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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

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
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