A note on Misiurewicz polynomials
Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 2, pp. 373-385.

Let f c,d (x)=x d +c[x]. The c 0 values for which f c 0 ,d has a strictly pre-periodic finite critical orbit are called Misiurewicz points. Any Misiurewicz point lies in ¯. Suppose that the Misiurewicz points c 0 ,c 1 ¯ are such that the polynomials f c 0 ,d and f c 1 ,d have the same orbit type. One classical question is whether c 0 and c 1 need to be Galois conjugates or not. Recently there has been partial progress on this question by several authors. In this note, we prove some new results when d is a prime. All the results known so far were in the cases of period at most 3. In particular, our work is the first to say something provable in the cases of period greater than 3.

Soit f c,d (x)=x d +c[x]. On appelle point de Misiurewicz une valeur c 0 pour laquelle f c 0 ,d a une orbite critique finie et strictement pré-périodique. Tout point de Misiurewicz appartient à ¯. Supposons que les points c 0 ,c 1 ¯ sont tels que les orbites de f c 0 ,d et de f c 1 ,d sont du même type. Une question classique est de savoir si c 0 et c 1 sont nécessairement conjugués sur . Récemment, certains progrès ont été réalisés par plusieurs auteurs pour répondre à cette question. Dans cette note, nous démontrons de nouveaux résultats dans le cas où d est un nombre premier. Tous les résultats connus jusqu’à présent portent sur des cas où la période est au plus 3. En particulier, notre travail est le premier à fournir des informations dans le cas de période plus grande que 3.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1126
Classification: 11R09,  37P15
Keywords: iteration, post-critically finite, Misiurewicz point
Vefa Goksel 1

1 Department of Mathematics and Statistics Lederle Graduate Research Tower, 1623D University of Massachusetts Amherst 710 N. Pleasant Street Amherst, MA 01003-9305, USA
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Vefa Goksel. A note on Misiurewicz polynomials. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 2, pp. 373-385. doi : 10.5802/jtnb.1126. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1126/

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