Equivalence among orbital equations of polynomial maps

Abstract

This paper shows that orbital equations generated by iteration of polynomial maps do not have necessarily a unique representation. Remarkably, they may be represented in an infinity of ways, all interconnected by certain nonlinear transformations. Five direct and five inverse transformations are established explicitly between a pair of orbits defined by cyclic quintic polynomials with real roots and minimum discriminant. In addition, infinite sequences of transformations generated recursively are introduced and shown to produce unlimited supplies of equivalent orbital equations. Such transformations are generic, valid for arbitrary dynamics governed by algebraic equations of motion.

Preperiodicity and systematic extraction of periodic orbits of the quadratic map

“…The three factors composing S 5 (σ) correspond to the three groups of orbits discriminated in Table 4. From numerically approximate orbits we obtain exact expressions for the pair of period-five clusters: c 5,1 (x) = x 10 + x 9 − 10 x 8 − 10 x 7 + 34 x 6 + 34 x 5 − 43 x 4 − 43 x 3 + 12 x 2 + 12 x + 1, (15) c 5,2 (x) = x 15 − x 14 − 14 x 13 + 13 x 12 + 78 x 11 − 66 x 10 − 220 x 9 + 165 x 8 + 330 x 7 −210 x 6 − 252 x 5 + 126 x 4 + 84…”

Polynomial interpolation as detector of orbital equation equivalence

“…As a first application, we apply polynomial interpolation to find direct and inverse transformations that establish the equivalence among a pair of cyclic quintics of minimum discriminant ∆ = 14641 = 11 4 originally considered by Cohn 19,14 , namely…”

Field discriminants of cyclotomic period equations

“…table, one finds four period equations whose discriminants are The first two period equations for primes p = ef + 1 with e = 21,23,25,27,29,33, 39, and field discriminants ∆e = p e−1 . Highlighted are (ℓ P , ℓ k ), the number of digits in D and ∆e.…”