Related papers: Every transformation is disjoint from almost every…

We prove ergodicity in a class of skew-product extensions of interval exchange transformations given by cocycles with logarithmic singularities. This, in particular, gives explicit examples of ergodic $\mathbb{R}$-extensions of minimal…

In this paper we introduce and explore the notion of rigidity group, associated with a collection of finitely many sequences, and show that this concept has many, somewhat surprising characterizations of algebraic, spectral, and unitary…

Let $(X, T)$ be a topological dynamical system. We show that if each invariant measure of $(X, T)$ gives rise to a measure-theoretic dynamical system that is either: a. rigid along a sequence of "bounded prime volume" or b. admits a…

We study the ergodic properties of compositions of interval exchange transformations and rotations. We show that for any interval exchange transformation T, there is a full measure set of \alpha in [0, 1) so that T composed with R_{\alpha}…

We define a diophantine condition for interval exchange transformations (i.e.t.s). When the number of intervals is two, that is for rotations on the circle, our condition coincides with classical Khinchin condition. We prove for i.e.t.s the…

The goal of this article is to show a rigidity property of conjugacies of generalized interval exchange transformations (GIETs). More precisely, we show that if two piecewise $C^3$ GIETs $f$ and $g$ of generic rotation number with…

For $n\geq 1$ we consider the class JP($n$) of dynamical systems whose every ergodic joining with a Cartesian product of $k$ weakly mixing automorphisms ($k\geq n$) can be represented as the independent extension of a joining of the system…

An element $f$ of a group $G$ is reversible if it is conjugated in $G$ to its own inverse; when the conjugating map is an involution, $f$ is called strongly reversible. We describe reversible maps in certain groups of interval exchange…

We study the class $Erg^\perp$ of automorphisms which are disjoint with all ergodic systems. We prove that the identities are the only multipliers of $Erg^\perp,$ that is, each automorphism whose every joining with an element of…

Recently, G\'{o}rska, Lema\'{n}czyk, and de la Rue characterized the class of automorphisms disjoint from all ergodic automorphisms. Inspired by their work, we provide several characterizations of systems that are disjoint from all minimal…

We prove that almost every interval exchange transformation, with an associated translation surface of genus $g\geq 2$, can be non-trivially and isometrically embedded in a family of piecewise isometries. In particular this proves the…

Motivated by Berg's notion of quasi-disjointness for ergodic systems, we introduce and investigate the concept of quasi-disjointness for minimal systems. Several equivalent characterizations are provided. We prove that quasi-disjointness is…

Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In…

We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space.…

In this note, we investigate the coboundaries of interval exchange transformations of 3 intervals (3-IETs). More precisely, we show that a differentiable function with absolutely continuous derivative with bounded variation, whose integral…

We introduce a novel method for proving ergodicity for skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by…

We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric $g$ is geodesically rigid: if any other complete metric $\bar g$ has the same (unparametrized) geodesics with $g$, then the Levi-Civita connections of $g$ and…

We generalize Berg's notion of quasi-disjointness to actions of countable groups and prove that every measurably distal system is quasi-disjoint from every measure preserving system. As a corollary we obtain easy to check necessary and…

We show that ergodic dynamical systems generated by infinitely divisible stationary processes are disjoint in the sense of Furstenberg with distally simple systems and systems whose maximal spectral type is singular with respect to the…

We prove a rigidity result for foliations on surfaces of genus two, which can be seen as a generalization to higher genus of Herman's theorem on circle diffeomorphisms and, correspondingly, flows on the torus. We prove in particular that,…