Ergodic Behavior of $$(2,2)$$-Rational Ising Mapping over $$\mathbb Q_2$$

Let $$f$$ be a $$2$$ -adic Ising mapping with parameter $$a\in\mathbb Q_2$$ . The dynamics of $$f$$ were investigated in the case $$a\equiv3(\operatorname{mod }8)$$ . It was shown that, in this case, the set of bad points of the function (denoted by $$\mathcal P(f)$$ ) consists of a single point, and that the study of the dynamical system can be restricted to the union of spheres $$\bigsqcup_{n=3}^\infty S_{\frac{1}{2^n}}(1)$$ . It was proven that the Ising function acts as a bijective isometry on each sphere $$S_{\frac{1}{2^n}}(1)$$ . Consequently, the function preserves the real-valued Haar measure on each such sphere. Moreover, necessary and sufficient conditions for the ergodicity of the function were established. In particular, it was shown that if $$a\equiv19(\operatorname{mod }32)$$ (or $$a\equiv3(\operatorname{mod }32)$$ ), then for each $$n>3$$ (resp. $$n=3$$ ), the sphere $$S_{\frac{1}{2^n}}(1)$$ decomposes into two disjoint open compact minimal components.