2.1 Linearization domain and good return times

Define a linearization domain of o to be an open neighborhood U of o such that there is an isomorphism $\phi : U\to {\mathbb D}$ sending o to $0$ , which conjugates $f|_{U_0}:U_0\to U$ to the morphism $z\mapsto \lambda z$ via $\phi $ , where $U_0=f^{-1}(U)\cap U$ and $\lambda =df(o)$ . We call such $\phi $ a linearization on U.

Define g to be the morphism $U\to U$ sending z to $\phi ^{-1}(\lambda ^{-1}\phi (z))$ . It is the unique endomorphism of U satisfiying $f\circ g=\mathrm {id}.$

The following lemma shows that for every repelling fixed point o, there are many homoclinic orbits.

2.2 Adjoint sequence of periodic points

Let U be a linearization domain, and let m be a good return time of U. We construct a sequence of periodic points $q_i, i\geq m$ as follows. By Remark 2.5, for every $i\geq m$ , $f^i|_{U_i}: U_i\to U$ is an isomorphism. Since $U_i\subset \subset U$ , the morphism $(f^i|_{U_i})^{-1}: U\to U_i$ is strictly attracting with respect to the hyperbolic metric on U. Hence, it has a unique fixed point $q_i\in U_i$ . Such $q_i$ is the unique i-periodic point of f which is contained in $U_i.$ Indeed, i is the smallest period of $q_i$ , and $q_i$ is repelling. We call such a sequence an adjoint sequence for the homoclinic orbit $o_i, i\geq 0$ with respect to the linearization domain U and the good return time m (we write $(U,m)$ for short). One can say that a sequence of points $q_i, i\geq 0$ is an adjoint sequence of the homoclinic orbit $o_i, i\geq 0$ if $q_i, i\geq m$ is an adjoint sequence for $o_i, i\geq 0$ with respect to some $(U,m).$ It is clear that for every adjoint sequence $q_i, i\geq 0$ of $o_i, i\geq 0$ , $\lim \limits _{i\to \infty } q_i=o.$ The following lemma shows that the adjoint sequences are unique modulo finite terms.

2.3 Poincaré’s linearization map

Set $\lambda :=df(o)\in {\mathbb C}$ . Since o is repelling, $|\lambda |>1.$ Let $(U,m)$ be the pair of linearization domain and good return time for $o_i, i\geq 0$ , and let $q_i, i\geq 0$ be an adjoint sequence.

A theorem of Poincaré [Reference MilnorMil11, Corollary 8.12] says that there is a morphism $\psi : {\mathbb C}\to {\mathbb P}^1({\mathbb C})$ such that $\psi |_{{\mathbb D}}$ gives an isomorphism between ${\mathbb D}$ and U and

(2.1) $$ \begin{align} f(\psi(z))=\psi(\lambda z) \end{align} $$

for every $z\in {\mathbb C}.$ In particular, $\psi |_{{\mathbb D}}^{-1}: U\to {\mathbb D}$ is a linearization of f on $U.$ Such a $\psi $ is called a Poincaré map.

The following criterion for exceptional endomorphisms using the Poincaré map $\psi $ is due to Ritt.

Ritt’s theorem can be easily generalized as following.

Proof. Let G be the group of affine automorphisms g of ${\mathbb C}$ satisfying $\psi \circ g=\psi .$ We have $h\in G$ . It takes form $h: z\mapsto az+b$ , $a\in {\mathbb C}^*$ and $b=h(0)\in {\mathbb C}^*.$ For every $z\in {\mathbb C}$ , we have

$$ \begin{align*}\psi(\lambda h(\lambda^{-1}z))=f\psi(h(\lambda^{-1}z))=f\psi(\lambda^{-1}z)=\psi(z).\end{align*} $$

Hence, the automorphism $g: z\mapsto \lambda h(\lambda ^{-1}z)=az+\lambda b$ is contained in $G.$ Then, $T:=h^{-1}\circ g: z\mapsto z+a^{-1}(\lambda -1)b$ is contained in $G.$ Since $b\neq 0$ and $|\lambda |>1$ , T is a nontrivial translation. We conclude the proof by Lemma 2.8.

Set $P:=\lambda ^m \psi |_{{\mathbb D}}^{-1}(o_m)$ and $V:=\lambda ^m(\psi |_{{\mathbb D}}^{-1}(U_m)).$ For $i\geq m,$ set $Q_i:=\psi |_{{\mathbb D}}^{-1}(q_i).$ One has $\psi (V)=U$ , $\psi (P)=o$ , and $\psi |_V: V\to U$ is an isomorphism. We set $T:=(\psi |_V)^{-1}\circ \psi |_{{\mathbb D}}:{\mathbb D}\to V$ . Then T is an isomorphism. Similar constructions of T appeared already in the works of Ritt [Reference RittRit22] and Eremenko-van Strien [Reference Eremenko and Van StrienEVS11]. We summarize our construction in the following figure.

We have $\psi \circ T=\psi $ and $T(0)=P.$ Moreover, by our construction, we have for every $i\geq m$ , $V= \lambda ^i (\psi |_{{\mathbb D}})^{-1}(U_i)$ . In particular, $\lambda ^i Q_i\in V$ . By (2.1) we have

$$ \begin{align*} \psi(\lambda^i Q_i)=f^i(\psi(Q_i))=f^i(q_i)=q_i. \end{align*} $$

This implies

(2.2) $$ \begin{align} T(Q_i)=\lambda^iQ_i. \end{align} $$

Since $\lim \limits _{i\to \infty }q_i=o$ , we have $\lim \limits _{i\to \infty }Q_i=0$ and

(2.3) $$ \begin{align} \lim_{i\to \infty}\lambda^iQ_i=P. \end{align} $$

By (2.1), we have for every $i\geq 1$ ,

(2.4) $$ \begin{align} df^i(\psi(z))d\psi(z)=\lambda^i d\psi(\lambda^i z), \end{align} $$

and by $\psi \circ T=\psi $ , we have

(2.5) $$ \begin{align} d\psi(T(z))T'(z)=d\psi(z). \end{align} $$

Set $z=Q_i$ . Combine (2.2), (2.4) and (2.5), and we get

$$ \begin{align*} df^i(q_i)d\psi(\lambda^i Q_i)T'(Q_i)=\lambda^i d\psi(\lambda^i Q_i). \end{align*} $$

Since zeros of a holomorphic function are isolated, as $\lambda ^i Q_i \to P$ , for i large enough, we have $d\psi (\lambda ^i Q_i)\neq 0$ . Hence, for i large enough,

(2.6) $$ \begin{align} \lambda^iT'(Q_i)^{-1}=df^i(q_i). \end{align} $$

The following observation will be used in the proof of Theorem 1.12.

Lemma 2.10. Set $\theta :=1/T': {\mathbb D}\to {\mathbb C}$ . We have

$$ \begin{align*}\lim_{i\to \infty}(df^i(q_i)-\lambda^i\theta(0))=P\theta'(0).\end{align*} $$

Proof. By (2.3) and (2.6),we have

$$ \begin{align*}\lim_{i\to \infty}(df^i(q_i)-\lambda^i\theta(0))/P=\lim_{i\to \infty}(df^i(q_i)-\lambda^i\theta(0))/\lambda^iQ_i\end{align*} $$

$$ \begin{align*}=\lim_{i\to \infty}(df^i(q_i)/\lambda^i-\theta(0))/Q_i=\lim_{i\to \infty}(\theta(Q_i)-\theta(0))/Q_i=\theta'(0),\end{align*} $$

which concludes the proof.

The following is the main result of this section, which characterizes exceptional endomorphisms by using the multipliers of adjoint sequence of a homoclinic orbit.

4.1 Analytification of the projective line

Let ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ be the analytification of ${\mathbb P}^{1}_{{\mathbf {k}}}$ in the sense of Berkovich, which is a compact topological space endowed with a structural sheaf of analytic functions. Only its topological structure will be used in this paper. We describe it briefly below.

The analytification ${\mathbb A}^{1,\mathrm {an}}_{{\mathbf {k}}}$ of the affine line ${\mathbb A}^1_{{\mathbf {k}}}$ is the space of all multiplicative semi-norms on ${\mathbf {k}}[z]$ whose restriction to ${\mathbf {k}}$ coincide with $|\cdot |$ , endowed with the topology of pointwise convergence. For any $x\in {\mathbb A}^{1,\mathrm {an}}_{{\mathbf {k}}}$ and $P\in {\mathbf {k}}[z]$ , it is customary to denote $|P(x)|:=|P|_x$ , where $|\cdot |_x$ is the semi-norm associated to x.

As a topological space, ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ is the one-point compactification of ${\mathbb A}_{{\mathbf {k}}}^{1,\mathrm { an}}$ . We write ${\mathbb P}_{{\mathbf {k}}}^{1,\mathrm {an}}={\mathbb A}_{{\mathbf {k}}}^{1,\mathrm {an}}\cup \{\infty \}$ . More formally, it is obtained by gluing two copies of ${\mathbb A}_{{\mathbf {k}}}^{1,\mathrm {an}}$ in the usual way via the transition map $z\mapsto z^{-1}$ on the punctured affine line $({\mathbb A}_{{\mathbf {k}}}^1\setminus \{0\})^{\mathrm {an}}$ .

The Berkovich projective line ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ is an ${\mathbb R}$ -tree in the sense that it is uniquely path-connected (see [Reference Jonsson, Nicaise, Ducros and FavreJon15, Section 2] for the precise definitions). In particular, for $x,y\in {\mathbb P}^{1,\mathrm { an}}_{{\mathbf {k}}}$ , there is a well-defined segment $[x,y]$ .

For $a\in {\mathbf {k}}$ and $r\in [0,+\infty )$ , we denote ${\mathbb D}(a,r)$ by the closed disk ${\mathbb D}(a,r):=\{x\in {\mathbb A}^{1,\mathrm { an}}_{{\mathbf {k}}}:\, |(z-a)(x)|\leq r\}.$ One may check that the norm $\sum _{i\geq 0}a_i(z-a)^i\mapsto \max \{|a_i|r^i, i\geq 0\}$ defines a point $\xi _{a,r}\in {\mathbb D}(a,r).$ One may set $x_{G}:=\xi _{0,1}$ and call it the Gauss point.

The group $\mathrm {PGL}\,_2({\mathbf {k}})$ acts on ${\mathbb P}^1_{{\mathbf {k}}}$ , and therefore on ${\mathbb P}^{1, \mathrm {an}}_{{\mathbf {k}}}$ .

4.2 Points in ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$

Let $\widehat {\overline {{\mathbf {k}}}}$ be the completion of the algebraic closure of ${\mathbf {k}}$ . It is still algebraically closed. By [Reference BerkovichBer90, Corollary 1.3.6], $\mathrm {Aut}(\widehat {\overline {{\mathbf {k}}}}/{\mathbf {k}})$ acts on ${\mathbb P}^{1,\mathrm {an}}_{\widehat {\overline {{\mathbf {k}}}}}$ and we have ${\mathbb P}^1_{\widehat {\overline {{\mathbf {k}}}}}/\mathrm { Aut}(\widehat {\overline {{\mathbf {k}}}}/{\mathbf {k}})= {\mathbb P}^1_{{\mathbf {k}}}.$ We denote by $\pi :{\mathbb P}^{1,\mathrm { an}}_{\widehat {\overline {{\mathbf {k}}}}}\to {\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ the quotient map. The points of ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ can be classified into 4 types:

1. (i) a type 1 point takes form $\pi (a)$ where $a\in \widehat {\overline {{\mathbf {k}}}}\cup \{\infty \}={\mathbb P}^{1,\mathrm { an}}_{\widehat {\overline {{\mathbf {k}}}}}$ ;

2. (ii) a type 2 point takes form $\pi (\xi _{x,r})$ where $x\in \widehat {\overline {{\mathbf {k}}}}$ and $r\in |\widehat {\overline {{\mathbf {k}}}}^*|$ ;

3. (iii) a type 3 point takes form $\pi (\xi _{x,r})$ where $x\in \widehat {\overline {{\mathbf {k}}}}$ and $r\in {\mathbb R}_{>0}\setminus |\hat {\overline {{\mathbf {k}}}}^*|$ ;

4. (iv) a type 4 point takes form $\pi (x)$ where x is the pointwise limit of $\xi _{x_i,r_i}$ such that the corresponding discs ${\mathbb D}(x_i,r_i)$ form a decreasing sequence with empty intersection.

See [Reference BerkovichBer90, Section 1.4.4] for further details when ${\mathbf {k}}$ is algebraically closed. See also [Reference KedlayaKed11, Proposition 2.2.7] and [Reference StevensonSte19, Section 2.1]. The set of type 1 (resp. type 2) points is dense in ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ . Points of type 4 exist only when ${\mathbf {k}}$ is not spherically complete. If we view ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ as a metric tree, then the end points have type 1 or 4.

For every $x\in {\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ , we can define an equivalence relation on the set ${\mathbb P}^{1,\mathrm { an}}_{{\mathbf {k}}}\setminus \{x\}$ as follows: $y\sim z$ if the paths $(x, y]$ and $(x,z]$ intersect. The tangent space $T_{x}$ at x is the set of equivalences classes of ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}\setminus \{x\}$ modulo $\sim $ . See [Reference Jonsson, Nicaise, Ducros and FavreJon15, Section 2.5] for details. If x is an end point (a point of type 1 or 4), then $|T_{x}|=1$ . If x is of type 3, then $|T_{x}|=2$ . If x is of type 2, then $|T_{x}|\geq 3.$ For a direction $v\in T_x$ , let $U(v)$ be the set of all $y\in {\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ such that the path $(x,y]$ presents $v.$ Then, $U(v)$ is an open subset such that $\partial {U(v)}=x$ .

4.3 Dynamics on ${\mathbb P}_{{\mathbf {k}}}^{1,\mathrm {an}}$

Let $f:{\mathbb P}^1_{{\mathbf {k}}}\to {\mathbb P}^1_{{\mathbf {k}}}$ be an endomorphism of degree $d\geq 2.$ We still denote by f the induced endomorphism on ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ .

4.3.1 The tangent map

For $x,y\in {\mathbb P}_{{\mathbf {k}}}^{1,\mathrm {an}}$ , if $f(x)=y$ , then $x,y$ have the same type. Moreover, f induces a tangent map $T_xf: T_x\to T_y$ sending $v\in T_x$ to the unique direction $w\in T_y$ such that for every $z\in U(v)$ , $(y,f(z)]\cap U(w)\neq \emptyset .$ We note that, in general, $f(U(v))$ may not be equal to $U(w).$ If $f(U(v))=U(w)$ , we say that v is a good direction. Otherwise, it is called a bad direction. If v is a bad direction, then $f(U(v))={\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ [Reference BenedettoBen, Theorem 7.34].

We may naturally identify $T_{x_{G}}$ with ${\mathbb P}^1(\tilde {{\mathbf {k}}})$ as follows. Consider the standard model ${\mathbb P}^1_{{\mathbf {k}}^{\circ }}$ of ${\mathbb P}_{{\mathbf {k}}}^{1,\mathrm {an}}$ . There is a reduction map $\mathrm {red}: {\mathbb P}_{{\mathbf {k}}}^{1,\mathrm {an}}\to {\mathbb P}^1_{\tilde {{\mathbf {k}}}}.$ The preimage of the generic point of ${\mathbb P}^1_{\tilde {{\mathbf {k}}}}$ is the Gauss point $x_G$ , and for every $y\in {\mathbb P}^1(\tilde {{\mathbf {k}}})$ , there is a unique $v_y\in T_{x_{G}}$ such that $U(v_y)=\mathrm {red}^{-1}(y).$ The map ${\mathbb P}^1(\tilde {{\mathbf {k}}})\to T_{x_{G}}$ sending y to $v_y$ is bijective. Let h be any endomorphism of ${\mathbb P}^1_{{\mathbf {k}}}$ such that $h(x_G)=x_G$ , and it extends to a rational self-map $h_{{\mathbf {k}}^{\circ }}$ of ${\mathbb P}^1_{{\mathbf {k}}^{\circ }}$ . We denote by $\tilde {h}: {\mathbb P}^1_{\tilde {{\mathbf {k}}}}\to {\mathbb P}^1_{\tilde {{\mathbf {k}}}}$ the restriction of h to the special fiber of ${\mathbb P}^1_{{\mathbf {k}}^{\circ }}$ and call it the reduction of $h.$ Then, $T_{x_G}h: T_{x_G}={\mathbb P}^1(\tilde {{\mathbf {k}}}) \to T_{x_G}$ is induced by $\tilde {h}$ . We define $\deg T_{x_G}h$ to be the degree of $\tilde {h}.$ We note that $\deg \tilde {h}\leq \deg h$ . The equality holds if and only if $h_{{\mathbf {k}}^{\circ }}$ is an endomorphism. In this case, we say that h has explicit good reduction.

More generally, for every $x,y\in \mathrm {PGL}\,_2({\mathbf {k}})\cdot x_{G}$ with $f(x)=y$ , we may define

$$ \begin{align*}\deg T_{x}f:=\deg T_{x_G}(h^{-1}fg)=\deg \widetilde{h^{-1}fg},\end{align*} $$

where $h,g\in \mathrm {PGL}\,_2({\mathbf {k}})$ with $g(x_G)=x$ and $h(x_G)=y.$ Then, $1\leq \deg T_{x_G}f\leq \deg f$ and $\deg T_{x_G}f$ does not depend on the choices of $g,h.$

Remark 4.4. Assume that ${\mathbf {k}}$ is algebraically closed. By Lemma 4.2, the set of type 2 points in ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ is exactly $\mathrm {PGL}\,_2({\mathbf {k}})\cdot x_G.$

5.1 Holomorphic families

Recall that $\Psi : \text {Rat}_d({\mathbb C})\to {\mathcal M}_d({\mathbb C})$ is the quotient morphism, where ${\mathcal M}_d({\mathbb C}):=\text {Rat}_d({\mathbb C})/\mathrm {PGL}\,_2({\mathbb C})$ is the moduli space.

Let $\Lambda $ be a complex manifold. We denote by ${\mathcal O}^{\mathrm {an}}(\Lambda )$ the ring of holomorphic functions on $\Lambda .$ Moreover, if $\Lambda $ is a complex algebraic variety, we denote by ${\mathcal O}(\Lambda )$ the ring of algebraic functions on $\Lambda .$

A holomorphic (resp. meromorphic) family on $\Lambda $ is an endomorphism (resp. meromorphic self-map) $f_{\Lambda }$ on ${\mathbb P}^1\times \Lambda $ such that $\pi _{\Lambda }\circ f_{\Lambda }=\pi _{\Lambda }$ , where $\pi _{\Lambda }:{\mathbb P}^1({\mathbb C})\times \Lambda \to \Lambda $ is the projection to $\Lambda .$ More concretely, one may write $f_{\Lambda }([x:y],t)=([P_t(x,y):Q_t(x,y)], t)$ where $P_t(x,y), Q_t(x,y)$ are homogenous polynomials of same degree d in ${\mathcal O}^{\mathrm {an}}(\Lambda )[x,y]$ without a common divisor. We say that $f_{\Lambda }$ is of degree d. Then, $f_{\Lambda }$ is holomorphic if there is no $(t,x,y)\in \Lambda \times {\mathbb C}^*\times {\mathbb C}^*$ such that $P_t(x,y)=Q_t(x,y)=0.$

For $t\in \Lambda $ , we denote by $f_t$ the restriction of $f_{\Lambda }$ to the fiber above $t.$ We denote by $I(f_{\Lambda })$ the indeterminacy locus of $f_{\Lambda }$ and $B(f_{\Lambda }):=\pi _{\Lambda }(I(f_{\Lambda })).$ Then, $I(f_{\Lambda })$ and $B(f_{\Lambda })$ are proper closed analytic subspaces of ${\mathbb P}^1\times \Lambda $ and $\Lambda $ , respectively. For every $t\in \Lambda \setminus B(f_{\Lambda })$ , we have $\deg f_t=d.$ When $\Lambda $ is connected, this is equivalent to say that $\deg f_t=d$ for one $t\in \Lambda \setminus B(f_{\Lambda }).$ A meromorphic family is holomorphic if and only if $B(f_{\Lambda })=\emptyset $ . So, giving a degree d holomorphic family $f_{\Lambda }$ on $\Lambda $ is equivalent to giving a holomorphic morphism $t\mapsto f_t=P_t/Q_t$ from $\Lambda $ to $\text {Rad}_d({\mathbb C}).$ We say that $f_{\Lambda }$ is algebraic if $\Lambda $ is a complex algebraic variety and $f_{\Lambda }: {\mathbb P}^1\times \Lambda \to {\mathbb P}^1\times \Lambda $ is algebraic (i.e., $P_t,Q_t \in {\mathcal O}(\Lambda )[x,y]).$ In other words, it means that the induced morphism $\Lambda \to \text {Rad}_d({\mathbb C})$ is algebraic.

For a degree d holomorphic family $f_{\Lambda }$ on $\Lambda $ , let $\Psi _{\Lambda }: \Lambda \to {\mathcal M}_d$ be the holomorphic morphism sending $t\in \Lambda $ to the class of $f_t$ in ${\mathcal M}_d({\mathbb C}).$ We say that $f_{\Lambda }$ is isotrivial if $\Psi _{\Lambda }: \Lambda \to {\mathcal M}_d$ is locally constant. More generally for degree d meromorphic family $f_{\Lambda }$ , we say that $f_{\Lambda }$ is isotrivial if $f|_{\Lambda \setminus B(f_{\Lambda })}$ is isotrivial.

5.2 Potentially good reduction

Assume that $\Lambda $ is a Riemann surface and $f_{\Lambda }$ is a meromorphic family of degree $d.$

For $b\in \Lambda $ , we say that $f_{\Lambda }$ has potentially good reduction at b if $\Phi _{\Lambda \setminus (B(f_{\Lambda })\cup \{b\})}: \Lambda \to {\mathcal M}_d$ extends to a holomorphic morphism on $(\Lambda \setminus B(f_{\Lambda }))\cup \{b\}.$ In particular, $f_{\Lambda }$ has potentially good reduction at every $b\in \Lambda \setminus B(f_{\Lambda }).$

Having potentially good reduction is a local property at b (i.e., for every open neighborhood U of b in $\Lambda $ , $f_{\Lambda }$ has potentially good reduction at b if and only if $f_{U}:=f_{\Lambda }|_{{\mathbb P}^1({\mathbb C})\times U}$ has potentially good reduction at b). Note that there is an open neighborhood U of b which is isomorphic to a disk ${\mathbb D}$ such that $f_{U\setminus \{b\}}$ is holomorphic. So we can focus on the case that $f_{{\mathbb D}}$ is a meromorphic family that is holomorphic on ${\mathbb D}^*.$ We will give another characterization of potentially good reduction via non-Archimedean dynamics.

5.3 Holomorphic family on puncture disk

Let $f_{{\mathbb D}}$ be a a meromorphic family of degree $d\geq 2$ that is holomorphic on ${\mathbb D}^*.$ Let t be the standard coordinate on ${\mathbb D}$ . We can relate $f_{{\mathbb D}}$ to some non-Archimedean dynamics on the field of Laurent’s series ${\mathbb C}((t)).$

Recall that on ${\mathbb C}((t))$ , there is a t-adic norm $|\cdot |$ : Given an element $z=\sum _{n\geq n_0} a_n t^n\neq 0$ , where $n_0\in {\mathbb Z}$ , $a_n\in {\mathbb C}$ and $a_{n_0}\neq 0$ , the t-adic norm of z is $|z|:=e^{-n_0}$ . This norm is non-Archimedean and ${\mathbb C}((t))$ is complete for $|\cdot |$ . Set ${\mathbb L}:=\widehat {\overline {{\mathbb C}((t))}}.$

Write

$$ \begin{align*}f([x:y],t)=([P_t(x,y):Q_t(x,y)], t)\end{align*} $$

where $P_t(x,y), Q_t(x,y)$ are homogenous polynomials of degree d in ${\mathcal O}^{\mathrm {an}}({\mathbb D})[1/t][x,y]$ without common divisors. Since ${\mathcal O}^{\mathrm {an}}({\mathbb D})[1/t]\subseteq {\mathbb C}((t))$ , $f_{{\mathbb D}}$ defines an endomorphism $f_{{\mathbb C}((t))}: [x,y]\mapsto [P_t(x,y):Q_t(x,y)]$ on ${\mathbb P}^1_{{\mathbb C}((t))}$ of degree $d.$ Set $f_{{\mathbb L}}:=f_{{\mathbb C}((t))}\hat {\otimes }_{{\mathbb C}((t))}{\mathbb L}:{\mathbb P}^1_{\mathbb L} \to {\mathbb P}^1_{\mathbb L}$ .

Recall that

(5.1) $$ \begin{align} \overline{{\mathbb C}((t))}=\cup_{n\geq 1}{\mathbb C}((t^{1/n})). \end{align} $$

To get endomorphisms over ${\mathbb C}((t^{1/n}))$ , we introduce some base changes of $f_{{\mathbb D}}$ as follows. Consider the morphism $\phi _n: U_n:={\mathbb D}\to {\mathbb D}$ sending t to $t^n.$ There is $u_n\in {\mathcal O}^{\mathrm {an}}(U_n)$ such that $u_n^n=\phi ^*t.$ Then, $u_n$ is a coordinate on $U_n$ , and we may identify ${\mathbb C}[u_n]$ with ${\mathbb C}[t^{1/n}]$ (hence, we may identify ${\mathbb C}((u_n))$ with ${\mathbb C}((t^{1/n}))$ ). Let $o\in U_n$ be the point defined by $u_n=0.$ The endomorphism on ${\mathbb P}^{1,\mathrm {an}}_{{\mathbb C}((u_n))}$ induced by $f_{U_n}$ is $f_{{\mathbb C}((u_n))}=f_{{\mathbb C}((t))}\hat {\otimes }_{{\mathbb C}((t))}{\mathbb C}((t^{1/n})).$

Proof of Lemma 5.2

By Theorem 4.6, there is $h\in \mathrm {PGL}\,_2({\mathbb L})$ such that ${\mathcal J}(f_{{\mathbb L}})=\{h(x_G)\}.$ Then, $h^{-1}\circ f_{{\mathbb L}}\circ h$ has explicit good reduction. By (5.1) and Remark 4.3, we may assume that $h\in \mathrm {PGL}\,_2({\mathbb C}((t^{1/n})))$ for some $n\geq 1.$ Since ${\mathbb C}(u_n)$ is dense in ${\mathbb C}((u_n))={\mathbb C}((t^{1/n}))$ , by Remark 4.3 again, we may assume that $h\in \mathrm {PGL}\,_2({\mathbb C}(u_n)).$ There is an open neighborhood V of o such that h and $h^{-1}$ are holomorphic on $V\setminus \{o\}$ (i.e., they define holomorphic families $h_{V\setminus \{o\}}$ and $h^{-1}_{V\setminus \{o\}}).$ We may assume further that $V\simeq {\mathbb D}.$ Consider the family $f_V:=h_V^{-1}\circ f_{U_n}|_V\circ h_V$ . Observe that

(5.2) $$ \begin{align} \Psi_{{\mathbb D}^*}\circ \phi|_{V\setminus \{o\}}=\Psi_{V\setminus\{o\}}. \end{align} $$

Then, $f_V$ induces an endomorphism $f_{{\mathbb C}((u))}=f_{{\mathbb C}((t))}\hat {\otimes }_{{\mathbb C}((t))}{\mathbb C}((u))$ on ${\mathbb P}^{1,\mathrm {an}}_{{\mathbb C}((u))}$ , which has good reduction. So $f_V$ is an endomorphism on ${\mathbb P}^1\times V$ . So $\Psi _{V\setminus \{o\}}$ extends to a holomorphic morphism $\Psi _V: V\to {\mathcal M}_d.$ By (5.2), $\Psi _{{\mathbb D}^*}$ is bounded in some neighborhood of $o.$ So $\Psi _{{\mathbb D}^*}$ extends to a holomorphic morphism on ${\mathbb D}$ , which means that $f_{{\mathbb D}}$ has potentially good reduction at $0$ .

The following definition was introduced by Kiwi.

Definition 5.4. [Reference KiwiKiw15] Let $f_{{\mathbb D}}$ be a meromorphic family of degree $d\geq 2$ which is holomorphic on ${\mathbb D}^*.$ We say an endomorphism g is a rescaling limit of $f_{{\mathbb D}}$ (or $f_{{\mathbb D}^*}$ ) (via $(q,M_{{\mathbb D}})$ ) if there is an integer $q\geq 1$ , a finite set $S\subset {\mathbb P}^1({\mathbb C})$ and a meromorphic family $M_{{\mathbb D}}$ of degree 1, such that $M_{{\mathbb D}}$ and $M^{-1}_{{\mathbb D}}$ are holomorphic on ${\mathbb D}^*$ and

$$ \begin{align*} M_t^{-1}\circ f_t^q\circ M_t(z)\to g(z) \end{align*} $$

when $t\to 0$ , uniformly on compact subsets of ${\mathbb P}^1({\mathbb C})\setminus S$ .

The following result was proved by Kiwi.

5.4 Endomorphisms without repelling type I periodic points

In general, the Julia set of an endomorphism $f_{{\mathbb L}}$ on ${\mathbb P}_{{\mathbb L}}^{1,\mathrm {an}}$ is a complicated object. The following theorem due to Favre-Rivera Letelier [Reference Favre and Rivera-LetelierFRL], and independently by Luo [Reference LuoLuo22, Proposition 11.4], classifies the case when $f_{{\mathbb L}}$ has no repelling type I periodic points.

By (v) of Proposition 4.5, ${\mathcal J}(f_{{\mathbb L}})\neq \emptyset .$ In the above theorem, if $f_{{\mathbb L}}$ does not have good reduction, then the segment cannot be a point. As a corollary, we get the following lemma.

The following theorem is the main result of this section.

7.1 Definition, examples and rigidity of CER

The following definition was introduced by Sullivan [Reference SullivanSul86].

The following is an important class of examples of CER.

Example 7.3. Assume V, $U_i$ , $1\leq i\leq k$ are connected open sets in ${\mathbb P}^1({\mathbb C})$ , $k\geq 2$ such that $\overline {U_i}\subset V$ , and there exists $m\geq 1$ such that $f^m: U_i\to V$ is an isomorphism. Then, we call

$$ \begin{align*} K:=\left\{ z\in \bigcup_{i=1}^k U_i \Bigg|\,\, f^{mn}(z)\in \bigcup_{i=1}^k U_i\;\text{for every}\;n\geq 0\right\} \end{align*} $$

a horseshoe of f. We check that K satisfies the three conditions in Definition 7.1. Let $V_0:= \cup _{i=1}^k U_i $ .

• Condition (i): It follows from the definition of K;

• Condition (ii): $f^m: V_0 \to V$ strictly expands the hyperbolic metric of V. This implies $f^m:K\to K$ is expanding;

• Condition (iii): Again, using $f^m: V_0 \to V$ strictly expands the hyperbolic metric of V. The maximal diameter of the connected components of $f^{-nm}(V_0)\cap V_0$ shrinks to $0$ when $n\to \infty $ . For each open set $W\subset K$ , there exist integer $n\geq 0$ and a connected component B of $f^{-nm}(V_0)\cap V_0$ such that $B\cap K\subset W$ . Since $f^{(n+1)m}(B\cap K)=K$ , we have $f^{(n+1)m}(W)=K$ . Hence, $f^m:K\to K$ is topologically exact.

Moreover, K is a Cantor set. In particular, K is not a finite set.

When f has degree at least $2$ , there are plenty of horseshoes. Following the terminology in section 2, we can construct a horseshoe associated to finite numbers of homoclinic orbits at o. We prove the following lemma which will be used in the proof of Theorem 1.8.