Shift operators and two applications to Fq[[T]]Fq[[T]]

This paper comprises two applications of shift operators to the characterization of continuous functions and ergodic functions defined on the integer ring of a non-Archimedean local field of positive characteristic. In the first part of the paper, we establish that digit expansion of shift operators becomes an orthonormal basis for the space of continuous functions on Fq[[T]]Fq[[T]], including a closed-form expression for expansion coefficients, and we establish that this is also true for p  -adic integers, excluding the coefficient formula. In the second part, we obtain the necessary and sufficient conditions for ergodicity of 1-Lipschitz functions represented on F2[[T]]F2[[T]] by digit shift operators, recalling the cases with the Carlitz polynomials and digit derivatives.