On dynamics of $(2,2)$-rational mapping over $\mathbb Q_2$

Abstract

This paper is devoted to the study of the discrete dynamical systems associated with a $(2,2)$-rational mapping $f_{a,b}(x) = \frac{ax^2 + bx + 1}{x^2 + bx + a}$ over the field of $2$-adic numbers $\mathbb{Q}_2$, where $a, b \in \mathbb{Q}_2$ are parameters. We fully characterize the sets of fixed and $2$-periodic points of the given mapping depending on the $2$-adic norms of the parameters. It is shown that the system can exhibit various dynamical behaviors: in the case $|a|_2 = |b|_2 = 1$, the system is regular and all trajectories converge to a unique fixed point; whereas for $|a|_2 = |b|_2 < 1$, the fixed point becomes a repeller and an attracting $2$-cycle emerges. Furthermore, for the case $|a|_2 = |b|_2 > 1$, we establish the existence of invariant spheres and provide necessary and sufficient conditions for the ergodicity of the mapping on these spheres with respect to the normalized Haar measure. These results contribute to the classification of $p$-adic rational dynamics for higher-degree mappings.