ADV MATH SCI JOURNAL Advances in Mathematics: Scientific Journal 12 (2023), no.10, 877–885 ISSN: 1857-8365 (printed); 1857-8438 (electronic) https://doi.org/10.37418/amsj.12.10.3 ON SOME ERGODIC RATIONAL FUNCTIONS ON Z 5 Jasmina Muminovi´ c Huremovi´ c ABSTRACT. In this paper, we considered ergodicity conditions of certain rational functions in Z 5 . It is given the case when numerator is transitive modulo 5, but not modulo 25, and the case when numerator is not transitive even modulo 5. 1. I NTRODUCTION Necessary and sufficient conditions for 1-Lipschitz functions that are uniformly differentiable modulo p on Z p , to be ergodic were studied in [15]. Ergodic poly- nomials were studied in [8], [9], [12], [13]. Besides, rectification of perturbed monomials to construct ergodic transformations was considered in [4], [16], and [14]. It is known that rational functions are not ergodic on the infinite measure set of p-adic numbers [6]. Ergodicity of rational functions on 2-adic spheres was studied in [11]. In Corollaries 2.1 and 2.2, a perturbation of some non-ergodic polynomials is obtained by division by a unit polynomial, to produce ergodic ra- tional functions on the ring of 5-adic integers. We recall some facts about the ring of p-adic integers Z p . Every x ∈ Z p has the p-adic representation x = ∞ ∑ i=0 x i p i , where for each nonnegative integer i, x i ∈ {0 ...,p − 1}. The p-adic valuation ν p (x) of any p-adic integer x is defined as the 2020 Mathematics Subject Classification. 11S82, 37A05. Key words and phrases. p-adic integers, uniformly differentiable functions, ergodic functions. Submitted: 26.09.2023; Accepted: 12.10.2023; Published: 13.10.2023. 877