Speaker: Sangtae Jeong (Inha University)
Time: May 22 (Wed), 2019 16:00-17:00.
Place: 317 Riemann

Title: Ergodic functions over the ring of p-adic integers

Abstract: In this talk, we present ergodicity criteria for $1$-Lipschitz functions on $\Zp$, in terms of the van der Put coefficients as well as the inherent data associated with the function. These criteria are applied to provide sufficient conditions for ergodicity of the $1$-Lipschitz $p$-adic functions with special features, such as everywhere/uniform differentiability with respect to the Mahler expansion. In particular, the ergodicity criteria are obtained for certain $1$-Lipschitz functions on $\Z_2$ and $\Z_3$, which are known as $\mathcal{B}$-functions, in terms of the Mahler and van der Put expansions. These functions are locally analytic functions of order $1$ (and therefore contain polynomials). For arbitrary primes $p\geq 5,$ an ergodicity criterion of $\mathcal{B}$-functions on $\Zp$ is introduced, which leads to an efficient and practical method of constructing ergodic polynomials on $\Z_p$ that realize a given unicyclic permutation modulo $p.$ Thus, a complete description of ergodic polynomials modulo $p^{\mu},$ which are reduced from all ergodic $\mathcal{B}$-functions on $\Zp,$ is provided where $\mu=\mu(p)$ = 3 for $p\in {2,3}$ and $\mu =2$ for $p\geq5.$