Computer simulat ions of dynamical systems can cont ain bot h discretizations, in which finite machine arithmet ic replaces cont inuum state spaces , and realizations , in which a continuous syste m is replaced by some approximation such as a computational method. In some circumstances, complicated theoret ical behavior collapses to t rivial and degenerate behavior. In others, the computation m...

In this paper we give sufficient conditions for weak isomorphism of Lebesgue measure-preserving expanding endomorphisms of S1.

Abstract. In this paper we recall a non-standard construction of the Borel sigma-algebraB in [0, 1] and construct a family of measures (in particular, Lebesgue measure) in B by a completely non-topological method. This approach, that goes back to Steinhaus, in 1923, is now used to introduce natural generalizations of the concept of normal numbers and explore their intrinsic probabilistic proper...

This paper deals with chain graphs under the classic Lauritzen-Wermuth-Frydenberg interpretation. We prove that the strictly positive discrete probability distributions with the prescribed sample space that factorize according to a chain graph G with dimension d have positive Lebesgue measure wrt R, whereas those that factorize according to G but are not faithful to it have zero Lebesgue measur...

Let D be a domain in C and φ a holomorphic automorphism of D. Let C be the measure class of the Lebesgue measure in D, i.e., the set of all positive regular Borel measures on D whose null sets coincide with the Lebesgue null sets. Let φ∗ be the automorphism of C given by (φ∗μ)(B) = μ(φ−1(B)) (μ ∈ C , B ∈ B(D)), where B(D) denotes the Borel σ-algebra of D. Adopting the terminology introduced in ...

We investigate the spectral properties of Schrödinger operators in `2(Z) with limit-periodic potentials. The perspective we take was recently proposed by Avila and is based on regarding such potentials as generated by continuous sampling along the orbits of a minimal translation of a Cantor group. This point of view allows one to separate the base dynamics and the sampling function. We show tha...

We prove that the free additive convolution of two Borel probability measures supported on the real line can have a component that is singular continuous with respect to the Lebesgue measure on R only if one of the two measures is a point mass. The density of the absolutely continuous part with respect to the Lebesgue measure is shown to be analytic wherever positive and finite. The atoms of th...

In this paper, Lebesgue decomposition type theorems for non-additive measure are shown under the conditions of null-additivity, converse null-additivity, weak null-additivity and σ-null-additivity, etc.. In our discussion, the monotone continuity of set function is not required.

This booklet is an exposition on the Lebesgue integral. I originally started it as a set of notes consolidating what I had learned on on Lebesgue integration theory, and published them in case somebody else may find them useful. Since there are already countless books on measure theory and integration written by professional mathematicians, that teach the same things on the basic level, you may...

We give two versions of the natural extension of a specific greedy β-transformation with deleted digits. We use the natural extension to obtain an explicit expression for the invariant measure, equivalent to the Lebesgue measure, of this β-transformation.