Related papers: A constructive Borel-Cantelli Lemma. Constructing …
The classical Birkhoff ergodic theorem in its most popular version says that the time average along a single typical trajectory of a dynamical system is equal to the space average with respect to the ergodic invariant distribution. This…
There are well-known analogs of the prime number theorem and Mertens' theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of…
All constructive methods employed in modern mathematics produce only countable sets, even when designed to transcend countability. We show that any constructive argument for uncountability -- excluding diagonalization techniques --…
In this paper we consider finite dimensional dynamical systems generated by a Lipschitz function. We prove a version of the Whitney's Extension Theorem on compact manifolds to obtain a version of the well-known Lambda Lemma for Lipschitz…
We study the computational problem of rigorously describing the asymptotic behaviour of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit,…
We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $\phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $\phi$…
The existence of a real linear-space structure on the set of observables of a quantum system -- i.e., the requirement that the linear combination of two generally non-commuting observables $A,B$ is an observable as well -- is a fundamental…
The predictions of quantum mechanics cannot be resolved with a completely classical view of the world. In particular, the statistics of space-like separated measurements on entangled quantum systems violate a Bell inequality. We put forward…
A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics. Namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics,…
We survey an area of recent development, relating dynamics to theoretical computer science. We discuss the theoretical limits of simulation and computation of interesting quantities in dynamical systems. We will focus on central objects of…
We study the averaging method for flows perturbed by a dynamical system preserving an infinite measure. Motivated by the case of perturbation by the collision dynamic on the finite horizon $\mathbb Z$-periodic Lorentz gas and in view of…
We study the strong Borel-Cantelli property both for events and for shifts on sequence spaces considering both a conventional and a nonconventional setups. Namely, under certain conditions on events $\Gamma_1,\Gamma_2,...$ we show that with…
We reprove the countable splitting lemma by adapting Nawrotzki's algorithm which produces a sequence that converges to a solution. Our algorithm combines Nawrotzki's approach with taking finite cuts. It is constructive in the sense that…
It can be argued that the ordinary description of the reversible quantum process between two one-to-one correlated measurement outcomes is incomplete because, by not specifying the direction of causality, it allows causal structures that…
The purpose of this paper is to provide a first class of explicit sufficient conditions for the central limit theorem and related results in the setup of non-uniformly (partially) expanding non iid random transformations, considered as…
Kim defined a very general combinatorial abstraction of the diameter of polytopes called subset partition graphs to study how certain combinatorial properties of such graphs may be achieved in lower bound constructions. Using Lov\'asz'…
We construct a smooth nontrivial mixed partially hyperbolic system and explicitly identify its skeleton. This example shares characteristics with the classical examples. Moreover, the support of each physical measure contains three fixed…
A sequence $\{x_{n}\}_1^\infty$ in $[0,1)$ is called Borel-Cantelli (BC) if for all non-increasing sequences of positive real numbers $\{a_n\}$ with $\underset{i=1}{\overset{\infty}{\sum}}a_i=\infty$ the set…
Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic…
We prove, for stably computably enumerable formal systems, direct analogues of the first and second incompleteness theorems of G\"odel. A typical stably computably enumerable set is the set of Diophantine equations with no integer…