The nature of generalized scales

The notion of a generalized scale emerged in recent joint work with Afsar–Brownlowe–Larsen on equilibrium states on [Formula: see text]-algebras of right Least Common Multiple (LCM) monoids, where it features as the key datum for the dynamics under investigation. This work provides the structure theory for such monoidal homomorphisms. We establish the uniqueness of the generalized scale and characterize its existence in terms of a simplicial graph arising from a new notion of irreducibility inside right LCM monoids. In addition, the method yields an explicit construction of the generalized scale if existent. We discuss applications for graph products as well as algebraic dynamical systems and reveal a striking connection to Saito’s degree map.

Entropy of MV-algebraic dynamical systems: An example

We give examples showing that the Kolmogorov-Sinai entropy generator theorem is false for both upper and lower Riesz entropy of MV-algebraic dynamical systems, both two sided (i.e., analogous to two sided Bernoulli shifts) and one sided (i.e., analogous to one sided Bernoulli shifts).

An almost mixing of all orders property of algebraic dynamical systems

We consider dynamical systems, consisting of $\mathbb{Z}^{2}$-actions by continuous automorphisms on shift-invariant subgroups of $\mathbb{F}_{p}^{\mathbb{Z}^{2}}$, where $\mathbb{F}_{p}$ is the field of order $p$. These systems provide natural generalizations of Ledrappier’s system, which was the first example of a 2-mixing $\mathbb{Z}^{2}$-action that is not 3-mixing. Extending the results from our previous work on Ledrappier’s example, we show that, under quite mild conditions (namely, 2-mixing and that the subgroup defining the system is a principal Markov subgroup), these systems are almost strongly mixing of every order in the following sense: for each order, one just needs to avoid certain effectively computable logarithmically small sets of times at which there is a substantial deviation from mixing of this order.

Surjunctivity and topological rigidity of algebraic dynamical systems

Let$X$be a compact metrizable group and let$\unicode[STIX]{x1D6E4}$be a countable group acting on$X$by continuous group automorphisms. We give sufficient conditions under which the dynamical system$(X,\unicode[STIX]{x1D6E4})$is surjunctive, i.e. every injective continuous map$\unicode[STIX]{x1D70F}:X\rightarrow X$commuting with the action of$\unicode[STIX]{x1D6E4}$is surjective.