# American Institute of Mathematical Sciences

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

## On the asymptotics of the scenery flow

 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.
Citation: Magnus Aspenberg, Fredrik Ekström, Tomas Persson, Jörg Schmeling. On the asymptotics of the scenery flow. Discrete & Continuous Dynamical Systems - A, 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, (1997).   Google Scholar [3] M. Gavish, Measures with uniform scaling scenery,, Ergod. Th. & Dynam. Sys., 31 (2011), 33.  doi: 10.1017/S0143385709000996.  Google Scholar [4] S. Graf, On Bandt's tangential distribution for self-similar measures,, Monatsh. Math., 120 (1995), 223.  doi: 10.1007/BF01294859.  Google Scholar [5] M. Hochman, Geometric rigidity of $\times m$ invariant measures,, J. Eur. Math. Soc., 14 (2012), 1539.  doi: 10.4171/JEMS/340.  Google Scholar [6] M. Hochman, Dynamics on fractals and fractal distributions,, preprint, (2013).   Google Scholar [7] M. Hochman, Erratum to "Geometric rigidity of $\times m$ invariant measures'',, J. Eur. Math. Soc., 15 (2013), 2463.  doi: 10.4171/JEMS/425.  Google Scholar [8] D. W. Stroock, Probability Theory, an Analytic View,, Cambridge University Press, (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, (1997).   Google Scholar [3] M. Gavish, Measures with uniform scaling scenery,, Ergod. Th. & Dynam. Sys., 31 (2011), 33.  doi: 10.1017/S0143385709000996.  Google Scholar [4] S. Graf, On Bandt's tangential distribution for self-similar measures,, Monatsh. Math., 120 (1995), 223.  doi: 10.1007/BF01294859.  Google Scholar [5] M. Hochman, Geometric rigidity of $\times m$ invariant measures,, J. Eur. Math. Soc., 14 (2012), 1539.  doi: 10.4171/JEMS/340.  Google Scholar [6] M. Hochman, Dynamics on fractals and fractal distributions,, preprint, (2013).   Google Scholar [7] M. Hochman, Erratum to "Geometric rigidity of $\times m$ invariant measures'',, J. Eur. Math. Soc., 15 (2013), 2463.  doi: 10.4171/JEMS/425.  Google Scholar [8] D. W. Stroock, Probability Theory, an Analytic View,, Cambridge University Press, (1993).   Google Scholar
 [1] Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118. [2] Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457 [3] Davit Karagulyan. Hausdorff dimension of a class of three-interval exchange maps. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1257-1281. doi: 10.3934/dcds.2020077 [4] V. V. Chepyzhov, A. A. Ilyin. On the fractal dimension of invariant sets: Applications to Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 117-135. doi: 10.3934/dcds.2004.10.117 [5] Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015 [6] Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140 [7] Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281 [8] Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437 [9] Michael Barnsley, James Keesling, Mrinal Kanti Roychowdhury. Special issue on fractal geometry, dynamical systems, and their applications. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : ⅰ-ⅰ. doi: 10.3934/dcdss.201908i [10] Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4647-4692. doi: 10.3934/dcds.2013.33.4647 [11] Joseph Squillace. Estimating the fractal dimension of sets determined by nonergodic parameters. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5843-5859. doi: 10.3934/dcds.2017254 [12] Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591 [13] Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405 [14] Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503 [15] Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293 [16] Yuanhong Chen, Chao Ma, Jun Wu. Moving recurrent properties for the doubling map on the unit interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2969-2979. doi: 10.3934/dcds.2016.36.2969 [17] Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940 [18] Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure & Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433 [19] Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098 [20] Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235

2018 Impact Factor: 1.143