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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 and 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.

[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.

[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.

[4]

N. D. Elkies, Mahler's theorem on continuous $p$-adic maps via generating functionshttp://www.math.harvard.edu/ elkies/Misc/mahler.pdf.

[5]

F. Q. Gouvêa, "$p$-adic Numbers, An Introduction," Universitext. Springer-Verlag, Berlin, 1993.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[13]

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

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.

[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.

[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.

[4]

N. D. Elkies, Mahler's theorem on continuous $p$-adic maps via generating functionshttp://www.math.harvard.edu/ elkies/Misc/mahler.pdf.

[5]

F. Q. Gouvêa, "$p$-adic Numbers, An Introduction," Universitext. Springer-Verlag, Berlin, 1993.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[13]

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

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