Joey Lupo | Searcharxiv

arXiv2022

Integrality and Thurston Rigidity for Bicritical PCF Polynomials

We give an algebraic proof of an important consequence of Thurston rigidity for bicritical PCF polynomials with periodic critical points under certain mild assumptions. The key result is that when the family of bicritical polynomials is parametrized using dynamical Belyi polynomials, the PCF solutions are integral at certain special primes, which we term ``index divisor free primes.'' We prove the existence of index divisor free primes in all but finitely many cases and conjecture the complete list of exceptions. These primes are then used to prove transversality.

math.DS↗

arXiv2022

New Normal Forms For Degree Three Polynomials and Rational Functions

When studying families in the moduli space of dynamical systems, choosing an appropriate representative function for a conjugacy class can be a delicate task. The most delicate questions surround rationality of the conjugacy class compared to rationality of the defining polynomials of the representation. We give a normal form for degree three polynomials which has the property that the set of fixed points is equal to the set of fixed point multipliers. This normal form is given in terms of moduli space invariants and, hence, has nice rationality properties. We further classify all degree three rational maps which can be conjugated to have a similar relationship between the fixed points and the fixed point multipliers.

math.DS↗