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This essay advocates the view that any problem that has a meaningful empirical content, can be formulated in constructive, more definitely, finite terms. We consider combinatorial models of dynamical systems and approaches to statistical…

We propose a new method that extends conservative explicit multirate methods to implicit explicit-multirate methods. We develop extensions of order one and two with different stability properties on the implicit side. The method is suitable…

In this paper we study ergodic theory of countable Markov shifts. These are dynamical systems defined over non-compact spaces. Our main result relates the escape of mass, the measure theoretic entropy, and the entropy at infinity of the…

We consider conservative ergodic measure preserving transformations on infinite measure spaces and investigate the asymptotic behaviour of distorted return time processes with respect to sets satisfying a type of Darling-Kac condition. We…

We present cut and project formalism based on measures and continuous weight functions of sufficiently fast decay. The emerging measures are strongly almost periodic. The corresponding dynamical systems are compact groups and homomorphic…

The Infinitesimal Calculus explores mainly two measurements: the instantaneous rates of change and the accumulation of quantities. This work shows that scientists, engineers, mathematicians, and teachers increasingly apply another change…

We re-address the problem of construction of new infinite-dimensional completely integrable systems on the basis of known ones, and we reveal a working mechanism for such transitions. By splitting the problem's solution in two steps, we…

It is well known that ergodic invariant measures for order preserving two-sided time random dynamical systems(RDS) on the real line $\mathbb R$ are Dirac. In the present note this is shown to hold also for one-sided time RDS.

Reciprocity is a fundamental principle follows the time reversal symmetry of physics. However, many practical applications require breaking time reversal symmetry, hence, are called nonreciprocal. This article aims at discussing time…

We prove almost sure ergodic theorems for a class of systems called quasistatic dynamical systems. These results are needed, because the usual theorem due to Birkhoff does not apply in the absence of invariant measures. We also introduce…

In these notes, we present an alternative version of discrete Dirac mechanics using Dirac structures. We first establish a notion of 'continuous Dirac system' and then propose a definition of discrete Dirac system, proving that it is…

We present a new paradigm for capturing the complementarity of two observables. It is based on the entanglement created by the interaction between the system observed and the two measurement devices used to measure the observables…

We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic…

We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics of rank two cluster algebras. These systems exhibit behavior associated with integrability, namely preservation of a symplectic form,…

Transseries expansions build upon ordinary power series methods by including additional basis elements such as exponentials and logarithms. Alternative summation methods can then be used to "resum" series to obtain more efficient…

Dirac's approach to gauge symmetries is discussed. We follow closely the steps that led him from his conjecture concerning the generators of gauge transformations {\it at a given time} --to be contrasted with the common view of gauge…

A simple construction is given of a class of Euclidean invariant, reflection positive measures on a compactification of the space of distributions. An unusual feature is that the regularizations used are not reflection positive.

A framework that structures the gravitational memory effects and which is consistent with gravitational electric-magnetic duality is presented. A correspondence is described between memory observables, particular subleading residual gauge…

We prove that multiple-recurrence and polynomial-recurrence of invertible infinite measure preserving transformations are both properties which pass to extensions.

We consider random dynamical systems such as groups of conformal transformations with a probability measure, or transversaly conformal foliations with a Laplace operator along the leaves, in which case we consider the holonomy pseudo-group.…