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

This article is cited in 17 scientific papers (total in 17 papers)

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.


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

Bibliographic databases:

UDC: 517.51
MSC: 26A16, 15A60, 28A80, 39B22
Received: 30.09.2003

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
\by V.~Yu.~Protasov
\paper On the regularity of de~Rham curves
\jour Izv. RAN. Ser. Mat.
\yr 2004
\vol 68
\issue 3
\pages 139--180
\jour Izv. Math.
\yr 2004
\vol 68
\issue 3
\pages 567--606

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    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  crossref  mathscinet  isi  scopus
    2. V. Yu. Protasov, “Fractal curves and wavelets”, Izv. Math., 70:5 (2006), 975–1013  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. Maesumi M., “Optimal norms and the computation of joint spectral radius of matrices”, Linear Algebra Appl., 428:10 (2008), 2324–2338  crossref  mathscinet  zmath  isi  elib  scopus
    4. Protasov V.Yu., “Extremal $L_p$-norms of linear operators and self-similar functions”, Linear Algebra Appl., 428:10 (2008), 2339–2356  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    7. Yu. A. Alpin, “Bounds for Joint Spectral Radii of a Set of Nonnegative Matrices”, Math. Notes, 87:1 (2010), 12–14  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    9. V. Yu. Protasov, “Invariant Functionals for Random Matrices”, Funct. Anal. Appl., 44:3 (2010), 230–233  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. A. S. Voynov, “Self-affine polytopes. Applications to functional equations and matrix theory”, Sb. Math., 202:10 (2011), 1413–1439  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. M. Ben Slimane, “The thermodynamic formalism for the de Rham function: increment method”, Izv. Math., 76:3 (2012), 431–445  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. Guglielmi N., Protasov V., “Exact Computation of Joint Spectral Characteristics of Linear Operators”, Found. Comput. Math., 13:1 (2013), 37–97  crossref  mathscinet  zmath  isi  elib  scopus
    13. V.Yu. Protasov, R.M. Jungers, “Lower and upper bounds for the largest Lyapunov exponent of matrices”, Linear Algebra and its Applications, 2013  crossref  mathscinet  isi  scopus
    14. Kazuki Okamura, “Singularity Results for Functional Equations Driven by Linear Fractional Transformations”, J Theor Probab, 2013  crossref  mathscinet  scopus
    15. J. Bochi, I. D. Morris, “Continuity properties of the lower spectral radius”, Proceedings of the London Mathematical Society, 2014  crossref  scopus
    16. Protasov V.Yu. Voynov A.S., “Matrix semigroups with constant spectral radius”, Linear Alg. Appl., 513 (2017), 376–408  crossref  mathscinet  zmath  isi  elib  scopus
    17. Barany B. Kiss G. Kolossvary I., “Pointwise Regularity of Parameterized Affine Zipper Fractal Curves”, Nonlinearity, 31:4 (2018), 1705–1733  crossref  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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