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 ...

We study the interaction between polynomial space randomness and a fundamental result of analysis, the Lebesgue differentiation theorem. We generalize Ko’s framework for polynomial space computability in Rn to define weakly pspace-random points, a new variant of polynomial space randomness. We show that the Lebesgue differentiation theorem holds for every weakly pspace-random point.

We study the interaction between polynomial space randomness and a fundamental result of analysis, the Lebesgue differentiation theorem. We generalize Ko’s framework for polynomial space computability in R to define weakly pspace-random points, a new variant of polynomial space randomness. We show that the Lebesgue differentiation theorem characterizes weakly pspace random points. That is, a po...

We denote by x a real variable and by n a positive integer variable. The reference measure on the real line R is the Lebesgue measure. In this note we will use only basic properties of the Lebesgue measure and integral on R. It is well known that the fact that a function tends to zero at infinity is a condition neither necessary nor sufficient for this function to be integrable. However, we hav...

In this paper, we shall establish some fixed point theorems for mappings with the contractive  condition of integrable type on complete intuitionistic fuzzy metric spaces $(X, M,N,*,lozenge)$. We also use Lebesgue-integrable mapping to obtain new results. Akram, Zafar, and Siddiqui introduced the notion of $A$-contraction mapping on metric space. In this paper by using the main idea of the work...

Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L2 and two constants c1, c2 > 0 such that for all x, y ∈ G we have ‖ f (x)− f (y)‖2 ≥ c1d(x, y)− c2. Tessara [16] proved that the Hilbert compression exponent of...

Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L2 and a constant c > 0 such that for all x, y ∈ G we have ‖ f (x) − f (y)‖2 ≥ cd(x, y). In [2] it was shown that the Hilbert compression exponent of the wreath ...