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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

James Kingsbery 1, , Alex Levin 2, , Anatoly Preygel 2, and  Cesar E. Silva 3,

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.
Keywords: $p$-adic., Measure-preserving, Bernoulli, shift, ergodic.
Mathematics Subject Classification: Primary 37A05; Secondary 37F1.
Citation: James Kingsbery, Alex Levin, Anatoly Preygel, Cesar E. Silva. Dynamics of the $p$-adic shift and applications. Discrete & Continuous Dynamical Systems, 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, Walter de Gruyter & Co., Berlin, 2009.  Google Scholar

[2]

D. K. Arrowsmith and F. Vivaldi, Some $p$-adic representations of the Smale horseshoe, Phys. Lett. A, 176 (1993), 292-294. 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), 212-232. 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, Berlin, 1993.  Google Scholar

[6]

A. Yu. Khrennikov and M. Nilson, "$p$-adic Deterministic and Random Dynamics," volume 574 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 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), 61-85. 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 "Analytic Number Theory (Russian)," Petrozavodsk. Gos. Univ., Petrozavodsk, (1986), 60-68, 92.  Google Scholar

[9]

A. M. Robert, "A Course in $p$-adic Analysis," volume 198 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.  Google Scholar

[10]

C. E. Silva, "Invitation to Ergodic Theory," volume 42 of Student Mathematical Library, American Mathematical Society, Providence, RI, 2008.  Google Scholar

[11]

J. H. Silverman, "The Arithmetic of Dynamical Systems," volume 241 of Graduate Texts in Mathematics, Springer, New York, 2007.  Google Scholar

[12]

D. Verstegen, $p$-adic dynamical systems, in "Number Theory and Physics (Les Houches, 1989)," volume 47 of Springer Proc. Phys., Springer, Berlin, (1990), 235-242.  Google Scholar

[13]

C. F. Woodcock and N. P. Smart, $p$-adic chaos and random number generation, Experiment. Math., 7 (1998), 333-342.  Google Scholar

show all references

References:
[1]

V. Anashin and A. Khrennikov, "Applied Algebraic Dynamics," volume 49 of de Gruyter Expositions in Mathematics, Walter de Gruyter & Co., Berlin, 2009.  Google Scholar

[2]

D. K. Arrowsmith and F. Vivaldi, Some $p$-adic representations of the Smale horseshoe, Phys. Lett. A, 176 (1993), 292-294. 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), 212-232. 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, Berlin, 1993.  Google Scholar

[6]

A. Yu. Khrennikov and M. Nilson, "$p$-adic Deterministic and Random Dynamics," volume 574 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 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), 61-85. 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 "Analytic Number Theory (Russian)," Petrozavodsk. Gos. Univ., Petrozavodsk, (1986), 60-68, 92.  Google Scholar

[9]

A. M. Robert, "A Course in $p$-adic Analysis," volume 198 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.  Google Scholar

[10]

C. E. Silva, "Invitation to Ergodic Theory," volume 42 of Student Mathematical Library, American Mathematical Society, Providence, RI, 2008.  Google Scholar

[11]

J. H. Silverman, "The Arithmetic of Dynamical Systems," volume 241 of Graduate Texts in Mathematics, Springer, New York, 2007.  Google Scholar

[12]

D. Verstegen, $p$-adic dynamical systems, in "Number Theory and Physics (Les Houches, 1989)," volume 47 of Springer Proc. Phys., Springer, Berlin, (1990), 235-242.  Google Scholar

[13]

C. F. Woodcock and N. P. Smart, $p$-adic chaos and random number generation, Experiment. Math., 7 (1998), 333-342.  Google Scholar

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