Uniform bounded-period conjecture for repelling periodic points of non-Archimedean rational maps

Let $d\ge2$. Consider a rational map $f$ of degree $d$ defined over a field of residual characteristic $0$, with Lyapunov exponent $\lambda(f)>0$. A rigid repelling periodic point is a rigid periodic point whose multiplier has absolute value greater than $1$.

Uniform bounded-period conjecture. There exists an integer $N(d)\ge1$ such that every such map $f$ admits a rigid repelling periodic point of period at most $N(d)$.

This conjecture proposes a bound on the period of a repelling rigid cycle depending only on the degree. The surrounding discussion indicates that the paper proves repelling periodic points of sufficiently large periods in the positive-Lyapunov-exponent case, while a degree-dependent uniform bound remains the asserted goal.