Back to Search
Start Over
Misiurewicz polynomials and dynamical units, part I.
- Source :
- International Journal of Number Theory; Jul2023, Vol. 19 Issue 6, p1249-1267, 19p
- Publication Year :
- 2023
-
Abstract
- We study the dynamics of the unicritical polynomial family f d , c (z) = z d + c ∈ ℂ [ z ]. The c -values for which f d , c has a strictly preperiodic postcritical orbit are called Misiurewicz parameters, and they are the roots of Misiurewicz polynomials. The arithmetic properties of these special parameters have found applications in both arithmetic and complex dynamics. In this paper, we investigate some new such properties. In particular, when d is a prime power and c is a Misiurewicz parameter, we prove certain arithmetic relations between the points in the postcritical orbit of f d , c . We also consider the algebraic integers obtained by evaluating a Misiurewicz polynomial at a different Misiurewicz parameter, and we ask when these algebraic integers are algebraic units. This question naturally arises from some results recently proven by Buff, Epstein, and Koch and by the second author. We propose a conjectural answer to this question, which we prove in many cases. [ABSTRACT FROM AUTHOR]
- Subjects :
- POLYNOMIALS
ORBITS (Astronomy)
INTEGERS
ARITHMETIC
ORBIT method
Subjects
Details
- Language :
- English
- ISSN :
- 17930421
- Volume :
- 19
- Issue :
- 6
- Database :
- Complementary Index
- Journal :
- International Journal of Number Theory
- Publication Type :
- Academic Journal
- Accession number :
- 164558264
- Full Text :
- https://doi.org/10.1142/S1793042123500616