13 research outputs found

    Quadratic maps with a periodic critical point of period 2

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    We provide a complete classification of possible graphs of rational preperiodic points of endomorphisms of the projective line of degree 2 defined over the rationals with a rational periodic critical point of period 2, under the assumption that these maps have no periodic points of period at least 7. We explain how this extends results of Poonen on quadratic polynomials. We show that there are 13 possible graphs, and that such maps have at most 9 rational preperiodic points. We provide data related to the analogous classification of graphs of endomorphisms of degree 2 with a rational periodic critical point of period 3 or 4.Comment: Updated theorem 2 to rule out the cases of quadratic maps with a rational periodic critical point of period 2 and a rational periodic point of period 5 or

    Roots and Dynamics of Octonion Polynomials

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    This paper is devoted to several new results concerning (standard) octonionpolynomials. The first is the determination of the roots of all right scalarmultiples of octonion polynomials. The roots of left multiples are alsodiscussed, especially over fields of characteristic not 2. We then turn tostudy the dynamics of monic quadratic real octonion polynomials, classifyingthe fixed points into attracting, repelling and ambivalent, and concluding witha discussion on the behavior of pseudo-periodic points

    Scarcity of cycles for rational functions over a number field

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    We provide an explicit bound on the number of periodic points of a rational function defined over a number field, where the bound depends only on the number of primes of bad reduction and the degree of the function, and is linear in the degree. More generally, we show that there exists an explicit uniform bound on the number of periodic points for any rational function in a given finitely generated semigroup (under composition) of rational functions of degree at least 2. We show that under stronger assumptions the dependence on the degree of the map in the bounds can be removed