coordinates have been fixed,
$\mathbb{A}_{k}^{2}=\operatorname{Spec}k[x,y]$
.
3.1 Dynamical invariants of polynomial mappings
The (algebraic) degree of a dominant polynomial endomorphism
$f=(F(x,y),G(x,y))$
defined on
$\mathbb{A}_{k}^{2}$
is defined by
$$\begin{eqnarray}\operatorname{deg}(f):=\max \{\operatorname{deg}(F),\operatorname{deg}(G)\}.\end{eqnarray}$$
It is not difficult to show that the sequence
$\operatorname{deg}(f^{n})$
is submultiplicative, so that the limit
$\unicode[STIX]{x1D706}_{1}(f):=\lim _{n\rightarrow \infty }(\operatorname{deg}(f^{n}))^{1/n}$
exists. It is referred to as the dynamical degree of
$f$
, and it is a theorem of Favre and Jonsson that
$\unicode[STIX]{x1D706}_{1}(f)$
is always a quadratic integer; see [Reference Favre and JonssonFJ11].
The (topological) degree
$\unicode[STIX]{x1D706}_{2}(f)$
of
$f$
is defined to be the number of preimages of a general closed point in
$\mathbb{A}^{2}(k)$
; one has
$\unicode[STIX]{x1D706}_{2}(fg)=\unicode[STIX]{x1D706}_{2}(f)\unicode[STIX]{x1D706}_{2}(g)$
.
It follows from Bézout’s theorem that
$\unicode[STIX]{x1D706}_{2}(f)\leqslant \operatorname{deg}(f)^{2}$
and hence
$$\begin{eqnarray}\unicode[STIX]{x1D706}_{1}(f)^{2}\geqslant \unicode[STIX]{x1D706}_{2}(f).\end{eqnarray}$$
The resonant case
$\unicode[STIX]{x1D706}_{1}(f)^{2}=\unicode[STIX]{x1D706}_{2}(f)$
is quite special and the following structure theorem for these maps is proven in [Reference Favre and JonssonFJ11].
Theorem 3.1. Any polynomial endomorphism
$f$
of
$\mathbb{A}_{k}^{2}$
such that
$\unicode[STIX]{x1D706}_{1}(f)^{2}=\unicode[STIX]{x1D706}_{2}(f)$
is proper,Footnote
2
and we are in exactly one of the following two exclusive cases.
-
(1)
$\operatorname{deg}(f^{n})\asymp \unicode[STIX]{x1D706}_{1}(f)^{n}$
; there exists a compactification
$X$
of
$\mathbb{A}_{k}^{2}$
to which
$f$
extends as a regular map
$f:X\rightarrow X$
. -
(2)
$\operatorname{deg}(f^{n})\asymp n\unicode[STIX]{x1D706}_{1}(f)^{n}$
; there exist affine coordinates
$x,y$
in which
$f$
takes the form where
$$\begin{eqnarray}f(x,y)=(x^{l}+a_{1}x^{l-1}+\cdots +a_{l},A_{0}(x)y^{l}+\cdots +A_{l}(x)),\end{eqnarray}$$
$a_{i}\in k$
and
$A_{i}\in k[x]$
with
$\operatorname{deg}A_{0}\geqslant 1$
, and
$l=\unicode[STIX]{x1D706}_{1}(f)$
.
Remark 3.2. Regular endomorphisms as in (1) have been classified in [Reference Favre and JonssonFJ11].
3.2 Valuative dynamics
Any dominant polynomial endomorphism
$f$
as in the previous section induces a natural map on the space of valuations at infinity in the following way.
For any
$v\in V_{\infty }$
, we set
$$\begin{eqnarray}d(f,v):=-\text{min}\{v(F),v(G),0\}\geqslant 0.\end{eqnarray}$$
In this way, we get a nonnegative continuous decreasing function on
$V_{\infty }$
. Observe that
$d(f,-\text{deg})=\operatorname{deg}(f)$
. It is a fact that
$f$
is proper if and only if
$d(f,v)>0$
for all
$v\in V_{\infty }$
.
We now set:
-
∙
$f_{\ast }v:=0$
if
$d(f,v)=0$
; -
∙
$f_{\ast }v(P)=v(f^{\ast }P)$
if
$d(f,v)>0$
.
In this way one obtains a valuation on
$k[x,y]$
(that may be trivial); we then get a continuous map
$$\begin{eqnarray}f_{\bullet }:\{v\in V_{\infty }\mid d(f,v)>0\}\rightarrow V_{\infty }\end{eqnarray}$$
defined by
$$\begin{eqnarray}f_{\bullet }(v):=d(f,v)^{-1}f_{\ast }v.\end{eqnarray}$$
This map extends to a continuous map
$f_{\bullet }:\overline{\{v\in V_{\infty }\mid d(f,v)>0\}}\rightarrow V_{\infty }$
. The image of any
$v\in \unicode[STIX]{x2202}\{v\in V_{\infty }\mid d(f,v)>0\}$
is a curve valuation defined by a rational curve with one place at infinity.
We now recall the following key result [Reference Favre and JonssonFJ11, Proposition 2.3, Theorem 2.4 and Proposition 5.3].
Theorem 3.3. There exists a valuation
$v_{\ast }$
such that
$\unicode[STIX]{x1D6FC}(v_{\ast })\geqslant 0\geqslant A(v_{\ast })$
and
$f_{\ast }v_{\ast }=\unicode[STIX]{x1D706}_{1}v_{\ast }$
.
If
$\unicode[STIX]{x1D706}_{1}(f)^{2}>\unicode[STIX]{x1D706}_{2}(f)$
, this valuation is unique.
If
$\unicode[STIX]{x1D706}_{1}(f)^{2}=\unicode[STIX]{x1D706}_{2}(f)$
, the set of such valuations is a closed segment in
$V_{\infty }$
.
This valuation
$v_{\ast }$
is called the eigenvaluation of
$f$
when
$\unicode[STIX]{x1D706}_{1}(f)^{2}>\unicode[STIX]{x1D706}_{2}(f)$
.
4 The Green function of
$f$
4.1 Subharmonic functions on
$V_{\infty }$
We refer to [Reference XieXie14, § 3] for details.
To any
$v\in V_{\infty }$
, we attach its Green function
$$\begin{eqnarray}Z_{v}(w):=\unicode[STIX]{x1D6FC}(v\wedge w).\end{eqnarray}$$
This is a decreasing continuous function taking values in
$[-\infty ,1]$
, satisfying
$g_{v}(-\text{deg})=1$
.
Given any positive Radon measure
$\unicode[STIX]{x1D70C}$
on
$V_{\infty }$
, we define
$$\begin{eqnarray}Z_{\unicode[STIX]{x1D70C}}(w):=\int _{V_{\infty }}Z_{v}(w)\,d\unicode[STIX]{x1D70C}(v).\end{eqnarray}$$
Observe that
$g_{v}(w)$
is always well defined as an element in
$[-\infty ,1]$
since
$g_{v}\leqslant 1$
for all
$v$
.
Then we recall the following result.
Theorem 4.1 [Reference XieXie14].
The map
$\unicode[STIX]{x1D70C}\mapsto Z_{\unicode[STIX]{x1D70C}}$
is injective.
One can thus make the following definition.
Definition 4.2. A function
$\unicode[STIX]{x1D719}:V_{\infty }\rightarrow \mathbb{R}\cup \{-\infty \}$
is said to be subharmonic if there exists a positive Radon measure
$\unicode[STIX]{x1D70C}$
such that
$\unicode[STIX]{x1D719}=Z_{\unicode[STIX]{x1D70C}}$
. In this case, we write
$\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D719}$
and call it the Laplacian of
$\unicode[STIX]{x1D719}$
.
4.2 Basic properties of the Green function of
$f$
We refer to [Reference XieXie15a, § 12] for details.
Let
$f$
be a dominant polynomial endomorphism on
$\mathbb{A}_{k}^{2}$
with
$\unicode[STIX]{x1D706}_{1}(f)^{2}>\unicode[STIX]{x1D706}_{2}(f)$
.
By [Reference XieXie15a, § 12], there exists a unique subharmonic function
$\unicode[STIX]{x1D703}^{\ast }$
on
$V_{\infty }$
such that:
-
(i)
$f^{\ast }\unicode[STIX]{x1D703}^{\ast }=\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D703}^{\ast }$
; -
(ii)
$\unicode[STIX]{x1D703}^{\ast }(v)\geqslant 0$
for all
$v\in V_{\infty }$
; -
(iii)
$\unicode[STIX]{x1D703}^{\ast }(-\text{deg})=1$
; -
(iv) for all
$v\in V_{\infty }$
satisfying
$\unicode[STIX]{x1D6FC}(v)>-\infty$
, we have
$\unicode[STIX]{x1D703}^{\ast }(v)>0$
if and only if
$d(f^{n},v)>0$
for all
$n\geqslant 0$
and
$$\begin{eqnarray}\lim _{n\rightarrow \infty }f_{\bullet }^{n}(v)=v_{\ast }.\end{eqnarray}$$
5 Metrics on projective varieties defined over a valued field
A field with an absolute value is called a valued field.
Definition 5.1. Let
$(K,|\cdot |_{v})$
be a valued field. For any integer
$n\geqslant 1$
, we define a metric
$d_{v}$
on the projective space
$\mathbb{P}^{n}(K)$
by
$$\begin{eqnarray}d_{v}([x_{0}:\cdots :x_{n}],[y_{0}:\cdots :y_{n}])=\frac{\max _{0\leqslant i,j\leqslant n}|x_{i}y_{j}-x_{j}y_{i}|_{v}}{\max _{0\leqslant i\leqslant n}|x_{i}|_{v}\max _{0\leqslant j\leqslant n}|y_{j}|_{v}}\end{eqnarray}$$
for any two points
$[x_{0}:\cdots :x_{n}],[y_{0}:\cdots :y_{n}]\in \mathbb{P}^{n}(K)$
.
Observe that when
$|\cdot |_{v}$
is Archimedean, then the metric
$d_{v}$
is not induced by a smooth Riemannian metric. However, it is equivalent to the restriction of the Fubini–Study metric on
$\mathbb{P}^{n}(\mathbb{C})$
or
$\mathbb{P}^{n}(\mathbb{R})$
to
$\mathbb{P}^{n}(K)$
induced by any embedding
$\unicode[STIX]{x1D70E}_{v}:K{\hookrightarrow}\mathbb{R}\text{ or }\mathbb{C}$
.
More generally, for a projective variety
$X$
defined over
$K$
, if we fix an embedding
$\unicode[STIX]{x1D704}:X{\hookrightarrow}\mathbb{P}^{n}$
, we may restrict the metric
$d_{v}$
on
$\mathbb{P}^{n}(K)$
to a metric
$d_{v,\unicode[STIX]{x1D704}}$
on
$X(K)$
. This metric depends on the choice of embedding
$\unicode[STIX]{x1D704}$
in general, but, for different embeddings
$\unicode[STIX]{x1D704}_{1}$
and
$\unicode[STIX]{x1D704}_{2}$
, the metrics
$d_{v,\unicode[STIX]{x1D704}_{1}}$
and
$d_{v,\unicode[STIX]{x1D704}_{2}}$
are equivalent. Since we are mostly interested in the topology induced by these metrics, we shall usually write
$d_{v}$
instead of
$d_{v,\unicode[STIX]{x1D704}}$
for simplicity.
Part II. The existence of Zariski dense orbits
The aim of this part is to prove Theorem 1.1.
6 The attracting case
In this section, we prove Theorem 1.1 in some special cases. In most of these cases, we find a Zariski dense orbit in some attracting locus. We also prove Theorem 1.1 when
$\unicode[STIX]{x1D706}_{1}^{2}=\unicode[STIX]{x1D706}_{2}>1$
.
Denote by
$k$
an algebraically closed field of characteristic
$0$
. Let
$f:\mathbb{A}^{2}\rightarrow \mathbb{A}^{2}$
be a dominant polynomial endomorphism defined over
$k$
. We have the following result.
Lemma 6.1. If
$\unicode[STIX]{x1D706}_{2}^{2}(f)>\unicode[STIX]{x1D706}_{1}(f)$
and the eigenvaluation
$v_{\ast }$
is not divisorial, then there exists a point
$p\in \mathbb{A}^{2}(k)$
whose orbit is Zariski dense in
$\mathbb{A}^{2}$
.
Proof of Lemma 6.1.
After replacing
$k$
by an algebraically closed subfield of
$k$
containing all the coefficients of
$f$
, we may suppose that the transcendence degree of
$k$
over
$\bar{\mathbb{Q}}$
is finite.
By [Reference Favre and JonssonFJ11, Theorem 3.1], there exist a compactification
$X$
of
$\mathbb{A}^{2}$
defined over
$k$
and a superattracting point
$q\in X\setminus \mathbb{A}_{k}^{2}$
such that for any valuation
$v\in V_{\infty }$
whose center in
$X$
is
$q$
, we have
$f_{\bullet }^{n}v\rightarrow v_{\ast }$
as
$n\rightarrow \infty$
.
By embedding
$k$
in
$\mathbb{C}$
, we endow
$X$
with the usual Euclidean topology. There exists a neighborhood
$U$
of
$q$
in
$X$
such that:
-
(i)
$I(f)\cap U=\emptyset$
; -
(ii)
$f(U)\subseteq U$
; -
(iii) for any point
$p\in U$
,
$f^{n}(p)\rightarrow q$
as
$n\rightarrow \infty$
.
Since
$\mathbb{A}^{2}(k)$
is dense in
$\mathbb{A}^{2}(\mathbb{C})$
, there exists a point
$p\in \mathbb{A}^{2}(k)\cap U$
. If the orbit of
$p$
is not Zariski dense, then its Zariski closure
$Z$
is a union of finitely many curves. Since
$f^{n}(p)\rightarrow q$
,
$p$
is not preperiodic. It follows that all the one-dimensional irreducible components of
$Z$
are periodic under
$f$
. Let
$C$
be a one-dimensional irreducible component of
$Z$
. Since there exists an infinite sequence
$\{n_{0}<n_{1}<\cdots \,\}$
such that
$f^{n_{i}}(p)\in C$
for
$i\geqslant 0$
, we have that
$C$
contains
$q=\lim _{i\rightarrow \infty }f^{n_{i}}(p)$
. Then there exists a branch
$C_{1}$
of
$C$
at infinity satisfying
$q\in C_{1}$
. Thus,
$v_{C_{1}}$
is periodic, which is a contradiction. It follows that
$O(p)$
is Zariski dense.◻
In many cases, for example
$\unicode[STIX]{x1D706}_{1}(f)^{2}>\unicode[STIX]{x1D706}_{2}(f)$
and
$v_{\ast }$
is divisorial, there exist a projective compactification
$X$
of
$\mathbb{A}^{2}$
and an irreducible component
$E$
of
$X\setminus \mathbb{A}^{2}$
satisfying
$f_{\bullet }(v_{E})=v_{E}$
. The following result proves Theorem 1.1 when
$f|_{E}$
is of infinite order.
Lemma 6.2. Let
$X$
be a projective compactification of
$\mathbb{A}^{2}$
defined over
$k$
. Then
$f$
extends to a rational selfmap on
$X$
. Let
$E$
be an irreducible component of
$X\setminus \mathbb{A}^{2}$
satisfying
$f_{\bullet }(v_{E})=v_{E}$
,
$d(f,v_{E})\geqslant 2$
and
$f^{n}|_{E}\neq \operatorname{id}$
for all
$n\geqslant 1$
. Then there exists a point
$p\in \mathbb{A}^{2}(k)$
whose orbit is Zariski dense in
$\mathbb{A}^{2}$
.
Proof of Lemma 6.2.
There exists a finitely generated
$\mathbb{Z}$
-subalgebra
$R$
of
$k$
such that
$X$
,
$E$
and
$f$
are defined over the fraction field
$K$
of
$R$
.
By [Reference BellBel06, Lemma 3.1], there exist a prime
$\mathfrak{p}\geqslant 3$
, an embedding of
$K$
into
$\mathbb{Q}_{\mathfrak{p}}$
and a
$\mathbb{Z}_{\mathfrak{p}}$
-scheme
${\mathcal{X}}_{Z_{\mathfrak{p}}}$
such that the generic fiber is
$X$
, the specialization
$E_{\mathfrak{p}}$
of
$E$
is isomorphic to
$\mathbb{P}_{\mathbb{F}_{\mathfrak{p}}}^{1}$
, the specialization
$f_{\mathfrak{p}}:X_{\mathfrak{p}}{\dashrightarrow}X_{\mathfrak{p}}$
of
$f$
at the prime ideal
$\mathfrak{p}$
of
$Z_{\mathfrak{p}}$
is dominant and
$\operatorname{deg}f_{\mathfrak{p}}|_{E_{\mathfrak{p}}}=\operatorname{deg}f_{E}$
.
Since there are only finitely many points in the orbits of
$I(f_{\mathfrak{p}})$
and the orbits of ramified points of
$f_{\mathfrak{p}}$
, by [Reference FakhruddinFak03, Proposition 5.5], there exists a closed point
$x\in E_{\mathfrak{p}}$
such that
$x$
is periodic,
$E_{\mathfrak{p}}$
is the unique irreducible component of
$X_{\mathfrak{p}}\setminus \mathbb{A}_{\mathfrak{p}}^{2}$
containing
$x$
,
$x\not \in I(f_{\mathfrak{p}}^{n})$
for all
$n\geqslant 0$
and
$f_{\mathfrak{p}}|_{E_{\mathfrak{p}}}$
is not ramified at any point on the orbit of
$x$
. After replacing
$\mathbb{Q}_{\mathfrak{p}}$
by a finite extension
$K_{\mathfrak{p}}$
, we may suppose that
$x$
is defined over
$O_{K_{\mathfrak{p}}}/\mathfrak{p}$
. After replacing
$f$
by a positive iterate, we may suppose that
$x$
is fixed by
$f_{\mathfrak{p}}$
.
The fixed point
$x$
of
$f_{\mathfrak{p}}$
defines an open and closed polydisc
$U$
in
$X(K_{\mathfrak{p}})$
with respect to the
$\mathfrak{p}$
-adic norm
$|\cdot |_{\mathfrak{p}}$
. We have
$f(U)\subseteq U$
. Observe that
$f^{\ast }E=d(f,v)E$
in
$U$
and
$d(f,v)\geqslant 2$
. So, for all points
$q\in U\cap \mathbb{A}^{2}(K_{\mathfrak{p}})$
, we have
$d_{\mathfrak{p}}(f^{n}(q),E)\rightarrow 0$
as
$n\rightarrow \infty$
.
Since
$f_{\mathfrak{p}}|_{E_{\mathfrak{p}}}$
is not ramified at
$x$
, after replacing
$f$
by some positive iterate, we may suppose that
$df_{\mathfrak{p}}|_{E_{\mathfrak{p}}}(x)=1$
. By [Reference PoonenPoo14, Theorem 1], we have that for any point
$q\in U\cap E$
, there exists a
$\mathfrak{p}$
-adic analytic map
$\unicode[STIX]{x1D6F9}:O_{K_{\mathfrak{p}}}\rightarrow U\cap E$
such that for any
$n\geqslant 0$
, we have
$f^{n}(q)=\unicode[STIX]{x1D6F9}(n)$
.
If there exists a preperiodic point
$q$
in
$U\cap E$
, then there exists
$m\geqslant 0$
such that
$f^{m}(q)$
is periodic. Then there are infinitely many
$n\in \mathbb{Z}^{+}\subseteq O_{K_{\mathfrak{p}}}$
such that
$\unicode[STIX]{x1D6F9}(n)=f^{m}(q)$
. The fact that
$O_{K_{\mathfrak{p}}}$
is compact shows that
$\unicode[STIX]{x1D6F9}$
is constant. It follows that
$q$
is fixed. Thus, all preperiodic points in
$U\cap E$
are fixed by
$f|_{E}$
.
Let
$S$
be the set of all fixed points in
$U\cap E$
. Since
$f^{n}|_{E}\neq \operatorname{id}$
for all
$n\geqslant 1$
,
$S$
is finite.
We first treat the case
$S=\emptyset$
. Pick a point
$p\in U\cap \mathbb{A}^{2}(\bar{\mathbb{Q}})$
. Then
$p$
is not preperiodic. If
$O(p)$
is not Zariski dense, we denote by
$Z$
its Zariski closure. Pick a one-dimensional irreducible component
$C$
of
$Z$
. We have
$C\cap E\cap U\neq \emptyset$
and, for all points
$q\in C\cap E\cap U$
,
$q$
is preperiodic under
$f|_{E}$
. This contradicts our assumption, so
$O(p)$
is Zariski dense.
Next, we treat the case
$S\neq \emptyset$
. Since
$|df|_{E}(q_{i})|_{\mathfrak{p}}=1$
for all
$i=1,\ldots ,m$
, we have
$df|_{E}(q_{i})\neq 0$
. By embedding
$K$
in
$\mathbb{C}$
, we endow
$X$
with the usual Euclidean topology. By [Reference AbateAba01, Theorem 3.1.4], for any
$i=1,\ldots ,m$
, there exists a unique complex analytic manifold
$W$
not contained in
$E$
such that
$f(W)=W$
. It follows that there are at most one irreducible algebraic curve
$C_{i}\neq E$
in
$X$
such that
$q_{i}\in C_{i}$
and
$f(C_{i})\subseteq C_{i}$
. For convenience, if such an algebraic curve does not exist, we define
$C_{i}$
to be
$\emptyset$
.
For any
$n\geqslant 1$
, by applying [Reference AbateAba01, Theorem 3.1.4] for
$f^{n}$
, if
$C$
is a curve satisfying
$q_{i}\in C$
and
$f(C)\subseteq C$
, then
$C=C_{i}$
. Moreover if
$C^{\prime }$
is an irreducible component of
$f^{-1}(C_{i})$
such that
$q\in C^{\prime }$
, then, for any point
$y\in C^{\prime }$
near
$q$
with respect to the Euclidean topology, we have
$f(p)\in C$
. Then, by [Reference AbateAba01, (iv) of Theorem 3.1.4], we have
$p\in C$
. It follows that
$C^{\prime }=C$
. Thus, there exists a small open and closed neighborhood
$U_{i}$
of
$q_{i}$
with respect to the norm
$|\cdot |_{\mathfrak{p}}$
such that for all
$j\neq i$
,
$q_{j}\not \in U_{i}$
,
$f(U_{i})\subseteq U_{i}$
and
$f^{-1}(C_{i}\cap U_{i})\cap U_{i}=C_{i}\cap U_{i}$
.
Observe that
$\mathbb{A}^{2}(\bar{\mathbb{Q}}\cap K_{\mathfrak{p}})$
is dense in
$\mathbb{A}^{2}(K_{\mathfrak{p}})$
with respect to
$|\cdot |_{\mathfrak{p}}$
.
There exists a
$\bar{\mathbb{Q}}$
-point
$p$
in
$U_{1}\setminus C_{1}(K_{\mathfrak{p}})$
. If the orbit
$O(p)$
of
$p$
is not Zariski dense, denote by
$Z$
its Zariski closure. Since
$d_{\mathfrak{p}}(f^{n}(p),E)\rightarrow 0$
as
$n\rightarrow \infty$
,
$p$
is not preperiodic. It follows that there exists a one-dimensional irreducible component
$C$
of
$Z$
which is periodic. There exist
$a,b>0$
such that
$f^{an+b}(p)\in C$
for all
$n\geqslant 0$
. Since
$d_{\mathfrak{p}}(f^{n}(p),E)\rightarrow 0$
as
$n\rightarrow \infty$
and
$U_{1}$
is closed, there exists a point
$q\in C\cap E\cap U_{1}$
. It follows that
$q$
is periodic under
$f|_{E}$
. Then
$q$
is fixed and
$q=q_{1}$
. This implies that
$C=C_{1}$
. Since
$f^{b}(p)\in C_{1}\cap V$
and
$f^{-1}(C_{1})\cap V=C_{1}$
, we have
$p\in C_{1}$
, which is a contradiction. It follows that
$O(p)$
is Zariski dense.◻
Proposition 6.3. If
$\unicode[STIX]{x1D706}_{2}^{2}(f)=\unicode[STIX]{x1D706}_{1}(f)>1$
, then Theorem 1.1 holds.
Proof of Proposition 6.3.
By [Reference Favre and JonssonFJ11, Proposition 5.1] and [Reference Favre and JonssonFJ11, Proposition 5.3], there exists a divisorial valuation
$v_{\ast }\in V_{\infty }$
satisfying
$f_{\bullet }(v_{\ast })=v_{\ast }$
and
$d(f,v_{\ast })=\unicode[STIX]{x1D706}_{1}\geqslant 2$
. Moreover, there exist a compactification
$X$
of
$\mathbb{A}^{2}$
and an irreducible component
$E$
in
$X\setminus \mathbb{A}^{2}$
satisfying
$v_{\ast }=v_{E}$
and
$\operatorname{deg}(f|_{E})=\unicode[STIX]{x1D706}_{1}\geqslant 2$
. We conclude our theorem by invoking Lemma 6.2.◻
7 Totally invariant curves
Let
$f:\mathbb{A}^{2}\rightarrow \mathbb{A}^{2}$
be a dominant polynomial endomorphism defined over an algebraically closed field
$k$
of characteristic
$0$
. Let
$X$
be a compactification of
$\mathbb{A}_{k}^{2}$
. Then
$f$
extends to a rational selfmap on
$X$
.
As in [Reference CantatCan10], a curve
$C$
in
$X$
is said to be
$totally$
$invariant$
if the strict transform
$f^{\#}C$
equals
$C$
.
If there are infinitely many irreducible totally invariant curves in
$\mathbb{A}_{k}^{2}$
, then [Reference CantatCan10, Theorem B] shows that
$f$
preserves a nontrivial fibration.
Proposition 7.1. If there are infinitely many irreducible curves in
$\mathbb{A}_{k}^{2}$
that are totally invariant under
$f$
, then there is a nonconstant rational function
$g$
satisfying
$g\circ f=g$
.
In this section, we give a direct proof of this result.
Proof of Proposition 7.1.
Let
$\{C_{i}\}_{i\geqslant 1}$
be an infinite sequence of distinct irreducible totally invariant curves in
$\mathbb{A}_{k}^{2}$
. Since the ramification locus of
$f$
is of dimension at most
$1$
, after replacing
$\{C_{i}\}_{i\geqslant 1}$
by an infinite subsequence, we may suppose that
$C_{i}$
is not contained in the ramification locus of
$f$
for any
$i\geqslant 0$
. Then we have
$\operatorname{ord}_{C_{i}}f^{\ast }C_{i}=1$
for all
$i\geqslant 0$
.
Let
$E_{1},\ldots ,E_{s}$
be the set of irreducible curves in
$\mathbb{A}^{2}$
contracted by
$f$
. Let
$V$
be the
$\mathbb{Q}$
-subspace in
$\operatorname{Div}(\mathbb{A}^{2})\otimes \mathbb{Q}$
spanned by
$E_{i}$
,
$i=1,\ldots ,s$
. Then we have
$f^{\ast }C_{i}=C_{i}+F_{i}$
, where
$F_{i}\in V$
for all
$i\geqslant 1$
. Set
$W_{i}:=\bigcap _{j\geqslant i}(\sum _{t\geqslant j}\mathbb{Q}F_{t})\subseteq V$
. We have
$W_{i+1}\subseteq W_{i}$
for
$i\geqslant 1$
. Since
$V$
is of finite dimension, there exists
$n_{0}\geqslant 1$
such that
$W_{i}=W_{n_{0}}$
for all
$i\geqslant n_{0}$
. We may suppose that
$n_{0}=1$
and set
$W:=W_{n_{0}}$
. Moreover, we may suppose that
$W$
is generated by
$F_{1},\ldots ,F_{l}$
, where
$l=\dim W$
.
For all
$i\geqslant l+1$
, we have
$F_{i}=\sum _{j=1}^{l}a_{j}^{i}F_{j}$
, where
$a_{j}^{i}\in \mathbb{Q}$
. Since
$F_{i}=f^{\ast }C_{i}-rC_{i}$
, we have
$f^{\ast }(C_{i}-\sum _{j=1}^{l}a_{j}^{i}C_{j})=r(C_{i}-\sum _{j=1}^{l}a_{j}^{i}C_{j})$
. There exists
$n_{i}\in \mathbb{Z}^{+}$
such that
$n_{i}a_{j}^{i}\in \mathbb{Z}$
for all
$j=1,\ldots ,l$
. Then we have
$f^{\ast }(n_{i}C_{i}-\sum _{j=1}^{l}n_{i}a_{j}^{i}C_{j})=r(n_{i}C_{i}-\sum _{j=1}^{l}n_{i}a_{j}^{i}C_{j})$
. Up to multiplication by a nonzero constant, there exists a unique
$g_{i}\in k(x,y)\setminus \{0\}$
such that
$\operatorname{Div}(g_{i})=n_{i}C_{i}-\sum _{j=1}^{l}n_{i}a_{j}^{i}C_{j}$
. It follows that
$f^{\ast }g_{i}=A_{i}g_{i}$
, where
$A_{i}\in k\setminus \{0\}$
. Since
$n_{i}C_{i}-\sum _{j=1}^{l}n_{i}a_{j}^{i}C_{j}\neq 0$
for
$i\geqslant l+1$
,
$g_{i}$
is nonconstant for
$i\geqslant l+1$
. This concludes the proof.◻
Corollary 7.2. If
$f$
is birational, then either there is a nonconstant rational function
$g$
satisfying
$g\circ f=g$
, or there exists a point
$p\in \mathbb{A}^{2}(k)$
with Zariski dense orbit.
Proof of Corollary 7.2.
There exists a finite generated
$\mathbb{Q}$
-subalgebra
$R$
of
$k$
such that
$f$
is defined over
$R$
. Denote by
$K$
the fraction field of
$R$
.
By [Reference BellBel06, Lemma 3.1], there exist a prime
$\mathfrak{p}\geqslant 3$
and an embedding of
$R$
into
$\mathbb{Z}_{\mathfrak{p}}$
such that all coefficients of
$f$
are of
$\mathfrak{p}$
-adic norm
$1$
. Denote by
$\mathbb{F}$
the algebraic closure of
$\mathbb{F}_{\mathfrak{p}}$
. Then the degree of the specialization
$f_{\mathfrak{p}}:\mathbb{A}_{\mathbb{F}}^{2}{\dashrightarrow}\mathbb{A}_{\mathbb{F}}^{2}$
of
$f$
equals
$\operatorname{deg}f$
.
By [Reference XieXie15b, Proposition 6.2], there exists a noncritical periodic point
$x\in \mathbb{A}^{2}(\mathbb{F})$
. After replacing
$\mathbb{Q}_{p}$
by a finite extension
$K_{\mathfrak{p}}$
, we may suppose that
$x$
is defined over
$O_{K_{\mathfrak{p}}}/\mathfrak{p}$
. Replacing
$f$
by a suitable iterate, we may suppose that
$x$
is fixed. Since
$x$
is noncritical,
$df_{\mathfrak{p}}(x)$
is invertible. After replacing
$f$
by a suitable iterate, we may suppose that
$df_{\mathfrak{p}}(x)=\operatorname{id}$
.
The fixed point
$x$
defines an open and closed neighborhood
$U$
in
$\mathbb{A}^{2}(K_{\mathfrak{p}})$
with respect to
$d_{\mathfrak{p}}$
such that
$f(U)\subseteq U$
. By applying [Reference PoonenPoo14, Theorem 1], we have that for any point
$q\in U$
, there exists a
$\mathfrak{p}$
-adic analytic map
$\unicode[STIX]{x1D6F9}:O_{K_{\mathfrak{p}}}\rightarrow U$
such that for any
$n\geqslant 0$
, we have
$f^{n}(q)=\unicode[STIX]{x1D6F9}(n)$
.
Arguing by contradiction, we suppose that the orbit
$O(q)$
of
$q$
is not Zariski dense for any
$q\in \mathbb{A}^{2}(k\cap K_{\mathfrak{p}})$
. As in the proof of Lemma 6.2, if
$q$
is preperiodic, then
$q$
is fixed. Suppose that
$q$
is nonpreperiodic. There exists an irreducible curve
$C$
such that
$f^{n}(q)\in C$
for infinitely many
$n\geqslant 0$
. Let
$P$
be a polynomial such that
$C$
is defined by
$P=0$
. Then
$P\circ \unicode[STIX]{x1D6F9}$
is an analytic function on
$O_{K_{\mathfrak{p}}}$
having infinitely many zeros. It follows that
$P\circ \unicode[STIX]{x1D6F9}\equiv 0$
and then
$f^{n}(q)\in C$
for all
$n\geqslant 0$
. Then we have
$f(C)=C$
. Since
$f$
is birational, a curve
$C$
is totally invariant by
$f$
if and only if
$f(C)=C$
. We may suppose that
$f\neq \operatorname{id}$
. Since
$\overline{\mathbb{Q}}\cap K_{\mathfrak{p}}\subseteq k$
and the
$\overline{\mathbb{Q}}\cap K_{\mathfrak{p}}$
-points in
$U$
are Zariski dense in
$\mathbb{A}_{K_{\mathfrak{p}}}^{2}$
, there are infinitely many irreducible totally invariant curves in
$\mathbb{A}^{2}$
. We conclude our corollary by invoking Proposition 7.1.◻
8 Proof of Theorem 1.1
Let
$f:\mathbb{A}^{2}\rightarrow \mathbb{A}^{2}$
be a dominant polynomial endomorphism defined over an algebraically closed field
$k$
of characteristic
$0$
.
After replacing
$k$
by an algebraically closed subfield which contains all the coefficients of
$f$
, we may suppose that the transcendence degree of
$k$
over
$\bar{\mathbb{Q}}$
is finite.
By Lemma 6.1, Proposition 6.3 and Corollary 7.2, we may suppose that
$\unicode[STIX]{x1D706}_{1}^{2}>\unicode[STIX]{x1D706}_{2}>1$
and
$v_{\ast }$
is divisorial. Suppose that
$v_{\ast }=v_{E}$
for some irreducible exceptional divisor
$E$
in some compactification
$X$
. If
$f^{n}|_{E}\neq \operatorname{id}$
for all
$n\geqslant 1$
, Lemma 6.2 concludes the proof. So, after replacing
$f$
by a suitable iterate, we may suppose that
$f|_{E}=\operatorname{id}$
.
By choosing a suitable compactification
$X\in {\mathcal{C}}_{0}$
, we may suppose that
$E\cap I(f)=\emptyset$
. There exists a subfield
$K$
of
$k$
which is finitely generated over
$\mathbb{Q}$
such that
$X,f,E,I(f)$
are defined over
$K$
. Moreover, we may suppose that
$E\simeq \mathbb{P}^{1}$
over
$K$
.
By [Reference BellBel06, Lemma 3.1], there exists a prime
$\mathfrak{p}\geqslant 3$
such that we can embed
$K$
into
$\mathbb{Q}_{\mathfrak{p}}$
. Further, there exists an open and closed set
$U$
of
$X(\mathbb{Q}_{\mathfrak{p}})$
with respect to the norm
$|\cdot |_{\mathfrak{p}}$
containing
$E$
and satisfying
$U\cap I(f)=\emptyset$
,
$f(U)\subseteq U$
and
$d_{\mathfrak{p}}(f^{n}(p),E)\rightarrow 0$
as
$n\rightarrow \infty$
for any
$p\in U$
.
By [Reference AbateAba01, Theorem 3.1.4], for any point
$q\in E(\bar{K}\cap Q_{\mathfrak{p}})$
, there is at most one irreducible algebraic curve
$C_{q}\neq E$
in
$X$
such that
$q\in C_{q}$
and
$f(C_{q})\subseteq C_{q}$
. For convenience, if such an algebraic curve does not exist, set
$C_{q}:=\emptyset$
. Further, if
$C_{q}\neq \emptyset$
,
$C_{q}$
is smooth at
$q$
and intersects
$E$
transitively.
If there are only finitely many points
$q\in E(\bar{K}\cap \mathbb{Q}_{\mathfrak{p}})$
such that
$C_{q}$
is an algebraic curve, there exist a point
$q\in E$
and an open and closed set
$V$
with respect to the norm
$|\cdot |_{\mathfrak{p}}$
containing
$q$
such that
$f(V)\subseteq V$
and
$C_{x}=\emptyset$
for all
$x\in V\cap E(\bar{K}\cap \mathbb{Q}_{\mathfrak{p}})$
. Pick a point
$p\in V\cap \mathbb{A}^{2}(\bar{K}\cap Q_{\mathfrak{p}})$
; then
$p$
is not preperiodic. If
$O(p)$
is not Zariski dense, we denote by
$Z$
its Zariski closure. There exists a one-dimensional irreducible component
$C$
of
$Z$
which is periodic under
$f$
. Since
$C\cap O(p)$
is infinite,
$C$
is defined over
$\bar{K}\cap Q_{\mathfrak{p}}$
. Since
$d_{v}(f^{n}(p),E)\rightarrow 0$
as
$n\rightarrow \infty$
, we have
$C\cap E(\bar{K}\cap Q_{\mathfrak{p}})\cap V\neq \emptyset$
. Pick
$q\in C\cap E(\bar{K}\cap Q_{\mathfrak{p}})\cap V$
; then we have that
$C_{q}=C$
is an algebraic curve, which contradicts our assumption. Thus,
$O(p)$
is Zariski dense.
Otherwise there exists an infinite sequence of points
$q_{i}$
,
$i\geqslant 1$
, such that
$C_{i}:=C_{q_{i}}$
is an algebraic curve. Since
$f|_{C_{i}}$
is an endomorphism of
$C_{i}$
of degree
$\unicode[STIX]{x1D706}_{1}>1$
, every
$C_{i}$
is rational and has at most two branches at infinity. We may suppose that
$E$
is the unique irreducible component of
$X\setminus \mathbb{A}^{2}$
containing
$q_{i}$
for all
$i\geqslant 1$
and
$C_{i}\neq C_{j}$
for
$i\neq j$
. We need the following result, which is proved below.
Lemma 8.1. After replacing
$\{C_{i}\}_{i\geqslant 1}$
by an infinite subsequence, we have that either
$\operatorname{deg}(C_{i})$
is bounded or Theorem 1.1 holds.
Suppose that
$\operatorname{deg}(C_{i})$
is bounded. Pick an ample line bundle
$L$
on
$X$
. Then there exists
$M>0$
such that
$(C_{i}\cdot L)\leqslant M$
for all
$i\geqslant 1$
.
There exist a smooth projective surface
$\unicode[STIX]{x1D6E4}$
, a birational morphism
$\unicode[STIX]{x1D70B}_{1}:\unicode[STIX]{x1D6E4}\rightarrow X$
and a morphism
$\unicode[STIX]{x1D70B}_{2}:\unicode[STIX]{x1D6E4}\rightarrow X$
satisfying
$f=\unicode[STIX]{x1D70B}_{2}\circ \unicode[STIX]{x1D70B}_{1}$
. We denote by
$f_{\ast }$
the map
$\unicode[STIX]{x1D70B}_{2\ast }\circ \unicode[STIX]{x1D70B}_{1}^{\ast }:\operatorname{Div}X\rightarrow \operatorname{Div}X$
. Let
$E_{\unicode[STIX]{x1D70B}_{1}}$
be the union of exceptional irreducible divisors of
$\unicode[STIX]{x1D70B}_{1}$
and
$\mathfrak{E}$
be the set of effective divisors in
$X$
supported by
$\unicode[STIX]{x1D70B}_{2}(E_{\unicode[STIX]{x1D70B}_{1}})$
. It follows that for any curve
$C$
in
$X$
, there exists
$D\in \mathfrak{E}$
such that
$f_{\ast }C=\operatorname{deg}(f|_{C})f(C)+D$
.
For any effective line bundle
$M\in \operatorname{Pic}(X)$
, the projective space
$H_{M}:=\mathbb{P}(H^{0}(M))$
parameterizes the curves
$C$
in the linear system
$|M|$
. Since
$\operatorname{Pic}^{0}(X)=0$
, for any
$l\geqslant 0$
, there are only finitely many effective line bundles satisfying
$(M\cdot L)\leqslant l$
.
Then
$H^{l}:=\coprod _{(M\cdot L)\leqslant l}H_{M}$
is a finite union of projective spaces and it parameterizes the curves
$C$
in
$X$
satisfying
$(C\cdot L)\leqslant l$
.
There exists
$d\geqslant 1$
such that
$dL-f^{\ast }L$
is nef. Then, for any curve
$C$
in
$X$
, we have
$(f_{\ast }C\cdot L)=(C\cdot f^{\ast }L)\leqslant d(C\cdot L)$
. It follows that
$f_{\ast }$
induces a morphism
$F:H^{l}\rightarrow H^{dl}$
by
$C\rightarrow f_{\ast }C$
for all
$l\geqslant 1$
. For all
$l\geqslant 1$
,
$a\in \mathbb{Z}^{+}$
and
$D\in \mathfrak{E}$
, there exists an embedding
$i_{a,D}:H_{l}\rightarrow H_{al+(D\cdot L)}$
by
$C\mapsto aC+D$
. Let
$Z_{1},\ldots ,Z_{m}$
be all irreducible components of the Zariski closure of
$\{C^{j}\}_{j\leqslant -1}$
in
$H^{M}$
of maximal dimension. For any
$i\in \{1,\ldots ,m\}$
, there exists
$l\leqslant M$
such that
$(C\cdot L)=l$
for all
$C\in Z_{i}$
. Let
$S$
be the finite set of pairs
$(a,D)$
, where
$a\in \mathbb{Z}^{+}$
,
$D\in \mathfrak{E}$
satisfying
$al+(D\cdot L)\leqslant dM$
. Then we have
$F(Z_{i})\subseteq \bigcup _{j=1,\ldots ,m}\bigcup _{(a,D)\in S}i_{a,D}(Z_{j})$
. It follows that there exist a unique
$j_{i}\in \{1,\ldots ,m\}$
and a unique
$(a,D)\in S$
such that
$F(Z_{i})=i_{a,D}(Z_{j_{i}})$
. Observe that the map
$i\mapsto j_{i}$
is a one to one map of
$\{1,\ldots ,m\}$
. After replacing
$f$
by a positive iterate, we may suppose that
$j_{i}=i$
and
$F(Z_{i})\subseteq i_{a_{Z_{i}},D_{Z_{i}}}Z_{i}$
for all
$i=1,\ldots ,m$
. Set
$Z:=Z_{1}$
,
$a=a_{Z_{1}}$
and
$D=D_{Z_{1}}$
. We may suppose that
$C_{i}\in Z$
for all
$i\geqslant 1$
. Since
$f(C_{i})=C_{i}$
for all
$i\geqslant 1$
, we have
$i_{a_{Z_{1}},D_{Z_{1}}}^{-1}\circ F|_{Y}=\operatorname{id}$
.
For any point
$t\in Z$
, denote by
$C^{t}$
the curve parameterized by
$t$
. Then
$f(C^{t})=C^{t}$
for all
$t\in Z$
. Since
$C_{i}$
intersects
$E$
transversely at at most two points, there exists
$s\in \{1,2\}$
such that
$C^{t}$
intersects
$E$
transversely at
$s$
points for a general
$t\in Z$
. It follows that, for a general point
$t\in Z$
, there are exactly
$s$
points
$q_{t}^{1},\ldots ,q_{t}^{s}\in E$
such that
$C^{t}=C_{q_{t}^{j}}$
for
$j=1,\ldots ,s$
.
Set
$Y:=\{(p,t)\in X\times Z\mid p\in C^{t}\}$
. Denote by
$\unicode[STIX]{x1D70B}_{1}:Y\rightarrow X$
the projection to the first coordinate and by
$\unicode[STIX]{x1D70B}_{2}:Y\rightarrow Z$
the projection to the second coordinate. Since
$C^{t}\cap E$
is not empty for general
$t\in Z$
, the map
$\unicode[STIX]{x1D70B}_{2}|_{\unicode[STIX]{x1D70B}_{1}^{\ast }E}$
is dominant. We see that
$f$
induces a map
$T:Y\rightarrow Y$
defined by
$(p,t)\rightarrow (f(p),t)$
. Since there are infinitely many points in
$E$
contained in
$\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D70B}_{1}^{\ast }E)$
,
$\unicode[STIX]{x1D70B}_{1}|_{\unicode[STIX]{x1D70B}_{1}^{\ast }E}:\unicode[STIX]{x1D70B}_{1}^{\ast }E\rightarrow E$
is dominant. For a general point
$t\in Z$
, there are exactly
$s$
points in
$\unicode[STIX]{x1D70B}_{1}^{\ast }E$
. It follows that the map
$\unicode[STIX]{x1D70B}_{2}|_{\unicode[STIX]{x1D70B}_{1}^{\ast }E}$
is generically finite of degree
$s$
. For a general point
$q\in E$
, there exists only one point
$(q,C_{q})\in \unicode[STIX]{x1D70B}_{1}^{\ast }E$
. Hence,
$\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D70B}_{1}^{\ast }E):\unicode[STIX]{x1D70B}_{1}^{\ast }E\rightarrow E$
is birational. It follows that
$Z$
is a rational curve and
$Y$
is a surface. Thus, the morphism
$\unicode[STIX]{x1D70B}_{1}:Y\rightarrow X$
is generically finite. Let
$p$
be a general point in
$\mathbb{A}^{2}$
. If
$\#\unicode[STIX]{x1D70B}_{1}^{-1}(p)\geqslant 2$
, then there are
$t_{1}\neq t_{2}\in Z$
such that
$p\in C^{t_{1}}\cap C^{t_{2}}$
. Since there exists
$M^{\prime }>0$
such that
$\operatorname{deg}C^{t}\leqslant M^{\prime }$
for all
$t\in Z$
, we have
$\#(C_{t_{1}}\cap C_{t_{2}})\leqslant {M^{\prime }}^{2}$
. It follows that there exist
$a<b\in \{0,\ldots ,{M^{\prime }}^{2}\}$
such that
$f^{a}(p)=f^{b}(p)$
. This contradicts the assumption that
$p$
is general. It follows that the morphism
$\unicode[STIX]{x1D70B}_{1}:Y\rightarrow X$
is birational. Identify
$Z$
with
$\mathbb{P}^{1}$
. Set
$g:=\unicode[STIX]{x1D70B}_{2}\circ \unicode[STIX]{x1D70B}_{1}^{-1}$
. Then we have
$g\circ f=g$
, which completes the proof.
Proof of Lemma 8.1.
If
$C_{i}$
has only one place at infinity for all
$i\geqslant 1$
, then
$\operatorname{deg}C_{i}=b_{E}$
. So, we may suppose that
$C_{i}$
has two places at infinity for all
$i\geqslant 1$
.
We first suppose that
$\#C_{i}\cap E=2$
for all
$i\geqslant 1$
. We may suppose that
$C_{i}\cap E\cap \operatorname{Sing}(X\setminus \mathbb{A}^{2})=\emptyset$
for all
$i\geqslant 1$
. Then, for all
$i\geqslant 1$
, we have
$\operatorname{deg}C_{i}=2b_{E}$
.
Then we may suppose that
$C_{i}\cap E=\{q_{i}\}$
for all
$i\geqslant 1$
. Let
$c_{i}$
be the unique branch of
$C$
at infinity centered at
$q_{i}$
and
$w_{i}$
the unique branch of
$C$
at infinity not centred at
$q_{i}$
. Since
$f(C_{i})=C_{i}$
, we have
$f_{\bullet }(c_{i})=c_{i}$
and
$f_{\bullet }(w_{i})=w_{i}$
.
We first treat the case
$\unicode[STIX]{x1D703}^{\ast }=Z_{v^{\ast }}$
, where
$v^{\ast }\in V_{\infty }$
is divisorial. It follows that
$\unicode[STIX]{x1D706}_{2}/\unicode[STIX]{x1D706}_{1}\in \mathbb{Z}^{+}$
. Observe that
$\operatorname{deg}(f|_{C_{i}})=\unicode[STIX]{x1D706}_{1}$
for
$i$
large enough.
If
$\unicode[STIX]{x1D706}_{1}=\unicode[STIX]{x1D706}_{2}$
, then the strict transform
$f^{\#}(C_{i})$
equals
$C_{i}$
for
$i$
large enough, and the lemma follows from Proposition 7.1.
Otherwise we have
$\unicode[STIX]{x1D706}_{2}/\unicode[STIX]{x1D706}_{1}\geqslant 2$
. Set
$v^{\ast }:=v_{E^{\prime }}$
. Then we have
$d(f,v^{\ast })=\unicode[STIX]{x1D706}_{2}/\unicode[STIX]{x1D706}_{1}\geqslant 2$
and
$\operatorname{deg}(f|_{E^{\prime }})=\unicode[STIX]{x1D706}_{1}>1$
. By Lemma 6.2, Theorem 1.1 holds.
Next, we treat the case
$\unicode[STIX]{x1D703}^{\ast }=Z_{v^{\ast }}$
, where
$v^{\ast }\in V_{\infty }$
is not divisorial. Since
$\unicode[STIX]{x1D6FC}(v^{\ast })=0$
,
$v^{\ast }$
cannot be irrational. Thus,
$v^{\ast }$
is infinitely singular and hence an end point in
$V_{\infty }$
. Then
$f$
is proper.
By [Reference XieXie15a, Proposition 15.2], there exists
$v_{1}<v^{\ast }$
such that for any valuation
$v\neq v^{\ast }$
in
$U:=\{w\in V_{\infty }\mid w>v_{1}\}$
, there exists
$N\geqslant 1$
such that
$f_{\bullet }^{n}(v)\not \in U$
for all
$n\geqslant N$
. It follows that there is no curve valuation in
$U$
which is periodic under
$f_{\bullet }$
. Let
$U_{E}$
be the open set in
$V_{\infty }$
consisting of all valuations whose center in
$X$
is contained
$E$
. It follows that
$w_{i}\not \in U\cup U_{E}$
for all
$i\geqslant 1$
. Hence,
$w_{i}\not \in U\cup f_{\bullet }^{-N}(U_{E})$
for all
$N\geqslant 0$
. Set
$W_{-4}:=\{v\in V_{\infty }\mid \unicode[STIX]{x1D6FC}(v)\geqslant -4\}$
. By [Reference XieXie15a, Proposition 11.6] and the fact that
$W_{-4}\setminus U$
is compact, there exists
$N\geqslant 0$
such that
$f_{\bullet }^{N}(W_{-4}\setminus U)\subseteq U_{E}$
. Then we have
$W_{-4}\subseteq U\cup f_{\bullet }^{-N}(U_{E})$
.
Since the boundary
$\unicode[STIX]{x2202}(V_{\infty }\setminus (U\cup f_{\bullet }^{-N}(U_{E})))$
of
$V_{\infty }\setminus (U\cup f_{\bullet }^{-N}(U_{E}))$
is finite and
$w_{i}\in V_{\infty }\setminus (U\cup f_{\bullet }^{-N}(U_{E}))$
for all
$i\geqslant 1$
, we may suppose that there exists
$w\in \unicode[STIX]{x2202}(V_{\infty }\setminus (U\cup f_{\bullet }^{-N}(U_{E})))$
satisfying
$w_{i}>w$
for all
$i\geqslant 1$
.
If, for all
$i\geqslant 1$
, we have
$(w_{i}\cdot l_{\infty })\leqslant 1/2\operatorname{deg}(C_{i})$
, then
$\operatorname{deg}C_{i}=(w_{i}\cdot l_{\infty })+(c_{i}\cdot l_{\infty })\leqslant 1/2\operatorname{deg}(C_{i})+b_{E}$
. It follows that
$\operatorname{deg}(C_{i})\leqslant 2b_{E}$
.
Thus, we may suppose that
$(w_{i}\cdot l_{\infty })\geqslant 1/2\operatorname{deg}(C_{i})$
for all
$i\geqslant 1$
. For any
$i\neq j$
, the intersection number
$(C_{i}\cdot C_{j})$
is the sum of the local intersection numbers at all points in
$C_{i}\cap C_{j}$
. Since all the local intersection numbers are positive, we have
$$\begin{eqnarray}\operatorname{deg}(C_{i})\operatorname{deg}(C_{j})\geqslant (c_{i}\cdot c_{j})+(w_{i}\cdot w_{j}).\end{eqnarray}$$
By the calculation in § 2.6, we have
$$\begin{eqnarray}\displaystyle (c_{i}\cdot c_{j})+(w_{i}\cdot w_{j}) & {\geqslant} & \displaystyle b_{E}^{2}(1-\unicode[STIX]{x1D6FC}(v_{E}))+(w_{i}\cdot l_{\infty })(w_{j}\cdot l_{\infty })(1-\unicode[STIX]{x1D6FC}(w))\nonumber\\ \displaystyle & {\geqslant} & \displaystyle b_{E}^{2}(1-\unicode[STIX]{x1D6FC}(v_{E}))+5/4\operatorname{deg}(C_{i})\operatorname{deg}(C_{j}).\nonumber\end{eqnarray}$$
Thus,
$\operatorname{deg}(C_{i})\operatorname{deg}(C_{j})\leqslant -4b_{E}^{2}(1-\unicode[STIX]{x1D6FC}(v_{E}))<0$
, which is a contradiction.
Finally, we treat the case when
$\#\operatorname{Supp}\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D703}^{\ast }\geqslant 2$
. By [Reference Favre and JonssonFJ11, Theorem 2.4], we have
$\unicode[STIX]{x1D703}^{\ast }>0$
on the set
$W_{0}:=\{v\in V_{\infty }\mid \unicode[STIX]{x1D6FC}(v)\geqslant 0\}$
. Set
$W_{-1}:=\{v\in V_{\infty }\mid \unicode[STIX]{x1D6FC}(v)\geqslant -1\}$
and
$Y:=\{v\in W_{-1}\mid \unicode[STIX]{x1D703}^{\ast }(v)=0\}$
. By [Reference XieXie15a, Proposition 11.2],
$Y$
is compact. For any point
$y\in Y$
, there exists
$w_{y}<y$
satisfying
$\unicode[STIX]{x1D6FC}(w_{y})\in (-1,0)$
. Set
$U_{y}:=\{v\in V_{\infty }\mid v>w_{y}\}$
. There are finitely many points
$y_{1},\ldots ,y_{l}$
such that
$Y\subseteq \bigcup _{i=1}^{l}U_{y_{i}}$
. Pick
$r:=1/2\min \{-\unicode[STIX]{x1D6FC}(w_{y_{i}})\}_{i=1,\ldots ,l}$
. Then
$r\in (0,1)$
and
$W_{-r}\cap (\bigcup _{i=1}^{l}U_{y_{i}})=\emptyset$
. It follows that there exists
$t>0$
such that
$\unicode[STIX]{x1D703}^{\ast }\geqslant t$
on
$W_{-r}$
. By [Reference XieXie15a, Proposition 11.6], there exists
$N\geqslant 0$
such that
$f_{\bullet }^{N}(W_{-r})\subseteq U_{E}$
. Then we have
$W_{-r}\subseteq f_{\bullet }^{-N}(U_{E})$
.
Since the boundary
$\unicode[STIX]{x2202}(V_{\infty }\setminus (U\cup f_{\bullet }^{-N}(U_{E})))$
of
$V_{\infty }\setminus (U\cup f_{\bullet }^{-N}(U_{E}))$
is finite and
$w_{i}\in V_{\infty }\setminus (U\cup f_{\bullet }^{-N}(U_{E}))$
for all
$i\geqslant 1$
, we may suppose that there exists
$w\in \unicode[STIX]{x2202}(V_{\infty }\setminus (U\cup f_{\bullet }^{-N}(U_{E})))$
satisfying
$w_{i}>w$
for all
$i\geqslant 1$
.
Pick
$\unicode[STIX]{x1D6FF}\in (0,r/2(1+r))$
. If, for all
$i\geqslant 1$
, we have
$(w_{i}\cdot l_{\infty })\leqslant (1-\unicode[STIX]{x1D6FF})\operatorname{deg}(C_{i})$
, then
$\operatorname{deg}C_{i}=(w_{i}\cdot l_{\infty })+(c_{i}\cdot l_{\infty })\leqslant (1-\unicode[STIX]{x1D6FF})\operatorname{deg}(C_{i})+b_{E}$
. It follows that
$\operatorname{deg}(C_{i})\leqslant b_{E}/\unicode[STIX]{x1D6FF}$
.
Thus, we may suppose that
$(w_{i}\cdot l_{\infty })\geqslant (1-\unicode[STIX]{x1D6FF})\operatorname{deg}(C_{i})$
for all
$i\geqslant 1$
. For any
$i\neq j$
, we have
$$\begin{eqnarray}\displaystyle \operatorname{deg}(C_{i})\operatorname{deg}(C_{j})\geqslant (c_{i}\cdot c_{j})+(w_{i}\cdot w_{j}) & {\geqslant} & \displaystyle b_{E}^{2}(1-\unicode[STIX]{x1D6FC}(v_{E}))+(w_{i}\cdot l_{\infty })(w_{j}\cdot l_{\infty })(1-\unicode[STIX]{x1D6FC}(w))\nonumber\\ \displaystyle & {\geqslant} & \displaystyle b_{E}^{2}(1-\unicode[STIX]{x1D6FC}(v_{E}))+(1-\unicode[STIX]{x1D6FF})^{2}(1+r)\operatorname{deg}(C_{i})\operatorname{deg}(C_{j}).\nonumber\end{eqnarray}$$
Set
$t:=(1-\unicode[STIX]{x1D6FF})^{2}(1+r)-1$
; then we have
$$\begin{eqnarray}t>(1-2\unicode[STIX]{x1D6FF})(1+r)-1>(1-r/(1+r))(1+r)-1=0\end{eqnarray}$$
and hence
$\operatorname{deg}(C_{i})\operatorname{deg}(C_{j})\leqslant -t^{-1}b_{E}^{2}(1-\unicode[STIX]{x1D6FC}(v_{E}))<0$
, which is a contradiction. This concludes the proof.◻
Acknowledgements
Part of this work was done during the PhD study of the author with Charles Favre at École Polytechnique. I would like to thank Charles Favre for his constant support during this time. The referee carefully read a first version of this paper and his/her comments greatly improved it. I also thank Serge Cantat and Stéphane Lamy for useful discussions. I thank Vladimir Popov and Dragos Ghioca for their comments on the first version of the paper. I finally thank Matteo Ruggiero for answering my question about the theorem of Hadamard–Perron [Reference AbateAba01, Theorem 3.1.4].
