Necklaces, permutations, and periodic critical orbits for quadratic polynomials

Let G n denote the n th Gleason polynomial , whose roots correspond to parameters c such that the critical point 0 is periodic of exact period n under iteration of z 2 + c , and let G ̄ n denote the reduction of G n modulo 2. Buff, Floyd, Koch, and Parry made the surprising observation that the number of real roots of G n is equal to the number of irreducible factors of G ̄ n for all n . We provide a bijective proof for this result by first providing explicit bijections between (a) the set of real roots of G n and the set N ̄ ( n ) of equivalence classes of primitive binary necklaces of length n under the inversion map swapping 0 and 1; and (b) the set of irreducible factors of G n modulo 2 and the set N ̃ + ( n ) of binary necklaces which are either primitive of length n with an even number of 1’s or primitive of length n / 2 with an odd number of 1’s. We then provide an explicit bijection, closely related to Milnor and Thurston’s kneading theory, between N ̄ ( n ) and N ̃ + ( n ) . In addition, we provide explicit bijections between N ̄ ( n ) , the set CUP ( n ) of cyclic unimodal permutations of { 1 , … , n } , and the set N − ( n ) of primitive binary necklaces of length n with an odd number of 1’s.