Unbounded Least Repelling Periods in Degree Four

Abstract

We construct degree-four rational maps of positive equilibrium Lyapunov exponent whose least classical repelling period is arbitrarily large. For every N ≥ 2, the map F_N(z) = z² − t^{2^{N−1}}z + t^{4·2^{N−1}+4}/z² has repelling cycles of exactly the periods n ≥ N. All its shortest cycles are defined over C((t)); over each fixed Q_p, p ≥ 5, it has rational shortest cycles. This disproves the proposed degree-dependent period bound in residue characteristic zero. We prove an exact threshold formula on a four-parameter valuation region. On each fixed-valuation affinoid domain, the shortest-cycle cover is an explicit connected Kummer cover whose algebra is generated by the multiplier. We determine the unramified point fields in the displayed examples and obtain complex meromorphic families with arbitrarily delayed simple multiplier poles, without a parameter base change.