Preperiodic points with local rationality conditions in the quadratic unicritical family

For rational numbers $c$, we present a trichotomy of the set of totally real (totally $p$-adic, respectively) preperiodic points for maps in the quadratic unicritical family $f_c(x)=x^2+c$. As a consequence, we classify quadratic polynomials $f_c$ with rational parameters $c\in\mathbb{Q}$ so that $f_c$ has only finitely many totally real (totally $p$-adic, respectively) preperiodic points. These results rely on an adelic Fekete-type theorem and dynamics of the filled Julia set of $f_c$. Moreover, using a numerical criterion introduced in [NP], we make explicit calculations of the set of totally real $f_c$-preperiodic points when $c=-1,0,\frac{1}{5}$ and $\frac{1}{4}.$


Introduction
Throughout this article, we study dynamics of a quadratic polynomial of the form for some integers 0 ≤ m < n, where f n c := f c • ... • f c denotes the n-fold decomposition of f c with itself.In other words, the (forward) orbit of α under the iteration of f c is finite.Denote by PrePer(f c ) the set of all preperiodic points of f c in Q.
Recall that an algebraic number α is totally real if its minimal polynomial over Q splits completely over R, or equivalently, if every complex embedding Q(α) ֒→ C has image contained in R. Let Q tr denote the collection of all totally real algebraic numbers.One may wonder, for which rational parameters c ∈ Q is the set of totally real f c -preperiodic points PrePer(f c ) ∩ Q tr finite.This question leads to the following result.
For some rational numbers c ∈ (−2, 1 4 ], we are able to compute totally real preperiodic points of f c explicitly by using the technique in [NP](which concerns totally real algebraic integers with all Galois conjugates lying in an interval of length less than 4).However, for Date: November 22, 2022.
arbitrary c ∈ Q the preperiodic points of f c are not always algebraic integers.Thus we need to get around this issue by making an adequate change of coordinates to f φ c by an affine automorphism φ.This gives PrePer(f φ c ) ⊆ Z, see Lemma 4.2.Hence we have the following full description the set PrePer(f c ) ∩ Q tr where c ∈ {−1, 0, 1 5 , 1 4 }.
Note that the technique presented in [NP] is not limited to rational parameters c ∈ {−1, 0, 1 5 , 1 4 }, but for other parameters c ∈ (−2, 1 4 ] ∩ Q the computation becomes more difficult.It would also be interesting to give a bound on PrePer(f c ) ∩ Q tr for additional parameters c, but we do not pursue it here.
In a similar spirit to Theorem 1.1, we next provide a trichotomy result for the set PrePer(f c ) ∩ Q tp , where Q tp denotes the collection of all totally p-adic algebraic numbers.Recall an algebraic number α is totally p-adic if its minimal polynomial over Q splits completely over Q p , or equivalently, if every embedding contains infinitely many totally p-adic f c -preperiodic points.
Again, the first part of Theorem 1.3 is an application of an adelic Fekete theorem.Whereas part (b) and part (c) are consequences of the shape of p-adic filled Julia set of f c due to Benedetto-Briend-Perdry [BBP,Théorème 1.1].In contrast to the totally real situation, the set PrePer(f c ) ∩ Q tp is not easily computed quantitatively.This article is organized as follows.We study dynamics of f c over archimedean and nonarchimedean fields in §2.We then provide the trichotomy of the set PrePer(f c ) ∩ Q tr and the set PrePer(f c ) ∩ Q tp in §3 and §5, respectively.In §4, we compute the set PrePer(f c ) ∩ Q tr for certain rational parameters c ∈ (−2, 1 4 ].

Preliminaries
In what follows, L is an arbitrary field which is complete with respect to an absolute value | • |.If | • | is an archimedean absolute value, we say that L is an archimedean field (e.g., L is either R or C).Otherwise, we call L a nonarchimedean field (e.g., C p or Q p endowed with | • | p ).Given a polynomial f (x) ∈ L[x] of degree at least 2. Then the corresponding filled Julia set of f is defined as We present a trichotomy of the shape of the filled Julia set of f c = x 2 +c over an archimedean valued field (L, | • |).
Proposition 2.1.Let c be a real number.
and it coincides with the Julia set of f −2 (x).
Proof.Let x be a real number.Since c > 1 4 , there is ε > 0 so that c = 1 4 + ε.Thus Inductively, we obtain that Letting n → ∞, it means that the orbit of x under f c is unbounded.The proof of (i) is complete.
In part (ii), we assert that The iteration of f c implies that f n c (x) ≥ x 0 + nδ 2 → +∞ as n → +∞.In the case x < −a c , we exploit the fact that f c is even to deduce that again To prove (iii), we follow the idea presented in [Mi,Lemma 7.1] and we include details here for completeness.We first assert that the complex Julia set of f c is contained in the real line when c ≤ −2.To see this, we consider the real interval [Mi,Corollary 4.13].In other words, ].It follows that, for any real number c ≤ −2, the Julia set J (f c ) must coincide with the filled Julia set K C (f c ) because J (f c ) = ∂K C (f c ) (i.e., the Julia set of polynomial map f c is the topological boundary of the filled Julia set of f c ) and K C (f c ) has empty interior.This follows from the fact that a closed set with empty interior is its own boundary, for example, see [Gi,Remark 9.3].Consequently, the filled Julia set When L is a nonarchimedean field, one obtains the following shapes of nonarchimedean filled Julia set of . This is also proved in [BBP], but we include the proof here for completeness.
Proposition 2.3.Let L be a complete field equipped with a nonarchimedean absolute value | • |.Then the filled Julia set K L (f c ) has the following dichotomy : Since |x| > 1, it follows that the orbit of x under applying the map f c grows without bounds.Therefore x does not belong to K L (f c ).This completes part (i).
To verify part (ii), suppose that |c| > 1.To obtain the desired inclusion, it suffices to show that if

Totally real preperiodic points
We review necessary background in archimedean and nonarchimedean potential theory.Let K be a number field.Then K v is the completion of K at each place v ∈ M K .Denote by C v the completion of an algebraic closure In this article, our absolute values | • | v coincide with usual real and p-adic absolute values when restricted to Q and this normalization is the same as described in [BH,§2].Then we define the n-diameter of E v at each place v of K to be The quantity d n (E v ) describes the supremum of geometric mean of pairwise distance among n-point in the bounded set E v .This sequence {d n (E v )} n≥2 is monotone decreasing and the limit exists, see [BH,§4].The limit is known as the transfinite diameter of

Thus the transfinite diameter of an adelic set
where the numbers [Ru,Example 5.2.16].
For each place v of K, the v-adic filled Julia set of f c , denoted by K Cv (f c ), is With this notation, Baker and Hsia [BH,Theorem 4.1] showed that the transfinite diameter of v-adic filled Julia set of the monic polynomial For each f (x) ∈ L[x] and φ ∈ Aff(L), we define f φ := φ −1 • f • φ.Then we say f and g are conjugate if g = f φ for some φ ∈ Aff(L).Recall the filled Julia set of f The following proposition is straightforward and follows from the above definitions.
Proposition 4.1.Let φ ∈ Aff(L) and f (x) ∈ L[x].Then the following statements hold : Note that preperiodic points of f c when c ∈ Q are not always algebraic integers.Thus we need to make a suitable change of coordinates of f c to f φ c for some φ ∈ Aut(C) so that all preperiodic points of f φ c are algebraic integers due to the following Lemma.
It remains to treat the case when p divides b.Then we have Recall that theorem 8 in [NP] provides a convenient method to bound the degree of algebraic integers having all of its conjugates lying in an interval of length less than 4.This tool is useful in our computation because we only have to search for low degree algebraic integers in the interval.With the strategy described, we are able to calculate the set PrePer .
Remark 4.4.It is interesting to point out that parameters in the set {−1, 0, 1 5 , 1 4 } are not the only rational numbers for which the corresponding real part of the filled Julia set of the conjugate map f φ c has length less than 4. The rational parameters c = − 1 2 and c = 1 6 also have this property, but the calculation in searching algebraic integers in K(f φ c ) ∩ R becomes more challenging.More precisely, [NP,Theorem 8] asserts that the real segment K(f φ c ) ∩ R contains algebraic integers of degree at most 97 when c ∈ {− 1 2 , 1 6 }.For additional rational parameters in (−2, 1 4 ], the length of K(f φ c ) ∩ R is either 4 or greater than 4 and we do not have a known tool to explicitly calculate PrePer(f c ) ∩ Q tr in this situation.
Proof.Part (a) is simple.The only preperiodic points of the squaring map are 0 and roots of unity.It is also clear that ±1 are the only totally real roots of unity.
In part (b), let φ(x) = 1 2 x.As in Lemma 4.2, we work with the conjugate map Thus the real segment of the filled Julia set of f φ is [−1, 1] by Proposition 4.1(ii) and totally real preperiodic points of f φ belonging to this interval.Moreover, all preperiodic points of f φ are algebraic integers by Lemma 4.2.Then applying Corollary 9 in [NP] (which states that algebraic integers in an interval of length < √ 5 are integers), it follows that −1, 0, and 1 are the only algebraic integers all of whose conjugates lie in the interval [−1, 1].But, 0 is not preperiodic for f φ because it does not belong to the 2-adic filled Julia set of f φ .This means the only totally real preperiodic points of f φ are ±1.Therefore ± 1 2 are the only totally real preperiodic points of f by Proposition 4.1(i).
To compute part (c), we let I denote the real segment of the filled Julia set of g(x) := f −1 (x) = x 2 − 1.Then Let θ be a totally real preperiodic point of g.We want to show that [Q(θ) : Q] < 7. To see this, we apply Theorem 8 in [NP] with Thus Theorem 8 in [NP] is applied with n 0 = 7 and so [Q(θ) : Q] < 7. Using Robinson's result [Ro] (in which he classified, up to degree 8, algebraic integers having all Galois conjugates containing in an interval of length less than 4), it is impossible that 3 ≤ [Q(θ) : Q] ≤ 6 because there is a zero of the minimal polynomial satisfied by θ that does not belong to I.
Thus the real interval I contains algebraic integers of degree at most 2. Again, using Table 1 in [Ro], we obtain algebraic integers of degree ≤ 2 having all of its conjugates in I and so (2) It is elementary to check that 9 algebraic integers in the set on the right side of (2) are g-preperiodic.This finishes the computation in part (c).
The idea in part (d) is (b) and (c) combined.Let φ(x) = 1 √ 5 x.After making a suitable change of coordinates of h(x) := f 1/5 (x) = x 2 + 1 5 to h φ (x) = 1 √ 5 (x 2 + 1), we have the real segment of the filled Julia set of h φ is I (as denoted in part (c)) by Proposition 4.1(ii).Thus the same argument shown in part (c) yields However, 0, ±1, and ± √ 2 do not belong to the 5-adic filled Julia set of h φ .It is easy to compute that −1± √ 5 2 and 1± √ 5 2 are h φ -preperiodic.Applying Proposition 4.1(i), the desired result follows.

Totally p-adic preperiodic points
Fix an odd prime p.In [BBP, Théorème 1.1], Benedetto-Briend-Perdry provided a trichotomy of the p-adic filled Julia set.The trichotomy plays a crucial role in understanding totally p-adic preperiodic points of f c in Theorem 5.2.
Theorem 5.1.[BBP] Let p be an odd prime and let φ(x) = x 2 + c be a quadratic polynomial defined on Q p .For c ∈ Q p , we have It is known that the transfinite diameter of the ring of integers Z p in Q p is less than 1 and it is computed precisely in [Ru,Example 5.2.17] Combining this fact, Proposition 3.1, and Theorem 5.1, it gives rise to the following trichotomy of the set of totally p-adic f c -preperiodic points.Recall that an algebraic number α is totally p-adic if its minimal polynomial over Q has all roots embedded into Q p .
Theorem 5.2.Fix an odd prime p.Let c be a rational number.
Hence the transfinite diameter of the adelic set E is Applying Proposition 3.1, it yields that there are only finitely many algebraic numbers all of whose algebraic conjugates lie in K Cq (f c ) at each place q = p, in Z p , and in K C (f c ) at the archimedean place ∞.The finiteness of PrePer(f c ) ∩ Q tp follows.This completes part (a).The proof of (b) follows immediately from Theorem 5.1 (ii).It remains to verify (c).We apply Theorem 5.1 (iii) to obtain If α is f c -preperiodic, then so are all of its algebraic conjugates, and therefore all preperiodic points are totally p-adic.Hence PrePer(f c ) ∩ Q tp is infinite.
Remark 5.3.We do not completely know if the set PrePer(f c ) ∩ Q tp is always nonempty when |c| p ≤ 1.However, f c has a totally p-adic fixed point when 1 − 4c is a square in Q p by applying Hensel's lemma to the polynomial f (X) = X 2 − (1 − 4c) ∈ Z p [X].