Ergodicity of isometries mapping a sphere into itself in the field of p-adic numbers

In this paper we study measure-preserving dynamical systems generated by isometries acting
on spheres in the field of p-adic numbers. We focus on the boundary case ρ(r) = |f(x)−x| p = r, which is not
covered by the proper invariant-ball argument. Anashin’s finite-level criterion is formulated for 1-Lipschitz
self-maps of Z p and therefore applies, after affine normalization, to dynamical systems on invariant balls.
For p ≥ 3, an invariant sphere is not a ball: it is a disjoint union of p − 1 balls of the next smaller radius.
Hence the criterion on Z p is not directly applicable to a map preserving the whole sphere. We establish an
intrinsic finite-factor criterion for isometries of arbitrary p-adic spheres: the system is ergodic if and only
if the permutations induced on all canonical partitions of the sphere consist of single cycles. We further
prove a metric cyclic-lifting criterion: once the first finite factor consists of a single cycle, cyclicity at every
subsequent level is equivalent to the return equalities |f jN n (x ∗ )−x ∗ | p = r/p n , j = 1,...,p−1, at one point
and at every level, where N n = (p − 1)p n−1 . We apply these criteria to the borderline invariant spheres of
(2,2)-rational functions with two fixed points and of (2,1)-rational functions with a unique fixed point. In
both families we give explicit ergodic examples over Q 3 . The resulting ergodic systems are also minimal and
uniquely ergodic.