Non-ergodic interval exchange transformations

We construct interval exchange transformations on four intervals satisfying a strong irrationality condition and having exactly two ergodic invariant probability measures. This shows that although Kronecker’s theorem remains true for interval exchange transformations, the Weyl equidistribution theorem is false even under the strongest irrationality assumptions.     

Random interval graphs

We consider models for random interval graphs that are based on stochastic service systems, with vertices corresponding to customers and edges corresponding to pairs of customers that are in the system simultaneously. The number N of vertices in a connected component thus corresponds to the number of customers arriving during a busy period, while the size K of the largest clique (which for interval graphs is equal to the chromatic number) corresponds to the maximum number of customers in the system during a busy period. We obtain the following results for both the M/D/∞ and the M/M/∞ models, with arrival rate λ per mean service time. The expected number of vertices is e^λ, and the distribution of the N/e^λ converges pointwise to an exponential distribution with mean 1 as λ tends to infinity. This implies that the distribution of log N−λ converges pointwise to a distribution with mean −γ (where γ is Euler's constant) and variance π²/6. The size K of the largest clique falls in the interval [eλ−2 log λ, eλ+1] with probability tending to 1 as λ tends to infinity. Thus the distribution of the ratio K/log N converges pointwise to that of the constant e, in contrast to the situation for random graphs generated by unbiased coin flips, in which the distribution of K/log N converges pointwise to that of the constant 2/log 2. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 361–380, 1998     

Random interval graphs

In this paper we introduce a notion ofrandom interval graphs: the intersection graphs of real, compact intervals whose end points are chosen at random. We establish results about the number of edges, degrees, Hamiltonicity, chromatic number and independence number of almost all interval graphs.     

General random perturbations of hyperbolic and expanding transformations

Small random perturbations of a general form of diffeomorphisms having hyperbolic invariant sets and expanding maps are considered. The convergence of invariant measures of perturbations to the Sinaî-Bowen-Ruelle measure in the case of a hyperbolic attractor and to the smooth invariant measure in the expanding case are proved. The convergence of corresponding entropy characteristics and the approximation of the topological pressure by means of perturbations is considered as well.     

Random covering of an interval and a variation of Kingman's coalescent

We consider the covering of [0, 1] by a large number of small random intervals. We show that a simple variation of Kingman's coalescent describes the emergence of macroscopic connected components. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004     

Random perturbations of 2-dimensional hamiltonian flows

We consider the motion of a particle in a periodic two dimensional flow perturbed by small (molecular) diffusion. The flow is generated by a divergence free zero mean vector field. The long time behavior corresponds to the behavior of the homogenized process - that is diffusion process with the constant diffusion matrix (effective diffusivity). We obtain the asymptotics of the effective diffusivity when the molecular diffusion tends to zero.     

Local complexity functions of interval exchange transformations

We studied numerically complexity functions for interval exchange transformations. We have shown that they grow linearly in time as well as the ?-complexity function. Moreover, we found out that they depend also linearly on ? where ? is the Lebesgue measure of a set of initial points. This allows us to hypothesize that the dimension of the measure related to the ?-complexity function could be determined by studying the dependence of local complexity functions on ?.     

Random C 0 homeomorphism perturbations of hyperbolic sets

In this note we consider random C ⁰ homeomorphism perturbations of a hyperbolic set of a C ¹ diffeomorphism. We show that the hyperbolic set is semi-stable under such perturbations, in particular, the topological entropy will not decrease under such perturbations.     

An entropy estimate for infinite interval exchange transformations

We determine upper bound estimates for minimal entropy growth rates in measure-preserving systems by utilizing the universal representability of such systems by means of interval exchange transformations. Using these estimates, we also establish several criteria for the identification of systems with vanishing entropy.     

On singular perturbations for differential inclusions on the infinite interval

We consider a differential inclusion subject to a singular perturbation, i.e., part of the derivatives are multiplied by a small parameter >0. We show that under some stability and structural assumptions, every solution of the singularly perturbed inclusion comes close to a solution of the degenerate inclusion (obtained for =0) when tends to 0. The goal of the present paper is to provide a new result of Tikhonov type on the time interval [0,+∞[.     

Cocycles over interval exchange transformations and multivalued Hamiltonian flows

We consider interval exchange transformations of periodic type and construct different classes of ergodic cocycles of dimension?1 over this special class of IETs. Then using Poincaré sections we apply this construction to obtain the recurrence and ergodicity for some smooth flows on non-compact manifolds which are extensions of multivalued Hamiltonian flows on compact surfaces.     

Random perturbations of dynamical systems and diffusion processes with conservation laws

In this paper we consider random perturbations of dynamical systems and diffusion processes with a first integral. We calculate, under some assumptions, the limiting behavior of the slow component of the perturbed system in an appropriate time scale for a general class of perturbations. The phase space of the slow motion is a graph defined by the first integral. This is a natural generalization of the results concerning random perturbations of Hamiltonian systems. Considering diffusion processes as the unperturbed system allows to study the multidimensional case and leads to a new effect: the limiting slow motion can spend non-zero time at some points of the graph. In particular, such delay at the vertices leads to more general gluing conditions. Our approach allows one to obtain new results on singular perturbations of PDEs. Mathematics Subject Classification (2001): 60H10; 34C29; 35B20     

On the commutator group of the group of interval exchange transformations

We study the group of interval exchange transformations and obtain several characterizations of its commutator group. In particular, it turns out that the commutator group is generated by elements of order 2.     

Almost periodic Schrödinger operators along interval exchange transformations

It is shown that Schrödinger operators, with potentials along the shift embedding of irreducible interval exchange transformations in a dense set, have pure singular continuous spectrum for Lebesgue almost all points of the interval. Such potentials are natural generalizations of the Sturmian case.     

Cantor singular continuous spectrum for operators along interval exchange transformations

It is shown that Schrödinger operators, with potentials along the shift embedding of Lebesgue almost every interval exchange transformations, have Cantor spectrum of measure zero and pure singular continuous for Lebesgue almost all points of the interval.