Related papers: Orbit equivalence for Cantor minimal Z^d-systems
We introduce the o-minimal LS-category of definable sets in o-minimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the o-minimal LS-category of definable groups. Along…
We investigate projections to odometers (group rotations over adic groups) of topological invertible dynamical systems with discrete time and compact Hausdorff phase space. For a dynamical system $(X, f)$ with a compact phase space we…
We study conditions under which the germinal groupoid associated to a minimal equicontinuous action of a countable group on a Cantor set has non-Hausdorff topology. We develop a new criterion, which serves as an obstruction for the \'etale…
We interpret the close link between the critical points of Mather's barrier functions and minimal homoclinic orbits with respect to the Aubry sets on $\mathbb{T}^n$. We also prove a critical point theorem for barrier functions, and the…
In this paper, we study the action of non abelian group G generated by affine homotheties on R^n. We prove that G satisfies one of the following properties: (i) there exist a subgroup F_{G} of R\{0} containing 0 in its closure, a…
In this paper, we study the action of non abelian group G generated by affine homotheties on R^n. We prove that G satisfies one of the following properties: (i) there exist a subgroup F_{G} of R\{0} containing 0 in its closure, a…
We give a new proof of quantifier elimination in the theory of all ordered abelian groups in a suitable language. More precisely, this is only "quantifier elimination relative to ordered sets" in the following sense. Each definable set in…
We study several aspects of higher-order regionally proximal relations for group actions. First, we develop an algebraic approach to study higher-order regionally proximal relations. To this end, we introduce a new topology on a subgroup of…
We introduce new systems that we call odomutants, built by distorting the orbits of an odometer. We use these transformations for flexibility results in quantitative orbit equivalence. It follows from the work of Kerr and Li that if the…
Schreier graphs of the actions of Thompson's group $F$ on the orbits of all points of the unit interval and of the Cantor set with respect to the standard generating set $\{x_0,x_1\}$ are explicitly constructed. The closure of the space of…
We extend the result of Nadel describing the relationship between approximations of canonical Scott sentences and admissible sets to the general case of orbit equivalence relations induced on an arbitrary Polish space by a Polish group…
Equationally compact subgroups of countable groups were introduced by Banaschewski. For all known cases the orbit closure of such a subgroup is a countable subset in the space of subgroups and has finite Cantor-Bendixson rank. We show that…
Motivated by Furstenberg's Theorem on sets in the circle invariant under multiplication by a non-lacunary semigroup, we define a general class of dynamical systems possessing similar topological dynamical properties. We call such systems…
Let $\Gamma$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $\Gamma$ on the circle is either trivial or semi-conjugate to a unique minimal action on the so-called simple circle.
In this paper, we completely solve the problem when a Cantor dynamical system $(X,f)$ can be embedded in $\mathbb{R}$ with vanishing derivative everywhere. For this purpose, we construct a refining sequence of marked clopen partitions of…
Let $(\mathcal{V},\mathcal{W};F)$ be a weakly reducible, unstabilized, Heegaard splitting of genus at least three in an orientable, irreducible $3$-manifold $M$. Then $Mod(M,F)$ naturally acts on the disk complex $\mathcal{D}(F)$ as a group…
Given an ergodic probability measure preserving dynamical system $\G\acts (X,\mu)$, where $\G$ is a finitely generated countable group, we show that the asymptotic growth of the number of finite models for the dynamics, in the sense of…
We construct an uncountable family of transversely Cantor laminations of compact spaces defined by free minimal actions of solvable groups, which are not affable and whose orbits are not quasi-isometric to Cayley graphs.
In this note we show that every definably connected, definably compact abelian definable group in an o-minimal expansion of a real closed field of dimension not 4 is definably homeomorphic to a torus of the same dimension. Moreover, in the…
It is shown that, modulo an equivalence relation induced by finite correspondences preserving Cantor rank, the class of topological spaces is an integral semi-ring on which the Cantor derivative is precisely a derivation.