 

Benjamin Hutz and Adam Towsley
Misiurewicz points for polynomial maps and transversality view print


Published: 
May 9, 2015

Keywords: 
Misiurewicz point, dynatomic polynomial, unicritical polynomial, postcritically finite 
Subject: 
Primary: 37P05, 37P35, 11B83; secondary: 37P45, 11R99 


Abstract
The behavior under iteration of the critical points of a polynomial map plays an essential role in understanding its dynamics.
We study the special case where the forward orbits of the critical points are finite.
Thurston's theorem tells us that fixing a particular critical point portrait and degree leads to only finitely many possible polynomials (up to equivalence) and that, in many cases, their defining equations intersect transversely.
We provide
explicit algebraic formulae for the parameters where the critical points of all unicritical polynomials and of cubic polynomials have a specified exact period. We pay particular attention to the parameters where the critical orbits are strictly preperiodic, called Misiurewicz points. Our main tool is the generalized dynatomic polynomial. We also study the discriminants of these polynomials to examine the failure of transversality in characteristic p>0 for the unicritical polynomials z^{d} + c.


Author information
Benjamin Hutz:
Department of Mathematics, Florida Institute of Technology, Melbourne, FL
bhutz@fit.edu
Adam Towsley:
Department of Mathematics, Elmira College, Elmira, NY
atowsley@elmira.edu

