On regularity for de Rham’s functional equations

Okamura, Kazuki

doi:10.1007/s00010-016-0439-6

Cited by 4 publications

(4 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem 3.9 generalizes a modified statement of [O16, Theorem 1] and is also related to [Ha85, Theorems 7.3 and 7.5] and [Z01,Theorems 6 and 7]. [Ha85,Z01,O16] deal with the case that X = [0, 1], however, our result is also applicable to the case that X is not [0, 1]. We deal with the case that X is the two-dimensional standard Sierpiński gasket.…”

Section: Introductionmentioning

confidence: 60%

Some results for conjugate equations

Okamura

2018

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper we consider a class of conjugate equations, which generalizes de Rham's functional equations. We give sufficient conditions for existence and uniqueness of solutions under two different series of assumptions. We consider regularity of solutions. In our framework, two iterated function systems are associated with a series of conjugate equations. We state local regularity by using the invariant measures of the two iterated function systems with a common probability vector. We give several examples, especially an example such that infinitely many solutions exists, and a new class of fractal functions on the two-dimensional standard Sierpiński gasket which are not harmonic functions or fractal interpolation functions. We also consider a certain kind of stability. Contents 1. Introduction 1 2. Existence and uniqueness 3 3. Regularity 6 4. Examples for existence and uniqueness 14 5. Examples for regularity 19 6. Stability 23 7. Open problems 26 References 27

show abstract

Section: Introductionmentioning

confidence: 60%

Some results for conjugate equations

Okamura

2018

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…(ii) The approaches in [Ha85,SLK04,Ok16] are different from the one used here. As an outline level, they are somewhat similar to each other.…”

Section: De Rham's Functional Equations Driven By N Linear Fractionalmentioning

confidence: 99%

Hausdorff Dimensions for Graph-Directed Measures Driven by Infinite Rooted Trees

Okamura¹

2020

Real Analysis Exchange

View full text Add to dashboard Cite

We give upper and lower bounds for the Hausdorff dimensions for a class of graph-directed measures when its underlying directed graph is the infinite N -ary tree. These measures are different from graph-directed self-similar measures driven by finite directed graphs and are not necessarily Gibbs measures. However our class contains several measures appearing in fractal geometry and functional equations, specifically, measures defined by restrictions of non-constant harmonic functions on the two-dimensional Sierpínski gasket, the Kusuoka energy measures on it, and, measures defined by solutions of de Rham's functional equations driven by linear fractional transformations.

show abstract

“…Protasov [34,35] proved in a more general context that the set of points x ∈ [0, 1] for which α(x) = β has full measure only if β = α, otherwise it has zero measure. Just recently, Okamura [31] bounds α(x) for Lebesgue typical points allowing in the definition (5.2) more than two functions and also non-linear functions under some conditions.…”

Section: An Example De Rham's Curvementioning

confidence: 99%

“…to study the possible values, which occur as (regular) pointwise Hölder exponents, and determine the magnitude of the sets, where it appears. This property was studied for several types of singular functions, for example for wavelets by Barral and Seuret [4], Seuret [38], for Weierstrass-type functions Otani [32], for complex analogues of the Takagi function by Jaerisch and Sumi [22] or for different functional equations by Coiffard, Melot and Willer [10], by Okamura [31] and by Slimane [7] etc.…”

Section: Introduction and Statementsmentioning

confidence: 99%

Pointwise regularity of parameterized affine zipper fractal curves

2018

View full text Add to dashboard Cite

We study the pointwise regularity of zipper fractal curves generated by affine mappings. Under the assumption of dominated splitting of index-1, we calculate the Hausdorff dimension of the level sets of the pointwise Hölder exponent for a subinterval of the spectrum. We give an equivalent characterization for the existence of regular pointwise Hölder exponent for Lebesgue almost every point. In this case, we extend the multifractal analysis to the full spectrum. In particular, we apply our results for de Rham's curve.

show abstract

On regularity for de Rham’s functional equations

Cited by 4 publications

References 13 publications

Some results for conjugate equations

Some results for conjugate equations

Hausdorff Dimensions for Graph-Directed Measures Driven by Infinite Rooted Trees

Pointwise regularity of parameterized affine zipper fractal curves

Contact Info

Product

Resources

About