On some dynamical systems in finite fields and residue rings

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

On some dynamical systems in finite fields and residue rings

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

Received December 2005 Revised November 2006 Published January 2007

Abstract
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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

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