Plotting basins of end points of rational maps with Sage

On Rational Maps with Two Critical Points

This is a preliminary investigation of the geometry and dynamics of rational maps with only two critical points.

Preperiodic points for families of rational maps

Let X be a smooth curve defined over Q̄, let a, b ∈ P(Q̄) and let fλ(x) ∈ Q̄(x) be an algebraic family of rational maps indexed by all λ ∈ X(C). We study whether there exist infinitely many λ ∈ X(C) such that both a and b are preperiodic for fλ. In particular, we show that if P,Q ∈ Q̄[x] such that deg(P ) 2 + deg(Q), and if a, b ∈ Q̄ such that a is periodic for P (x)/Q(x), but b is not preperiodic f...

Portraits of Preperiodic Points for Rational Maps

Let K be a function field over an algebraically closed field k of characteristic 0, let φ ∈ K(z) be a rational function of degree at least equal to 2 for which there is no point at which φ is totally ramified, and let α ∈ K. We show that for all but finitely many pairs (m,n) ∈ Z≥0×N there exists a place p of K such that the point α has preperiod m and minimum period n under the action of φ. Thi...

Distribution of Rational Points: A Survey

Fixed Points of Maps on the Space of Rational Functions

Given integers s, t, define a function φs,t on the space of all formal series expansions by φs,t( ∑ anx ) = ∑ asn+tx . For each function φs,t, we determine the collection of all rational functions whose Taylor expansions at zero are fixed by φs,t. This collection can be described as a subspace of rational functions whose basis elements correspond to certain s-cyclotomic cosets associated with t...