Algebraic $\mathbb {Z}^d$-actions of entropy rank one

Abstract

Abstract. We investigate algebraic Z d -actions of entropy rank one, namely those for which each element has finite entropy. Such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery. This leads to numerous alternative characterizations of entropy rank one, both geometric and algebraic. We then compute the measure entropy of a class of skew products, where the fiber maps are elements from an algebraic Z d -action of entropy rank one. This leads, via the relative variational principle, to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures. We use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms.

Flexibility of statistical properties for smooth systems satisfying the central limit theorem

“…Generalized T, T −1 transformation is a classical subject (see [56,78,93] and reference therein for the early work on this topic) with a rich range of applications in probability and ergodic theory. In fact, generalized T, T −1 transformations were used to exhibit examples of systems with unusual limit laws [64,27], central limit theorem with non standard normalization [11], K but non Bernoulli systems in abstract [57] and smooth setting in various dimensions [60,88,59], very weak Bernoulli but not weak Bernoulli partitions [29], slowly mixing systems [30, 76,33], systems with multiple Gibbs measures [43,77]. Here, we exhibit further ergodic and statistical properties of these systems.…”

Entropies of commuting transformations on Hilbert spaces

“…Unlike the classical entropy for N 2 -actions, Friedland's entropy is positive when the generators have finite entropy as single transformations. From the known results about Friedland's entropy, we can see that it is not an easy task to compute it, even for some "simple" examples (see for example [5,6,4]).…”

The dynamical zeta function for commuting automorphisms of zero-dimensional groups

“…The commutative algebra used to study a Z d -action α by continuous automorphisms of a compact abelian group X is now familiar (see, for example, Schmidt's monograph [24] and Einsiedler and Lind's paper [8] which is particularly useful when each automorphism α n has finite entropy, in which case α is said to be of entropy rank one. The starting point is to notice that the Pontryagin dual of X, denoted M = X, becomes a module over the Laurent polynomial ring R d = Z[u ±1 1 , .…”

“…Since M corresponds to an entropy rank one action, so does every submodule L ⊂ M and quotient M/L. Also, since M can be expressed as a countable increasing union of Noetherian submodules, [8,Prop. 4.4 and Prop.…”

“…3.4] and in many subsequent studies of algebraic Z d -actions. For example, it is a prototypical example of a Z d -action of a compact abelian zero-dimensional group where every element of the action has finite entropy, that is, an entropy rank one action [8]. If α is an entropy rank one action, then clearly h(α) = 0 if d 2.…”