Mahler measure and entropy for commuting automorphisms of compact groups

Abstract

SummaryWe compute the joint entropy of d commuting automorphisms of a compact metrizable group. Let R d = Z[u( 1 ..... uf 1] be the ring of Laurent polynomials in d commuting variables, and M be an Ra-module. Then the dual group X M of M is compact, and multiplication on M by each of the d variables corresponds to an action a M of 7/d by automorphisms of Xu. Every action of 7/d by automorphisms of a compact abelian group arises this way. Iffe R d, our main formula shows that the topological entropy of ~R,/</> is given bywhere M(f) is the Mahler measure off. This reduces to the classical result for toral automorphisms via Jensen's formula. While the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer, this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7((3)/4n 2 occur. If p is a non-principal prime ideal, we show h(CtR,ip ) = 0. Using an analogue of the Yuzvinskii-Thomas addition formula, we compute h(~u) for arbitrary Rd-modules M, and then the joint entropy for an action of 7/a on a (not necessarily abelian) compact group. Using a result of Boyd, we characterize those ~t M which have completely positive entropy in terms of the prime ideals associated to M, and show this condition implies that ct M is mixing of all orders. We also establish an analogue of Berg's theorem, proving that if ct M has finite entropy then Haar measure is the unique measure of maximal entropy if and only if a M has completely positive entropy. Finally, we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy.

Homoclinic points of algebraic $\mathbb {Z}^d$-actions

“…Hence by Theorem 6.1 we see that α R/f 2 R is measurably isomorphic to α R/f R × α R/f R . The existence of an isomorphism between these actions (although not of the precise form given by Theorem 6.1) also follows from the deeper facts that they are both Bernoulli Z d -actions [15] with the same entropy [8].…”

“…We recall the following results from [14], [8], and [17,Lemma 4.5] (cf. also [15], [4], and [16]), which show that the dynamical properties of α M are largely controlled by the prime ideals associated to M .…”

Almost block independence for the three dot ℤ2 dynamical system

“…The action is mixing (by Theorem 11.2(4) of [7]), has a dense set of periodic points (Theorem 7.2 of [7]), and has trivial Pinsker algebra (Theorem 6.13 of [11]). It follows that the action is mixing of all orders (Corollary 6.7 of [11]).…”

“…The action is mixing (by Theorem 11.2(4) of [7]), has a dense set of periodic points (Theorem 7.2 of [7]), and has trivial Pinsker algebra (Theorem 6.13 of [11]). It follows that the action is mixing of all orders (Corollary 6.7 of [11]). By Theorem 6.14 of [11], Haar measure is maximal for α, and the topological entropy of α is given in [11], Example 5.1:…”

“…This action is an example of a Z d action on a compact abelian group, and these have been systematically studied in [7], [11], [18], and [19]. The action is mixing (by Theorem 11.2(4) of [7]), has a dense set of periodic points (Theorem 7.2 of [7]), and has trivial Pinsker algebra (Theorem 6.13 of [11]).…”