# American Institute of Mathematical Sciences

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

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.
Citation: Igor E. Shparlinski. On some dynamical systems in finite fields and residue rings. Discrete & Continuous Dynamical Systems, 2007, 17 (4) : 901-917. doi: 10.3934/dcds.2007.17.901
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