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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Symbolic dynamics on free groups

Pages: 725 - 738, Volume 20, Issue 3, March 2008      doi:10.3934/dcds.2008.20.725

 
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Steven T. Piantadosi - Department of Mathematics, University of North Carolina at Chapel Hill, CB #3250, Phillips Hall, Chapel Hill, NC 27599, United States (email)

Abstract: We study nearest-neighbor shifts of finite type (NNSOFT) on a free group $\G$. We determine when a NNSOFT on $\G$ admits a periodic coloring and give an example of a NNSOFT that does not allow a periodic coloring. Then, we find an expression for the entropy of the golden mean shift on $\G$. In doing so, we study a new generalization of Fibonacci numbers and analyze their asymptotics with a one-dimensional iterated map that is related to generalized continued fractions.

Keywords:  Symbolic dynamics, free group, generalized continued fraction, generalized Fibonacci numbers, golden mean shift, entropy, hard core model.
Mathematics Subject Classification:  Primary: 37B10, 37B40; Secondary: 37B50.

Received: December 2006;      Revised: November 2007;      Available Online: December 2007.