Expansions of quadratic maps in prime fields
نویسنده
چکیده
Let f(x) = ax+bx+c ∈ Z[x] be a quadratic polynomial with a ̸≡ 0 mod p. Take z ∈ Fp and let Oz = {fi(z)}i∈Z+ be the orbit of z under f , where fi(z) = f(fi−1(z)) and f0(z) = z. For M < |Oz|, We study the diameter of the partial orbit OM = {z, f(z), f2(z), . . . , fM−1(z)} and prove that there exists c1 > 0 such that diam OM & min { Mp c1 , 1 log p M 4 5p 1 5 ,M 1 13 log logM } . For a complete orbit C, we prove that diam C & min{p 5c1 , e T/4 }, where T is the period of the orbit. Introduction. This paper belongs to the general theme of dynamical systems over finite fields. Let p be a prime and Fp the finite field of p elements, represented by ∗2000 Mathematics Subject Classification.Primary 11B50, 37A45, 11B75; Secondary 11T23, 37F10 11G99. †
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