The asymptotic shape theorem for the frog model on finitely generated abelian groups

We study the frog model on Cayley graphs of groups with polynomial growth rate $D \geq 3$. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one of these particles is active when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices. We prove that the activation time of particles grows at least linearly and we show that in the abelian case with any finite generator set the set of activated sites has a limiting shape.

Krieger’s finite generator theorem for actions of countable groups III

We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving (p.m.p.) actions of countablegroups introduced in Part I [B. Seward. Krieger’s finite generator theorem for actions of countable groups I. Invent. Math. 215(1) (2019), 265–310]. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semicontinuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov–Sinai entropy. In particular, using Rokhlin entropy we give a new proof of the fact that ergodic actions with positive sofic entropy have finite stabilizers.

An Estimation of the Logarithmic Timescale in Ergodic Dynamics

An estimation of the logarithmic timescale in quantum systems having an ergodic dynamics in the semiclassical limit, is presented. The estimation is based on an extension of the Krieger’s finite generator theorem for discretized [Formula: see text]-algebras and using the time rescaling property of the Kolmogorov–Sinai entropy. The results are in agreement with those obtained in the literature but with a simpler mathematics and within the context of the ergodic theory. Moreover, some consequences of the Poincaré’s recurrence theorem are also explored.

A remark to a division algorithm in the proof of Oka's first coherence theorem

The problem is the locally finite generation of a relation sheaf ℛ(τ1,…,τq) in 𝒪Cn. After τj reduced to Weierstrass' polynomials in zn, it is the key for applying an induction on n to show that elements of ℛ(τ1,…,τq) are expressed as a finite linear sum of zn-polynomial-like elements of degree at most p = max j deg zn τj over 𝒪Cn. In that proof one is used to use a division by τj of the maximum degree, deg zn τj = p (Oka (1948); Cartan (1950); Hörmander (1966); Narasimhan (1966); Nishino (1996), etc.) Here we shall confirm that the division above works by making use of τk of the minimum degree, min j deg zn τj, and show that there is a degree structure in the locally finite generator system. This proof is naturally compatible with the simple case when some τj is a unit, and gives some improvement in the degree estimate of generators.

Two facts concerning the transformations which satisfy the weak Pinsker property

We show that every ergodic, finite entropy transformation which satisfies the weak Pinsker property possesses a finite generator whose two-sided tail field is exactly the Pinsker algebra. This is proved by exhibiting a generator endowed with a block structure quite analogous to the one appearing in the construction of the Ornstein–Shields examples of non Bernoulli K-automorphisms. We also show that, given two transformations T1 and T2 in the previous class (i.e. satisfying the weak Pinsker property), and a Bernoulli shift B, if T1×B is isomorphic to T2×B, then T1 is isomorphic to T2. That is, one can ‘factor out’ Bernoulli shifts.

Entropy and finite generators for locally compact subshifts

We introduce transition entropy and periodic entropy for locally compact subshifts. Finiteness of both characterizes the existence of a finite generator. Finiteness of the transition entropy characterizes the existence of a generator with bounded degree. We extend Krieger's embedding theorem and characterize the locally compact non-dense subsystems of a (compact) mixing shift of finite type.