1. Motivation and introduction. In [1], the concept of random entropy associated with a Zd random group action was introduced and studied. Every such Zd random group action is generated via a cocycle. (For the readers with a probabilistic background, a cocycle is a generalization of an ordinary random walk, the main difference being the fact that cocycles are generally indexed by Zd rather than...

In this paper we use the description of free group factors as the von Neumann algebras of Berezin’s deformation of the upper half-plane, modulo PSL (2, Z). The derivative, in the deformation parameter, of the product in the corresponding algebras, is a positive Hochschild 2-cocycle, defined on a dense subalgebra. By analyzing the structure of the cocycle we prove that there is a generator L for...

Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study algebraic structures with replicating operations. To understand their combinatorial nature, we first use rooted forests multiple decoration sets to construct free Hochschild 1-cocycle conditions. Applying universal property underlying operated method Gr\"obner-Shirshov bases, then obj...

We will show that any two outer actions of Z on O2 are cocycle conjugate.

We will show that any two outer actions of Z on O2 are cocycle conjugate.

Given α in some set Σ of total (Haar) measure in T = R/Z, and A ∈ C(T, SL(2,R)) which is homotopic to the identity, we prove that if the fibered rotation number of the skew-product system (α,A) : T × SL(2,R) → T × SL(2,R), (α,A)(θ, y) = (θ + α,A(θ)y) is diophantine with respect to α and if the fibered products are uniformly bounded in the C0-topology then the cocycle (α,A) is C∞reducible –that ...

In this paper wewill consider SL(2,R)-valued cocycles over aminimal homeomorphism f : K → K of a compact set K. Such a cocycle can be defined as a pair ( f ,A) where A : K → SL(2,R) is continuous. The cocycle acts on K ×R2 by (x, y) 7→ ( f (x),A(x) · y). The iterates of the cocycle are denoted ( f ,A) = ( f ,An). We say that ( f ,A) is uniformly hyperbolic if there exists ε > 0 and N > 0 such t...

We complete a description of additive partition entropies. This time we do not assume I to be continous. In the process we solve a 2-cocycle functional equation for certain subsets of convex cones. 1. Introduction. We assume all definitions and results from [3]. In the current follow-up paper we drop the assumption of the continuity of the additve partition entropy. Our aim is to prove a theore...

We show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates. Then we consider sets Σ in GL(d, R) with the property that any cocycle with values in Σ has a dominated splitting. We characterize these sets in terms of existence of invariant multicones, thus extending a 2-dimensional result by Avila, Bochi, and Yoccoz. W...

We use braided groups to introduce a theory of ∗-structures on general inhomogeneous quantum groups, which we formulate as quasi-∗ Hopf algebras. This allows the construction of the tensor product of unitary representations up to a quantum cocycle isomorphism, which is a novel feature of the inhomogeneous case. Examples include q-Poincaré quantum group enveloping algebras in R-matrix form appro...