Uniform Bounds on Pre-Images under Quadratic Dynamical Systems

Details Uniform Bounds on Pre-Images under Quadratic Dynamical Systems Uniform Bounds on Pre-Images under Quadratic Dynamical Systems

2008-05-05

arxiv.org/abs/0805.0441v2

Mathematics

Number Theory

15 pages; very minor changes from v1

Scientific paper

For any elements b,c of a number field K, let G(b,c) denote the backwards orbit of b under the map f_c: C-->C given by f_c(x)=x^2+c. We prove an upper bound on the number of elements of G(b,c) whose degree over K is at most some constant B. This bound depends only on b, [K:Q], and B, and is valid for all b outside an explicit finite set. We also show that, for any N>3 and any b in K outside a finite set, there are only finitely many pairs of complex numbers (y,c) for which [K(y,c):K]<2^(N-3) and the value of the N-th iterate of f_c(x) at x=y is b. Moreover, the bound 2^(N-3) in this result is optimal.

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