American Institute of Mathematical Sciences

Advanced
October  2007, 17(4): 901-917. doi: 10.3934/dcds.2007.17.901

On some dynamical systems in finite fields and residue rings

Igor E. Shparlinski 1,

1. 

Department of Computing, Macquarie University, Sydney, NSW 2109, Australia

Received  December 2005 Revised  November 2006 Published  January 2007

We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show that some conjectures are wrong. We also analyze several other conjectures of V. I. Arnold related to the orbit length of similar dynamical systems in residue rings and outline possible ways to prove them. We also show that some of them require further tuning.
Keywords: finite fields and rings, uniformity of distribution, orbit length., Character sums.
Mathematics Subject Classification: 11K38, 11N37, 37A45, 37E1.
Citation: Igor E. Shparlinski. On some dynamical systems in finite fields and residue rings. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 901-917. doi: 10.3934/dcds.2007.17.901
[1]

Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395
[2]

Anderson Silva, C. Polcino Milies. Cyclic codes of length $ 2p^n $ over finite chain rings. Advances in Mathematics of Communications, 2020, 14 (2) : 233-245. doi: 10.3934/amc.2020017
[3]

Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181
[4]

Katayun Barmak, Eva Eggeling, Maria Emelianenko, Yekaterina Epshteyn, David Kinderlehrer, Richard Sharp, Shlomo Ta'asan. An entropy based theory of the grain boundary character distribution. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 427-454. doi: 10.3934/dcds.2011.30.427
[5]

David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131
[6]

Xiao-Song Yang. Index sums of isolated singular points of positive vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1033-1039. doi: 10.3934/dcds.2009.25.1033
[7]

Aicha Batoul, Kenza Guenda, T. Aaron Gulliver. Some constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2016, 10 (4) : 683-694. doi: 10.3934/amc.2016034
[8]

Somphong Jitman, San Ling, Patanee Udomkavanich. Skew constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2012, 6 (1) : 39-63. doi: 10.3934/amc.2012.6.39
[9]

M. DeDeo, M. Martínez, A. Medrano, M. Minei, H. Stark, A. Terras. Spectra of Heisenberg graphs over finite rings. Conference Publications, 2003, 2003 (Special) : 213-222. doi: 10.3934/proc.2003.2003.213
[10]

Jiarong Peng, Xiangyong Zeng, Zhimin Sun. Finite length sequences with large nonlinear complexity. Advances in Mathematics of Communications, 2018, 12 (1) : 215-230. doi: 10.3934/amc.2018015
[11]

Łukasz Struski, Jacek Tabor, Tomasz Kułaga. Cone-fields without constant orbit core dimension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3651-3664. doi: 10.3934/dcds.2012.32.3651
[12]

Florian Luca, Igor E. Shparlinski. On finite fields for pairing based cryptography. Advances in Mathematics of Communications, 2007, 1 (3) : 281-286. doi: 10.3934/amc.2007.1.281
[13]

Thomas Westerbäck. Parity check systems of nonlinear codes over finite commutative Frobenius rings. Advances in Mathematics of Communications, 2017, 11 (3) : 409-427. doi: 10.3934/amc.2017035
[14]

Thomas Gauthier, Gabriel Vigny. Distribution of postcritically finite polynomials Ⅱ: Speed of convergence. Journal of Modern Dynamics, 2017, 11: 57-98. doi: 10.3934/jmd.2017004
[15]

Steven T. Dougherty, Joe Gildea, Adrian Korban, Abidin Kaya. Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020037
[16]

François Ledrappier, Omri Sarig. Fluctuations of ergodic sums for horocycle flows on $\Z^d$--covers of finite volume surfaces. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 247-325. doi: 10.3934/dcds.2008.22.247
[17]

Stefania Fanali, Massimo Giulietti, Irene Platoni. On maximal curves over finite fields of small order. Advances in Mathematics of Communications, 2012, 6 (1) : 107-120. doi: 10.3934/amc.2012.6.107
[18]

Jean-François Biasse, Michael J. Jacobson, Jr.. Smoothness testing of polynomials over finite fields. Advances in Mathematics of Communications, 2014, 8 (4) : 459-477. doi: 10.3934/amc.2014.8.459
[19]

Robert Granger, Thorsten Kleinjung, Jens Zumbrägel. Indiscreet logarithms in finite fields of small characteristic. Advances in Mathematics of Communications, 2018, 12 (2) : 263-286. doi: 10.3934/amc.2018017
[20]

Shengtian Yang, Thomas Honold. Good random matrices over finite fields. Advances in Mathematics of Communications, 2012, 6 (2) : 203-227. doi: 10.3934/amc.2012.6.203

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences