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Based on T.Tao's result of norm convergence of multiple ergodic averages for commut-ing transformation, we obtain there is a subsequence which converges almost everywhere. Meanwhile, the ergodic behaviour, which the time average is equal to…
This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure…
We employ an extension of ergodic theory to the random setting to investigate the existence of random periodic solutions of random dynamical systems. Given that a random dynamical system has a dissipative structure, we proved that a random…
There is a fundamental relationship between belief propagation and maximum a posteriori decoding. A decoding algorithm, which we call the Maxwell decoder, is introduced and provides a constructive description of this relationship. Both, the…
We study the ergodic properties (recurrence, discrepancy, diffusion coefficients and ergodicity itself) of a class of $\mathbb Z$-extensions over infinite interval exchange transformations called rotated odometers. The choice of a…
We introduce a family of dynamical systems that generate negative $\beta$-expansions and study the support of the invariant measure which is absolutely continuous with respect to Lebesgue measure. We give a characterization of the set of…
This paper contains results on geometric Routh reduction and it is a continuation of a previous paper where a new class of transformations is introduced between Lagrangian systems obtained after Routh reduction. In general, these reduced…
We examine some of the properties of uniformly rigid transformations, and analyze the compatibility of uniform rigidity and (measurable) weak mixing along with some of their asymptotic convergence properties. We show that on Cantor space,…
In this paper, based on a construction by J. Fickenscher, we construct a family of non-uniquely ergodic interval exchange transformations on $n$ intervals with the maximal possible number of measures, $\left\lfloor \frac{n}{2}…
Dissipative phenomena manifest in multiple mechanical systems. In this dissertation, different geometric frameworks for modelling non-conservative dynamics are considered. The objective is to generalize several results from conservative…
In this paper we study in detail, both analytically and numerically, the dynamical properties of the triangle map, a piecewise parabolic automorphism of the two-dimensional torus, for different values of the two independent parameters…
The structural invariant subspaces of the discrete-time singular Hamiltonian system are used in 1] to give an analytic nonrecursive expression of all the admissible trajectories. A deeper insight into the features of these subspaces,…
We define extension maps as maps that extend a system (through adding ancillary systems) without changing the state in the original system. We show, using extension maps, why a completely positive operation on an initially entangled system…
Measure structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure structured deformation is defined via relaxation departing either from energies associated with…
We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and study its properties. Further, we study how the invariant measures depend on the…
Taking into account the recent developments associated with duality in physics, this article is focused on investigating the properties of a tensor generalization of the electrodynamics dual to the standard vector model even considering the…
Holomorphic (nondegenerate) mappings between complex manifolds of the same dimension are of special interest. For example, they appear as coverings of complex manifolds. At the same time they have very strong "extra" extension properties in…
Shortly after Szemer\'edi's proof that a set of positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof of this theorem using ergodic theory. This gave rise to the field of ergodic Ramsey…
Nonreciprocal transport in superconducting systems serves as a powerful probe of symmetry-breaking mechanisms, with the superconducting diode effect emerging as a key manifestation enabling cryogenic rectification. While theoretical models…
We study some new dynamical systems where the corresponding piecewise linear flow is neither time reversible nor measure preserving. We create a dissipative system by starting with a finite polysquare translation surface, and then modifying…