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July  2015, 35(7): 2797-2815. doi: 10.3934/dcds.2015.35.2797

On the asymptotics of the scenery flow

Magnus Aspenberg 1, , Fredrik Ekström 1, , Tomas Persson 1, and  Jörg Schmeling 2,

1. 

Centre for Mathematical Sciences, Lund University, Box 118, 22 100 Lund, Sweden, Sweden, Sweden
2. 

Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-221 00 Lund

Received  June 2014 Revised  December 2014 Published  January 2015

We study the asymptotics of the scenery flow. We give corrected versions with proofs of a certain lemma by Hochman, and study some related phenomena.
Keywords: interval map, invariant measure, fractal geometry, Hausdorff dimension., Scenery flow.
Mathematics Subject Classification: Primary: 37A05, 28A8.
Citation: Magnus Aspenberg, Fredrik Ekström, Tomas Persson, Jörg Schmeling. On the asymptotics of the scenery flow. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2797-2815. doi: 10.3934/dcds.2015.35.2797
References:
[1]

C. Bandt, The Tangent Distribution for Self-Similar Measures, Lecture at the 5th Conference on Real Analysis and Measure Theory, 1992. Google Scholar

[2]

K. Falconer, Techniques in Fractal Geometry, John Wiley & sons, Chichester, 1997.  Google Scholar

[3]

M. Gavish, Measures with uniform scaling scenery, Ergod. Th. & Dynam. Sys., 31 (2011), 33-48. doi: 10.1017/S0143385709000996.  Google Scholar

[4]

S. Graf, On Bandt's tangential distribution for self-similar measures, Monatsh. Math., 120 (1995), 223-246. doi: 10.1007/BF01294859.  Google Scholar

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M. Hochman, Geometric rigidity of $\times m$ invariant measures, J. Eur. Math. Soc., 14 (2012), 1539-1563. doi: 10.4171/JEMS/340.  Google Scholar

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M. Hochman, Dynamics on fractals and fractal distributions, preprint, arXiv:1008.3731, 2013. Google Scholar

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M. Hochman, Erratum to "Geometric rigidity of $\times m$ invariant measures'', J. Eur. Math. Soc., 15 (2013), 2463-2464. doi: 10.4171/JEMS/425.  Google Scholar

[8]

D. W. Stroock, Probability Theory, an Analytic View, Cambridge University Press, Cambridge, 1993.  Google Scholar

show all references

References:
[1]

C. Bandt, The Tangent Distribution for Self-Similar Measures, Lecture at the 5th Conference on Real Analysis and Measure Theory, 1992. Google Scholar

[2]

K. Falconer, Techniques in Fractal Geometry, John Wiley & sons, Chichester, 1997.  Google Scholar

[3]

M. Gavish, Measures with uniform scaling scenery, Ergod. Th. & Dynam. Sys., 31 (2011), 33-48. doi: 10.1017/S0143385709000996.  Google Scholar

[4]

S. Graf, On Bandt's tangential distribution for self-similar measures, Monatsh. Math., 120 (1995), 223-246. doi: 10.1007/BF01294859.  Google Scholar

[5]

M. Hochman, Geometric rigidity of $\times m$ invariant measures, J. Eur. Math. Soc., 14 (2012), 1539-1563. doi: 10.4171/JEMS/340.  Google Scholar

[6]

M. Hochman, Dynamics on fractals and fractal distributions, preprint, arXiv:1008.3731, 2013. Google Scholar

[7]

M. Hochman, Erratum to "Geometric rigidity of $\times m$ invariant measures'', J. Eur. Math. Soc., 15 (2013), 2463-2464. doi: 10.4171/JEMS/425.  Google Scholar

[8]

D. W. Stroock, Probability Theory, an Analytic View, Cambridge University Press, Cambridge, 1993.  Google Scholar

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