Differential and integral calculus on time scales allows to develop a theory of dynamic equations in order to unify and extend the usual differential equations and difference equations. For single variable differential and integral calculus on time scales, we refer the reader to the textbooks [4, 5] and the references given therein. Multivariable calculus on time scales was developed by the aut...

We investigate the prevalence of Li-Yorke pairs for C and C multimodal maps f with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If f is topologically mixing and has no Cantor attractor, then typical (...

Many modern multiobjective evolutionary algorithms (MOEAs) store the points discovered during optimization in an external archive, separate from the main population, as a source of innovation and/or for presentation at the end of a run. Maintaining a bound on the size of the archive may be desirable or necessary for several reasons, but choosing which points to discard and which to keep in the ...

We consider the sigma-finite measures in the space of vector-valued distributions on the manifold X with characteristic functional

Using techniques introduced by C. Güntürk, we prove that the attractors of a family of overlapping self-affine iterated function systems contain a neighbourhood of zero for all parameters in a certain range. This corresponds to giving conditions under which a single sequence may serve as a ‘simultaneous β-expansion’ of different numbers in different bases.

• The exterior measure of any set A ⊂ R is μ∗(A) = inf E⊂A μ(E) where the infimum is taken over all elementary sets. • A is Lebesgue measurable if for all > 0 there exists an open set O ⊃ A such that μ∗(O \A) ≤ . • The (Lebesgue) measure of a measurable set is μ(A) = μ∗(A). A function f is measurable if the sets {t | f(t) ≤ a} are measurable for all a ∈ R. Definition 2 (Lebesgue integral) • For...

We calculate the Lebesgue–measures of the stability sets of Nashequilibria in pure coordination games. The results allow us to observe that the ordering induced by the Lebesgue–measure of stability sets upon strict Nashequilibria does not necessarily agree with the ordering induced by risk–dominance. Accordingly, an equilibrium selection theory based on the Lebesgue–measure of stability sets wo...

We shall start by giving the definition of the entropy of dynamical system. Consider dynamical systems with discrete time. The phase space of dynamical system is denoted by M . It is equipped with σ-algebra M and a probability measure μ defined on M. In the general ergodic theory dynamics is given by a measurable transformation T of M onto itself preserving the measure μ. It is enough for many ...

In this note we shall give examples of rational maps on the Riemann sphere and also of polynomial interval maps which are transitive but not ergodic with respect to Lebesgue measure. In fact, these maps have two disjoint compact attrac-tors whose attractive basins arèintermingled', each having a positive Lebesgue measure in every open set. In addition, we show that there exists a real bi-modal ...