Related papers: Cocycles over interval exchange transformations an…
Hamiltonian systems are a classical example in the ergodic theory of flows with an invariant measure. In this matter, we present a brief introduction to measure theory and prove the Poincare recurrence theorem to present the conditions for…
Considerable research has led to ergodic isothermal dynamics which can replicate Gibbs' canonical distribution for simple ( small ) dynamical problems. Adding one or two thermostat forces to the Hamiltonian motion equations can give an…
We construct geometrically infinite hyperbolic surfaces supporting horocycles with tailored recurrence properties. In particular, we obtain the first examples of non-trivial minimal horocyclic orbit closures and of infinite locally-finite…
To an ergodic, essentially free and measure-preserving action of a non-amenable Baumslag-Solitar group on a standard probability space, a flow is associated. The isomorphism class of the flow is shown to be an invariant of such actions of…
Properties of Hamiltonian symmetry flows on hyperbolic Euler-type Liouvillean equations E' are analyzed. Description of their Noether symmetries assigned to the integrals for these equations is obtained. The integrals provide Miura…
In our paper we study periodic geodesic motion on multidimensional ellipsoids with elastic impacts along confocal quadrics. We show that the method of isoperiodic deformation is applicable.
We obtain expansions of ergodic integrals for $\Z^d$-covers of compact self-similar translation flows, and as a consequence we obtain a form of weak rational ergodicity with optimal rates. As examples, we consider the so-called self-similar…
We obtain a description of the Poincar\'e recurrences of chaotic systems in terms of the ergodic theory of transient chaos. It is based on the equivalence between the recurrence time distribution and an escape time distribution obtained by…
Many authors have constructed different, but related, linear group cocycles that are usually referred to as ``Eisenstein cocycles.'' The main goal of this work is to describe a topological construction that is a common source for all these…
We present the first explicit example of an interval exchange transformation with flips (FIET) possessing three distinct invariant ergodic measures. The proof is based on a generalization of M. Keane's method, using the Rauzy induction…
We analyse the intersection of positively and negatively sectional-hyperbolic sets for flows on compact manifolds. First we prove that such an intersection is hyperbolic if the intersecting sets are both transitive (this is false without…
We define a notion of equivariant non-degeneracy of $G$-maps to introduce the class of equivariantly non-degenerate flows on smooth compact manifolds with compact Lie group action. We prove genericity of this class and use this result to…
We consider impulsive semiflows and establish sufficient conditions to the existence of invariant measures. Namely, the impulsive set and its image are both submanifolds of codimension one that are transversal to the flow direction.…
We give a criterion which allows to prove non-ergodicity for certain infinite periodic billiards and directional flows on Z-periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with…
We study the evolution, under convex Hamiltonian flows on cotangent bundles of compact manifolds, of certain distinguished subsets of the phase space. These subsets are generalizations of Lagrangian graphs, we call them pseudographs. They…
We construct a Poincar\'e section for the horocycle flow on the modular surface $SL(2, \R)/SL(2, \Z)$, and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We…
We introduce a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds using pseudoholomorphic curve techniques from symplectic…
We study dynamics of the horocycle flow on strata of translation surfaces, introduce new invariants for ergodic measures, and analyze the interaction of the horocycle flow and real Rel surgeries. We use this analysis to complete and extend…
The main result of this work is the following: for volume preserving flows on compact manifolds with the $C^r$ topology, $1 \leqq r \leqq \infty$ , the closure of every invariant manifold of periodic orbits and singularities is a chain…
We describe an averaging procedure on a Dirac manifold, with respect to a class of compatible actions of a compact Lie group. Some averaging theorems on the existence of invariant realizations of Poisson structures around (singular)…