On the lebesgue measure of the periodic points of a contact manifold

Abstract In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

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A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

Mathematics ◽

10.3390/math9131546 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1546

Author(s):

Mohsen Soltanifar

Keyword(s):

Hausdorff Dimension ◽

Lebesgue Measure ◽

Virtual World ◽

Fractal Dimensions ◽

Definition Of ◽

The Given

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

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Rational Periodic Points of the Quadratic Function Q c (x) = x 2 + c

American Mathematical Monthly ◽

10.2307/2975624 ◽

1994 ◽

Vol 101 (4) ◽

pp. 318 ◽

Cited By ~ 3

Author(s):

Ralph Walde ◽

Paula Russo

Keyword(s):

Quadratic Function ◽

Periodic Points

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Consistency conditions for a finite set of projections of a function

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410005550x ◽

1979 ◽

Vol 85 (1) ◽

pp. 61-68 ◽

Cited By ~ 5

Author(s):

K. J. Falconer

Keyword(s):

Lebesgue Measure ◽

Convex Subset ◽

Compact Convex ◽

Unit Vector ◽

Compact Convex Subset ◽

Finite Collection ◽

Finite Set ◽

Almost All ◽

Dimensional Lebesgue Measure ◽

Unit Vectors

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

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A topological lens for a measure-preserving system

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385709000984 ◽

2010 ◽

Vol 31 (1) ◽

pp. 49-75 ◽

Cited By ~ 1

Author(s):

E. GLASNER ◽

M. LEMAŃCZYK ◽

B. WEISS

Keyword(s):

Topological Entropy ◽

Periodic Points ◽

Positive Entropy ◽

Topological Properties ◽

Topologically Transitive ◽

Topological System ◽

Zero Entropy ◽

Weakly Mixing ◽

Dynamical Properties

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

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The Lebesgue measure of the algebraic difference of two random Cantor sets

Indagationes Mathematicae ◽

10.1016/s0019-3577(09)80007-4 ◽

2009 ◽

Vol 20 (1) ◽

pp. 131-149 ◽

Cited By ~ 3

Author(s):

Péter Móra ◽

Károly Simon ◽

Boris Solomyak

Keyword(s):

Lebesgue Measure ◽

Cantor Sets ◽

Random Cantor Sets ◽

Algebraic Difference

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Powers of the ideal of Lebesgue measure zero sets

Journal of Symbolic Logic ◽

10.2307/2274906 ◽

1991 ◽

Vol 56 (1) ◽

pp. 103-107

Author(s):

Maxim R. Burke

Keyword(s):

Partial Order ◽

Lebesgue Measure ◽

Regular Cardinal ◽

Measure Zero ◽

Lebesgue Measure Zero ◽

Zero Sets ◽

Covering Lemma ◽

Inner Model ◽

The Ideal

AbstractWe investigate the cofinality of the partial order κ of functions from a regular cardinal κ into the ideal of Lebesgue measure zero subsets of R. We show that when add () = κ and the covering lemma holds with respect to an inner model of GCH, then cf (κ) = max{cf(κκ), cf([cf()]κ)}. We also give an example to show that the covering assumption cannot be removed.

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