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A note on measure-geometric Laplacians

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Abstract

We consider the measure-geometric Laplacians \(\Delta ^{\mu }\) with respect to atomless compactly supported Borel probability measures \(\mu \) as introduced by Freiberg and Zähle (Potential Anal. 16(1):265–277, 2002) and show that the harmonic calculus of \(\Delta ^{\mu }\) can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of \(\Delta ^{\mu }\). Further, it is shown that there exists a measure-geometric Laplacian whose eigenfunctions are the Chebyshev polynomials and illustrate our results through specific examples of fractal measures, namely inhomogeneous self-similar Cantor measures and Salem measures.

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References

  1. Arzt, P.: Measure theoretic trigonometric functions. J. Fractal Geom. 2, 115–169 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bandt, C., Barnsley, M., Hegland, M., Vince, A.: Conjugacies provided by fractal transformations I: Conjugate measures, Hilbert spaces, orthogonal expansions, and flows, on self-referential spaces (2014). arXiv:1409.3309

  3. Barlow, M.T., Perkins, E.A.: Brownian motion on the Sierpiński gasket. Probab. Theory Relat. Fields 79(4), 543–623 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bird, E.J., Ngai, S.-M., Teplyaev, A.: Fractal Laplacians on the unit interval. Ann. Sci. Math. Quebec 27, 135–168 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Denker, M., Sato, H.: Sierpiński gasket as a Martin boundary I Martin kernels. Potential Anal. 14(3), 211–232 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Denker, M., Sato, H.: Sierpiński gasket as a Martin boundary II. The intrinsic metric. Publ. Res. Inst. Math. Sci. 35(5), 769–794 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications, 3rd edn. Wiley, New York (2014)

    MATH  Google Scholar 

  8. Feller, W.: Generalized second order differential operators and their lateral conditions. Ill. J. Math. 1, 459–504 (1957)

    MathSciNet  MATH  Google Scholar 

  9. Freiberg, U.: Spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets. Forum Math. 17, 87–104 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Freiberg, U., Zähle, M.: Harmonic calculus on fractals: a measure geometric approach I. Potential Anal. 16(1), 265–277 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Goldstein, S.: Random walks and diffusions on fractals. In: Percolation Theory and Ergodic Theory of Infinite Particle Systems (Minneapolis, Minn., 1984–1985), vol. 8 of IMA vol. Math. Appl., pp. 121–129. Springer, New York (1987)

  12. Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jordan, T., Kesseböhmer, M., Pollicott, M., Stratmann, B.O.: Sets of non-differentiability for conjugacies between expanding interval maps. Fundam. Math. 206, 161–183 (2009)

    Article  MATH  Google Scholar 

  14. Ju, H., Lau, K.-S., Wang, X.-Y.: Post-critically finite fractal and Martin boundary. Trans. Am. Math. Soc. 364(1), 103–118 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kac, I.S., Kreĭn, G.: On the spectral functions of the string. Am. Math. Soc. Transl. 103(1), 19–102 (1974)

    Article  MATH  Google Scholar 

  16. Kesseböhmer, M., Stratmann, B.O.: Hölder-differentiability of Gibbs distribution functions. Math. Proc. Cambridge Philos. Soc. 147(2), 489–503 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kigami, J.: Analysis on Fractals. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  18. Kusuoka, S.: A Diffusion Process on a Fractal. Probabilistic Methods in Mathematical Physics (Katata/Kyoto, 1985), pp. 251–274. Academic Press, Cambridge (1987)

    Google Scholar 

  19. Lindstrøm, T.: Brownian motion on nested fractals. Memoirs of the American Mathematical Society, vol. 83, no 420. American Mathematical Society, Providence, RI (1990)

  20. Saad, Y.: Numerical Methods for Large Eigenvalue Problems. Algorithms and Architectures for Advanced Scientific Computing. Wiley, New York (1992)

    Google Scholar 

  21. Salem, R.: On some singular monotonic functions which are strictly increasing. Trans. Am. Math. Soc. 53, 427–439 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  22. Strichartz, R.S.: Differential Equations on Fractals: A Tutorial. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  23. Teschl, G.: Ordinary differential equations and dynamical systems. Graduate Studies in Mathematics, vol. 140. American Mathematical Society, Providence, RI (2012)

  24. Zähle, M.: Harmonic calculus on fractals: a measure geometric approach II. Trans. Am. Math. Soc. 357(9), 3407–3423 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. FB3, Mathematik, Universität Bremen, Bibliothekstr. 1, 28359, Bremen, Germany

    M. Kesseböhmer & H. Weyer

  2. Mathematics Department, California Polytechnic State University, San Luis Obispo, CA, 93407-0403, USA

    T. Samuel

Authors
  1. M. Kesseböhmer
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  2. T. Samuel
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  3. H. Weyer
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Correspondence to T. Samuel.

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Communicated by A. Constantin.

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Kesseböhmer, M., Samuel, T. & Weyer, H. A note on measure-geometric Laplacians. Monatsh Math 181, 643–655 (2016). https://doi.org/10.1007/s00605-016-0906-0

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