Periodic Orbits of Quadratic Polynomials

Abstract

It is known that quadratic polynomials do not have real rational orbits of period four. By using a two-dimensional model for the quadratic family, this result is generalized for complex rational orbits.

Uniqueness of the Sum of Points of the Period-Five Cycle of Quadratic Polynomials

“…After this we concentrate on the period five case and show that the sum of period five cycle points is at most three-valued. The next result shows the relation between the sums of cycle points of the (x, y)-plane [7] and the (u, v)-plane [9]:…”

“…The dynamics of quadratic polynomials is commonly studied by using the family of maps f c (x) = x 2 + c, where c ∈ C and x i+1 = f c (x i ) = x 2 i + c. In the article [9] we presented the corresponding iterating system on a new coordinate plane using the change of variables (1.1) u = x + y = x 0 + x 1 v = x + y 2 + y − x 2 = x 0 + x 2 1 + x 1 − x 2 0 to the (x, y)-plane model (see [7]). In this new (u, v)-plane model, equations of periodic curves are of remarkably lower degree than in earlier models.…”

“…In complex dynamics, the sum of period cycle points has been a commonly used parameter in many connections (see, for example, [3], [6], [7], [8] and [10]). In the article [8] Giarrusso and Fisher used it to the parameterization of the period 3 hyperbolic components of the Mandelbrot set.…”

“…In the article [8] Giarrusso and Fisher used it to the parameterization of the period 3 hyperbolic components of the Mandelbrot set. Later, in the article [7], Erkama studied the case of the period 3 − 4 hyperbolic components of the Mandelbrot set on the (x, y)-plane and completely solved both cases.…”

Preperiodic points for quadratic polynomials with small cycles over quadratic fields

“…If G(f c , K) is strongly admissible, then the cycle structure of G(f c , K) is (1, 1), ( 2), (3), or (1, 1, 2) Remark 5.2. It was shown by Erkama in [14] that if c ∈ Q(i), then f c cannot have Q(i)-rational points of period 4. His proof uses different techniques from ours, including an interesting 2dimensional dynamical system that models iteration of the family f c .…”