Simultaneously preperiodic integers for quadratic polynomials

New York Journal of Mathematics

New York J. Math.27(2021) 363–378.

Simultaneously preperiodic integers for quadratic polynomials

Valentin Huguin

Abstract. In this article, we study the set of parameters c ∈ C for which two given complex numbersa andbare simultaneously preperi- odic for the quadratic polynomialfc(z) =z²+c. Combining complex- analytic and arithmetic arguments, Baker and DeMarco showed that this set of parameters is infinite if and only ifa² =b². Recently, Buff answered a question of theirs, proving that the set of parametersc∈C for which both 0 and 1 are preperiodic forfc is equal to {−2,−1,0}.

Following his approach, we complete the description of these sets when aandbare two given integers with|a| 6=|b|.

Contents

1. Introduction 363

2. The dynamics of the quadratic polynomials 367

3. Back to the parameter space 373

References 378

1. Introduction

Forc∈C, let fc:C→Cbe the complex quadratic map f_c:z7→z²+c.

Given a pointz∈C, we study the sequence (f_c^◦n(z))_n≥0 of iterates of f_c atz. The set{f_c^◦n(z) :n≥0}is called theforward orbit of z underfc.

The point z is said to be periodic for f_c if there exists an integer p ≥ 1 such that fc^◦p(z) = z. The least such integer p is called the period of z.

The point z is said to be preperiodic for f_c if its forward orbit is finite or, equivalently, if there is an integer k≥0 such that f_c^◦k(z) is periodic for f_c. The smallest integer kwith this property is called thepreperiod of z.

Definition 1.1. Fora∈C, let S_a be the set defined by S_a={c∈C:ais preperiodic forfc}.

Received October 15, 2019.

2010Mathematics Subject Classification. Primary 37P05; Secondary 37F45, 37P35.

Key words and phrases. Preperiodic points, quadratic polynomials, unlikely intersections.

ISSN 1076-9803/2021

363

In this paper, we wish to examine these sets of parameters.

Forn≥0, let Fn∈Z[c, z] be the polynomial given by Fn(c, z) =f_c^◦n(z) .

The sequence (Fn)_n≥0 satisfies F0(c, z) =z and the recursion formulas F_n(c, z) =Fn−1 c, z²+c

=Fn−1(c, z)²+c for n≥1 .

In particular, when n≥1, the polynomial Fn is monic in c of degree 2ⁿ⁻¹ and monic inz of degree 2ⁿ.

Now, given a point a∈C, define – fork≥0 and p≥1 – the set S_a^k,p ={c∈C:F_k+p(c, a) =F_k(c, a)}.

For allk≥0 andp≥1, the setS_a^k,p contains at most 2^k+p−1 elements and consists of the parameters c∈Cfor which the point ais preperiodic for f_c with preperiod less than or equal to kand period dividingp.

In particular, it follows that the set S_a= [

k≥0, p≥1

S_a^k,p

is countable. Moreover, we have the following (see [BaD11, Lemma 3.5];

when a= 0, also compare [HT15, Theorem 1.1]):

Proposition 1.2. For every a∈C, the setS_a is infinite.

Proof. To obtain a contradiction, suppose that S_a contains finitely many elements. Then, since the sequence

S_a^n,1

n≥0 is increasing with respect to set inclusion, there exists an integer N ≥0 such that S_a^n+1,1 =S_a^n,1 for all n≥N. Now, note that, for everyn≥0, we have

F_n+2(c, a)−F_n+1(c, a) = (F_n+1(c, a)−F_n(c, a)) (F_n+1(c, a) +F_n(c, a)) . It follows that, if n ≥ N and γ is a root of the polynomial Fn+1(c, a) + F_n(c, a), then

F_n+1(γ, a)−F_n(γ, a) =F_n+1(γ, a) +F_n(γ, a) = 0 ,

and henceF_n+1(γ, a) =F_n(γ, a) = 0, which yieldsγ = 0. Therefore, we have Fn(0, a) = 0 andFn+1(c, a) +Fn(c, a) =c²ⁿ for alln≥N. In particular, we get

∂(FN+2+FN+1)

∂c (0, a) = 2∂FN+1

∂c (0, a)F_N+1(0, a) + 2∂F_N

∂c (0, a)F_N(0, a) + 2

= 2 ,

which contradicts the fact thatF_N+2(c, a) +F_N+1(c, a) =c^2N+1.

a forc=−a²+a, f_c

a −a forc=−a²−a,

f_c f_c

a −a−1 forc=−a²−a−1, f_c

fc

a a−1 −a forc=−a²+a−1.

f_c f_c

fc

Figure 1. Some parametersc∈Cfor which a given complex numbera is preperiodic forfc.

Remark 1.3. Note that, if a∈C, then fc(a) =fc(−a) for allc∈C. Conse- quently, we have S_a=S−a andS_a^k,p=S_−a^k,p for all k≥1 andp≥1.

Example 1.4. Assume that a∈C. Then (see Figure 1) we have S_a^0,1 =

−a²+a , S_a^1,1 =

−a²−a,−a²+a , S_a^0,2 =

−a²−a−1,−a²+a , S_a^1,2 =

−a²−a−1,−a²−a,−a²+a−1,−a²+a .

Here, the problem we are interested in is the description of the setsS_a∩S_b when aand bare two given complex numbers.

Example 1.5. Suppose that a∈C. Then (see Figure 2) we have

−a²−a−1 =−(a+ 1)²+ (a+ 1)−1∈ S_a^0,2∩ S_a+1^1,2 and

−a²−a=−(a+ 1)²+ (a+ 1)∈ S_a^1,1∩ S_a+1^0,1 .

Example 1.6. We have−2∈ S₀^2,1∩S₁^1,1,−1∈ S₀^0,2∩S₁^1,2 and 0∈ S₀^0,1∩S₁^0,1 (see Figure 3).

Since the sets S_a are countably infinite (see Proposition 1.2), we may wonder whether the sets S_a∩ S_b are infinite. This question was answered by Baker and DeMarco in [BaD11]. Using potential theory and an equidis- tribution result for points of small height with respect to an adelic height function, they proved that the set S_a∩ S_b is infinite if and only if a² =b².

a+ 1 a −a−1 forc=−a²−a−1,

f_c f_c

fc

a −a and a+ 1 forc=−a²−a.

fc f_c fc

Figure 2. Two parameters c∈Cfor whichaand a+ 1 are simultaneously preperiodic for f_c whena is a given complex number.

0 −2 2 and 1 −1

f−2 f−2 f−2 f−2 f−2

1 0 −1

f−1 f−1

f−1

0 and 1

f₀ f₀

Figure 3. Three parameters c∈C for which both 0 and 1 are preperiodic for fc.

As they pointed out, their proof is not effective and does not provide any estimate on the cardinality of these sets when they are finite. In their article, Baker and DeMarco conjectured that −2, −1 and 0 were the only parameters c ∈ C for which 0 and 1 are simultaneously preperiodic for fc

(see Example 1.6). Using localization properties of the set of parameters c ∈ C for which both 0 and 1 have bounded forward orbit under fc and the fact that 0 is the only parameter c ∈ C that is contained in the main cardioid of the Mandelbrot set and for which 0 is preperiodic for f_c, Buff gave an elementary proof of their conjecture in [Bu18].

Following his approach, we complete the description of the sets S_a∩ S_b whenaand bare two given integers with|a| 6=|b|. More precisely, we prove the following theorem, which asserts that Example 1.5 and Example 1.6 present all the parameters c ∈ C for which two given distinct and non- opposite integers are simultaneously preperiodic for the polynomialf_c: Theorem 1.7. Assume thata andb are two integers with |b|>|a|. Then

• eithera= 0, |b|= 1 andS_a∩ S_b ={−2,−1,0},

• or a= 0, |b|= 2 and S_a∩ S_b ={−2},

• or |a| ≥1, |b|=|a|+ 1and S_a∩ S_b =

−a²− |a| −1,−a²− |a| ,

• or |b|>max{2,|a|+ 1} and S_a∩ S_b=∅.

Our proof is elementary and uses only basic analytic and arithmetic argu- ments. In particular, the reader does not need to be familiar with complex dynamics.

In Section 2, we reprove some well-known results on the dynamics of the polynomialsfc. In Section 3, we go back to the study of the parameter space and give a proof of Theorem 1.7.

Acknowledgments. The author would like to thank his Ph.D. advisors, Xavier Buff and Jasmin Raissy, for helpful discussions without which this paper would not exist and the anonymous referee for his comments.

2. The dynamics of the quadratic polynomials

We shall investigate here the dynamics of the quadratic mapsfc:C→C. Given a parameterc∈C, let X_c be the set

X_c={z∈C:z is preperiodic forf_c} , and, fork≥0 and p≥1, let X_c^k,p be the set

X_c^k,p={z∈C:F_k+p(c, z) =F_k(c, z)} .

For all k ≥ 0 and p ≥ 1, the set X_c^k,p contains at most 2^k+p elements, is invariant underfcand consists of the preperiodic points forfcwith preperiod less than or equal to kand period dividingp. In particular, we have

X_c= [

k≥0, p≥1

X_c^k,p.

Moreover, the set X_c is completely invariant under f_c – that is, for every z∈C,fc(z)∈ X_c if and only if z∈ X_c.

Remark 2.1. Note that, if c∈C, thenf_c(z) =f_c(−z) for all z∈C. There- fore, the setsX_candX_c^k,p, withk≥1 andp≥1, are symmetric with respect to the origin.

Proposition 2.2. For every c∈C, we have X_c⊂ \

n≥0

{z∈C:|f_c^◦n(z)| ≤ρc} ,

where ρ_c= ¹⁺

√1+4|c|

2 .

Proof. For every z ∈C, we have |f_c(z)| ≥ |z|²− |c|, and |z|² − |c|>|z|if and only if|z|> ρc. It follows by induction that, ifz ∈C satisfies|z|> ρc, then

fc^◦(k+p)(z) >

f_c^◦k(z)

for all k ≥ 0 and p ≥ 1, and hence z is not preperiodic for fc. As the set X_c is invariant under fc, this completes the

proof of the proposition.

Now, let us study the dynamics of the polynomial f_c when c is a real parameter. Suppose that c ∈ −∞,¹₄

. Then the map fc:R → R is even and strictly increasing onR≥0, has two fixed pointsα_c≤β_c– with equality if and only if c= ¹₄ – given by

α_c= 1−√ 1−4c

2 and β_c= 1 +√ 1−4c 2

and satisfiesf_c(z)> z for all z∈(β_c,+∞). In particular, we have fc([−β_c, βc]) = [c, βc]

and the sequence (f_c^◦n(z))_n≥0 diverges to +∞ for all z ∈ (−∞,−β_c) ∪ (βc,+∞).

It follows that, ifc∈

−2,¹₄ , then

f_c([−β_c, β_c])⊂[−β_c, β_c] ,

and hence, for everyz∈R, the pointz has bounded forward orbit underf_c if and only if z∈[−β_c, βc].

Remark 2.3. Note that, for every c∈C, we haveρc=β−|c|.

Let us examine more thoroughly the dynamics of the map fc when c ∈ (−∞,−2]. It is related to the dynamics of the shift map in the space of sign sequences.

Let σ:{−1,1}^Z≥0 → {−1,1}^Z≥0 denote the shift map, which sends any sequence ε= (_n)_n≥0 of ±1 to the sequence (_n+1)_n≥0.

A sign sequence ε is said to beperiodic with period p ≥1 if σ^◦p(ε) =ε andpis the least such integer. The sequenceεis said to bepreperiodic with preperiod k≥0 if the sequenceσ^◦k(ε) is periodic andkis minimal with this property.

Fork≥0 and p≥1, define Σ^k,p=n

ε∈ {−1,1}^Z≥0 :σ^◦(k+p)(ε) =σ^◦k(ε)o

to be the set of all preperiodic sign sequences with preperiod less than or equal tok and period dividingp, and define

Σ= [

k≥0, p≥1

Σ^k,p

to be the collection of all preperiodic sign sequences. For allk≥0 andp≥1, the setΣ^k,pcontains exactly 2^k+p elements – each of them being completely determined by the choice of its first k+p terms – and is invariant under the shift map. Moreover, the set Σ is completely invariant under the shift map – that is, any sign sequenceεis preperiodic if and only if the sequence σ(ε) is preperiodic.

Theorem 2.4. For everyc∈(−∞,−2], there exists a unique map ψc:Σ→R

that makes the diagram below commute and satisfies ₀ψ_c(ε) ≥ 0 for all ε∈Σ.

Σ Σ

R R

σ

fc

ψc ψc

Furthermore, for every ε∈Σ, we have ₀ψ_c(ε)∈hp

−β_c−c, β_ci ,

for all c∈(−∞,−2], and the map ζε: (−∞,−2]→Rdefined by ζ_ε(c) =ψ_c(ε)

is continuous.

Before proving Theorem 2.4, observe thatc ≤ −β_c for all c∈(−∞,−2], with equality if and only if c = −2. Consequently, for c ∈ (−∞,−2] and =±1, the partial inverse g_c: [c,+∞)→R off_c given by

g_c(z) =√ z−c is well defined on [−β_c, βc], and we have

g_c([−β_c, β_c]) =h p

−β_c−c, β_ci

⊂[−β_c, β_c] .

Lemma 2.5. For all c ∈ (−∞,−2] and all ε = (₀, . . . , p−1) ∈ {−1,1}^p, with p≥1, the map g^ε_c: [−β_c, βc]→[−β_c, βc] defined by

g_c^ε(z) =g_c⁰◦ · · · ◦gc^p−1(z) has a unique fixed point z_ε(c).

Moreover, for every finite sequence εof ±1, the mapc7→z_ε(c) is contin- uous.

Claim 2.6. Ifc∈(−∞,−2],ε∈ {−1,1}^p, withp≥1, and zis a fixed point of g_c^ε, then z∈ X_c^0,p and_jfc^◦j(z)>0 for all j∈ {0, . . . , p−1}.

Proof of Claim 2.6. We have fc^◦p(z) = z and the set X_c^0,p is invariant underf_c. Therefore, for allj∈ {0, . . . , p−1}, we have

f_c^◦j(z) =g_c^j◦ · · · ◦g_c^p−1(z)∈g_c^j([−β_c, β_c])∩ X_c^0,p, which yields

jf_c^◦j(z)∈p

−β_c−c, βc

i

⊂R>0

sincej

√−β_c−c is preperiodic forfcwith preperiod 2 and period 1.

Proof of Lemma 2.5. Fix c ∈ (−∞,−2] and p ≥ 1. For every ε ∈ {−1,1}^p, the map g^ε_c has a fixed point z_ε(c) by the intermediate value the- orem. Now, note that z_ε(c) is not a fixed point of g_c^ε0 whenever ε 6= ε⁰ ∈ {−1,1}^p by Claim 2.6. Therefore, the points z_ε(c), with ε ∈ {−1,1}^p, are pairwise distinct, and, since X_c^0,p contains at most 2^p elements, it follows that

X_c^0,p={z_ε(c) :ε∈ {−1,1}^p} .

Thus, for everyε∈ {−1,1}^p,z_ε(c) is the unique fixed point of the mapg_c^ε. Now, fix p ≥ 1, ε = (0, . . . , p−1) ∈ {−1,1}^p and c ∈ (−∞,−2]. It remains to verify that the map c⁰ 7→ z_ε(c⁰) is continuous at c. For each c⁰ ∈(−∞,−2], choose ε_c⁰ ∈ {−1,1}^p such that

z_ε(c)−z_ε

c0(c⁰)

is minimal.

Then we have

z_ε(c)−z_ε

c0 c⁰ ≤



 Y

ε⁰∈{−1,1}^p

z_ε(c)−z_ε⁰ c⁰





1 2p

=

Fp c⁰,z_ε(c)

−z_ε(c)

1 2p

for all c⁰ ∈ (−∞,−2], and so z_ε

c0(c⁰) tends to z_ε(c) as c⁰ approaches c.

By Claim 2.6, it follows that, whenever c⁰ is close enough to c, we have _jf_c^◦j0 z_ε

c0(c⁰)

>0 for allj ∈ {0, . . . , p−1}, which yieldsε_c⁰ =ε. Thus, the limit of z_ε(c⁰) as c⁰ approachescis z_ε(c), and the lemma is proved.

We may now deduce Theorem 2.4 from Lemma 2.5.

Proof of Theorem 2.4. Fix c∈(−∞,−2]. Assume thatψc:Σ→R is a map that satisfiesf_c◦ψ_c=ψ_c◦σ and ₀ψ_c(ε)≥0 for all ε∈Σ. Then, for all ε∈Σand all n≥0, we have

ψ_c(ε) =g_c⁰◦ · · · ◦g_cⁿ ψ_c

σ^◦(n+1)(ε) .

It follows that, ifεis a periodic sign sequence with periodp≥1, thenψc(ε) is a fixed point of the mapgc^εp, whereεp= (0, . . . , p−1)∈ {−1,1}^p, and hence ψc(ε) =z_εp(c). Therefore, for everyε∈Σwith preperiod k≥0 and period p≥1, we have ψ_c(ε) =g^ε_c^pp z_εp(c)

, whereε_pp = (₀, . . . , k−1)∈ {−1,1}^k and εp = (k, . . . , k+p−1) ∈ {−1,1}^p, adopting the convention that g_c^∅ denotes the identity map of [−β_c, βc]. In particular, there is at most one mapψ_c:Σ→Rthat satisfies the conditions above.

For ε = (n)_n≥0 a preperiodic sign sequence with preperiod k ≥ 0 and periodp≥1, defineε_pp = (₀, . . . , k−1)∈ {−1,1}^k,ε_p= (_k, . . . , k+p−1)∈ {−1,1}^p and ψ_c(ε) = gc^εpp z_εp(c)

. If ε is a periodic sign sequence with periodp≥1, then f_c◦ψ_c(ε) is a fixed point of the mapg^σ(ε)c ^p sinceσ(ε)_p= (1, . . . , p−1, 0), and hence fc◦ ψc(ε) = ψc◦ σ(ε). Similarly, if ε ∈ Σ has preperiod k ≥ 1 and period p ≥ 1, then fc◦ψc(ε) = ψc◦σ(ε) since

σ(ε)_pp = (₁, . . . , k−1) and σ(ε)_p = ε_p. Moreover, for all ε ∈Σ, we have ψc(ε)∈g_c⁰([−β_c, βc]), which yields

0ψc(ε)∈hp

−β_c−c, βc

i

⊂R≥0.

Thus, the map ψ_c:Σ→R so defined has the required properties.

Furthermore, for every ε∈Σ, the map ζε:c 7→ψc(ε) is clearly continu-

ous.

Remark 2.7. Observe that, ifc ∈(−∞,−2] and ε,ε⁰ ∈ Σsatisfy ₀ = −⁰₀ and σ(ε) =σ(ε⁰), thenψ_c(ε) =−ψ_c(ε⁰).

Note that the proof of Theorem 2.4 provides explicit formulas for the maps ζεwithε∈Σ^k,1 and k≥0, which are defined in the statement of the theorem.

Example 2.8. Suppose that =±1. Then

• forε∈Σ^1,1 given by 0 =and 1=−1, we have ζ_ε:c7→ψ_c(ε) =−α_c;

• forε∈Σ^1,1 given by 0 =and 1= 1, we have ζε:c7→ψc(ε) =βc;

• forε∈Σ^2,1 given by ₀ =,₁= 1 and ₂=−1, we have ζε:c7→ψc(ε) =√

−α_c−c;

• forε∈Σ^2,1 given by ₀ =,₁=−1 and ₂ = 1, we have ζ_ε:c7→ψ_c(ε) =p

−β_c−c.

Proposition 2.9. Assume that c∈(−∞,−2]. Then we have X_c^k,p=ψ_c

Σ^k,p

⊂[−β_c, β_c] for all k≥0 and p≥1 (see Figure 4).

Furthermore, ifc∈(−∞,−2), then the map ψc:Σ→Ris injective.

Proof. For alln≥0, we havef_c^◦n◦ψ_c=ψc◦σ^◦n. Consequently,ψc Σ^k,p

⊂ X_c^k,p for all k≥0 andp≥1.

Now, suppose that c∈(−∞,−2). Then, for all ε∈Σand all n≥0, we have

_nf_c^◦n(ψ_c(ε))∈hp

−β_c−c, β_ci

⊂R>0.

Therefore, the map ψ_c is injective, and, since X_c^k,p contains at most 2^k+p elements, it follows thatψc Σ^k,p

=X_c^k,p, for all k≥0 andp≥1.

−

β

β

c

β

c β

z

c

−

β

β

w

Figure 4. Graphs of the maps z 7→ Fn(c, z), with n ∈ {0, . . . ,3}, when c∈(−∞,−2].

It remains to prove that X₋₂^k,p ⊂ψ−2 Σ^k,p

for all k≥0 and p≥1. Fix k ≥0 and p≥ 1, and suppose thatz ∈ X₋₂^k,p. Then, for allc ∈(−∞,−2), we have

min

ε∈Σ^k,p

|z−ψc(ε)| ≤



 Y

ε∈Σ^k,p

|z−ψc(ε)|





1 2k+p

=|F_k+p(c, z)−Fk(c, z)|^2k+p1 . As the maps ζε, with ε ∈ Σ^k,p, are continuous at −2, it follows that z ∈ ψ−2 Σ^k,p

. Thus, the proposition is proved.

Remark 2.10. Applying Montel’s theorem, it follows from Proposition 2.9 that, for every c∈(−∞,−2], the filled-in Julia set off_c– that is, the set of points z∈C that have bounded forward orbit underfc – is also contained in [−β_c, βc].

Note that the map ψ−2 is not injective. More precisely, we have the following:

Proposition 2.11. For all ε 6= ε⁰ ∈ Σ, ψ−2(ε) = ψ−2(ε⁰) if and only if there exists an integer k ≥ 2 such that ε,ε⁰ ∈ Σ^k,1, j = ⁰_j for all j∈ {0, . . . , k−3}, k−2 =−⁰_k−2, k−1=⁰_k−1 =−1 andk=⁰_k= 1.

Proof. Suppose that ε 6= ε⁰ ∈ Σ satisfy ψ−2(ε) = ψ−2(ε⁰). Then, for all n≥0, we have

nf₋₂^◦n(ψ−2(ε))≥0 and ⁰_nf₋₂^◦n(ψ−2(ε))≥0 .

Since ε 6= ε⁰, it follows that there is an integer k ≥ 0, which we may assume minimal, such that f₋₂^◦k(ψ−2(ε)) = 0. For all j ∈ {0, . . . , k−1}, the inequalities above are strict, and hence j = ⁰_j. Moreover, we have f₋₂^◦(k+1)(ψ−2(ε)) =−2 and f₋₂^◦n(ψ−2(ε)) = 2 for all n≥k+ 2, which yields k+1 = ⁰_k+1 = −1 and n = ⁰_n = 1 for all n ≥ k+ 2. Thus, the sign sequencesεand ε⁰ have the desired form.

Conversely, observe that, forε∈Σ^2,1 with₁ =−1 and₂= 1, we have ψ−2(ε) =₀p

−β₋₂−(−2) = 0 .

Therefore, if k≥2 andε∈Σ^k,1 satisfies k−1 =−1 and_k= 1, then ψ−2(ε) =g⁽₋₂^0,...,k−3)

ψ−2

σ^◦(k−2)(ε)

=g₋₂^{(0,...,k−3)}(0)

does not depend on k−2. This completes the proof of the proposition.

Remark 2.12. It follows from Proposition 2.9 and Proposition 2.11 that, for all k≥0 and p≥1, the set X₋₂^k,p contains exactly 2^p elements ifk= 0 and 2^k+p−2^k−1+ 1 elements if k≥1.

Remark 2.13. Note that we can actually describe the map ψ−2: Σ → R explicitly. Forε∈Σ, define the sequence (δn(ε))_n≥0 ∈ {0,1}^Z≥0 by

δ_n(ε) =

(δn−1(ε) ifn= 1 1−δn−1(ε) if_n=−1 ,

whereδ−1(ε) = 0 by convention. Then the mapψ−2:Σ→Ris given by ψ−2(ε) = 2 cos π

+∞

X

n=0

δn(ε) 2ⁿ⁺¹

! .

3. Back to the parameter space

We shall now exploit the statements given in Section 2 to get results concerning the parameter space.

Remark 3.1. By definition, for every pointa∈Cand every parameterc∈C, c ∈ S_a if and only if a ∈ X_c and, for all k ≥0 and p ≥1, c ∈ S_a^k,p if and only if a∈ X_c^k,p.

Proposition 3.2. For every a∈C, we have S_a⊂ {c∈C:|c| ≤Ra}, where R_a=|a|²+p

|a|²+ 1 + 1.

Proof. Suppose that c∈ S_a. Then, by Proposition 2.2, we have

|c| − |a|² ≤ |f_c(a)| ≤ρc, and henceϕ(|c|)≤ |a|², where ϕ:R≥0→R is given by

ϕ(x) =x− 1 +√ 1 + 4x

2 .

The map ϕ is strictly increasing and satisfies ϕ(Ra) = |a|². Thus, the

proposition is proved.

Now, let us give a more extensive description ofS_a whena∈(−∞,−2]∪ [2,+∞).

Given=±1, letΣ^k,p – with k≥0 andp≥1 – be the set defined by Σ^k,p =

n

ε= (n)_n≥0 ∈Σ^k,p:0 = o

, and letΣ be the set defined by

Σ = [

k≥0, p≥1

Σ^k,p ={ε∈Σ:₀=} .

For all k ≥ 0 and p ≥ 1, the set Σ^k,p contains exactly 2^k+p−1 elements – each of them being completely determined by the choice of its terms with index in{1, . . . , k+p−1}.

Suppose that a∈(−∞,−2]∪[2,+∞). Then

• forε∈Σ^2,1_sgn(a) given by1 =−1 and2 = 1, the map sgn(a)ζ_ε:c7→p

−β_c−c

is strictly decreasing on (−∞,−2] and we have ζε(c⁻_a) = a, where c⁻_a is the parameter defined by

c⁻_a =−a²−p

a²+ 1−1∈ S_a^2,1;

• forε∈Σ^1,1_sgn(a) given by₁ = 1, the map sgn(a)ζ_ε:c7→β_c

is strictly decreasing on (−∞,−2] and we have ζ_ε(c⁺_a) = a, where c⁺_a is the parameter defined by

c⁺_a =−a²+|a| ∈ S_a^1,1.

Remark 3.3. Note that, for every a∈C with|a| ≥2, we haveR_a=−c⁻_|a|.

c

c

2

c ₀

2 a z

Figure 5. Graphs of the maps ζε, with ε∈ Σ^2,1_sgn(a), when a∈[2,+∞).

Theorem 3.4. Assume thata∈(−∞,−2]∪[2,+∞). Then there is a unique map

γ_a:Σ_sgn(a)→(−∞,−2]

that satisfies ζ_ε(γ_a(ε)) =a for allε∈Σ_sgn(a) (see Figure 5).

Furthermore, we have

S_a^k,p=γ_a

Σ^k,p_sgn(a)

⊂ c⁻_a, c⁺_a

,

for all k≥0 and p≥1, (see Figure 6) and the mapγ_a is injective.

Claim 3.5. Ifa∈(−∞,−2]∪[2,+∞) andγ∈(−∞,−2], thenahas at most one preimage underψγ.

Proof of Claim 3.5. If γ ∈(−∞,−2), then the mapψ_γ is injective.

Ifγ =−2 and ε∈Σsatisfies ψγ(ε) =a, then we have 2≤ |a|=|ψ₋₂(ε)| ≤β−2 = 2 ,

so ψ−2(ε) = sgn(a)β−2, and, by Proposition 2.11, it follows that ε is the sign sequence in Σ^1,1_sgn(a) given by 1= 1. Thus, the claim is proved.

Proof of Theorem 3.4. For every ε∈Σ_sgn(a), we have sgn(a)ζε c⁻_a

≥ q

−β_c⁻

a −c⁻a =|a| and sgn(a)ζε c⁺_a

≤β_c⁺

a =|a|, and hence, by the intermediate value theorem, there exists γ_a(ε)∈[c⁻_a, c⁺_a] such that ζ_ε(γ_a(ε)) = a. Now, note that, if ε∈ Σ^k,p_sgn(a) – with k ≥0 and

c

c

c ₀

a z

Figure 6. Graphs of the maps c 7→ Fn(c, a), with n ∈ {0, . . . ,3}, when a∈[2,+∞).

p ≥ 1 – and γ ∈ (−∞,−2] satisfy ζε(γ) = a, then ε is a preimage of a under ψγ, and in particular γ ∈ S_a^k,p. Therefore, by Claim 3.5, the map γ_a so defined is injective, and, as S_a^k,p contains at most 2^k+p−1 elements, it follows that γa

Σ^k,p_sgn(a)

= S_a^k,p, for all k ≥ 0 and p ≥ 1. Thus, for everyε∈Σ_sgn(a),γ_a(ε) is the unique parameterγ ∈(−∞,−2] that satisfies ζ_ε(γ) =a. This completes the proof of the theorem.

Remark 3.6. Applying Montel’s theorem, it follows from Theorem 3.4 that, for every a ∈ (−∞,−2]∪[2,+∞), the set of parameters c ∈ C for which the pointahas bounded forward orbit underf_cis also contained in the line segment [c⁻_a, c⁺_a].

Note that, when a is an integer, the set S_a has the following arithmetic property (whena= 0, compare [HT15, Corollary 3.4]):

Proposition 3.7. For every a ∈ Z, the set S_a is contained in the set of algebraic integers and is invariant under the action of Gal Q/Q

.

Proof. For allk≥0 andp≥1, the polynomialFk+p(c, a)−F_k(c, a) is monic with integer coefficients sincea∈Z. Thus, the proposition is proved.

We shall now prove Theorem 1.7, which we recall below.

Theorem 1.7. Assume thata andb are two integers with |b|>|a|. Then

• eithera= 0, |b|= 1 andS_a∩ S_b ={−2,−1,0},

• or a= 0, |b|= 2 and S_a∩ S_b ={−2},

• or |a| ≥1, |b|=|a|+ 1and S_a∩ S_b =

−a²− |a| −1,−a²− |a| ,

• or |b|>max{2,|a|+ 1} and S_a∩ S_b=∅.

Lemma 3.8. Assume that m ∈ Z and c is an algebraic integer whose all Galois conjugates lie in the interval(m−2, m]. Then c=m−1 or c=m.

Proof of Lemma 3.8. Setα =c−m+ 1. Then α is an algebraic integer whose all Galois conjugatesα₁, . . . , α_dlie in the interval (−1,1]. Therefore, we have

d

Y

j=1

α_j ∈(−1,1]∩Z={0,1},

and it follows that eitherα_j = 0 for somej ∈ {1, . . . , d}, which yieldsα= 0, orαj = 1 for allj ∈ {1, . . . , d}. Thus, either c=m−1 or c=m.

Proof of Theorem 1.7. For a proof of the casea= 0 and|b|= 1, we refer the reader to [Bu18, Proposition 6].

Thus, we may assume that |b| ≥2. By Proposition 3.2, Theorem 3.4 and Proposition 3.7, the setS_a∩ S_b is contained in the set of algebraic integers, is invariant under the action of Gal Q/Q

and satisfies S_a∩ S_b ⊂ {c∈C:|c| ≤R_a} ∩

c⁻_b , c⁺_b . Suppose that a= 0. Then we have

c⁺_b =−b²+|b| ≤ −2 =−R_a,

with equality if and only if |b| = 2. Therefore, S_a∩ S_b ⊂ {−2} if |b| = 2 and S_a∩ S_b =∅otherwise. Conversely, observe that −2∈ S_a^2,1∩ S_b^1,1 when

|b|= 2.

Now, suppose that |a| ≥1. Then we have c⁺_b −2<−R_a=−a²−p

a²+ 1−1<−a²− |a|=c⁺_b if |b|=|a|+ 1 and

c⁺_b =−b²+|b|<−a²−p

a²+ 1−1 =−R_a if |b| ≥ |a|+ 2 . Therefore,S_a∩ S_b⊂

−a²− |a| −1,−a²− |a| if|b|=|a|+ 1 by Lemma 3.8 andS_a∩S_b =∅otherwise. Conversely, observe that−a²−|a|−1∈ S_a^1,2∩S_b^1,2 and−a²−|a| ∈ S_a^1,1∩S_b^1,1when|b|=|a|+1. Thus, the theorem is proved.

References

[BaD11] Baker, Matthew; DeMarco, Laura. Preperiodic points and unlikely inter- sections.Duke Math. J. 159(2011), no. 1, 1–29. MR2817647, Zbl 1242.37062, arXiv:0911.0918, doi: 10.1215/00127094-1384773. 364, 365

[Bu18] Buff, Xavier. On postcritically finite unicritical polynomials. New York J.

Math.24(2018), 1111–1122. MR3890968, Zbl 1412.37084. 366, 377

[HT15] Hutz, Benjamin; Towsley, Adam. Misiurewicz points for polynomial maps and transversality. New York J. Math. 21 (2015), 297–319. MR3358544, Zbl 1391.37071, arXiv:1309.4048. 364, 376

(Valentin Huguin) Institut de Math´ematiques de Toulouse, UMR 5219, Univer- sit´e de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France

[email protected]

This paper is available via http://nyjm.albany.edu/j/2021/27-13.html.