Degenerations of complex dynamical systems II: analytic and algebraic stability

Abstract

We study pairs $(f, \Gamma )$ consisting of a non-Archimedean rational function f and a finite set of vertices $\Gamma$ in the Berkovich projective line, under a certain stability hypothesis. We prove that stability can always be attained by enlarging the vertex set $\Gamma$. As a byproduct, we deduce that meromorphic maps preserving the fibers of a rationally-fibered complex surface are algebraically stable after a proper modification. The first article in this series examined the limit of the equilibrium measures for a degenerating 1-parameter family of rational functions on the Riemann sphere. Here we construct a convergent countable-state Markov chain that computes the limit measure. A classification of the periodic Fatou components for non-Archimedean rational functions, due to Rivera-Letelier, plays a key role in the proofs of our main theorems. The appendix contains a proof of this classification for all tame rational functions.

Appendix A: Rivera domains, by Jan Kiwi

Facultad de Matemáticas, Pontificia Universidad Católica de Chile.

Let k be a characteristic zero algebraically closed and complete non-Archimedean field with respect to a nontrivial absolute value $| \cdot |$. Denote by $\mathbf {P}^{1}_{k}$ the Berkovich projective line over k. Throughout this appendix Berkovich type I points will be also called rigid points.

Let $f : \mathbf {P}^{1}_{k}\rightarrow \mathbf {P}^{1}_{k}$ be a rational map of degree at least 2. By definition, a point $\zeta \in \mathbf {P}^{1}_{k}$ belongs to the Julia set $\mathbf {J}(f)$ if for every neighborhood U of $\zeta$, the union of images $\bigcup _n f^n(U)$ omits at most finitely many points of $\mathbf {P}^{1}_{k}$. The Fatou set is the complement of the Julia set. A connected component of the Fatou set will be simply called a Fatou component. Each Fatou component maps onto a Fatou component under f. A Fatou component U is called fixed if $f(U)=U$. An attracting type I fixed point $x_0$ (i.e., a fixed point $x_0$ such that $| f'(x_0)| < 1)$ belongs to the Fatou set. The component U that contains $x_0$ is fixed and it is called the immediate basin of $x_0$.

The ramification locus of f is formed by all $x \in \mathbf {P}^{1}_{k}$ such that the local degree $m_f(x)$ satisfies $m_f(x) \ge 2$. We say that $f: \mathbf {P}^{1}_{k}\rightarrow \mathbf {P}^{1}_{k}$ is tame if its ramification locus is contained in the convex hull of the rigid critical points of f.

In his doctoral thesis Rivera-Letelier ([21], Théorème 3) classified periodic Fatou components of rational maps for $k= \mathbb {C}_p$. The aim of this appendix is to provide a proof of this classification for the case of tame rational maps.

Theorem A.1

(Rivera-Letelier) Let $f: \mathbf {P}^{1}_{k}\rightarrow \mathbf {P}^{1}_{k}$ be a tame rational map of degree $d \ge 2$. If U is a fixed Fatou component, then exactly one of the following holds:

• 1. U is the immediate basin of attraction of a type I fixed point.

• 2. The map $f: U \rightarrow U$ is a bijection and $\partial U$ is a union of at most $d-1$ type II periodic orbits.

The proof relies on the general topological fact that maps such as f must have a fixed point $x_0$ in U (see Sect.  A.3). When the fixed point is an attracting rigid (type I) point we are in case (1). Otherwise, the local dynamics at non-rigid Fatou fixed points (see Sect. A.1) allows us to spread the periodic behavior from $x_0$ to the directions containing boundary points of U. In order to be able to reach the boundary of U with this periodic behavior we employ some properties of injective maps (see Sect. A.2). Finally, a counting argument establishes that $\partial U$ is finite and we conclude that $f:U \rightarrow U$ is a bijection.

To ease notation, in the sequel we drop the subscript k. We identify an element $\mathbf {v}$ in $T \mathbf {P}^{1}_x$ with the corresponding open disk $D=D(\mathbf {v})$ defining $\mathbf {v}$. Thus, we abuse notation and regard simultaneously D as an element of $T \mathbf {P}^{1}_x$ and as a subset of $\mathbf {P}^{1}$. For short, we say that D is a direction at x. Also, we let ${\mathbb {H}}= \mathbf {P}^{1}\setminus \mathbb {P}^1(k)$.

1.1 A.1. Periodic points

Theorem 2.1 in [19] says:

Proposition A.2

Let g be a rational map over k of degree at least 2, and let $x \in {\mathbb {H}}$ be a fixed point. The fixed point x belongs to the Julia set $\mathbf {J}(g)$ if and only if one of the following occurs:

• 1. $m_g(x) \ge 2$ where $m_g(x)$ denotes the local degree of g at x.

• 2. There exists a direction D at x with infinite forward orbit under $T_x g : T \mathbf {P}^{1}_x \rightarrow T \mathbf {P}^{1}_x$ such that $g(D) = \mathbb {P}^1$.

Corollary A.3

If $x \in {\mathbb {H}}$ is a Fatou fixed point of g and D is a direction at x with infinite forward orbit under $T_x g$, then D is contained in the Fatou set of g.

Proof

From Proposition A.2, it follows that at a Fatou fixed point x every direction D with infinite forward orbit under $T_xg$ has zero surplus multiplicity, that is, g(D) is a direction at x. In particular, $g^n(D)$ omits $T_xg^{-1}(D)$ for all $n \ge 0$. Thus, D is contained in the Fatou set. $\square$

1.2 A.2. Injective maps

For convenience we identify $\mathbb {P}^1= \mathbb {P}^1(k)$ with $k \cup \{ \infty \}$ via the map $[z:1] \mapsto z$. As usual we denote by $\tilde{k}$ the residue field of k (i.e. the ring of integers $\mathfrak {O}=\{ z \in k : |z|\le 1\}$ modulo its maximal ideal $\mathfrak {M}=\{ z \in k : |z| < 1\}$).

The diameter ${\text {diam}}\,B$ of a Berkovich closed ball $B \subset \mathbf {P}^{1}\setminus \{\infty \}$ is by definition the diameter of $B \cap k$ with respect to the metric in k induced by $| \cdot |$. For all $x \in \mathbf {P}^{1}$ such that $x \ne \infty$ we set

$$\begin{aligned} {\text {diam}}\,x := \inf \{ {\text {diam}}\,B \ : \ x \in B, B \text { closed} \}. \end{aligned}$$

Recall that for $z \in k$ the Berkovich open (resp. closed) ball of diameter $r \in |k^\times |$ containing z is denoted by $\mathcal {D}(z,r)^-$ (resp. $\mathcal {D}(z,r)$). The $\sup$ norm on $\{w \in k \ : \ |w-z|\le r \}$ is the unique boundary point of these Berkovich balls.

Lemma A.4

(Rivera’s approximation Lemma) Let $\zeta _g \in \mathbf {P}^{1}$ denote the Gauss point and $D= \mathcal {D}(0,1)^-$ the Berkovich open unit ball containing the origin. Consider a rational map g that fixes the Gauss point $\zeta _g$ and such that $T_{\zeta _g} g (D) =D$. Assume that for a closed ball B contained in D we have that $g: D \setminus B \rightarrow D$ is injective. Then there exists an injective analytic map $h : D \rightarrow D$ such that $h(x) = g(x)$ for all x with ${\text {diam}}\,x \ge {\text {diam}}\,B$.

Proof

Lemme d’Approximation in Sect. 5 of [21]. $\square$

Lemma A.5

(Constant tangent map) Let D be the Berkovich open unit ball containing the origin and let $\mathfrak {M} = D \cap k$ denote the maximal ideal.

Let $g : D \rightarrow D$ be a bijective analytic map. For all rigid points $z_0$ and $\rho$ in D we have that

$$\begin{aligned} g(z_0 + \rho z + \rho \mathfrak {M}) = g(z_0) + g'(0) \cdot (\rho z) + \rho \mathfrak {M}, \end{aligned}$$

for all $|z| \le 1$.

In other words, for all $x \in D$ with diameter $r= |\rho | <1$, the tangent map $T_x g$ is multiplication by $\lambda =\widetilde{g'(0)} \in \tilde{k}$, in the “coordinates” of $T \mathbf {P}^{1}_x$ and $T\mathbf {P}^{1}_{g(x)}$ determined by the choice of $z_0$ and $\rho$. These coordinates assign $\tilde{z} \in \tilde{k}$ to the direction $z_0 + \rho z + \rho \mathfrak {M}$ at x and $\tilde{w} \in \tilde{k}$ to the direction $g(z_0) + g'(0) \, (\rho w) + \rho \mathfrak {M}$ at g(x).

Proof

Write the series of g:

$$\begin{aligned} g(z) = a_0 + a_1 z + a_2 z^2 + \cdots . \end{aligned}$$

It follows that $|a_0| <1, |a_1|=1, |a_j| \le 1$ for all $j \ge 2$ since $g:D \rightarrow D$ is bijective. For all $|z| \le 1$,

$$\begin{aligned} \rho ^{-1} ( g(z_0 + \rho z) - g(z_0) ) = a_1 z + a_2(2 z z_0 + \rho z^2) + \cdots \end{aligned}$$

which is congruent to $a_1z = g'(0) z$, modulo $\mathfrak {M}$. $\square$

Lemma 2.14 in [21] in the language of Berkovich spaces reads as follows:

Lemma A.6

(Injectivity domain) Let $g : \mathbf {P}^{1}\rightarrow \mathbf {P}^{1}$ be a rational map. Let V be a connected component of

$$\begin{aligned} \mathbf {P}^{1}\setminus \{ x \in \mathbf {P}^{1}\ : \ m_g(x) \ge 2 \}. \end{aligned}$$

Then $g: V \rightarrow g(V)$ is a bijection.

1.3 A.3. Fixed point

Lemma A.7

Let $f: \mathbf {P}^{1}\rightarrow \mathbf {P}^{1}$ be a tame rational map of degree at least 2. If U is a fixed Fatou component, then U contains a rigid attracting fixed point or a type II fixed point.

Proof

Note that $f: \overline{U} \rightarrow \overline{U}$ is a continuous self-map of a compact, Hausdorff acyclic and locally connected tree in the sense of Wallace [23]. Thus it has a fixed point $x_0$.

Assume that U contains no rigid attracting fixed point. We must conclude that U contains a type II fixed point.

If $x_0$ is an indifferent rigid point (i.e., $|f'(0)|=1$), then there exist arbitrarily small rigid closed balls around it which are fixed under f. The associated Berkovich type II points are fixed and we may choose one of these points in U.

If $x_0$ is a repelling rigid point, then let V be a small Berkovich open ball about $x_0$ such that f(V) compactly contains V. Denote by $x_V$ the boundary point of V. Consider the continuous map $F: \overline{U} \setminus V \rightarrow \overline{U} \setminus V$ defined by $F(x) =f(x)$ if $f(x) \notin f(V)$ and $F(x)=f(x_V)$ otherwise. It follows that F has a fixed point which is not in V and hence it is a fixed point of f.

Since there are at most finitely many rigid fixed points, we may now assume that there exists $x_0 \in \partial U \cap {\mathbb {H}}$ which is a fixed point of f. (Although not needed below, we observe that, from Proposition A.2, the fixed point $x_0 \in \mathbf {J}(f)$ is of type II.) It follows that the direction of U at $x_0$ is fixed under $T_{x_0} f$. If the degree in this direction is 1, then, as in the rigid indifferent case, there exists a type II fixed point in that direction. If the degree in that direction is $\ge 2$ we proceed as in the rigid repelling case and remove a small Berkovich open ball. Since the boundary of an open and connected set U has at most finitely many intersections with the ramification locus of f (points of degree at least 2 in $\mathbf {P}^{1}$), after finitely many removals we obtain a type II fixed point in U. $\square$

1.4 A.4. Proof of Theorem A.1

We work under the hypothesis of Theorem A.1 and assume that U is not the immediate basin of attraction of a rigid fixed point. We must prove that (2) in the statement of the theorem holds.

By Lemma A.7, we may assume that the Gauss point $\zeta _g$ is a fixed point which lies in U. From Proposition A.2 the local degree of f at $\zeta _g$ is 1 (i.e., $m_f(\zeta _g)=1$).

Lemma A.8

Let U be a fixed Fatou component and assume that $\zeta _g \in U$ is a fixed point. For all $y_0 \in \partial U$, the arc $[y_0,\zeta _g]$ joining $y_0 \in \partial U$ and $\zeta _g$ is periodic. That is, there exists $m \ge 1$ such that $f^m (y) =y$ for all $y \in [y_0,\zeta _g]$.

Proof

Changing coordinates we may assume that $y_0$ lies in the Berkovich open unit ball D containing the origin. By Corollary A.3 and passing to an iterate of f, we may suppose that the direction D is fixed under $T_{\zeta _g} f$.

Let

$$\begin{aligned} r = \inf \{ {\text {diam}}\,y \ : \ y \in [y_0,\zeta _g], [y,\zeta _g] \text{ is } \text{ periodic } \text{ under } f \}. \end{aligned}$$

The lemma follows from continuity of f once we show that $r = {\text {diam}}\,y_0$ and that the period of $y \in ]y_0,\zeta _g]$ has an uniform upper bound m.

Let $y_0' \in [y_0,\zeta _g]$ be such that ${\text {diam}}\,y_0'=r$. Note that for all $y \in ]y'_0, \zeta _g]$ we have that $[y,\zeta _g]$ is a periodic interval, say of period p. That is, $f^p: [y,\zeta _g] \rightarrow [y,\zeta _g]$ is the identity. Moreover, $m_{f^p} (z) =1$ for all $z \in [y,\zeta _g]$, for otherwise z would be a Julia periodic point, by Proposition A.2.

Let $\Gamma$ be the convex hull of the set obtained as the union of the ramification locus of f and the Gauss point $\zeta _g$. By tameness, $\Gamma$ is contained in the the convex hull of the set formed by the rigid critical points of f and $\zeta _g$. Consider R such that $r< R <1$ and no vertex of $\Gamma$ has diameter in ]r, R].

Let $x_0' \in [y_0', \zeta _g]$ be the point of diameter R. Observe that $x_0'$ is periodic under f, say of period $p_0'$. By Corollary A.3, the direction of $y_0'$ at $x_0'$ is periodic under $T_{x_0'}f^{p'_0}$ since it contains $y_0 \in \mathbf {J}(f)$. Without loss of generality we assume that this direction is fixed under $T_{x_0'}f^{p'_0}$. Since $f^{p'_0}: [x_0', \zeta _g] \rightarrow [x_0', \zeta _g]$ is the identity, the direction of $\infty$ is also fixed. Thus, in an appropriate coordinate of $T \mathbf {P}^{1}_{x_0'} \equiv \tilde{k} \cup \{ \infty \}$, the map $T_{x_0'}f^{p'_0}$ becomes $z \mapsto \lambda z$ for some $\lambda \in \tilde{k}$ where 0 corresponds to the direction $D_0$ of $y_0'$ at $x_0'$ and $\infty$ corresponds to the direction of $\zeta _g$.

Let $x'_n = f^n(x'_0)$ and $D_n =T_{x_0'}f^{n} (D_0) \in T_{x_n'} \mathbf {P}^{1}$, subscripts $\pmod {p_0'}$. Note that $[x_n',\zeta _g]$ maps isometrically onto $[x_{n+1}',\zeta _g]$ since it is a periodic interval. Thus, $D_n$ is not the direction of $\infty$ and ${\text {diam}}\,x'_n = R$ for all n. If $\Gamma \cap D_n \ne \emptyset$, then there exists a unique element w of $\Gamma \cap D_n$ of diameter $r'$ for all $r \le r' \le R$ (by the choice of R). Let $B_n$ be the closed (possibly degenerate) ball contained in $D_n$ with boundary point being the unique point in $\Gamma \cap D_n$ of diameter r. If $\Gamma \cap D_n = \emptyset$, then let $B_n$ be any ball of diameter r in $D_n$. By the choice of R the ramification locus is either disjoint from $D_n \setminus B_n$ or its intersection with $D_n \setminus B_n$ is the open interval joining the boundary point of $B_n$ with $x'_n$. The latter is impossible since the multiplicity at $x_n'$ is 1. By Lemma A.6, we conclude that $f: D_n \setminus B_n \rightarrow D_{n+1}$ is injective. By Rivera’s approximation Lemma A.4, there exists an analytic bijection $g_n : D_n \rightarrow D_{n+1}$ which agrees with f for all $y \in D_n$ such that ${\text {diam}}\,y \ge r$.

Consider any $y \in ]y_0' , x_0']$. Since $[y,\zeta _g]$ is a periodic interval, ${\text {diam}}\,f^n (y) = {\text {diam}}\,y > r$ for all n. In particular, $f^n(y) \in D_n \setminus B_n$ for all n $\pmod {p_0'}$. Hence y is also periodic under $G=g_{p_0'-1} \circ \cdots \circ g_0$.

The direction of $y_0'$ at $x_0'$ is fixed under G. Thus, $[y,x_0'] \cap G([y,x_0'])= [x_1, x_0']$ where $x_1$ is fixed under G. If $x_1 \ne y$, then $T_{x_1} G$ has finite order $>1$, since y is contained in a periodic direction which is not fixed and $\zeta _g$ is in a fixed direction. Say that the order of $T_{x_1} G$ is q. From the constant tangent map Lemma A.5, it follows that $q p_0'$ is the order of $\lambda$. Now $[y,x_1] \cap G^q ([y,x_1]) = [x_2,x_1]$. It follows that $x_2$ is fixed under $G^q$ and $x_2 \ne x_1$. If $x_2 \ne y$, then $T_{x_2} G^q$ has finite order $>1$ which is a contradiction with the fact that $\lambda$ has order $q p_0'$ and the constant tangent map Lemma A.5. Therefore $x_2 = y$ and the period of y is the order of $\lambda$. We have shown that for all $y \in ]y_0' , x_0']$, the period of y is bounded above by $p_0'$ if $x'_1 = y$ and by the order of $\lambda$ otherwise. By continuity, it follows that $[y_0',x_0']$ is a periodic interval.

Since $y_0'$ is periodic, from Corollary A.3 we have that the direction of $y_0$ at $y_0'$ is periodic. Therefore $y_0 = y_0'$, for otherwise we may find a periodic sub-interval of $[y_0, y_0']$ including $y_0'$ which contradicts the definition of r. $\square$

To finish the proof of Theorem A.1 we first show that U has finitely many boundary points, all periodic and all of type II and then proceed to show that $f:U \rightarrow U$ is a bijection.

From the previous lemma we have that every point $y_0 \in \partial U$ is a periodic point in the Julia set. Hence, it is a rigid point or a type II point. However, $y_0$ is not a rigid point, for otherwise $y_0$ would be a repelling periodic point. From the previous lemma $[y_0,x_0]$ would be a periodic interval, which would be incompatible with the repelling nature of $y_0$. Thus, $y_0$ is a type II point.

Let $\mathcal {O}= \{y_0, \dots , y_{p-1}\}$ be the orbit of $y_0 \in \partial U$. Let $B_0, \dots , B_{p-1}$ be the complement of the direction of U at $y_0, \dots , y_{p-1}$, respectively. Then, at least one of these balls $B_j$ must map onto $\mathbf {P}^{1}$ (otherwise all $B_j$ and therefore $\partial B_j$ would be contained in the Fatou set). Denote such a ball by $B(\mathcal {O})$. For distinct periodic orbits $\mathcal {O}_0, \mathcal {O}_1$ contained in $\partial U$, the corresponding closed balls $B(\mathcal {O}_0)$ and $B(\mathcal {O}_1)$ are disjoint and each contains a preimage of U. Therefore, $\partial U$ consists of at most $d-1$ type II periodic orbits.

It remains to show that $f: U \rightarrow U$ is a bijection. By Lemma A.6 it is sufficient to prove that the ramification locus is disjoint from U. We proceed by contradiction and let $\gamma \subset U$ be a connected component of the ramification locus. The convex hull $\Gamma _U \subset \overline{U}$ of $\partial U$ is fixed pointwise by an iterate of f. Passing to this iterate we may assume that $\Gamma _U$ is pointwise fixed. Thus, $\Gamma _U \cap \gamma = \emptyset$ since $\Gamma _U \setminus \partial U$ consists of Fatou fixed points. Let $a \in \gamma$ and $b \in \Gamma _U$ be such that $]a,b[ \subset \mathbf {P}^{1}$ is an arc disjoint from $\gamma \cup \Gamma _U$. Without loss of generality, replacing $\gamma$ if necessary, we may assume that the multiplicity of f along ]a, b[ is 1. It follows that $f(]a,b]) = ]f(a),b]$ and the direction of b at a maps under $T_a f$ onto the direction $D_b$ of b at f(a) with multiplicity 1. Since the multiplicity at a is at least 2 there exists a direction $D_a$ at a not containing b which is also mapped under $T_a f$ onto the direction $D_b$ of b at f(a). The connected graph $\Gamma _U$ is disjoint from $D_a$ and contained in $D_b$, therefore $D_a \subset U$ which implies that $f(D_a) \subset U$. But $f(D_a) \supset D_b \supset \Gamma _U \supset \partial U$ which is a contradiction. This completes the proof of Theorem A.1.