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Scenery Flow, Conical Densities, and Rectifiability

  • Conference paper
Fractal Geometry and Stochastics V

Part of the book series: Progress in Probability ((PRPR,volume 70))

Abstract

We present an application of the recently developed ergodic theoretic machinery on scenery flows to a classical geometric measure theoretic problem in Euclidean spaces. We also review the enhancements to the theory required in our work. Our main result is a sharp version of the conical density theorem, which we reduce to a question on rectifiability.

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References

  1. T. Bedford, A.M. Fisher, On the magnification of Cantor sets and their limit models. Monatsh. Math. 121(1–2), 11–40 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. T. Bedford, A.M. Fisher, Ratio geometry, rigidity and the scenery process for hyperbolic Cantor sets. Ergod. Theory Dyn. Syst. 17(3), 531–564 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. T. Bedford, A.M. Fisher, M. Urbański, The scenery flow for hyperbolic Julia sets. Proc. Lond. Math. Soc. (3) 85(2), 467–492 (2002)

    Google Scholar 

  4. A.S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points. Math. Ann. 98(1), 422–464 (1928)

    Article  MathSciNet  Google Scholar 

  5. A.S. Besicovitch, On linear sets of points of fractional dimension. Math. Ann. 101(1), 161–193 (1929)

    Article  MathSciNet  MATH  Google Scholar 

  6. A.S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points II. Math. Ann. 115, 296–329 (1938)

    Article  MathSciNet  Google Scholar 

  7. A. Ferguson, J. Fraser, T. Sahlsten, Scaling scenery of (×m, ×n) invariant measures. Adv. Math. 268, 564–602 (2015)

    Google Scholar 

  8. A.M. Fisher, Small-scale structure via flows, in Fractal Geometry and Stochastics III. Volume 57 of Progress in Probability (Birkhäuser, Basel, 2004), pp. 59–78

    Google Scholar 

  9. H. Furstenberg, Intersections of Cantor sets and transversality of semigroups, in Problems in Analysis (Sympos. Salomon Bochner, Princeton University, Princeton, 1969) (Princeton University Press, Princeton, 1970), pp. 41–59

    Google Scholar 

  10. H. Furstenberg, Ergodic fractal measures and dimension conservation. Ergod. Theory Dyn. Syst. 28(2), 405–422 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Gavish, Measures with uniform scaling scenery. Ergod. Theory Dyn. Syst. 31(1), 33–48 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Hochman, Dynamics on fractals and fractal distributions (2010, preprint). Available at http://arxiv.org/abs/1008.3731

  13. M. Hochman, P. Shmerkin, Equidistribution from fractal measures. Invent. Math. (Published Online)

    Google Scholar 

  14. A. Käenmäki, On upper conical density results, in Recent Developments in Fractals and Related Fields, ed. by J.J. Benedetto, J. Barral, S. Seuret. Applied and Numerical Harmonic Analysis (Birkhäuser, Boston, 2010), pp. 45–54

    Google Scholar 

  15. A. Käenmäki, T. Sahlsten, P. Shmerkin, Dynamics of the scenery flow and geometry of measures. Proc. Lond. Math. Soc. 110(5), 1248–1280 (2015)

    Article  MathSciNet  Google Scholar 

  16. A. Käenmäki, T. Sahlsten, P. Shmerkin, Structure of distributions generated by the scenery flow. J. Lond. Math. Soc. 91(2), 464–494 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. J.M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. Lond. Math. Soc. (3) 4, 257–301 (1954)

    Google Scholar 

  18. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability (Cambridge University Press, Cambridge, 1995)

    Book  MATH  Google Scholar 

  19. P. Mörters, D. Preiss, Tangent measure distributions of fractal measures. Math. Ann. 312(1), 53–93 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. D. Preiss, Geometry of measures in R n: distribution, rectifiability, and densities. Ann. Math. (2) 125(3), 537–643 (1987)

    Google Scholar 

  21. U. Zähle, Self-similar random measures and carrying dimension, in Proceedings of the Conference: Topology and Measure, V, Binz, 1987. Wissenschaftliche Beiträge der Ernst-Moritz-Arndt Universität Greifswald (DDR) (Greifswald, 1988), pp. 84–87

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Jyväskylä, 35 (MaD), FI-40014, Jyväskylä, Finland

    Antti Käenmäki

Authors
  1. Antti Käenmäki
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Correspondence to Antti Käenmäki .

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Editors and Affiliations

  1. Institut für Mathematik und Informatik, Universität Greifswald, Greifswald, Germany

    Christoph Bandt

  2. Mathematical Institute, University of St Andrews, St Andrews, United Kingdom

    Kenneth Falconer

  3. Institut für Mathematik, Friedrich-Schiller-Universität Jena, Jena, Germany

    Martina Zähle

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Käenmäki, A. (2015). Scenery Flow, Conical Densities, and Rectifiability. In: Bandt, C., Falconer, K., Zähle, M. (eds) Fractal Geometry and Stochastics V. Progress in Probability, vol 70. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-18660-3_2

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