Characterization of ergodic rational functions on the set of 2-adic units

We give an explicit characterization of ergodicity of rational functions on the set of units of the [Formula: see text]-adic group by means of the coefficients in their nominators and denominators. Although rational functions are not ergodic on the set of p-adic numbers, this was proved by Diao and Silva, we study their ergodicity on the 2-adic unit sphere if they only contain integer coefficients. The first results of this paper provide isometricity tests of polynomials and rational functions on the set of 2-adic units with 2-adic integer coefficients. Then, follow results on ergodicity tests on general isometric functions, then on rational functions on the set of 2-adic units with 2-adic integer coefficients.

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