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Dynamics of the $p$-adic shift and applications

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  • There is a natural continuous realization of the one-sided Bernoulli shift on the $p$-adic integers as the map that shifts the coefficients of the $p$-adic expansion to the left. We study this map's Mahler power series expansion. We prove strong results on $p$-adic valuations of the coefficients in this expansion, and show that certain natural maps (including many polynomials) are in a sense small perturbations of the shift. As a result, these polynomials share the shift map's important dynamical properties. This provides a novel approach to an earlier result of the authors.
    Mathematics Subject Classification: Primary 37A05; Secondary 37F10.


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