Mathematics – Dynamical Systems
Scientific paper
2004-09-15
Mathematics
Dynamical Systems
Scientific paper
If $\alpha $ is an irreducible nonexpansive ergodic automorphism of a compact abelian group $X$ (such as an irreducible nonhyperbolic ergodic toral automorphism), then $\alpha $ has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts in $X$. In spite of this we are able to construct a symbolic space $V$ and a class of shift-invariant probability measures on $V$ each of which corresponds to an $\alpha$-invariant probability measure on $X$. Moreover, every $\alpha$-invariant probability measure on $X$ arises essentially in this way. The last part of the paper deals with the connection between the two-sided beta-shift $V_\beta $ arising from a Salem number $\beta $ and the nonhyperbolic ergodic toral automorphism $\alpha $ arising from the companion matrix of the minimal polynomial of $\beta $, and establishes an entropy-preserving correspondence between a class of shift-invariant probability measures on $V_\beta $ and certain $\alpha $-invariant probability measures on $X$. This correspondence is much weaker than, but still quite closely modelled on, the connection between the two-sided beta-shifts defined by Pisot numbers and the corresponding hyperbolic ergodic toral automorphisms.
Lindenstrauss Elon
Schmidt Klaus
No associations
LandOfFree
Symbolic representations of nonexpansive group automorphisms does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Symbolic representations of nonexpansive group automorphisms, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Symbolic representations of nonexpansive group automorphisms will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-621860