Although Sarnak's conjecture holds for compact group rotations (irrational rotations, odometers), it is not even known whether it holds for all Jewett-Krieger models of such rotations. In this paper we show that it does, as long as the model is at the same a topological extension, via the same map that establishes the isomorphism, of an equicontinuous model. In particular, we recover (after [AKL]) that regular Toeplitz systems satisfy Sarnak's conjecture, and, as another consequence, so do...

We compare self-joining and embeddability properties. In particular, we prove that a measure preserving flow ${\left(T_t\right)}_{t\in ℝ}$ with T₁ ergodic is 2-fold quasi-simple (resp. 2-fold distally simple) if and only if T₁ is 2-fold quasi-simple (resp. 2-fold distally simple). We also show that the Furstenberg-Zimmer decomposition for a flow ${\left(T_t\right)}_{t\in ℝ}$ with T₁ ergodic with respect to any flow factor is the same for ${\left(T_t\right)}_{t\in ℝ}$ and for T₁. We give an example of a 2-fold quasi-simple flow disjoint from simple flows and whose time-one map is...

We define the concept of directional entropy for arbitrary ${\mathbb{Z}}^2$-actions on a Lebesgue space, we examine its basic properties and consider its behaviour in the class of product actions and rigid actions.

We show that for any cellular automaton (CA) ℤ²-action Φ on the space of all doubly infinite sequences with values in a finite set A, determined by an automaton rule $F=F_{[l,r]}$, l,r ∈ ℤ, l ≤ r, and any Φ-invariant Borel probability measure, the directional entropy $h_{v⃗}\left(\Phi \right)$, v⃗= (x,y) ∈ ℝ², is bounded above by $max\left(\right|z_l|,|z_r\left|\right)logA$ if $z_l{z}_r\ge 0$ and by $|z_r-z_l|$ in the opposite case, where $z_l=x+ly$, $z_r=x+ry$. We also show that in the class of permutative CA-actions the bounds are attained if the measure considered is uniform Bernoulli.

We show that for a finitely generated group of C² circle diffeomorphisms, the entropy of the action equals the entropy of the restriction of the action to the non-wandering set.

The Hudetz correction of the fuzzy entropy is applied to the $g$-entropy. The new invariant is expressed by the Hudetz correction of fuzzy entropy.

For a continuous map T of a compact metrizable space X with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the context of entropy structures and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if M is a metrizable Choquet simplex,...