Related papers: Affine transformation with zero entropy and nilsys…
We provide an estimate of the amenable category of oriented closed connected complete affine manifolds whose fundamental group contains an infinite amenable normal subgroup. As an application we show that all such manifolds have zero…
Under the assumption of the gluing orbit property, equivalent conditions to having zero topological entropy are investigated. In particular, we show that a dynamical system has the gluing orbit property and zero topological entropy if and…
Ray nil-affine geometries are defined on nilpotent spaces. They occur in every parabolic geometry and in those cases, the nilpotent space is an open dense subset of the corresponding flag manifold. We are interested in closed manifolds…
We present a method to compute the group of affine transformations of a homogeneous $G$-space under specific conditions: when the group $G$ and the homogeneous $G$-space admit linear connections so that the natural projection is affine, and…
In this paper, we show that a compact affine manifold endowed with an Affine Anosov transformation is finitely covered by a complete affine nilmanifold.
In this paper we study topological rigidity of affine actions on compact connected metrizable abelian groups. We also classify one-parameter flows of translations upto orbit equivalence and discrete group actions by translations upto…
Let X_1 and X_2 be mixing connected algebraic dynamical systems with the Descending Chain Condition. We show that every equivariant continuous map X_1 to X_2 is affine (that is, X_2 is topologically rigid) if and only if the system X_2 has…
Affine manifolds are called integral if there is an atlas such that all transition maps are affine transformations with integer matrices of linear parts. In this paper we describe all complete integral affine structures on compact…
The topological entropy of piecewise affine maps is studied. It is shown that singularities may contribute to the entropy only if there is angular expansion and we bound the entropy via the expansion rates of the map. As a corollary we…
Extending our results in "Entropy conjecture for continuous maps of nilmanifolds", to appear in Israel Jour. of Math., we confirm that Entropy Conjecture holds for every continuous self-map of a compact $K(\pi,1)$ manifold with the…
We discuss conditions under which an affine automorphism of a compact nilmanifold is almost automorphic, and the structure of such automorphisms from a dynamical point of view.
For Euclidean pure point diffractive Delone sets of finite local complexity and with uniform patch frequencies it is well known that the patch counting entropy computed along the closed centred balls is zero. We consider such sets in the…
We classify affine varieties with an action of a connected, reductive algebraic group such that the group is isomorphic to an open orbit in the variety. This is accomplished by associating a set of one-parameter subgroups of the group to…
We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view,…
This note is purely expository. A subset N of the plane is affine ambient homogeneous if for each x,y in N there exists an affine transformation taking x to y and N to itself. The result of D. Repovs, E. V. Scepin and the author on such…
The relevance of the algebraic entropy in the study of birational discrete time dynamical systems highlights the need to relate it to other characteristics of these systems. In this letter, two complementary proofs are given that the…
In this paper we construct an equivariant embedding of the affine space $\mathbb{A}^n$ with the translation group action into a complete non-projective algebraic variety $X$ for all $n \geq 3$. The theory of toric varieties is used as the…
We show that for a $\mathbb{Z}^{l}$-action (or $(\N\cup\{0\})^l$-action) on a non-empty compact metrizable space $\Omega$, the existence of a affine space dense in the set of continuous functions on $\Omega$ constituted by elements…
In this paper, we show that if a group acts isometrically on a good hyperbolic space of finite volume entropy through a non-elementary action, then it admits an affine action on some $L^p$ -space with an unbounded orbit for sufficiently…
We prove that any compact complex manifold with finite fundamental group and algebraic dimension zero admits no holomorphic affine connection.