A Note on the History of the Cantor Set and Cantor Function

1994 ◽  
Vol 67 (2) ◽  
pp. 136 ◽  
Author(s):  
Julian F. Fleron
Keyword(s):  
Cantor Set ◽  
Cantor Function ◽  
History Of

scholarly journals Characterization of the Local Growth of Two Cantor-Type Functions

2019 ◽  
Vol 3 (3) ◽  
pp. 45 ◽  
Author(s):  
Dimiter Prodanov
Keyword(s):  
Lebesgue Measure ◽  
Cantor Set ◽  
Positive Lebesgue Measure ◽  
Singular Function ◽  
Local Growth ◽  
Cantor Function ◽  
Physical Phenomena

The Cantor set and its homonymous function have been frequently utilized as examples for various physical phenomena occurring on discontinuous sets. This article characterizes the local growth of the Cantor’s singular function by means of its fractional velocity. It is demonstrated that the Cantor function has finite one-sided velocities, which are non-zero of the set of change of the function. In addition, a related singular function based on the Smith–Volterra–Cantor set is constructed. Its growth is characterized by one-sided derivatives. It is demonstrated that the continuity set of its derivative has a positive Lebesgue measure of 1/2.

scholarly journals On Hausdorff dimension of invariant sets for expanding maps of a circle

1986 ◽  
Vol 6 (2) ◽  
pp. 295-309 ◽  
Author(s):  
Mariusz Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Topological Entropy ◽  
Cantor Set ◽  
Invariant Sets ◽  
Expanding Maps ◽  
Cantor Function ◽  
The Family ◽  
Expanding Mapping

AbstractGiven an orientation preserving C2 expanding mapping g: S1 → Sl of a circle we consider the family of closed invariant sets Kg(ε) defined as those points whose forward trajectory avoids the interval (0, ε). We prove that topological entropy of g|Kg(ε) is a Cantor function of ε. If we consider the map g(z) = zq then the Hausdorff dimension of the corresponding Cantor set around a parameter ε in the space of parameters is equal to the Hausdorff dimension of Kg(ε). In § 3 we establish some relationships between the mappings g|Kg(ε) and the theory of β-transformations, and in the last section we consider DE-bifurcations related to the sets Kg(ε).

ANALYSIS ON A FRACTAL SET

Fractals ◽  
2009 ◽  
Vol 17 (01) ◽  
pp. 45-52 ◽  
Author(s):  
SANTANU RAUT ◽  
DHURJATI PRASAD DATTA
Keyword(s):  
Logarithmic Derivative ◽  
Asymptotic Properties ◽  
Cantor Set ◽  
Constant Function ◽  
Real Analysis ◽  
Scale Invariant ◽  
Absolute Value ◽  
Zero Measure ◽  
Cantor Function ◽  
Ordinary Derivative

The formulation of a new analysis on a zero measure Cantor set C(⊂I = [0,1]) is presented. A non-Archimedean absolute value is introduced in C exploiting the concept of relative infinitesimals and a scale invariant ultrametric valuation of the form log ε-1 (ε/x) for a given scale ε > 0 and infinitesimals 0 < x < ε, x ∈ I\C. Using this new absolute value, a valued (metric) measure is defined on C and is shown to be equal to the finite Hausdorff measure of the set, if it exists. The formulation of a scale invariant real analysis is also outlined, when the singleton {0} of the real line R is replaced by a zero measure Cantor set. The Cantor function is realized as a locally constant function in this setting. The ordinary derivative dx/dt in R is replaced by the scale invariant logarithmic derivative d log x/d log t on the set of valued infinitesimals. As a result, the ordinary real valued functions are expected to enjoy some novel asymptotic properties, which might have important applications in number theory and in other areas of mathematics.

scholarly journals NON-ARCHIMEDEAN SCALE INVARIANCE AND CANTOR SETS

Fractals ◽  
2010 ◽  
Vol 18 (01) ◽  
pp. 111-118 ◽  
Author(s):  
SANTANU RAUT ◽  
DHURJATI PRASAD DATTA
Keyword(s):  
Scale Invariance ◽  
Cantor Set ◽  
Constant Function ◽  
Cantor Sets ◽  
Scale Invariant ◽  
Absolute Value ◽  
Closed Intervals ◽  
Cantor Function ◽  
Invariant Analysis ◽  
Definition Of

The framework of a new scale invariant analysis on a Cantor set C ⊂ I = [0,1], presented recently1 is clarified and extended further. For an arbitrarily small ε > 0, elements [Formula: see text] in I\C satisfying [Formula: see text], x ∈ C together with an inversion rule are called relative infinitesimals relative to the scale ε. A non-archimedean absolute value [Formula: see text], ε → 0 is assigned to each such infinitesimal which is then shown to induce a non-archimedean structure in the full Cantor set C. A valued measure constructed using the new absolute value is shown to give rise to the finite Hausdorff measure of the set. The definition of differentiability on C in the non-archimedean sense is introduced. The associated Cantor function is shown to relate to the valuation on C which is then reinterpreated as a locally constant function in the extended non-archimedean space. The definitions and the constructions are verified explicitly on a Cantor set which is defined recursively from I deleting q number of open intervals each of length [Formula: see text] leaving out p numbers of closed intervals so that p + q = r.

Hausdorff dimension of sets of non-differentiability points of Cantor functions

1995 ◽  
Vol 117 (1) ◽  
pp. 185-191 ◽  
Author(s):  
Richard Darst
Keyword(s):  
Distribution Function ◽  
Hausdorff Dimension ◽  
Probability Measure ◽  
Relative Length ◽  
Cantor Set ◽  
Uniform Probability ◽  
Cantor Function ◽  
Lower Derivative

AbstractEach number a in the segment (0, ½) produces a Cantor set, Ca, by putting b = 1 − 2a and recursively removing segments of relative length b from the centres of the interval [0, 1] and the intervals which are subsequently generated. The distribution function of the uniform probability measure on Ca is a Cantor function, fa. When a = 1/3 = b, Ca is the standard Cantor set, C, and fa is the standard Cantor function, f. The upper derivative of f is infinite at each point of C and the lower derivative of f is infinite at most points of C in the following sense: the Hausdorff dimension of C is ln(2)/ln(3) and the Hausdorff dimension of S = {x ∈ C: the lower derivative of f is finite at x} is [ln(2)/ln(3)]2. The derivative of f is zero off C, the derivative of f is infinite on C — S, and S is the set of non-differentiability points of f. Similar results are established in this paper for all Ca: the Hausdorff dimension of Ca is ln (2)/ln (1/a) and the Hausdorff dimension of Sa is [ln (2)/ln (1/a)]2. Removing k segments of relative length b and leaving k + 1 intervals of relative length a produces a Cantor set of dimension ln(k + l)/ln(1/a); the dimension of the set of non-differentiability points of the corresponding Cantor function is [ln (k + l)/ln (1/a)]2.

scholarly journals On Differentiable Functions having an Everywhere Dense set of Intervals of Constancy

1965 ◽  
Vol 8 (1) ◽  
pp. 73-76 ◽  
Author(s):  
A. M. Bruckner ◽  
John L. Leonard
Keyword(s):  
Open Interval ◽  
Cantor Set ◽  
List Type ◽  
Type Number ◽  
Absolutely Continuous ◽  
Differentiable Functions ◽  
Cantor Function ◽  
Set Of Points ◽  
Increasing Function ◽  
Dense Set

The Cantor function C [2; p. 213], which appears in analysis as a simple example of a continuous increasing function which is not absolutely continuous, has the following properties:(i)C is defined on [0,1], with C(0) = 0, C (l) = l;(ii)C is continuous and non-decreasing on [0,1];(iii)C is constant on each interval contiguous to the perfect Cantor set P;(iv)C fails to be constant on any open interval containing points of P;(v)The set of points at which C is non-differentiable is non-denumerable.

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