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 Izv. RAN. Ser. Mat., 2004, Volume 68, Issue 3, Pages 139–180 (Mi izv489)

On the regularity of de Rham curves

V. Yu. Protasov

M. V. Lomonosov Moscow State University

Abstract: De Rham curves are obtained from a polygonal arc by passing to the limit in repeatedly cutting off the corners: at each step, the segments of the arc are divided into three pieces in the ratio $\omega:(1-2\omega):\omega$, where $\omega\in(0,1/2)$ is a given parameter. We find explicitly the sharp exponent of regularity of such a curve for any $\omega$. Regularity is understood in the natural parametrization using the arclength as a parameter. We also obtain a formula for the local regularity of a de Rham curve at each point and describe the sets of points with given local regularity. In particular, we characterize the sets of points with the largest and the smallest local regularity. The average regularity, which is attained almost everywhere in the Lebesgue measure, is computed in terms of the Lyapunov exponent of certain linear operators. We obtain an integral formula for the average regularity and derive upper and lower bounds.

DOI: https://doi.org/10.4213/im489

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English version:
Izvestiya: Mathematics, 2004, 68:3, 567–606

Bibliographic databases:

UDC: 517.51
MSC: 26A16, 15A60, 28A80, 39B22

Citation: V. Yu. Protasov, “On the regularity of de Rham curves”, Izv. RAN. Ser. Mat., 68:3 (2004), 139–180; Izv. Math., 68:3 (2004), 567–606

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv489
• https://doi.org/10.4213/im489
• http://mi.mathnet.ru/eng/izv/v68/i3/p139

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Protasov V., “Applications of the joint spectral radius to some problems of functional analysis, probability and combinatorics”, 44th IEEE Conference on Decision and Control & European Control Conference, 2005, 3025–3030
2. V. Yu. Protasov, “Fractal curves and wavelets”, Izv. Math., 70:5 (2006), 975–1013
3. Maesumi M., “Optimal norms and the computation of joint spectral radius of matrices”, Linear Algebra Appl., 428:10 (2008), 2324–2338
4. Protasov V.Yu., “Extremal $L_p$-norms of linear operators and self-similar functions”, Linear Algebra Appl., 428:10 (2008), 2339–2356
5. Jungers R.M., Protasov V.Yu., Blondel V.D., “Computing the growth of the number of overlap-free words with spectra of matrices”, Latin 2008: Theoretical Informatics, Lecture Notes in Computer Science, 4957, 2008, 84–93
6. Jungers R.M., Protasov V.Y., Blondel V.D., “Overlap-free words and spectra of matrices”, Theoret. Comput. Sci., 410:38-40 (2009), 3670–3684
7. Yu. A. Alpin, “Bounds for Joint Spectral Radii of a Set of Nonnegative Matrices”, Math. Notes, 87:1 (2010), 12–14
8. Protasov V.Y., Jungers R.M., Blondel V.D., “Joint Spectral Characteristics of Matrices: A Conic Programming Approach”, SIAM J. Matrix Anal. Appl., 31:4 (2010), 2146–2162
9. V. Yu. Protasov, “Invariant Functionals for Random Matrices”, Funct. Anal. Appl., 44:3 (2010), 230–233
10. A. S. Voynov, “Self-affine polytopes. Applications to functional equations and matrix theory”, Sb. Math., 202:10 (2011), 1413–1439
11. M. Ben Slimane, “The thermodynamic formalism for the de Rham function: increment method”, Izv. Math., 76:3 (2012), 431–445
12. Guglielmi N., Protasov V., “Exact Computation of Joint Spectral Characteristics of Linear Operators”, Found. Comput. Math., 13:1 (2013), 37–97
13. V.Yu. Protasov, R.M. Jungers, “Lower and upper bounds for the largest Lyapunov exponent of matrices”, Linear Algebra and its Applications, 2013
14. Kazuki Okamura, “Singularity Results for Functional Equations Driven by Linear Fractional Transformations”, J Theor Probab, 2013
15. J. Bochi, I. D. Morris, “Continuity properties of the lower spectral radius”, Proceedings of the London Mathematical Society, 2014
16. Protasov V.Yu. Voynov A.S., “Matrix semigroups with constant spectral radius”, Linear Alg. Appl., 513 (2017), 376–408
17. Barany B. Kiss G. Kolossvary I., “Pointwise Regularity of Parameterized Affine Zipper Fractal Curves”, Nonlinearity, 31:4 (2018), 1705–1733
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