scholarly journals On some dynamical systems in finite fields and residue rings

2007 ◽  
Vol 17 (4) ◽  
pp. 901-917 ◽  
Author(s):  
Igor E. Shparlinski ◽  
Keyword(s):  
Dynamical Systems ◽  
Finite Fields

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

scholarly journals Monomial dynamical systems of dimension one over finite fields

2011 ◽  
Vol 148 (4) ◽  
pp. 309-331 ◽  
Author(s):  
Min Sha ◽  
Su Hu
Keyword(s):  
Dynamical Systems ◽  
Finite Fields ◽  
Dimension One
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