The trajectory of the turning point is dense for almost all tent maps

1996 ◽  
Vol 16 (6) ◽  
pp. 1173-1183 ◽  
Author(s):  
Karen Brucks ◽  
Michal Misiurewicz
Keyword(s):  
Lebesgue Measure ◽  
Turning Point ◽  
Tent Map ◽  
Tent Maps ◽  
Almost All

AbstractWe prove that for almost every (with respect to the Lebesgue measure) a ∈ [√2, 2], the forward trajectory of the turning point of the tent map fa with slope a is dense in the interval of transitivity of fa.

TRAJECTORY OF THE TURNING POINT IS DENSE FOR ALMOST ALL TENT MAPS

1993 ◽  
Vol 19 (1) ◽  
pp. 15
Author(s):  
Brucks ◽  
Misiurewicz
Keyword(s):  
Turning Point ◽  
Tent Maps ◽  
Almost All

scholarly journals A point is normal for almost all maps βx+α mod 1 or generalized β-transformations

2009 ◽  
Vol 29 (5) ◽  
pp. 1529-1547 ◽  
Author(s):  
B. FALLER ◽  
C.-E. PFISTER
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Critical Orbit ◽  
Maximal Entropy ◽  
Tent Map ◽  
Almost All ◽  
Analytic Curves ◽  
Measure Of Maximal Entropy

AbstractWe consider the map Tα,β(x):=βx+α mod 1, which admits a unique probability measure μα,β of maximal entropy. For x∈[0,1], we show that the orbit of x is μα,β-normal for almost all (α,β)∈[0,1)×(1,∞) (with respect to Lebesgue measure). Nevertheless, we construct analytic curves in [0,1)×(1,∞) along which the orbit of x=0 is μα,β-normal at no more than one point. These curves are disjoint and fill the set [0,1)×(1,∞). We also study the generalized β-transformations (in particular, the tent map). We show that the critical orbit x=1 is normal with respect to the measure of maximal entropy for almost all β.

Consistency conditions for a finite set of projections of a function

1979 ◽  
Vol 85 (1) ◽  
pp. 61-68 ◽  
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.

A Family of Generalized Riesz Products

1996 ◽  
Vol 48 (2) ◽  
pp. 302-315 ◽  
Author(s):  
A. H. Dooley ◽  
S. J. Eigen
Keyword(s):  
Lebesgue Measure ◽  
Spectral Measure ◽  
Probability Space ◽  
Rank One ◽  
Almost All ◽  
Riesz Products

AbstractGeneralized Riesz products similar to the type which arise as the spectral measure for a rank-one transformation are studied. A condition for the mutual singularity of two such measures is given. As an application, a probability space of transformations is presented in which almost all transformations are singular with respect to Lebesgue measure.

‘Mangaldeep’: Spreading Fragrance in India

2016 ◽  
Vol 5 (1) ◽  
pp. 108-115
Author(s):  
Bijoylaxmi Sarmah ◽  
Zillur Rahman
Keyword(s):  
Turning Point ◽  
Sources Of Information ◽  
Know How ◽  
Secondary Sources ◽  
Societal Needs ◽  
To Come ◽  
Societal Problems ◽  
Almost All ◽  
Corporate Social ◽  
Indian Tobacco

This case highlights Indian Tobacco Corporation (ITC)’s journey from being a pure leaf tobacco selling company to a reputed conglomerate with popular brands in diversified areas. ITC’s corporate social responsibility (CSR) and sustainability activities taking a turning point with the company taking an immense interest in integrating societal problems in its company’s policies and strategies. These transformations can be seen in almost all the business divisions of ITC. Mangaldeep division, an incense stick division is not an exception to this change. However, the authors are trying to analyze the activities of ITC–Mangaldeep Business unit from different perspectives such as CSR, sustainability and shared value initiatives. Considering the resource constraint and the demand to meet the societal needs, it will be quite interesting to know how both these two challenges are met by a conglomerate like ITC simultaneously in the days to come. The case uses both primary and secondary sources of information to develop this teaching case.

Decidability of the “almost all” theory of degrees

1972 ◽  
Vol 37 (3) ◽  
pp. 501-506 ◽  
Author(s):  
John Stillwell
Keyword(s):  
Lebesgue Measure ◽  
Measure Theory ◽  
Random Sequence ◽  
Recursion Theory ◽  
Sharp Contrast ◽  
Standard Theory ◽  
Simple Manner ◽  
Fubini’S Theorem ◽  
Almost All ◽  
Random Pair

Ever since Spector's brilliant application of measure theory to recursion theory in 1958 [6] it has been realized that measure theory promotes sweeping simplifications in the theory of degrees. Results previously thought to be pathological were shown by Spector, and later Sacks [4], [5], to hold for almost all degrees (“almost all” in the sense of Lebesgue measure), often with much simpler proofs. Good examples of this phenomenon are Spector's demonstration that almost all pairs of sets are of incomparable degree (as an immediate consequence of Fubini's theorem) and Sacks' exquisitely simple deduction from this result that almost every degree is the join of two incomparable degrees (for if a random sequence is decomposed into its even and odd parts, the result is a random pair).The present paper attempts to vindicate the feeling that almost all degrees behave in a simple manner by showing that if the quantifier in the theory of degrees with ′(jump), ∪ (join) and ∩ (meet) is taken to be (almost ∀a) instead of (∀a) then the theory is decidable. We are able to use ∩ because it will be shown that if t1, t2 are any terms built from degree variables a1, …, am with ′ and ∪ then t1 ∩ t2 exists for almost all a1, …, am. Thus the “almost all” theory presents a sharp contrast to the standard theory, where ∩ is not always defined (Kleene-Post [1]) and which is known to be undecidable (Lachlan [2]).

Some theorems in the metric theory of diophantine approximation

1986 ◽  
Vol 99 (3) ◽  
pp. 385-394 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Additional Constraint ◽  
Number Of Solutions ◽  
Initial Question ◽  
Image Position ◽  
Almost All

An excellent introduction to the metric theory of diophantine approximation is provided by [19], where, in chapter 1·7, the reader may find a discussion of the first two problems considered in this paper. Our initial question concerns the number of solutions of the inequalityfor almost all α(in the sense of Lebesgue measure on ℝ). Here ∥ ∥ denotes distance to a nearest integer, {βr}, {ar} are given sequences of reals and distinct integers respectively, and f is a function taking values in [0, ½] and with Σf(r) divergent (for convenience we write ℱ for the set of all such functions). It is reasonable to expect that, for almost all α and with some additional constraint on f, the number of solutions of (1) is asymptotically equal toas k tends to infinity.

Similarities and Differences between West and East German Primers, 1945–1949

2018 ◽  
Vol 11 (1) ◽  
pp. 73-96
Author(s):  
Verena Stürmer
Keyword(s):  
Comparative Analysis ◽  
Everyday Life ◽  
Turning Point ◽  
History Of Reading ◽  
Children's Play ◽  
Children’S Play ◽  
History Of ◽  
East German ◽  
Similarities And Differences ◽  
Almost All

The ban on almost all previously approved textbooks in occupied Germany in 1945 brought about a turning point in the history of reading primers in this country. This article examines the requirements that textbooks had to fulfill in order to be approved by the authorities of the various occupation zones. In spite of differing sociopolitical and pedagogical attitudes and conditions, reading primersin all occupied zones shared the theme of children’s play and harmonious everyday life. However, a comparative analysis of the primers reveals significant differences that cannot be explained exclusively as a consequence of influence exerted by occupying powers. Rather, these differences resulted from the context in which each primer appeared.

scholarly journals An alternative approach to the ergodic theory of measured foliations on surfaces

1981 ◽  
Vol 1 (4) ◽  
pp. 461-488 ◽  
Author(s):  
Mary Rees
Keyword(s):  
Projective Space ◽  
Lebesgue Measure ◽  
Ergodic Theory ◽  
Diffeomorphism Group ◽  
Alternative Approach ◽  
Interval Exchanges ◽  
Almost All ◽  
Uniquely Ergodic

AbstractWe consider measured foliations on surfaces, and interval exchanges. We give alternative proofs of the following theorems first proved by Masur and (independently) Veech. The action of the diffeomorphism group of the surface on the projective space of measured foliations (with respect to a natural ‘Lebesgue’ measure) is ergodic. Almost all measured foliations are uniquely ergodic. Almost all interval exchanges (again, with respect to a natural ‘Lebesgue’ measure) are uniquely ergodic.

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