# The degeneration of a family of rational surface automorphisms

Qitong Jiang  Department of Mathematics, The Pennsylvania State University,
University Park, PA 16801, USA<EMAIL_ADDRESS>
###### Abstract.

We consider a one-dimensional family of rational surfaces with automorphisms
discussed in [7, 2, 3]. In a degeneration of this family, the limiting map is
the identity map on a special fiber. We check that the map on the total space
of the family has indeterminacy in the special fiber. However, we show that
after blowing-up at an indeterminate curve, there is an induced birational map
on the exceptional divisor over the indeterminate curve. Moreover, we show
that this map has dynamical degree $\lambda=16$.

##  0\. Introduction

Given an automorphism $f:X\to X$ of an algebraic variety $X$, if we blow-up
$X$ at a subvariety $V$ with $f(V)=V$, then $f$ induces an automorphism
$\widetilde{f}|_{E}:E\to E$ of the exceptional divisor $E$. This map does not
have any new dynamical properties, as it fibers over the original map
$f|_{V}:V\to V$. On the other hand, if $f(V)=V^{\prime}\neq V$, then the lift
$\widetilde{f}$ on $\operatorname{Bl}_{V}{X}$ simply contracts $E$.

In this paper, we will be concerned with the situation when
$f:X\dashrightarrow X$ is a birational map and $V$ is contained in
$\operatorname{Ind}(f)$. In this case, interesting dynamics is possible on
$E\subset\operatorname{Bl}_{V}(X)$. We particularly focus on cases related to
degenerations of rational surface automorphisms. Let $T$ be an elliptic curve
and $X=\mathbb{P}^{2}\times T$. Given $t\in T$, there is an involution
$\iota_{t}$ associated to $t$ and $T$ constructed in [7, 2, 3]. Fix a non-
trivial $2$-torsion point $q\in T$. We have the map $\varphi:X\dashrightarrow
X$ defined by

$(p,t)\mapsto(\iota_{q}(p),t),$

the map $\psi:X\dashrightarrow X$ by

$(p,t)\mapsto(\iota_{t}(p),t),$

and the map $\chi:X\dashrightarrow X$ by

$(p,t)\mapsto(\iota_{-t}(p),t),$

where $-t$ is the inverse of $t$. The map
$F=\varphi\circ\chi\circ\varphi\circ\psi$ can be viewed as a family of
birational maps of $\mathbb{P}^{2}$:

$\\{\iota_{q}\circ\iota_{-t}\circ\iota_{q}\circ\iota_{t}:t\in T\\}.$

For a general $t\in T$, the map $F_{t}:X_{t}\dashrightarrow X_{t}$ can be
regularized to an automorphism on some higher birational model of $X$, and the
dynamical degree of $F_{t}$ is $\lambda(F_{t})=17+12\sqrt{2}$; whereas the
dynamical degree of $F_{q}$ is $\lambda(F_{q})=1$ as
$F_{q}=(\iota_{q})^{4}=\operatorname{id}_{\mathbb{P}^{2}}$. As the family
degenerates to the map on the fiber $X_{q}=\mathbb{P}^{2}\times\\{q\\}$, $F$
develops indeterminate curves inside the fiber $X_{q}$. In fact, the
interesting dynamics does not completely disappear as the maps degenerate, but
becomes concentrated along the indeterminate curves. To see this, we blow-up
$X$ at the indeterminate curves and find that the induced map on the
exceptional divisor has dynamical degree $\lambda=16$.

Here is the outline of the discussion:

  * •

In Section 2, we will exhibit families of rational surface birational maps
whose general member can be regularized as automorphisms. We will study the
indeterminate curves, that are in the fiber over $q$, of the corresponding
threefold map by factoring the map as a sequence of automorphisms and Atiyah
flops.

  * •

In Section 3, we will find induced self-maps of the exceptional divisors over
the identified indeterminate curves in the special fiber over $q$.

  * •

In Section 4, we will study the dynamical properties of the induced self-maps
of the exceptional divisors.

In particular, we show the following result.

######  Theorem 1.

There exists a curve $\mathfrak{q}\subset X_{q}$ which is indeterminate for
the map $\varphi\circ\chi\circ\varphi\circ\psi$. Let
$\operatorname{Bl}_{\mathfrak{q}}{X}$ be the blow-up of
$X=\mathbb{P}^{2}\times T$ at $\mathfrak{q}$ with exceptional divisor
$E_{\mathfrak{q}}$. There is an induced birational map on $E_{\mathfrak{q}}$
by $\varphi\circ\chi\circ\varphi\circ\psi$, and the map has dynamical degree
$\lambda=16$.

As a warm-up, let’s consider several examples of birational self-maps of
$\mathbb{P}^{3}$ where the induced maps on exceptional divisors can be easily
obtained.

######  Example 1.

Given $b\in\mathbb{C}$, let $f:\mathbb{P}^{3}\to\mathbb{P}^{3}$ be the
automorphism defined by the formula

$[w,x,y,z]\mapsto[bw,x,y,z].$

Let $L$ be the line $\\{y=z=0\\}$ in $\mathbb{P}^{3}$. Then $f(L)=L$, and the
exceptional divisor $E$ over $L$ is isomorphic to
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ On $\operatorname{Bl}_{L}\mathbb{P}^{3}$.
The induced map $\widetilde{f}|_{E}:E\to E$ can be written as

$([u,v],[s,t])\mapsto([bu,v],[s,t]).$

In the affine chart $\mathbb{A}^{1}\times\mathbb{A}^{1}$ where $v,t\neq 0$,
this is a dilation in the $u$-diration:

$(u,s)\mapsto(bu,s).$

######  Example 2.

For a slightly more complicated example, let
$f:\mathbb{P}^{3}\dashrightarrow\mathbb{P}^{3}$ be the birational map given by

$[w,x,y,z]\mapsto[x^{2}w,x^{3},x^{2}y,x^{2}y-ayw^{2}+x^{2}z],$

for some $a\in\mathbb{C}$. Then $f$ is defined on a dense open subset of the
line $L$. The induced map $\widetilde{f}|_{E}:E\dashrightarrow E$ is
birational in this case:

$([u,v],[s,t])\mapsto([u,v],[v^{2}t-au^{2}t+v^{2}s,v^{2}t]).$

We note that in the affine chart where $v,t\neq 0$, this is an “area-
preserving” bend:

$(u,s)\mapsto(u,1-au^{2}+s).$

######  Example 3.

If $f(V)\neq V$, then the lifted map simply contracts $E$. Let
$f:\mathbb{P}^{3}\to\mathbb{P}^{3}$ be the automorphism

$[w,x,y,z]\mapsto[y,z,w,x].$

The lifted map
$\widetilde{f}:\operatorname{Bl}_{L}\mathbb{P}^{3}\dashrightarrow\operatorname{Bl}_{L}\mathbb{P}^{3}$
will contract the exceptional surface $E$ to the strict transform of the line
$\\{w=x=0\\}\subset\mathbb{P}^{3}$.

######  Example 4.

We suppose that $V\subset\operatorname{Ind}(f)$, and we blow-up $X$ at $V$. We
consider when there will be an induced birational map on the exceptional
divisor. Let $f:\mathbb{P}^{3}\dashrightarrow\mathbb{P}^{3}$ be the
composition of the automorphism

$[w,x,y,z]\mapsto[y,z,w,x]$

and the standard involution of $\mathbb{P}^{3}$. The line $L=\\{y=z=0\\}$ is
contained in the indeterminacy locus of $f$. In this case, $f$ induces a
(regular) birational map $\widetilde{f}|_{E}:E\to E$, whose formula can be
written as

$([u,v],[s,t])\mapsto([s,t],[u,v]).$

We note that, in the affine chart where $v,t\neq 0$, this map is the
reflection across the line $u=s$:

$(u,s)\mapsto(s,u).$

######  Example 5.

We can actually obtain a Hénon map on the exceptional divisor by composing the
maps introduced in Examples 1, 2 and 4. The composition $f$ is undefined along
the line $L=\\{y=z=0\\}$, but the induced map on the exceptional divisor $E$
over $L$ is a Hénon map.

For our main discussion, we need to cover some background knowledge.

##  1\. Groundwork

We assume that the base field $K$ is algebraically closed of characteristic
$\operatorname{char}{K}=0$.

###  1.1. Relative numerical equivalence

Let $X$ be a smooth variety of dimension $d$ over $K$, and let $p:X\to S$ be a
proper morphism onto a variety $S$. We define $Z_{1}(X/S)$ to be the free
abelian group generated by reduced irreducible curves which are mapped to
points by $p$. There is a numerical equivalence $\equiv_{S}$ in
$\operatorname{Pic}(X)$ defined by

$D\equiv_{S}D^{\prime}\quad\text{if and only if}\quad D\cdot C=D^{\prime}\cdot
C\text{ for all }C\in Z_{1}(X/S).$

We write
$N^{1}(X/S):=(\operatorname{Pic}(X)/\equiv_{S})\otimes_{\mathbb{Z}}\mathbb{R}$.
When $S=\operatorname{Spec}{K}$, we drop $/S$ from the notation.

###  1.2. Blow-up and exceptional locus

The blow-up of a variety $X$ at a subvariety $V$ will be written as
$\operatorname{Bl}_{V}X$. Let $\pi:\operatorname{Bl}_{V}X\to X$ be the
projection map. The exceptional divisor $\pi^{-1}(V)$ is isomorphic to
$\mathbb{P}(N_{V/X})$, where $N_{V/X}$ is the normal bundle to $V$ in $X$. We
denote $\pi^{-1}(V)$ by $E_{V}$. We can thus identify the points on $E_{V}$
with elements in $\mathbb{P}(N_{V/X})$.

We will frequently consider sequences of blow-ups throughout the paper. To
simplify notations, we use the same notation for a subvariety $V$ and all of
its strict transforms.

###  1.3. Dynamical degree and algebraic stability

Let $X$ be a smooth projective variety of dimension $k$, and
$f:X\dashrightarrow X$ a dominant rational map. Let
$H\in\operatorname{Div}(X)$ be a big and nef divisor. We define the
($H$-)degree of $f$ to be

$\deg_{H}(f):=H^{k-1}\cdot f^{*}(H).$

The (first) dynamical degree of $f$ is defined to be

$\lambda(f):=\lim_{n\to\infty}\left(\deg_{H}(f^{n})\right)^{1/n}.$

The fact that the above limit exists and is independent of the choices of $H$
is proved in [5] [10].

Now, let $S$ be a quasi-projective surface and $f:S\dashrightarrow S$ a
birational map. The map $f$ is said to be algebraically stable if

$(f^{n})^{*}=(f^{*})^{n}$

for any integer $n\geq 0$. The following theorem is useful in practice.

######  Theorem 2 (Theorem 1.14 in [6]).

Let $S$ be a quasi-projective surface and $f:S\dashrightarrow S$ a birational
map. $f$ is algebraically stable if and only if there is no curve $C\subset S$
such that such that $f^{k}(C)\subset\operatorname{Ind}(f)$ for some integer
$k\geq 0$.

Moreover, we can use the following Theorem to compute the dynamical degree for
an algebraically stable birational surface map.

######  Theorem 3 (Corollary 1.19 in [6]).

Let $S$ be a quasi-projective surface and $f:S\dashrightarrow S$ an
algebraically stable birational map. Then $\lambda(f)$ coincides with the
spectral radius of $f_{*}:N^{1}(S)\to N^{1}(S)$.

###  1.4. Atiyah flop

Let $X$ be a quasi-projective threefold, and let $B$ and $C$ be two curves on
$X$ intersecting only at a point $p\in X$ with distinct tangent directions.
Let

$\displaystyle\widetilde{X}:$
$\displaystyle=\operatorname{Bl}_{C}\operatorname{Bl}_{B}X\overset{\pi_{2}}{\longrightarrow}\operatorname{Bl}_{B}X\overset{\pi_{1}}{\longrightarrow}X$
$\displaystyle\widetilde{X}^{+}:$
$\displaystyle=\operatorname{Bl}_{B}\operatorname{Bl}_{C}X\overset{\pi_{2}^{+}}{\longrightarrow}\operatorname{Bl}_{C}X\overset{\pi_{1}^{+}}{\longrightarrow}X$

be the blow-ups of $X$ at $B$ and $C$ in different orders. Finally, we take
the preimage of the point $p$ on $\operatorname{Bl}_{B}{X}$, which is a curve
isomorphic to $\mathbb{P}^{1}$; and we let $\mathfrak{b}$ be the strict
transform of the curve on $\widetilde{X}$.

######  Proposition 4.

The normal bundle $N_{\mathfrak{b}/\widetilde{X}}$ is isomorphic to
$\mathcal{O}_{\mathbb{P}^{1}}(-1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-1)$, and
the natural birational map
$\phi:\widetilde{X}\dashrightarrow\widetilde{X}^{+}$ is an Atiyah flop with
the flopping curve $\mathfrak{b}$.

In the case where a pseudo-automorphism $f$ of a threefold can be factored
into a sequence of Atiyah flops and automorphisms, the image of a divisor
class can be computed recursively. We need the following lemma.

######  Lemma 5 (cf. Lemma 8 in [9]).

Let $\phi:X\dashrightarrow X^{+}$ be an Atiyah flop with the flopping curve
$c\subset X$. Let $c^{+}$ denote the flopping curve of $\phi^{-1}$. Suppose
that $D^{+}$ is a divisor on $X^{+}$ and that $D$ is the pullback of $D^{+}$
under $\phi$. Then

$\operatorname{mult}_{c}(D)=\operatorname{mult}_{c^{+}}(D^{+})+D^{+}\cdot
c^{+}.$

Let $p:X\to T$ be a proper and flat family of surfaces parametrized by a curve
$T$. Let $\phi:X\dashrightarrow X^{+}$ be an Atiyah flop with flopping curve
$c$ lying in the fiber $F:=X_{q}$ for some $q\in T$. Let $c^{+}$ denote the
flopped curve, and let $F^{+}:=\phi(F)\subset X^{+}$. For a divisor $D^{+}\in
N^{1}(X^{+}/T)$, we have

(1.1) $\phi^{*}(D^{+})|_{F}=(\phi|_{F})^{*}(D^{+}|_{F^{+}})+(D^{+}\cdot
c^{+})c.$

Indeed, by Lemma 5, we have

$\displaystyle\operatorname{mult}_{c}(\phi^{*}(D^{+})|_{F})=\operatorname{mult}_{c}(D)=\operatorname{mult}_{c^{+}}(D^{+})+D^{+}\cdot
c^{+}=\operatorname{mult}_{c^{+}}(D^{+}|_{F^{+}})+D^{+}\cdot c^{+}.$

Since $\phi|_{F}:F\to F^{+}$ is an isomorphism,

$\operatorname{mult}_{c}((\phi|_{F})^{*}(D^{+}|_{F^{+}}))=\operatorname{mult}_{c^{+}}(D^{+}|_{F^{+}}).$

The equation (1.1) follows.

Inductively, we obtain the following result: Let $X$ be a threefold and
$f:X\dashrightarrow X$ a pseudo-automorphism of $X$ preserving the fibration
$p:X\to T$ with indeterminate curves lying inside the fiber $F$. Suppose $f$
factors into a sequence of flops and isomorphisms

$f:X=X_{1}\overset{\phi_{1}}{\dashrightarrow}X_{1}^{+}\overset{f_{1}}{\longrightarrow}X_{2}\overset{\phi_{2}}{\dashrightarrow}\dotsm\overset{f_{n-1}}{\longrightarrow}X_{n}=X,$

where $\phi_{i}$ are Atiyah flops, and $f_{i}$ are the isomorphisms. Let
$c_{i}$ be the flopping curve of $\phi_{i}$ with flopped curve $c_{i}^{+}$. We
write

$g_{i}:=f_{n-1}\circ\dotsm\circ f_{n-i}:X_{n-i}^{+}\dashrightarrow
X_{n},i=1,\dots,n-1;$

and write

$h_{i}:=f_{n-i-1}\circ\dotsm\circ\phi_{1}:X_{1}\dashrightarrow
X_{n-i},i=1,\dots,n-2,\quad\text{and}\quad h_{n-1}:=\operatorname{id}_{X}.$

######  Proposition 6.

For a divisor $D\in N^{1}(X/T)$,

$f^{*}(D)|_{F}=\left(f|_{F}\right)^{*}(D|_{F})+\sum_{i=1}^{n-1}(g_{i}^{*}(D)\cdot
c_{n-i}^{+})h_{i}^{*}(c_{n-i}).$

In particular, the difference

$(f^{n})^{*}(D)|_{F}-(f^{n}|_{F})^{*}(D|_{F})$

is an effective divisor supported on flopping curves and their pullbacks under
$f|_{F}$.

Finally, we observe that there is a natural birational map between two
birational models. Let $B$ and $C$ be two curves in a threefold $X$
intersecting at a point $p$ with distinct tangent directions. There are
several ways to blow-up $X$:

  1. (1)

The first sequence of blow-ups (at the obvious centers) is

${\operatorname{Bl}_{C}\operatorname{Bl}_{B}\operatorname{Bl}_{p}{X}}$${\operatorname{Bl}_{B}\operatorname{Bl}_{p}{X}}$${\operatorname{Bl}_{p}{X}}$${X}$$\scriptstyle{\pi_{C}}$$\scriptstyle{\pi_{B}}$$\scriptstyle{\pi}$

with exceptional divisors are $E_{p}$, $E_{B}$, $E_{C}$ over $p,B,C$,
respectively.

  2. (2)

The second sequence of blow-ups is

${\operatorname{Bl}_{C}\operatorname{Bl}_{\mathfrak{f}^{\prime}}\operatorname{Bl}_{B}{X}}$${\operatorname{Bl}_{\mathfrak{f}^{\prime}}\operatorname{Bl}_{B}{X}}$${\operatorname{Bl}_{B}{X}}$${X}$$\scriptstyle{\pi^{\prime}_{C}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\pi^{\prime}_{B}}$

with exceptional divisors are $E_{B}^{\prime}$,
$E_{\mathfrak{f}^{\prime}}^{\prime}$, $E_{C}^{\prime}$ over $B,p,C$,
respectively. The blow-up center of $\pi^{\prime}$ is the fiber
$\mathfrak{f}^{\prime}$ of $E_{B}^{\prime}$ over $p$.

  3. (3)

The third sequence of blow-ups is

${\operatorname{Bl}_{\mathfrak{f}^{\prime\prime}}\operatorname{Bl}_{C}\operatorname{Bl}_{B}{X}}$${\operatorname{Bl}_{C}\operatorname{Bl}_{B}{X}}$${\operatorname{Bl}_{B}{X}}$${X}$$\scriptstyle{\pi^{\prime\prime}}$$\scriptstyle{\pi^{\prime\prime}_{C}}$$\scriptstyle{\pi^{\prime\prime}_{B}}$

with exceptional divisors are $E_{B}^{\prime\prime}$, $E_{C}^{\prime\prime}$,
$E_{\mathfrak{f}^{\prime\prime}}^{\prime\prime}$ over $B,C,p$, respectively.
The blow-up center of $\pi^{\prime\prime}$ is the flopping curve
$\mathfrak{f}^{\prime\prime}$ of the Atiyah flop
$\phi:\operatorname{Bl}_{C}\operatorname{Bl}_{B}{X}\dashrightarrow\operatorname{Bl}_{B}\operatorname{Bl}_{C}{X}$.

######  Proposition 7.

There is an isomorphism (induced by the identity map)

$\operatorname{Bl}_{C}\operatorname{Bl}_{B}\operatorname{Bl}_{p}{X}\overset{\cong}{\longrightarrow}\operatorname{Bl}_{C}\operatorname{Bl}_{\mathfrak{f}^{\prime}}\operatorname{Bl}_{B}{X},$

which identifies $E_{p}$ and $E_{\mathfrak{f}^{\prime}}^{\prime}$.
Furthermore, there is an Atiyah flop (induced by the identity map)

$\operatorname{Bl}_{C}\operatorname{Bl}_{B}\operatorname{Bl}_{p}{X}\cong\operatorname{Bl}_{C}\operatorname{Bl}_{\mathfrak{f}^{\prime}}\operatorname{Bl}_{B}{X}\dashrightarrow\operatorname{Bl}_{\mathfrak{f}^{\prime\prime}}\operatorname{Bl}_{C}\operatorname{Bl}_{B}{X}$

that sends $E_{p}\cong\operatorname{Bl}_{2}\mathbb{P}^{2}$ to
$E_{\mathfrak{f}^{\prime\prime}}^{\prime\prime}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$.

We have acquired all the necessary tools for further discussion.

##  2\. Family of Maps

We present families of rational surface birational self-maps in this section
and examine their properties.

###  2.1. Cubic involution of $\mathbb{P}^{2}$

We recall the following construction in [7, 2, 3]: Let $T$ be a smooth cubic
curve in $\mathbb{P}^{2}$, and let $q\in T$ be a point. For a general point
$p\in\mathbb{P}^{2}$, the line $L_{q,p}$ connecting $q,p$ intersects $T$ at
three points $q,a,b$. There is a unique involution $i_{q,p}$ of $L_{q,p}$ with
$i_{q,p}(a)=a$ and $i_{q,p}(b)=b$. Therefore, given a smooth cubic curve $T$
and $q\in T$, we can define a cubic involution
$\iota_{q}:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^{2}$ by

$p\mapsto\iota_{q}(p):=i_{q,p}(p).$

We first note that the cubic involution fixes (a dense open subset of) the
cubic curve pointwise by construction. Now, let $q$ be a point in $T$ such
that there are four distinct lines
$\mathfrak{l}_{1},\mathfrak{l}_{2},\mathfrak{l}_{3},\mathfrak{l}_{4}$ passing
through $q$ and being tangent to $T$ at $q_{1},q_{2},q_{3},q_{4}$
respectively. The points $q_{1},q_{2},q_{3},q_{4}$ will be sometimes referred
to as the (four) tangent points associated to $q$. The indeterminacy locus of
$\iota_{q}$ is

$\operatorname{Ind}(\iota_{q})=\\{q,q_{1},q_{2},q_{3},q_{4}\\}.$

In fact, the cubic involution $\iota_{q}$ contracts the conic $\mathfrak{q}$
passing through $q,q_{1},q_{2},q_{3},q_{4}$ to $q$ and contracts the tangent
line $\mathfrak{l}_{i}$ to $q_{i}$ for each $i$. The (affine) picture of the
cubic curve, the conic and the tangent lines is in Figure 1.

$\mathfrak{q}$$\mathfrak{l}_{3}$$\mathfrak{l}_{4}$$\mathfrak{l}_{2}$$\mathfrak{l}_{1}$$q_{3}$$q_{4}$$q_{2}$$q_{1}$$q$
Figure 1. Curves contracted by $\iota_{q}$

We will see that the sequence
$\\{\deg\left((\iota_{q}\circ\iota_{t})^{n}\right)\\}$ grows quadratically for
general $t\neq q$. So, by varying $t\in T$, we get a family of birational maps
of $\mathbb{P}^{2}$, whose general member has quadratic degree growth, but the
special map $\iota_{q}\circ\iota_{q}$ is the identity of $\mathbb{P}^{2}$.
Furthermore, the map
$\iota_{q}\circ\iota_{t^{\prime}}\circ\iota_{q}\circ\iota_{t}$ has dynamical
degree greater than $1$ for general distinct $t,t^{\prime},q$. We can obtain a
family of birational maps of $\mathbb{P}^{2}$, whose general member has
$\lambda>1$, but again, the special member when $t=t^{\prime}=q$ is the
identity.

###  2.2. Involution of $\mathbb{P}^{2}\times T$

We define the families of maps formally as birational maps of a threefold. Let
$X$ be the threefold $\mathbb{P}^{2}\times
T\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$. Fix a point $q\in T$ such that
$\\#\operatorname{Ind}(\iota_{q})=5$. We define an involution
$\varphi:X\dashrightarrow X$ using $\iota_{q}$ by

$(p,t)\mapsto(\iota_{q}(p),t).$

We note that $\varphi$ can also be viewed as a family of birational
involutions of $\mathbb{P}^{2}$, i.e., $\\{\phi_{t}:=\iota_{q}:t\in T\\}$.
Next, we further let $q$ be a nontrivial $2$-torsion point with respect to the
group law on $T$. We define the involution $\psi:X\dashrightarrow X$ by

$(p,t)\mapsto(\iota_{t}(p),t)$

and the involution $\chi:X\dashrightarrow X$ by

$(p,t)\mapsto(\iota_{-t}(p),t),$

where $-t$ is the inverse of $t$ with respect to the group law on $T$. So, we
have two other families of maps $\\{\psi_{t}:=\iota_{t}:t\in T\\}$ and
$\\{\chi_{t}:=\iota_{-t}:t\in T\\}$.

Let us take the open subset of $T$ consisting of the point $q$ and points
$t\in T$ such that

$\\#\operatorname{Ind}(\iota_{t})=\\#\operatorname{Ind}(\iota_{-t})=5,$

and that

$\operatorname{Ind}(\iota_{t})\cap\operatorname{Ind}(\iota_{q})=\operatorname{Ind}(\iota_{-t})\cap\operatorname{Ind}(\iota_{q})=\operatorname{Ind}(\iota_{t})\cap\operatorname{Ind}(\iota_{-t})=\varnothing;$

and replace $T$ with this open subset. We still keep the same notations for
the threefold $X=\mathbb{P}^{2}\times T$ and restricted involutions
$\varphi,\psi,\chi$. We are now ready to state the indeterminacy locus of the
maps $\varphi$, $\psi$ and $\chi$.

The indeterminacy locus of $\varphi$ contains five curves

$B=\\{q\\}\times T\quad\text{and}\quad B_{i}=\\{q_{i}\\}\times T$

for $i=1,\dots,4$. The indeterminacy locus of $\psi$ contains two irreducible
curves, namely, the curve

$C=\\{(p,t):p=t\\}$

and the curve $C^{\prime}$. For $t\in T$, the restriction of the curve
$C^{\prime}$ to the fiber $\mathbb{P}^{2}\times\\{t\\}$ consists of the
tangent points $t_{1},t_{2},t_{3},t_{4}$ associated to $t$. The indeterminacy
locus of $\chi$ contains the curve

$D=\\{(p,t):p=-t\\}$

and the curve $D^{\prime}$ whose restriction to $\mathbb{P}^{2}\times\\{t\\}$
are the four tangent points associated to $-t$. We note that

$B\cap C\cap D=\\{(q,q)\\}\quad\text{and}\quad B_{i}\cap C^{\prime}\cap
D^{\prime}=\\{(q_{i},q)\\}$

for $i=1,\dots,4$. These are the only intersection points of the indeterminate
curves.

######  Example 6.

Let $T$ be an elliptic curve over a field $K$ of characteristic
$\operatorname{char}{K}\neq 2$ on $\mathbb{A}^{2}$ given by

$b^{2}=a(a-\alpha_{1})(a-\alpha_{2}),$

where $\alpha_{1},\alpha_{2}$ are distinct nonzero elements of $K$, and
$q=(0,0)\in T$. We take the open subset of $T$ as discussed above and still
call it $T$. Let $X=\mathbb{A}^{2}\times
T\subset\mathbb{A}^{2}\times\mathbb{A}^{2}$ with coordinates $((x,y),(a,b))$.

The curve $B$ is defined (in $X$) by

$\displaystyle x=y=0.$

The curve $C$ is defined by

$x-a=y-b=0.$

The curve $D$ is defined by

$x-a=y+b=0.$

For $B_{1},\dots,B_{4}$, $C^{\prime}$, and $D^{\prime}$, we need explicit
formulae for the tangent points associated to a given point on the elliptic
curve, and we will follow the calculation in [1]. For example, for $q=(0,0)$,
the curves $B_{1},$ is defined in $X$ by

$x+\sqrt{-\alpha_{1}}\sqrt{-\alpha_{2}}=y-(-\sqrt{-\alpha_{1}}+\sqrt{-\alpha_{2}})x=0.$

The curves $C^{\prime}$ and $D^{\prime}$ can be obtained as the two
irreducible components of the following reducible curve defined by

$\displaystyle
x^{4}-4ax^{3}+4(\alpha_{1}+\alpha_{2})ax^{2}-2\alpha_{1}\alpha_{2}x^{2}-4\alpha_{1}\alpha_{2}ax+\alpha_{1}^{2}\alpha_{2}^{2}=y^{2}-x(x-\alpha_{1})(x-\alpha_{2})=0.$

Using $\varphi,\psi$ and $\chi$, we can define families of birational self-
maps. We will first study the composition $\varphi\circ\psi:X\dashrightarrow
X$, i.e., the family of maps $\\{\iota_{q}\circ\iota_{t}:t\in T\\}$. We will
then take the composition $\varphi\circ\chi\circ\varphi\circ\psi$, which is
the family of maps
$\\{\iota_{q}\circ\iota_{-t}\circ\iota_{q}\circ\iota_{t}:t\in T\\}$.

###  2.3. Indeterminate curves of $\varphi\circ\psi$

Let us write $\mathcal{B}:=B\cup B_{1}\cup B_{2}\cup B_{3}\cup B_{4}$, and
$\mathcal{C}:=C\cup C^{\prime}$. The map $\varphi$ will be regularized on
$\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$ and hence on
$\operatorname{\operatorname{Bl}_{\mathcal{C}}}\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$
as $\varphi$ fixes the curves in $\mathcal{C}$ pointwise; and that the map
$\psi$ will be regularized on
$\operatorname{\operatorname{Bl}_{\mathcal{C}}}{X}$ and hence on
$\operatorname{\operatorname{Bl}_{\mathcal{B}}}\operatorname{\operatorname{Bl}_{\mathcal{C}}}{X}$
as $\psi$ fixes the curves in $\mathcal{B}$ pointwise (cf. [4]). Let
$\tilde{\varphi}$ and $\tilde{\psi}$ be the lifts of $\varphi$ and $\psi$ on
$\operatorname{\operatorname{Bl}_{\mathcal{C}}}\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$.
The map $\tilde{\varphi}\circ\tilde{\psi}$ has no indeterminacy on each fiber
over $t$ for $t\neq q$. However, since
$\operatorname{\operatorname{Bl}_{\mathcal{C}}}\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$
and
$\operatorname{\operatorname{Bl}_{\mathcal{B}}}\operatorname{\operatorname{Bl}_{\mathcal{C}}}{X}$
differ by a finite compositions of Atiyah flops with pairwise disjoint
flopping curves in the fiber over $q$, we will see that
$\tilde{\varphi}\circ\tilde{\psi}$ will have indeterminate curves within the
fiber over $q$.

Let us fix some notations for the rest of the discussion. Given a threefold
$X$ with a fibration $X\to T$, let $X_{t}$ denote the fiber of $X$ over $t\in
T$. For a map $f:X\dashrightarrow X$ preserving $X\to T$, let $f_{t}$ denote
the restriction $f|_{X_{t}}$.

#### Blow-up, first round

We denote by $\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$ the blow-up
of $X$ at $\mathcal{B}$ with exceptional divisors $E_{B}$ and $E_{B_{i}}$ over
$B$ and $B_{i}$ respectively. For $t\in T$, let
$(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{t}$ be the fiber of
$\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$ over $t$. Then
$(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{t}$ is the blow-up of
$X_{t}\cong\mathbb{P}^{2}$ at the five points

$(t,t),(t_{1},t),(t_{2},t),(t_{3},t),(t_{4},t)\in
X_{t}=\mathbb{P}^{2}\times\\{t\\},$

where $t,t_{1},t_{2},t_{3},t_{4}$ are the indeterminate points of $\iota_{t}$
as a birational involution of $\mathbb{P}^{2}$. The involution $\varphi$ lifts
to a regular involution $\hat{\varphi}$ on
$\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$. On each fiber
$(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{t}$,
$\hat{\varphi}|_{(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{t}}:(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{t}\to(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{t}$
swaps the exceptional curve over $(q,t)$ and the strict transform of the conic
$\mathfrak{q}\times\\{t\\}$, and exchanges the other four exceptional curves
and the strict transforms of the four tangent lines respectively. In
particular, on the fiber
$(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{q}$, we have the five
exceptional curves

$\mathfrak{b}:=E_{B}\cap(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{q}\quad\text{and}\quad\mathfrak{b}_{i}:=E_{B_{i}}\cap(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{q}$

for $i=1,\dots,4$. Moreover, we still denote by $\mathfrak{q}$ the strict
transform of the curve $\mathfrak{q}\times\\{q\\}\subset X_{q}$ on
$\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$, and by $\mathfrak{l}_{i}$
the strict transforms of the curves $\mathfrak{l}_{i}\times\\{q\\}\subset
X_{q}$ on $\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$ for
$i=1,\dots,4$. We have

$\hat{\varphi}(\mathfrak{b})=\mathfrak{q}\quad\text{and}\quad\hat{\varphi}(\mathfrak{b}_{i})=\mathfrak{l}_{i}$

for $i=1,\dots,4$.

Let $\operatorname{\operatorname{Bl}_{\mathcal{C}}}{X}$ be the blow-up of $X$
at $\mathcal{C}$ with the exceptional divisors $E_{C}^{+}$ and
$E_{C^{\prime}}^{+}$ over $C$ and $C^{\prime}$ respectively. The map $\psi$
lifts to an automorphism $\hat{\psi}^{+}$ on
$\operatorname{\operatorname{Bl}_{\mathcal{C}}}{X}$. On the fiber
$(\operatorname{\operatorname{Bl}_{\mathcal{C}}}{X})_{q}$ over $q$, there are
the exceptional curves

$\mathfrak{c}^{+}=E_{C}^{+}\cap(\operatorname{\operatorname{Bl}_{\mathcal{C}}}{X})_{q}\quad\text{and}\quad\mathfrak{c}_{1}^{+}\sqcup\mathfrak{c}_{2}^{+}\sqcup\mathfrak{c}_{3}^{+}\sqcup\mathfrak{c}_{4}^{+}=E_{C^{\prime}}^{+}\cap(\operatorname{\operatorname{Bl}_{\mathcal{C}}}{X})_{q};$

and there are the strict transform $\mathfrak{q}^{+}$ of the conic and the
strict transforms $\mathfrak{l}_{i}^{+}$ of the tangent lines for
$i=1,\dots,4$. We have

$\hat{\psi}^{+}(\mathfrak{c}^{+})=\mathfrak{q}^{+}\quad\text{and}\quad\hat{\psi}^{+}(\mathfrak{c}_{i}^{+})=\mathfrak{l}_{i}^{+}$

for $i=1,\dots,4$.

#### Blow-up, second round

We want to relate $\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$ and
$\operatorname{\operatorname{Bl}_{\mathcal{C}}}{X}$.

We can blow-up $\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$ at the
strict transform of $\mathcal{C}$ to get
$\operatorname{\operatorname{Bl}_{\mathcal{C}}}\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$.
We still denote by $E_{B}$ and $E_{B_{i}}$ the exceptional divisors over $B$
and $B_{i}$, respectively, on
$\operatorname{\operatorname{Bl}_{\mathcal{C}}}\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$;
and we denote by $E_{C}$ and $E_{C^{\prime}}$ the exceptional divisors over
$C$ and $C^{\prime}$, respectively, on
$\operatorname{\operatorname{Bl}_{\mathcal{C}}}\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$.
We write
$\widetilde{X}:=\operatorname{\operatorname{Bl}_{\mathcal{C}}}\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$.

In the second round of blow-ups, we actually blow-up each fiber of
$\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X}$ further at five additional
points. For $t\neq q$, the fiber of $\widetilde{X}$ over $t$ is isomorphic to
the blow-up of $X_{t}\cong\mathbb{P}^{2}$ at the ten distinct points
$(\mathcal{B}\cup\mathcal{C})\cap X_{t}$. However, since the curves
$\mathcal{B}$ and $\mathcal{C}$ intersect in the fiber $X_{q}$ at the five
points

$(q,q),(q_{1},q),(q_{2},q),(q_{3},q),(q_{4},q),$

the fiber $\widetilde{X}_{q}$ is the blow-up of $X_{q}\cong\mathbb{P}^{2}$ at
five points and then at five infinitely near points. More specifically,
$(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{q}$ is the blow-up of
$X_{q}$ at $(q,q),(q_{1},q),(q_{2},q),(q_{3},q),(q_{4},q)$ with exceptional
curves
$\mathfrak{b},\mathfrak{b}_{1},\mathfrak{b}_{2},\mathfrak{b}_{3},\mathfrak{b}_{4}$.
The curves
$\mathfrak{b},\mathfrak{b}_{1},\mathfrak{b}_{2},\mathfrak{b}_{3},\mathfrak{b}_{4}$
intersect the strict transforms of $C,C^{\prime}$ transversally at the points
$(q^{\prime},q),(q_{1}^{\prime},q),(q_{2}^{\prime},q),(q_{3}^{\prime},q),(q_{4}^{\prime},q)$
in $(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{q}$ as shown in
Figure 2. Therefore, $\widetilde{X}_{q}$ is the blow-up of
$(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{q}$ at such points.

$(\operatorname{\operatorname{Bl}_{\mathcal{B}}}{X})_{q}$$\mathfrak{b}$$\mathfrak{b}_{3}$$\mathfrak{b}_{4}$$\mathfrak{b}_{2}$$\mathfrak{b}_{1}$$C$$C^{\prime}$$C^{\prime}$$C^{\prime}$$C^{\prime}$$q_{3}^{\prime}$$q_{4}^{\prime}$$q_{2}^{\prime}$$q_{1}^{\prime}$$q^{\prime}$

$\widetilde{X}_{q}$$\mathfrak{b}$$\mathfrak{b}_{3}$$\mathfrak{b}_{4}$$\mathfrak{b}_{2}$$\mathfrak{b}_{1}$$E_{C}$$E_{C^{\prime}}$$E_{C^{\prime}}$$E_{C^{\prime}}$$E_{C^{\prime}}$$\mathfrak{c}$$\mathfrak{c}_{3}$$\mathfrak{c}_{4}$$\mathfrak{c}_{2}$$\mathfrak{c}_{1}$

Figure 2. Blow-ups of $X_{q}$

Let

$\mathfrak{c}:=E_{C}\cap\widetilde{X}_{q}\quad\text{and}\quad\mathfrak{c}_{1}\sqcup\mathfrak{c}_{2}\sqcup\mathfrak{c}_{3}\sqcup\mathfrak{c}_{4}:=E_{C^{\prime}}\cap\widetilde{X}_{q}.$

We still write $\mathfrak{b}$ for the strict transform of $\mathfrak{b}$ in
$\widetilde{X}$, and $\mathfrak{b}_{i}$ for the strict transform of
$\mathfrak{b}_{i}$ in $\widetilde{X}$ for each $i$. Since the map
$\widehat{\varphi}$ fixes the strict transform $C$ and $C^{\prime}$ pointwise,
it lifts to an automorphism $\tilde{\varphi}$ on $\widetilde{X}$, and
$\tilde{\varphi}$ will preserve the new exceptional divisors
$E_{C},E_{C^{\prime}}$ as $C$ and $C^{\prime}$ are fixed curves by $\varphi$.
On $\widetilde{X}_{q}$, we have

$\tilde{\varphi}(\mathfrak{c})=\mathfrak{c}\quad\text{and}\quad\tilde{\varphi}(\mathfrak{c}_{i})=\mathfrak{c}_{i}$

for $i=1,\dots,4$, and

$\tilde{\varphi}(\mathfrak{b})=\mathfrak{q}\quad\text{and}\quad\tilde{\varphi}(\mathfrak{b}_{i})=\mathfrak{l}_{i},$

where we use the same notations for the strict transforms of the conic and
tangent lines on $\widetilde{X}$.

On the other hand, we can blow-up
$\operatorname{\operatorname{Bl}_{\mathcal{C}}}{X}$ at the strict transform of
$\mathcal{B}$ to get
$\widetilde{X}^{+}:=\operatorname{\operatorname{Bl}_{\mathcal{B}}}\operatorname{\operatorname{Bl}_{\mathcal{C}}}{X}$.
The exceptional divisors over $B$ and $B_{i}$ in $\widetilde{X}^{+}$ are
$E_{B}^{+}$ and $E_{B_{i}}^{+}$ respectively, and the exceptional divisors
over $C$ and $C^{\prime}$ in $\widetilde{X}^{+}$ are $E_{C}^{+}$ and
$E_{C^{\prime}}^{+}$ respectively. On the fiber $(\widetilde{X}^{+})_{q}$, we
have

$\mathfrak{b}^{+}:=E_{B}^{+}\cap\widetilde{X}^{+}_{q}\quad\text{and}\quad\mathfrak{b}_{i}^{+}=E_{B_{i}}^{+}\cap\widetilde{X}^{+}_{q}$

for $i=1,\dots,4$; and we again write $\mathfrak{c}^{+}$ and
$\mathfrak{c}_{i}^{+}$ for the strict transforms of $\mathfrak{c}^{+}$ and
$\mathfrak{c}_{i}^{+}$ in $\widetilde{X}^{+}$ for each $i$. The map
$\hat{\psi}^{+}$ lifts to an automorphism $\tilde{\psi}^{+}$ on
$\widetilde{X}^{+}$, which preserves the new exceptional divisors
$E_{B}^{+},E_{B_{1}}^{+},E_{B_{2}}^{+},E_{B_{3}}^{+},E_{B_{4}}^{+}$. In
particular, we have

$\tilde{\psi}^{+}(\mathfrak{b}^{+})=\mathfrak{b}^{+}\quad\text{and}\quad\tilde{\psi}^{+}(\mathfrak{b}_{i}^{+})=\mathfrak{b}_{i}^{+},$

for $i=1,\dots,4$, and

$\tilde{\psi}^{+}(\mathfrak{c}^{+})=\mathfrak{q}^{+}\quad\text{and}\quad\tilde{\psi}^{+}(\mathfrak{c}_{i}^{+})=\mathfrak{l}_{i}^{+},$

for $i=1,\dots,4$.

#### Factorization of $\tilde{\varphi}\circ\tilde{\psi}$

Now the blow-ups $\widetilde{X}$ and $\widetilde{X}^{+}$ are related by a map
$\sigma:\widetilde{X}\dashrightarrow\widetilde{X}^{+}$, which is a composition
of five Atiyah flops. The five pairwise disjoint flopping curves for $\sigma$
are
$\mathfrak{b},\mathfrak{b}_{1},\mathfrak{b}_{2},\mathfrak{b}_{3},\mathfrak{b}_{4}$;
and the five pairwise disjoint flopped curves, which are the flopping curves
for $\sigma^{-1}$, are
$\mathfrak{c}^{+},\mathfrak{c}^{+}_{1},\mathfrak{c}^{+}_{2},\mathfrak{c}^{+}_{3},\mathfrak{c}^{+}_{4}$.

Let us denote by $\tilde{\psi}$ the lift of $\psi$ on $\widetilde{X}$. We
factorize the map $\tilde{\varphi}\circ\tilde{\psi}$ in terms of flops and
isomorphisms as follows:

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Although $\tilde{\psi}^{+}$ is an automorphism on $\widetilde{X}^{+}$,
$\tilde{\psi}$ is a pseudo-automorphism on $\widetilde{X}$, which is conjugate
to $\tilde{\psi}^{+}$ by $\sigma$. We note that $\sigma$ is undefined at
$\mathfrak{b},\mathfrak{b}_{1},\mathfrak{b}_{2},\mathfrak{b}_{3},\mathfrak{b}_{4}$,
and $\sigma^{-1}$ is undefined at
$\mathfrak{c}^{+},\mathfrak{c}^{+}_{1},\mathfrak{c}^{+}_{2},\mathfrak{c}^{+}_{3},\mathfrak{c}^{+}_{4}$.
So the composition
$\tilde{\psi}=\sigma^{-1}\circ\tilde{\psi}^{+}\circ\sigma:\widetilde{X}\dashrightarrow\widetilde{X}$
is undefined along the ten curves

$\mathfrak{b},\mathfrak{b}_{1},\mathfrak{b}_{2},\mathfrak{b}_{3},\mathfrak{b}_{4},\mathfrak{q},\mathfrak{l}_{1},\mathfrak{l}_{2},\mathfrak{l}_{3},\mathfrak{l}_{4}.$

Since $\tilde{\varphi}$ is a regular involution exchanging
$\mathfrak{b},\mathfrak{b}_{1},\mathfrak{b}_{2},\mathfrak{b}_{3},\mathfrak{b}_{4}$
and
$\mathfrak{q},\mathfrak{l}_{1},\mathfrak{l}_{2},\mathfrak{l}_{3},\mathfrak{l}_{4}$,
the composition
$\tilde{\varphi}\circ\tilde{\psi}:\widetilde{X}\dashrightarrow\widetilde{X}$
is also a pseudo-automorphism that is undefined at the same ten curves.

###  2.4. Dynamics on fibers

For any divisor, we use the same notation to denote its class in the
(relative) Néron-Severi group.

Let $\pi_{1}:\mathbb{P}^{2}\times E\to\mathbb{P}^{2}$ be the first projection,
and let $\pi:\widetilde{X}\to X$ be the blow-up map. For a hyperplane divisor
$h\in N^{1}(\mathbb{P}^{2})$, we write $H:=\pi^{*}(\pi_{1}^{*}(h))\in
N^{1}(\widetilde{X}/T)$. For any $t\neq q$, we write $h_{t}$ for
$H|_{\widetilde{X}_{t}}$. To distinguish from the exceptional curves in the
fiber over $q$, we denote by $b_{t}:=E_{B}|_{\widetilde{X}_{t}}$,
$c_{t}:=E_{C}|_{\widetilde{X}_{t}}$,
$b_{t,i}:=E_{B_{i}}|_{\widetilde{X}_{t}}$,
$c_{t,i}:=E_{C_{i}}|_{\widetilde{X}_{t}}$, $i=1,\dots,4$, the classes of the
exceptional curves of $\widetilde{X}_{t}$. Together with $h_{t}$, they form a
set of basis elements for $N^{1}(\widetilde{X}_{t})$:

$\\{h_{t},b_{t},b_{t,1},b_{t,2},b_{t,3},b_{t,4},c_{t},c_{t,1},c_{t,2},c_{t,3},c_{t,4}\\}.$

We note that $\tilde{\varphi}_{t}=\tilde{\varphi}|_{\widetilde{X}_{t}}$ and
$\tilde{\psi}_{t}=\tilde{\psi}|_{\widetilde{X}_{t}}$ are the lifts of
$\iota_{q}$ and $\iota_{t}$, respectively, on the blow-up of $\mathbb{P}^{2}$
at the ten indeterminate points. These maps are both regular, and as computed
in [4], we have

######  Proposition 8.

For $t\neq q$, the action of $\tilde{\varphi}_{t}$ on
$N^{1}(\widetilde{X}_{t})$ is given by

$M_{\varphi}:=\left(\begin{array}[]{rrrrrrrrrrr}3&2&1&1&1&1&0&0&0&0&0\\\
-2&-1&-1&-1&-1&-1&0&0&0&0&0\\\ -1&-1&-1&0&0&0&0&0&0&0&0\\\
-1&-1&0&-1&0&0&0&0&0&0&0\\\ -1&-1&0&0&-1&0&0&0&0&0&0\\\
-1&-1&0&0&0&-1&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&0&0&1&0\\\
0&0&0&0&0&0&0&0&0&0&1\end{array}\right),$

and the action of $\tilde{\psi}_{t}$ on $N^{1}(\widetilde{X}_{t})$ is given by

$M_{\psi}:=\left(\begin{array}[]{rrrrrrrrrrr}3&0&0&0&0&0&2&1&1&1&1\\\
0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0&0\\\
0&0&0&0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0&0&0\\\
-2&0&0&0&0&0&-1&-1&-1&-1&-1\\\ -1&0&0&0&0&0&-1&-1&0&0&0\\\
-1&0&0&0&0&0&-1&0&-1&0&0\\\ -1&0&0&0&0&0&-1&0&0&-1&0\\\
-1&0&0&0&0&0&-1&0&0&0&-1\end{array}\right).$

######  Corollary 9.

The product of $M_{\psi}$ and $M_{\varphi}$ has a $3\times 3$ Jordan block
associated with the eigenvalue $\lambda=1$. Hence, the sequence
$\\{\deg((\tilde{\varphi}\circ\tilde{\psi})_{t}^{n})\\}_{n}$ grows
quadratically for $t\neq q$.

On the other hand, the map $(\tilde{\varphi}\circ\tilde{\psi})_{q}$ is the
identity map of the fiber $\widetilde{X}_{q}$. Write
$\mathfrak{h}:=H|_{\widetilde{X}_{q}}$. The difference (Proposition 6)

$(\tilde{\varphi}\circ\tilde{\psi})^{*}(H)|_{\widetilde{X}_{q}}-((\tilde{\varphi}\circ\tilde{\psi})_{q})^{*}(\mathfrak{h})=2\mathfrak{q}+\sum\mathfrak{l}_{i}+4\mathfrak{b}+2\sum\mathfrak{b}_{i}$

is a sum of the flopping curves (indeterminate curves); the indeterminate
curves contribute the difference between

$\deg_{\mathfrak{h}}((\tilde{\varphi}\circ\tilde{\psi})_{q}^{n})\quad\text{and}\quad\deg_{h_{t}}((\tilde{\varphi}\circ\tilde{\psi})_{t}^{n})$

for $t\neq q$. In the rest of the paper, we will show that the dynamical
complexity persists but is concentrated around the indeterminacy locus. Once
we blow-up at the indeterminate curves and then look at the induced maps on
the exceptional divisors, the interesting dynamics will be seen again.

As for the composition of four involutions
$\varphi\circ\chi\circ\varphi\circ\psi$, i.e., the family
$\\{\iota_{q}\circ\iota_{-t}\circ\iota_{q}\circ\iota_{t}:t\in T\\}$, the map
$\iota_{q}\circ\iota_{-t}\circ\iota_{q}\circ\iota_{t}$ can be regularizaed and
has dynamical degree $17+12\sqrt{2}$ for $t\neq q$, while the limiting map is
again the identity map when $t=q$. The fact that dynamical degree drops on the
special fiber in a family of birational maps is consistent with the lower
semicontinuity theorem on dynamical degree by [8]. We will also blow-up at the
indeterminate curves of $\varphi\circ\chi\circ\varphi\circ\psi$ in the fiber
over $q$. The interesting dynamics will be recovered on the exceptional
divisors too.

##  3\. Induced Map on Divisors

In this section we aim to blow-up $\widetilde{X}$ at $\mathfrak{q}$ and obtain
(a map which is conjugate to) the birational self-map
$E_{\mathfrak{q}}\dashrightarrow E_{\mathfrak{q}}$ induced by
$\tilde{\varphi}\circ\tilde{\psi}$.

###  3.1. Some reductions

To look at the exceptional divisor $E_{\mathfrak{q}}$, it suffices to blow-up
$X$ at $\mathfrak{q}=\mathfrak{q}\times\\{q\\}\subset X$ and find the map
induced by $\varphi\circ\psi$ on
$E_{\mathfrak{q}}\subset\operatorname{Bl}_{\mathfrak{q}}{X}$.

Next, we note that the induced map can be broken down into the composition of
maps

${E_{\mathfrak{q}}}$${E_{q}}$${E_{\mathfrak{q}}}$$\scriptstyle{\bar{\psi}^{-1}}$$\scriptstyle{\bar{\varphi}}$

induced by $\varphi$ and $\psi$, where $E_{q}$ is the exceptional divisor of
the blow-up of $X$ at the point $(q,q)\in X$. In fact, by Proposition 7, there
is a birational map from $E_{q}$ to $E_{\mathfrak{b}}$ (the exceptional
divisor over the flopping curve $\mathfrak{b}$) induced by the identity map.

Finally, to see the dynamical property of the induced map
$\bar{\varphi}\circ\bar{\psi}^{-1}$, we will analyze the map
$\Phi:=\bar{\psi}^{-1}\circ\bar{\varphi}:E_{q}\dashrightarrow E_{q}$, which is
birationally conjugate to
$\bar{\varphi}\circ\bar{\psi}^{-1}:E_{\mathfrak{q}}\dashrightarrow
E_{\mathfrak{q}}$ by $\bar{\varphi}$. As $E_{q}$ is isomorphic to
$\mathbb{P}^{2}$ and $E_{\mathfrak{q}}$ is isomorphic to
$\mathfrak{q}\times\mathbb{P}^{1}$, it is simpler to study the map
$\bar{\psi}^{-1}\circ\bar{\varphi}$ than the map
$\bar{\varphi}\circ\bar{\psi}^{-1}$.

The map $\varphi\circ\chi\circ\varphi\circ\psi$ induces a birational self-map
on $E_{\mathfrak{q}}\subset\operatorname{Bl}_{\mathfrak{q}}{X}$ as well, and
it can be broken down into the composition

${E_{\mathfrak{q}}}$${E_{q}}$${E_{\mathfrak{q}}}$${E_{q}}$${E_{\mathfrak{q}}.}$$\scriptstyle{\bar{\psi}^{-1}}$$\scriptstyle{\bar{\varphi}}$$\scriptstyle{\bar{\chi}^{-1}}$$\scriptstyle{\bar{\varphi}}$

Again, we will study the map $\Psi:=\bar{\chi}^{-1}\circ\bar{\varphi}$ which
is birationally conjugate to $\bar{\varphi}\circ\bar{\chi}^{-1}$ by
$\bar{\varphi}$. Therefore, the map $\Psi\circ\Phi$ is birationally conjugate
to $\bar{\varphi}\circ\bar{\chi}^{-1}\circ\bar{\varphi}\circ\bar{\psi}^{-1}$
by $\bar{\varphi}$.

###  3.2. Explicit formulae for cubic involution

In order to obtain simpler formulae, we now fix the base field
$K=\overline{\mathbb{Q}}$, a specific cubic curve $T$ given by

$y^{2}z=x^{3}+5x^{2}z+4xz^{2}$

in $\mathbb{P}^{2}$ and the point $q=[0,0,1]\in T$. We note that this is an
elliptic curve of the form

$y^{2}z=x(x-\alpha_{1}z)(x-\alpha_{2}z)$

with $\alpha_{1}=-1$ and $\alpha_{2}=-4$ in Example 6. One of the affine
pieces of $T$ is depicted in Figure 1.

The cubic involution associated to $T$ and $q$ is
$\iota_{q}:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^{2}$ defined by

$[x,y,z]\mapsto[5x^{3}-xy^{2}+8x^{2}z,5x^{2}y-y^{3}+8xyz,-2x^{3}-5x^{2}z+y^{2}z].$

The involution $\iota_{q}$ contracts the four tangent lines
$\mathfrak{l}_{1},\dots,\mathfrak{l}_{4}$ through $q$ to the points
$q_{1},\dots,q_{4}$ respectively. The conic through $q,q_{1},\dots,q_{4}$ is

$\mathfrak{q}:y^{2}=5x^{2}+8xz,$

and $\iota_{q}$ contracts $\mathfrak{q}$ to $q$.

Now let $\mathbb{P}^{2}\times\mathbb{P}^{2}$ be the product of two projective
planes with coordinates $([x,y,z],[a,b,c])$ and $X=\mathbb{P}^{2}\times T$ the
threefold defined by the equation

$b^{2}c=a^{3}+5a^{2}c+4ac^{2}$

in $\mathbb{P}^{2}\times\mathbb{P}^{2}$. We define the involution
$\varphi:X\dashrightarrow X$ by

$\left([x,y,z],[a,b,c]\right)\mapsto\left(\iota_{q}([x,y,z]),[a,b,c]\right).$

We will use the map $\varphi$ to explain how to find induced maps.

###  3.3. Procedure for finding induced maps

It suffices to make local calculations for finding birational maps. Let
$\mathbb{A}^{2}\times\mathbb{A}^{2}$ be the affine space with coordinates
$((x,y),(a,b))$. One of the affine pieces of the above threefold $X$ is
defined by the equation

$b^{2}=a^{3}+5a^{2}+4a$

in $\mathbb{A}^{2}\times\mathbb{A}^{2}$. We still use $X$ to denote this
affine piece. We also keep the same notation for the affine counterpart of
each of the aforementioned curves and points correspondingly. Moreover, we
keep writing $\varphi$ for the involution of the affine threefold $X$.

#### Blow-up of $X$ at conic

We explicitly define (one of the affine pieces of) the blow-up
$\operatorname{Bl}_{\mathfrak{q}}X$ of $X$ at
$\mathfrak{q}=\mathfrak{q}\times\\{q\\}$ in
$\mathbb{A}^{2}\times\mathbb{A}^{2}\times\mathbb{A}^{2}$ with coordinates
$((u,v),(c,d),(r,s))$, and write the projection map
$\pi:\operatorname{Bl}_{\mathfrak{q}}X\to X$ and inverse of the projection
maps $\pi^{-1}:X\dashrightarrow\operatorname{Bl}_{\mathfrak{q}}X$ in
coordinates. The exceptional divisor over $\mathfrak{q}$ is denoted by
$E_{\mathfrak{q}}$. We note in particular that (one of the affine pieces of)
$E_{\mathfrak{q}}$ is subvariety

$\\{(u,v,c,d,r,s):v^{2}=5u^{2}+8u,c=d=0,s=0\\}\subset(\mathbb{A}^{2})^{3}.$

So there is an isomorphism

$E_{q}\cong\mathfrak{q}\times\mathbb{A}^{1}=\\{((u,v),r):v^{2}=5u^{2}+8u\\}\subset\mathbb{A}^{2}\times\mathbb{A}^{1}.$

#### Blow-up of $X$ at point

Let $\operatorname{Bl}_{q}X$ be the blow-up of $X$ at $(q,q)$. We want to
represent the exceptional divisor $E_{q}$ over $(q,q)$ using curves, which can
be considered as elements in $\mathbb{P}(N_{(q,q)/X})$. We take parametrized
curves through the point $(q,q)=((0,0),(0,0))\in X$ in different tangent
directions. For example, given $(z,w)\in\mathbb{A}^{2}$, the parametrized
curve $\gamma_{(z,w)}:T\to X$ with
$T:=\\{\beta^{2}=\alpha^{3}+5\alpha^{2}+4\alpha\\}\subset\mathbb{A}_{\alpha,\beta}^{2}$
given by

$(\alpha,\beta)\mapsto((\beta z,\beta w),(\alpha,\beta)).$

has tangent vector $((z,w),(0,1))$ at $((0,0),(0,0))$. In this way, the curve
$\gamma_{(z,w)}$ corresponds to a point $p_{(z,w)}\in E_{q}$. Different curves
$\gamma_{(z,w)},\gamma_{(z^{\prime},w^{\prime})}$ have different tangent
vectors at $(q,q)$ and therefore correspond to different points
$p_{(z,w)},p_{(z^{\prime},w^{\prime})}$ in $E_{q}$. Taking all different
points $(z,w)$ of $\mathbb{A}^{2}$, we obtain a family of parametrized curves
$\Gamma:\mathbb{A}^{2}\times T\to X$ given by

$((z,w),(\alpha,\beta))\mapsto((\beta z,\beta w),(\alpha,\beta)),$

which correspond to a dense open subset of $E_{q}$. We identify the
$\mathbb{A}^{2}$ with (a dense open subset of) $E_{q}$, and now $\Gamma$ has
parameter domain $E_{q}\times T$.

#### Induced map

The composition

$E_{q}\times
T\overset{\Gamma}{\longrightarrow}X\overset{\varphi}{\longrightarrow}X\overset{\pi^{-1}}{\longrightarrow}\operatorname{Bl}_{\mathfrak{q}}X$

is a family (parametrized by $E_{q}$) of parameterizations (all parameterized
by $T$) of curves, each of them passing through a point on the exceptional
divisor $E_{\mathfrak{q}}$.

Regard $E_{\mathfrak{q}}$ as $\mathfrak{q}\times\mathbb{A}^{1}$ with
coordinates $((u,v),r)$. The correspondence between a pair $(z,w)$ and a point
on $E_{\mathfrak{q}}$, hence the induced map
$\bar{\varphi}:E_{q}\dashrightarrow E_{\mathfrak{q}}$, is given by

$\displaystyle(z,w)\mapsto$
$\displaystyle\left(\left(\frac{8z^{2}}{-5z^{2}+w^{2}},\frac{8zw}{-5z^{2}+w^{2}}\right),\frac{25z^{4}-10w^{2}z^{2}+w^{4}}{72z^{5}-80w^{2}z^{3}+8w^{4}z}\right).$

The inverse $\bar{\varphi}^{-1}:E_{\mathfrak{q}}\to E_{q}$ is given by

$\left((u,v),r\right)\mapsto\left(\frac{(25u^{4}-10u^{2}v^{2}+v^{4})u}{8u(9u^{4}-10u^{2}v^{2}+v^{4})r},\frac{(25u^{4}-10u^{2}v^{2}+v^{4})v}{8u(9u^{4}-10u^{2}v^{2}+v^{4})r}\right).$

###  3.4. Projectivized induced maps

Given the threefold $X=\mathbb{P}^{2}\times T$ defined by
$b^{2}c=a^{3}+5a^{2}c+4ac^{2}$ in $\mathbb{P}^{2}\times\mathbb{P}^{2}$, we
have the involution $\varphi:X\dashrightarrow X$ by

$\left([x,y,z],[a,b,c]\right)\mapsto\left([5x^{3}-xy^{2}+8x^{2}z,5x^{2}y-y^{3}+8xyz,-2x^{3}-5x^{2}z+y^{2}z],[a,b,c]\right).$

We can also explicitly spell out the defining equations of
$\psi:X\dashrightarrow X$:

$\left([x,y,z],[a,b,c]\right)\mapsto\left([P_{1}(x,y,z;a,b,c),P_{2}(x,y,z;a,b,c),P_{3}(x,y,z;a,b,c)],[a,b,c]\right),$

where

$\displaystyle P_{1}$
$\displaystyle=(x^{3}+2y^{2}z-4xz^{2})a+(-2xyz)b+(5x^{3}-xy^{2}+8x^{2}z)c,$
$\displaystyle P_{2}$
$\displaystyle=(3x^{2}y+10xyz+4yz^{2})a+(-2x^{3}-10x^{2}z-8xz^{2})b+(5x^{2}y-y^{3}+8xyz)c,$
$\displaystyle P_{3}$
$\displaystyle=(3x^{2}z+10xz^{2}+4z^{3})a+(-2yz^{2})b+(-2x^{3}-5x^{2}z+y^{2}z)c.$

For the elliptic curve we have chosen, the inverse of the point $[a,b,c]$ has
coordinates $[a,-b,c]$. Therefore, the involution $\chi:X\dashrightarrow X$ is
defined by

$\left([x,y,z],[a,b,c]\right)\mapsto\left([P_{1}(x,y,z;a,-b,c),P_{2}(x,y,z;a,-b,c),P_{3}(x,y,z;a,-b,c)],[a,b,c]\right).$

The process in Section LABEL:subsec:process above yields three induced maps
$\bar{\varphi},\bar{\psi},\bar{\chi}:E_{q}\dashrightarrow E_{\mathfrak{q}}$
with inverses
$\bar{\varphi}^{-1},\bar{\psi}^{-1},\bar{\chi}^{-1}:E_{\mathfrak{q}}\dashrightarrow
E_{q}$. To analyze the dynamical property of these maps, we projectivize the
exceptional divisors and homogenize the defining equations. Now we have

$E_{q}\cong\mathbb{P}^{2}\quad\text{and}\quad
E_{\mathfrak{q}}\cong\mathfrak{q}\times\mathbb{P}^{1},$

and, for example, $\overline{\varphi}:E_{q}\dashrightarrow E_{\mathfrak{q}}$
is given by

$[x,y,z]\mapsto\left([8x^{2},8xy,(-5x^{2}+y^{2})],[(5x^{2}-y^{2})^{2}z,8x(x-y)(x+y)(3x-y)(3x+y)]\right).$

##  4\. Dynamics of Induced Maps

In the following discussion, if no confusion arises, we always use the same
notation for a point and the corresponding point on any blow-up; and we always
use the same notation for a subvariety and all of its strict transforms.

###  4.1. The map induced by $\varphi\circ\psi$

Recall that the map that is conjugate to the map induced by $\varphi\circ\psi$
is
$\Phi:=\bar{\psi}^{-1}\circ\bar{\varphi}:E_{q}\cong\mathbb{P}^{2}\dashrightarrow\mathbb{P}^{2}\cong
E_{q}$, and it is defined by

$\displaystyle[x,y,z]\mapsto\begin{bmatrix}9x^{5}-10x^{3}y^{2}+xy^{4}+10x^{3}yz-2xy^{3}z,\\\
9x^{4}y-10x^{2}y^{3}+y^{5}+9x^{4}z-y^{4}z,\\\
9x^{4}z-10x^{2}y^{2}z+y^{4}z\end{bmatrix}^{T}.$

The inverse of $\Phi$ is
$\Phi^{-1}:=\bar{\varphi}^{-1}\circ\bar{\psi}:E_{q}\dashrightarrow E_{q}$. We
will show that the sequence $\\{\deg(\Phi^{n})\\}$ grows quadratically.

A direct computation shows that the indeterminate points of $\Phi$ are

$p:=[0,0,1],p_{1}:=[1,1,0],p_{2}:=[1,-1,0],p_{3}:=[1,3,0],p_{3}:=[1,-3,0].$

The indeterminate points of $\Phi^{-1}$ are

$p^{\prime}:=[0,1,1],p_{1}:=[1,1,0],p_{2}:=[1,-1,0],p_{3}:=[1,3,0],p_{4}:=[1,-3,0].$

We write $P:=\\{p,p_{1},p_{2},p_{3},p_{4},p^{\prime}\\}$.

Let $\operatorname{Bl}_{P}E_{q}$ be the blow-up of $E_{q}$ at $P$ with
exceptional divisors $e_{p}$, $e_{p_{1}}$, $e_{p_{2}}$, $e_{p_{3}}$,
$e_{p_{4}}$, $e_{p^{\prime}}$. Both the lift of $\Phi$ and the lift of
$\Phi^{-1}$ on $\operatorname{Bl}_{P}E_{q}$ will still have four indeterminate
points $p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime},p_{4}^{\prime}$, one on
each of the exceptional divisors $e_{p_{1}},e_{p_{2}},e_{p_{3}},e_{p_{4}}$. We
write
$P^{\prime}:=\\{p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime},p_{4}^{\prime}\\}$.
In fact, the pullback of a general line in $E_{q}\cong\mathbb{P}^{2}$ under
$\Phi$ passes through the points $p_{1},p_{2},p_{3},p_{4}$ with tangent lines

$\displaystyle l_{1}:2x-2y+z$ $\displaystyle=0,$ $\displaystyle l_{2}:2x+2y-z$
$\displaystyle=0,$ $\displaystyle l_{3}:6x-2y+z$ $\displaystyle=0,$
$\displaystyle l_{4}:6x+2y-z$ $\displaystyle=0$

at the points respectively. The same claim holds for $\Phi^{-1}$.

In fact, the map $\Phi$ preserves a fibration on $E_{q}\cong\mathbb{P}^{2}$ of
singular quartic curves with two nodal points at $p,p^{\prime}$ and with
tangent lines $l_{1},\dots,l_{4}$ at $p_{1},\dots,p_{4}$, respectively. Now we
blow-up $\operatorname{Bl}_{P}E_{q}$ at $P^{\prime}$ to get
$\operatorname{Bl}_{P^{\prime}}\operatorname{Bl}_{P}E_{q}$ with exceptional
divisors
$e_{p_{1}^{\prime}},e_{p_{2}^{\prime}},e_{p_{3}^{\prime}},e_{p_{4}^{\prime}}$
at the corresponding points. Both $\Phi$ and $\Phi^{-1}$ lift to an
automorphism on $\operatorname{Bl}_{P^{\prime}}\operatorname{Bl}_{P}E_{q}$.

###  4.2. Quadratic degree growth of $\Phi$

Given the sequence of blow-ups

$\operatorname{Bl}_{P^{\prime}}\operatorname{Bl}_{P}E_{q}\to\operatorname{Bl}_{P}E_{q}\to
E_{q},$

we write $\hat{\Phi}$ for the lift of $\Phi$ on $\operatorname{Bl}_{P}E_{q}$,
and write $\tilde{\Phi}$ for the lift of $\Phi$ on
$\operatorname{Bl}_{P^{\prime}}\operatorname{Bl}_{P}E_{q}$. To see the degree
growth of $\Phi$, we examine the action of $\tilde{\Phi}$ on
$N^{1}\left(\operatorname{Bl}_{P^{\prime}}\operatorname{Bl}_{P}E_{q}\right)$.
Let $h$ be a hyperplane class. A set of (ordered) basis elements for
$N^{1}(\operatorname{Bl}_{P^{\prime}}\operatorname{Bl}_{P}E_{q})$ is

$\left\\{h,e_{p},e_{p_{1}},e_{p_{2}},e_{p_{3}},e_{p_{4}},e_{p^{\prime}},e_{p_{1}^{\prime}},e_{p_{2}^{\prime}},e_{p_{3}^{\prime}},e_{p_{4}^{\prime}}\right\\}.$

######  Proposition 10.

The action of $\tilde{\Phi}$ on
$N^{1}\left(\operatorname{Bl}_{P^{\prime}}\operatorname{Bl}_{P}E_{q}\right)$
is given by

$M:=\left(\begin{array}[]{rrrrrrrrrrr}5&4&0&0&0&0&0&1&1&1&1\\\
0&0&0&0&0&0&1&0&0&0&0\\\ -1&-1&1&0&0&0&0&-1&0&0&0\\\
-1&-1&0&1&0&0&0&0&-1&0&0\\\ -1&-1&0&0&1&0&0&0&0&-1&0\\\
-1&-1&0&0&0&1&0&0&0&0&-1\\\ -4&-3&0&0&0&0&0&-1&-1&-1&-1\\\
-2&-2&0&0&0&0&0&-1&0&0&0\\\ -2&-2&0&0&0&0&0&0&-1&0&0\\\
-2&-2&0&0&0&0&0&0&0&-1&0\\\ -2&-2&0&0&0&0&0&0&0&0&-1\end{array}\right).$

###### Proof.

The image of a general hyperplane under $\Phi$ is a degree-$5$ curve passing
through $p^{\prime}=[0,1,1]$ with multiplicity $4$ and
$p_{1},p_{2},p_{3},p_{4}$ with tangent lines $l_{1},l_{2},l_{3},l_{4}$
respectively. On $\operatorname{Bl}_{P}E_{q}$, we have

$\hat{\Phi}_{*}(h)=5h-4e_{p^{\prime}}-\sum e_{p_{i}}.$

On $\operatorname{Bl}_{P^{\prime}}\operatorname{Bl}_{P}E_{q}$, we have

$\displaystyle\tilde{\Phi}_{*}(h)$
$\displaystyle=5h-4e_{p^{\prime}}-\sum(e_{p_{i}}+e_{p_{i}^{\prime}})-\sum
e_{p_{i}^{\prime}}$ $\displaystyle=5h-4e_{p^{\prime}}-\sum e_{p_{i}}-2\sum
e_{p_{i}^{\prime}}$

$\Phi^{-1}$ contracts a degree-$4$ curve to $p=[0,0,1]$; moreover, the curve
passes through the point $p^{\prime}=[0,1,1]$ with multiplicity $3$ and
through $p_{1},p_{2},p_{3},p_{4}$ with tangent lines $l_{1},l_{2},l_{3},l_{4}$
respectively. We have

$\hat{\Phi}_{*}(e_{p})=4h-3e_{p^{\prime}}-\sum e_{p_{i}},$

and so

$\displaystyle\tilde{\Phi}_{*}(e_{p})$
$\displaystyle=4h-3e_{p^{\prime}}-\sum(e_{p_{i}}+e_{p_{i}^{\prime}})-\sum
e_{p_{i}^{\prime}}$ $\displaystyle=4h-3e_{p^{\prime}}-\sum e_{p_{i}}-2\sum
e_{p_{i}^{\prime}}.$

We note that $\Phi(0,1,1)=[0,0,1]$, and it can be verified that

$\tilde{\Phi}_{*}(e_{p^{\prime}})=e_{p}.$

Next, we observe that the image of a general line passing through
$p_{1}=[1,1,0]$ is a degree-$4$ curve passing through the point
$p^{\prime}=[0,1,1]$ with multiplicity $3$. The image also passes through the
point $p_{1}$ with tangent line not equal to $l_{1}:2x-2y+z=0$ and through
$p_{2},p_{3},p_{4}$ with tangent lines $l_{2},l_{3},l_{4}$ respectively. Thus,

$\hat{\Phi}_{*}(h-e_{p_{1}})=4h-3e_{p^{\prime}}-\sum e_{p_{i}},$

and hence

(4.1)
$\tilde{\Phi}_{*}(h-e_{p_{1}}-e_{p_{1}^{\prime}})=4h-3e_{p^{\prime}}-\sum
e_{p_{i}}-2\sum e_{p_{i}^{\prime}}+e_{p_{1}^{\prime}}.$

In particular, the image of the line $l_{1}:2x-2y+z=0$ is a degree- $3$ curve
passing through $p^{\prime}=[0,1,1]$ with multiplicity $2$ and
$p_{2},p_{3},p_{4}$ with tangent lines $l_{2},l_{3},l_{4}$ respectively. So

(4.2)
$\displaystyle\begin{split}\tilde{\Phi}_{*}(h-e_{p_{1}}-2e_{p_{1}^{\prime}})=3h-2e_{p^{\prime}}-\sum
e_{p_{i}}-2\sum e_{p_{i}^{\prime}}+e_{p_{1}}+2e_{p_{1}^{\prime}}.\end{split}$

Combining (4.1) and (4.2) we get

(4.3)
$\tilde{\Phi}_{*}(e_{p_{1}^{\prime}})=h-e_{p^{\prime}}-e_{p_{1}}-e_{p_{1}^{\prime}}.$

Similarly,

$\tilde{\Phi}_{*}(e_{p_{i}^{\prime}})=h-e_{p^{\prime}}-e_{p_{i}}-e_{p_{i}^{\prime}},$

for $i=2,3,4$.

Finally, substituting (4.3) back into (4.1) and rearranging terms, we get

$\displaystyle\tilde{\Phi}_{*}(e_{p_{1}})$
$\displaystyle=\tilde{\Phi}_{*}(h)-\tilde{\Phi}_{*}(e_{p})-\tilde{\Phi}_{*}(e_{p_{1}^{\prime}})-e_{p_{1}^{\prime}}$
$\displaystyle=h-e_{p^{\prime}}-e_{p_{1}^{\prime}}-(h-e_{p^{\prime}}-e_{p_{1}}-e_{p_{1}^{\prime}})$
$\displaystyle=e_{p_{1}}.$

Similarly, we calculate that

$\tilde{\Phi}_{*}(e_{p_{i}})=e_{p_{i}},$

for $i=2,3,4$. ∎

######  Corollary 11.

$M$ has a $3\times 3$ Jordan block associated with the eigenvalue $1$. Hence
the sequence $\\{\deg(\Phi^{n})\\}_{n}$ grows quadratically.

Hence, the map $E_{\mathfrak{q}}\dashrightarrow E_{\mathfrak{q}}$ induced by
$\varphi\circ\psi$ on the exceptional divisor over the curve $\mathfrak{q}$ in
the fiber over $q$ has a quadratic degree growth. Recall that the map
$(\varphi\circ\psi)_{t}$ on a general fiber has a quadratic degree growth, but
the map $(\varphi\circ\psi)_{q}$ is the identity map. In this way, we found
the interesting dynamics on the exceptional divisor $E_{\mathfrak{q}}$.

Moreover, we recall that that the map
$(\varphi\circ\chi\circ\varphi\circ\psi)_{t}$ on a general fiber has dynamical
degree $>1$. We expect the induced map to have $\lambda>1$ as well.

###  4.3. The map induced by $\varphi\circ\chi\circ\varphi\circ\psi$

Using the maps $\bar{\chi}$ and $\bar{\varphi}$ (and their inverses), we
obtain another map
$\Psi:=\bar{\chi}^{-1}\circ\bar{\varphi}:E_{q}\dashrightarrow E_{q}$ with the
inverse map
$\Psi^{-1}:=\bar{\varphi}^{-1}\circ\bar{\chi}^{-1}:E_{q}\dashrightarrow
E_{q}$. $\Psi\circ\Phi$ is conjugate to the map induced by
$\varphi\circ\chi\circ\varphi\circ\psi$. We will see that
$\lambda(\Psi\circ\Phi)=16$, but we need to understand the map $\Psi$ at
first.

The map $\Psi$ behaves almost the same as $\Phi$. The indeterminate points of
$\Psi$ are

$p=[0,0,1],p_{1}=[1,1,0],p_{2}=[1,-1,0],p_{3}=[1,3,0],p_{4}=[1,-3,0].$

Moreover, the indeterminate points of $\Psi^{-1}$ are

$p^{\prime\prime}:=[0,-1,1],p_{1}=[1,1,0],p_{2}=[1,-1,0],p_{3}=[1,3,0],p_{4}=[1,-3,0].$

The pullback of a general line in $E_{q}$ under both $\Psi$ passes through the
points $p_{1},p_{2},p_{3},p_{4}$ with tangent lines

$\displaystyle 2x-2y-z$ $\displaystyle=0$ $\displaystyle 2x+2y+z$
$\displaystyle=0$ $\displaystyle 6x-2y-z$ $\displaystyle=0$ $\displaystyle
6x+2y+z$ $\displaystyle=0$

at $p_{1},p_{2},p_{3},p_{4}$, respectively. The same claim holds for
$\Psi^{-1}$. Therefore, if we blow-up $E_{q}$ at
$Q:=\\{p,p^{\prime\prime},p_{1},p_{2},p_{3},p_{4}\\}$, then both the lifts
$\hat{\Psi}$ and $\hat{\Psi}^{-1}$ on $\operatorname{Bl}_{Q}E_{q}$ will have
an indeterminate point $p_{i}^{\prime\prime}$ at the exceptional divisors
$e_{p_{i}}$ for each $i=1,\dots,4$. If we blow-up again at
$Q^{\prime}:=\\{p_{1}^{\prime\prime},p_{2}^{\prime\prime},p_{3}^{\prime\prime},p_{4}^{\prime\prime}\\}$,
then $\hat{\Psi}$ and $\hat{\Psi}^{-1}$ will be regularized on
$\operatorname{Bl}_{Q^{\prime}}\operatorname{Bl}_{Q}E_{q}$ and lift to
$\tilde{\Psi}$ and $\tilde{\Psi}^{-1}$. The action of $\tilde{\Psi}$ on
$N^{1}(\operatorname{Bl}_{Q^{\prime}}\operatorname{Bl}_{Q}E_{q})$ can be
represented by the same matrix (with respect to a different basis, of course).

Now we are ready to compute the dynamical degree of $\Psi\circ\Phi$. Let us
blow-up $E_{q}$ at

$R:=\\{p,p^{\prime},p^{\prime\prime},p_{1},p_{2},p_{3},p_{4}\\}$

to get $\operatorname{Bl}_{R}\mathbb{P}^{2}$ and then at

$R^{\prime}:=\\{p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime},p_{4}^{\prime},p_{1}^{\prime\prime},p_{2}^{\prime\prime},p_{3}^{\prime\prime},p_{4}^{\prime\prime}\\}$

to get $\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q}$. Then
$N^{1}\left(\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q}\right)$
has rank $16$ with an ordered basis

$\left\\{h,e_{p},e_{p^{\prime}},e_{p^{\prime\prime}},e_{p_{1}},e_{p_{2}},e_{p_{3}},e_{p_{4}},e_{p_{1}^{\prime}},e_{p_{2}^{\prime}},e_{p_{3}^{\prime}},e_{p_{4}^{\prime}},e_{p_{1}^{\prime\prime}},e_{p_{2}^{\prime\prime}},e_{p_{3}^{\prime\prime}},e_{p_{4}^{\prime\prime}}\right\\}.$

Let $\widetilde{\Phi}$ and $\widetilde{\Psi}$ denote the lifts of
$\tilde{\Phi}$ and $\tilde{\Psi}$ on
$\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q}$. The maps
$\tilde{\Phi}$ (resp. $\tilde{\Psi}$) is an isomorphism (which contracts no
curves) on $\operatorname{Bl}_{P^{\prime}}\operatorname{Bl}_{P}E_{q}$ (resp.
$\operatorname{Bl}_{Q^{\prime}}\operatorname{Bl}_{Q}E_{q}$). Nevertheless,
some curves on $\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q}$ will
be contracted by $\widetilde{\Phi}$ and $\widetilde{\Psi}$. The map
$\widetilde{\Phi}$ contracts the curves
$e_{p^{\prime\prime}},e_{p_{1}^{\prime\prime}},e_{p_{2}^{\prime\prime}},e_{p_{3}^{\prime\prime}},e_{p_{4}^{\prime\prime}}$,
and the map $\widetilde{\Psi}$ contracts the curves
$e_{p^{\prime}},e_{p_{1}^{\prime}},e_{p_{2}^{\prime}},e_{p_{3}^{\prime}},e_{p_{4}^{\prime}}$.

######  Proposition 12.

For any curve $C$ contracted by $\widetilde{\Phi}$, we have
$\widetilde{\Phi}(C)\not\in\operatorname{Ind}(\widetilde{\Psi})$. We conclude
that

$(\widetilde{\Psi}\circ\widetilde{\Phi})_{*}=\widetilde{\Psi}_{*}\circ\widetilde{\Phi}_{*}.$

###### Proof.

Since $\Phi(p^{\prime\prime})=\Phi(0,-1,1)=[0,-2,1]=:p_{-2}$,
$\widetilde{\Phi}(e_{p^{\prime\prime}})=p_{-2}$ on
$\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q}$. Moreover,
$\widetilde{\Psi}$ is defined at $p_{-2}$, as $\Psi$ is defined at $[0,-2,1]$.

What happens at $e_{p_{i}^{\prime\prime}}$, for $i=1,\dots,4$, is similar, and
we present only the local computations around $p_{1}$. The blow-up
$\operatorname{Bl}_{p_{1}}E_{q}$ of $E_{q}$ at $p_{1}=[1,1,0]$ can be defined
in $E_{q}\times\mathbb{P}^{1}\cong\mathbb{P}^{2}\times\mathbb{P}^{1}$ (with
coordinates $([x,y,z],[s,t])$) by the equation

$(x-y)s=zt$

with the exceptional divisor

$e_{p_{1}}=\left\\{([1,1,0],[s,t])\right\\}\subset\operatorname{Bl}_{p_{1}}E_{q}.$

In terms of the coordinates on $e_{p_{1}}$,
$p^{\prime\prime}_{1}=([1,1,0],[2,1])$. Let $\Phi_{1}$ and $\Psi_{1}$ be the
lifts of $\Phi$ and $\Psi$ on $\operatorname{Bl}_{p_{1}}E_{q}$. The
restriction $\Phi_{1}|_{e_{p_{1}}}:e_{p_{1}}\to e_{p_{1}}$ is given by

$([1,1,0],[s,t])\mapsto([1,1,0],[2s,2t-s]).$

So $\Phi_{1}([1,1,0],[2,1])=([1,1,0],[1,0])$. This implies that
$\widetilde{\Phi}$ contracts $e_{p_{1}^{\prime\prime}}$ to the point
corresponding to $([1,1,0],[1,0])$. Moreover, $\Psi_{1}$ is defined at
$([1,1,0],[1,0])$, so $\widetilde{\Psi}$ is defined at the point corresponding
to $([1,1,0],[1,0])$. ∎

Consequently, $\widetilde{\Psi}\circ\widetilde{\Phi}$ contracts the curves

$e_{p^{\prime\prime}},e_{p_{1}^{\prime\prime}},e_{p_{2}^{\prime\prime}},e_{p_{3}^{\prime\prime}},e_{p_{4}^{\prime\prime}},(\widetilde{\Phi}^{-1})(e_{p^{\prime}}),(\widetilde{\Phi}^{-1})(e_{p_{1}^{\prime}}),(\widetilde{\Phi}^{-1})(e_{p_{2}^{\prime}}),(\widetilde{\Phi}^{-1})(e_{p_{3}^{\prime}}),(\widetilde{\Phi}^{-1})(e_{p_{4}^{\prime}}).$

on $\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q}$, wher
$(\widetilde{\Phi}^{-1})(C)$ means the proper transform of a curve $C$ under
the inverse of $\widetilde{\Phi}$.

###  4.4. Algebraic stability of $\widetilde{\Psi}\circ\widetilde{\Phi}$

We prove the algebraic stability of $\widetilde{\Psi}\circ\widetilde{\Phi}$ on
$\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q}$.

######  Proposition 13.

The map $\widetilde{\Psi}\circ\widetilde{\Phi}$ is algebraically stable on
$\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q}$.

###### Proof.

By Theorem 2, it suffices to show that for any curve $C$ contracted by
$\widetilde{\Psi}\circ\widetilde{\Phi}$,
$(\widetilde{\Psi}\circ\widetilde{\Phi})(C)$ is a fixed point of
$\widetilde{\Psi}\circ\widetilde{\Phi}$. We first deal with the curves
$e_{p^{\prime\prime}},e_{p_{1}^{\prime\prime}},e_{p_{2}^{\prime\prime}},e_{p_{3}^{\prime\prime}},e_{p_{4}^{\prime\prime}}$.

For $e_{p^{\prime\prime}}$, we note that
$(\Psi\circ\Phi)(p^{\prime\prime})=p^{\prime\prime}$. So
$(\widetilde{\Psi}\circ\widetilde{\Phi})(e_{p^{\prime\prime}})\subset
e_{p^{\prime\prime}}$. We can check that there is a curve (other than
$e_{p^{\prime\prime}}$) whose image under
$\widetilde{\Psi}\circ\widetilde{\Phi}$ is $e_{p^{\prime\prime}}$. So we
cannot have
$(\widetilde{\Psi}\circ\widetilde{\Phi})(e_{p^{\prime\prime}})=e_{p^{\prime\prime}}$
as $\widetilde{\Psi}\circ\widetilde{\Phi}$ is birational. Thus
$(\widetilde{\Psi}\circ\widetilde{\Phi})(e_{p^{\prime\prime}})$ is a point on
$e_{p^{\prime\prime}}$.

To see the points $p_{i}^{\prime\prime}$, we need to blow-up $\mathbb{P}^{2}$
at $p_{i}$ for $i=1,\dots,4$. Let $\widehat{\Phi}$ and $\widehat{\Psi}$ (resp.
$\widehat{\Phi}^{-1}$ and $\widehat{\Psi}^{-1}$) be the lifts of $\Psi$ and
$\Phi$ (resp. $\widehat{\Phi}^{-1}$ and $\widehat{\Psi}^{-1}$) on
$\operatorname{Bl}_{R}E_{q}$. We first claim that
$(\widehat{\Psi}\circ\widehat{\Phi})(p_{i}^{\prime\prime})=p_{i}^{\prime\prime}$.
Again, we exhibit the calculation around $p_{1}$ and omit the similar local
calculations at $p_{2},p_{3},p_{4}$. Recall that in the proof of Proposition
12, we have $p_{1}^{\prime}=([1,1,0],[-2,1])$ in terms of the coordinates on
$e_{p_{1}}$ and the restriction $\Phi_{1}|_{e_{p_{1}}}:e_{p_{1}}\to e_{p_{1}}$
is given by

$([1,1,0],[s,t])\mapsto([1,1,0],[2s,2t-s]).$

Moreover, we have $p_{1}^{\prime\prime}=([1,1,0],[2,1])$ and the restriction
$\Psi_{1}|_{e_{p_{1}}}:e_{p_{1}}\to e_{p_{1}}$ is given by

$([1,1,0],[s,t])\mapsto([1,1,0],[2s,2t+s]).$

Therefore, $\Psi_{1}\circ\Phi_{1}$ fixes each point on $e_{p_{1}}$ as long as
it is defined at the point. In particular,
$(\Psi_{1}\circ\Phi_{1})(p^{\prime\prime}_{1})=p^{\prime\prime}_{1}$.

Now, for each $i=1,\dots,4$, we have
$(\widetilde{\Psi}\circ\widetilde{\Phi})(e_{p_{i}^{\prime\prime}})\subset
e_{p_{i}^{\prime\prime}}$. There is a curve (other than
$e_{p_{i}^{\prime\prime}}$) whose image under
$(\widetilde{\Psi}\circ\widetilde{\Phi})$ is $e_{p_{i}^{\prime\prime}}$. So
$(\widetilde{\Psi}\circ\widetilde{\Phi})(e_{p_{i}^{\prime\prime}})$ is a point
on $e_{p_{i}^{\prime\prime}}$.

We next consider the curves
$(\widetilde{\Phi}^{-1})(e_{p^{\prime}}),(\widetilde{\Phi}^{-1})(e_{p_{1}^{\prime}}),(\widetilde{\Phi}^{-1})(e_{p_{2}^{\prime}}),(\widetilde{\Phi}^{-1})(e_{p_{3}^{\prime}}),(\widetilde{\Phi}^{-1})(e_{p_{4}^{\prime}})$.
We note that $\widetilde{\Psi}\circ\widetilde{\Phi}$ fixes the strict
transform of the line $\\{x=0\\}$ and exceptional divisors
$e_{p_{1}^{\prime}},e_{p_{2}^{\prime}},e_{p_{3}^{\prime}},e_{p_{4}^{\prime}}$
pointwise (wherever $\widetilde{\Psi}\circ\widetilde{\Phi}$ is defined).
Moreover, $\widetilde{\Psi}\circ\widetilde{\Phi}$ contracts
$(\widetilde{\Phi}^{-1})(e_{p^{\prime}})$ to a point on the strict transform
of $\\{x=0\\}$ and $(\widetilde{\Phi}^{-1})(e_{p_{i}^{\prime}})$ to a point on
$e_{p_{i}^{\prime}}$ for each $i=1,\dots,4$. ∎

###  4.5. Dynamical degree of $\Psi\circ\Phi$

The blow-up $\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q}$ is a
birational model on which $\widetilde{\Psi}\circ\widetilde{\Phi}$ is
algebraically stable. In order to see the action of
$\widetilde{\Psi}\circ\widetilde{\Phi}$ on
$N^{1}(\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q})$, we compute
the matrices that represent the action of $\widetilde{\Phi}$ and
$\widetilde{\Psi}$ and then take their product.

######  Proposition 14.

The action of $\widetilde{\Phi}$ on
$N^{1}(\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q})$ is given by

$M_{\Phi}:=\left(\begin{array}[]{rrrrrrrrrrrrrrrr}5&4&0&0&0&0&0&1&1&1&1&0&0&0&0&0\\\
0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0\\\ -1&-1&1&0&0&0&0&-1&0&0&0&0&0&0&0&0\\\
-1&-1&0&1&0&0&0&0&-1&0&0&0&0&0&0&0\\\ -1&-1&0&0&1&0&0&0&0&-1&0&0&0&0&0&0\\\
-1&-1&0&0&0&1&0&0&0&0&-1&0&0&0&0&0\\\ -4&-3&0&0&0&0&0&-1&-1&-1&-1&0&0&0&0&0\\\
-2&-2&0&0&0&0&0&-1&0&0&0&0&0&0&0&0\\\ -2&-2&0&0&0&0&0&0&-1&0&0&0&0&0&0&0\\\
-2&-2&0&0&0&0&0&0&0&-1&0&0&0&0&0&0\\\ -2&-2&0&0&0&0&0&0&0&0&-1&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ -1&-1&1&0&0&0&0&-1&0&0&0&0&0&0&0&0\\\
-1&-1&0&1&0&0&0&0&-1&0&0&0&0&0&0&0\\\ -1&-1&0&0&1&0&0&0&0&-1&0&0&0&0&0&0\\\
-1&-1&0&0&0&1&0&0&0&0&-1&0&0&0&0&0\end{array}\right),$

and the action of $\widetilde{\Psi}$ on
$N^{1}(\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q})$ is given by

$M_{\Psi}:=\left(\begin{array}[]{rrrrrrrrrrrrrrrr}5&4&0&0&0&0&0&0&0&0&0&0&1&1&1&1\\\
0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0\\\ -1&-1&1&0&0&0&0&0&0&0&0&0&-1&0&0&0\\\
-1&-1&0&1&0&0&0&0&0&0&0&0&0&-1&0&0\\\ -1&-1&0&0&1&0&0&0&0&0&0&0&0&0&-1&0\\\
-1&-1&0&0&0&1&0&0&0&0&0&0&0&0&0&-1\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
-1&-1&1&0&0&0&0&0&0&0&0&0&-1&0&0&0\\\ -1&-1&0&1&0&0&0&0&0&0&0&0&0&-1&0&0\\\
-1&-1&0&0&1&0&0&0&0&0&0&0&0&0&-1&0\\\ -1&-1&0&0&0&1&0&0&0&0&0&0&0&0&0&-1\\\
-4&-3&0&0&0&0&0&0&0&0&0&0&-1&-1&-1&-1\\\ -2&-2&0&0&0&0&0&0&0&0&0&0&-1&0&0&0\\\
-2&-2&0&0&0&0&0&0&0&0&0&0&0&-1&0&0\\\ -2&-2&0&0&0&0&0&0&0&0&0&0&0&0&-1&0\\\
-2&-2&0&0&0&0&0&0&0&0&0&0&0&0&0&-1\end{array}\right).$

###### Proof.

We will show the action of $\widetilde{\Phi}$ is given by $M_{\Phi}$, and the
proof for $\widetilde{\Psi}$ is similar.

First, we note that

$\widetilde{\Phi}_{*}(e_{p^{\prime\prime}})=\widetilde{\Phi}_{*}(e_{p_{i}^{\prime\prime}})=0$

for $i=1,\dots,4$.

Continuing the arguments in the proof of Proposition 10, we have

$\displaystyle\widetilde{\Phi}_{*}(h)$ $\displaystyle=5h-4e_{p^{\prime}}-\sum
e_{p_{i}}-2\sum e_{p_{i}^{\prime}}-\sum e_{p_{i}^{\prime\prime}}$
$\displaystyle\widetilde{\Phi}_{*}(e_{p})$
$\displaystyle=4h-3e_{p^{\prime}}-\sum e_{p_{i}}-2\sum e_{p_{i}^{\prime}}-\sum
e_{p_{i}^{\prime\prime}}$ $\displaystyle\widetilde{\Phi}_{*}(e_{p^{\prime}})$
$\displaystyle=e_{p}$
$\displaystyle\widetilde{\Phi}_{*}(e_{p_{i}}+e_{p_{i}^{\prime\prime}})$
$\displaystyle=e_{p_{i}}+e_{p_{i}^{\prime\prime}}$
$\displaystyle\widetilde{\Phi}_{*}(e_{p_{i}^{\prime}})$
$\displaystyle=h-e_{p^{\prime}}-e_{p_{i}}-e_{p_{i}^{\prime}}-e_{p_{i}^{\prime\prime}}.$

Also

$\widetilde{\Phi}_{*}(e_{p_{i}})=e_{p_{i}}+e_{p_{i}^{\prime\prime}}-\widetilde{\Phi}_{*}(e_{p_{i}^{\prime\prime}})=e_{p_{i}}+e_{p_{i}^{\prime\prime}}.$

∎

Proposition 12 suggests that we can use the product of $M_{\Psi}$ and
$M_{\Phi}$ to represent the action of $\widetilde{\Psi}\circ\widetilde{\Phi}$
on $N^{1}(\operatorname{Bl}_{R^{\prime}}\operatorname{Bl}_{R}E_{q})$. The
spectral radius of $M_{\Psi}M_{\Phi}$ is $16$. By Theorem 3, we obtain the
following result:

######  Proposition 15.

The induced map $\Psi\circ\Phi$ on the exceptional divisor by
$\chi\circ\varphi\circ\psi\circ\varphi$ has $\lambda(\Psi\circ\Phi)=16$. In
particular, $\Psi\circ\Phi$ is not regularizable.

Thus, the conjugate map $E_{\mathfrak{q}}\dashrightarrow E_{\mathfrak{q}}$
induced by $\varphi\circ\chi\circ\varphi\circ\psi$ has $\lambda=16>1$. This
proves the Theorem 1, and we again found the interesting dynamics on the
exceptional divisor $E_{\mathfrak{q}}$ in the fiber over $q$.

In the attached SageMath code, we compute an explicit equation for the map
$\Psi\circ\Phi$. We show that the dynamical degree $\lambda=16$ given here is
consistent with the direct approximations of the $\lambda$ given by assuming
the Kawaguchi-Silverman conjecture and computing the height growth along a few
orbits.

## References

  * [1] B. M. Bekker and Yu G. Zarhin, _Division by 2 of rational points on elliptic curves_ , St. Petersburg mathematical journal 29 (2018), no. 4, 683–713 (English).
  * [2] Jérémy Blanc, _On the inertia group of elliptic curves in the Cremona group of the plane_ , Michigan Mathematical Journal 56 (2008), no. 2, 315 – 330.
  * [3] Jérémy Blanc, _Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces_ , Indiana University Mathematics Journal 62 (2013), no. 4, 1143–1164.
  * [4] Anna Bot, _On automorphisms of rational surfaces_ , Master’s thesis, ETH Zürich, Zürich, Switzerland, 2021.
  * [5] Nguyen-Bac Dang, _Degrees of iterates of rational maps on normal projective varieties_ , Proceedings of the London Mathematical Society 121 (2020), no. 5, 1268–1310.
  * [6] Jeffery Diller and Charles Favre, _Dynamics of bimeromorphic maps of surfaces_ , American Journal of Mathematics 123 (2001), 1135 – 1169.
  * [7] M. H. Gizatullin, _The decomposition, inertia and ramification groups in birational geometry_ , Vieweg+Teubner Verlag, Wiesbaden, 1994.
  * [8] Xie Junyi, _Periodic points of birational transformations on projective surfaces_ , Duke Mathematical Journal 164 (2015), no. 5, 903 – 932.
  * [9] John Lesieutre, _A pathology of asymptotic multiplicity in the relative setting_ , Mathematical Research Letters 23 (2016), no. 5, 1433–1451.
  * [10] Tuyen Trung Truong, _Relative dynamical degrees of correspondences over a field of arbitrary characteristic_ , Journal für die reine und angewandte Mathematik (Crelles Journal) 2020 (2020), no. 758, 139–182.

## Acknowledgement

I extend my sincere gratitude to my advisor, John Lesieutre, for proposing
this problem and for his invaluable discussions and guidance throughout this
work. I also thank Anna Bot, Junyi Xie, Luen-Chau Li, Tatiana Bandman, and
Yuri Zarhin for their valuable discussions. This work was partially supported
by NSF-DMS grant 2142966.