Dynamics of the \(p\)-adic shift and applications (Q536128)

The shift map \(T(b_0+b_1p+b_2p^2+\cdots)= b_1+b_2p+b_3p^2+\cdots\) on the \(p\)-adic integers \(\mathbb Z_p\) gives a natural realization of an abstract one-sided Bernoulli shift as a continuous map on \(\mathbb Z_p\). By developing detailed results on the \(p\)-norms of the coefficients of the Mahler expansion \(T(x)=\sum_{n\geq0}a_n\binom{x}{n}\), the authors show that certain natural maps, including many polynomials, may be viewed as small perturbations of \(T\). These maps as a result share many dynamical properties of \(T\), and this recovers an earlier result of the authors [Trans. Am. Math. Soc. 361, No. 1, 61--85 (2009; Zbl 1155.37003)].