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April  2011, 30(1): 209-218. doi: 10.3934/dcds.2011.30.209

## Dynamics of the $p$-adic shift and applications

 1 400 E 71 St. Apt. 5B, New York, NY 10021, United States 2 Massachusetts Institute of Technology, Cambridge, MA 02139, United States, United States 3 Williams College, Williamstown, MA 01267, United States

Received  March 2010 Revised  July 2010 Published  February 2011

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.
Citation: James Kingsbery, Alex Levin, Anatoly Preygel, Cesar E. Silva. Dynamics of the $p$-adic shift and applications. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 209-218. doi: 10.3934/dcds.2011.30.209
##### References:
 [1] V. Anashin and A. Khrennikov, "Applied Algebraic Dynamics,", volume 49 of de Gruyter Expositions in Mathematics, 49 (2009).   Google Scholar [2] D. K. Arrowsmith and F. Vivaldi, Some $p$-adic representations of the Smale horseshoe,, Phys. Lett. A, 176 (1993), 292.  doi: 10.1016/0375-9601(93)90920-U.  Google Scholar [3] J. Bryk and C. E. Silva, Measurable dynamics of simple $p$-adic polynomials,, Amer. Math. Monthly, 112 (2005), 212.  doi: 10.2307/30037439.  Google Scholar [4] N. D. Elkies, Mahler's theorem on continuous $p$-adic maps via generating functions,, , ().   Google Scholar [5] F. Q. Gouvêa, "$p$-adic Numbers, An Introduction,", Universitext. Springer-Verlag, (1993).   Google Scholar [6] A. Yu. Khrennikov and M. Nilson, "$p$-adic Deterministic and Random Dynamics,", volume 574 of Mathematics and its Applications, 574 (2004).   Google Scholar [7] J. Kingsbery, A. Levin, A. Preygel, and C. E. Silva, On measure-preserving $C^1$ transformations of compact-open subsets of non-Archimedean local fields,, Trans. Amer. Math. Soc., 361 (2009), 61.  doi: 10.1090/S0002-9947-08-04686-2.  Google Scholar [8] A. V. Mikhaĭlov, The central limit theorem for a $p$-adic shift. I,, in, (1986), 60.   Google Scholar [9] A. M. Robert, "A Course in $p$-adic Analysis,", volume 198 of Graduate Texts in Mathematics, 198 (2000).   Google Scholar [10] C. E. Silva, "Invitation to Ergodic Theory,", volume 42 of Student Mathematical Library, 42 (2008).   Google Scholar [11] J. H. Silverman, "The Arithmetic of Dynamical Systems,", volume 241 of Graduate Texts in Mathematics, 241 (2007).   Google Scholar [12] D. Verstegen, $p$-adic dynamical systems,, in, 47 (1990), 235.   Google Scholar [13] C. F. Woodcock and N. P. Smart, $p$-adic chaos and random number generation,, Experiment. Math., 7 (1998), 333.   Google Scholar

show all references

##### References:
 [1] V. Anashin and A. Khrennikov, "Applied Algebraic Dynamics,", volume 49 of de Gruyter Expositions in Mathematics, 49 (2009).   Google Scholar [2] D. K. Arrowsmith and F. Vivaldi, Some $p$-adic representations of the Smale horseshoe,, Phys. Lett. A, 176 (1993), 292.  doi: 10.1016/0375-9601(93)90920-U.  Google Scholar [3] J. Bryk and C. E. Silva, Measurable dynamics of simple $p$-adic polynomials,, Amer. Math. Monthly, 112 (2005), 212.  doi: 10.2307/30037439.  Google Scholar [4] N. D. Elkies, Mahler's theorem on continuous $p$-adic maps via generating functions,, , ().   Google Scholar [5] F. Q. Gouvêa, "$p$-adic Numbers, An Introduction,", Universitext. Springer-Verlag, (1993).   Google Scholar [6] A. Yu. Khrennikov and M. Nilson, "$p$-adic Deterministic and Random Dynamics,", volume 574 of Mathematics and its Applications, 574 (2004).   Google Scholar [7] J. Kingsbery, A. Levin, A. Preygel, and C. E. Silva, On measure-preserving $C^1$ transformations of compact-open subsets of non-Archimedean local fields,, Trans. Amer. Math. Soc., 361 (2009), 61.  doi: 10.1090/S0002-9947-08-04686-2.  Google Scholar [8] A. V. Mikhaĭlov, The central limit theorem for a $p$-adic shift. I,, in, (1986), 60.   Google Scholar [9] A. M. Robert, "A Course in $p$-adic Analysis,", volume 198 of Graduate Texts in Mathematics, 198 (2000).   Google Scholar [10] C. E. Silva, "Invitation to Ergodic Theory,", volume 42 of Student Mathematical Library, 42 (2008).   Google Scholar [11] J. H. Silverman, "The Arithmetic of Dynamical Systems,", volume 241 of Graduate Texts in Mathematics, 241 (2007).   Google Scholar [12] D. Verstegen, $p$-adic dynamical systems,, in, 47 (1990), 235.   Google Scholar [13] C. F. Woodcock and N. P. Smart, $p$-adic chaos and random number generation,, Experiment. Math., 7 (1998), 333.   Google Scholar
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