The scenery flow of self-similar measures with weak separation condition. (English) Zbl 1507.37002

For a Radon measure \(\mu\) on \(\mathbb{R}^d\) the scenery flow \((\mu_{x,t})_{t\geqslant0}\) at a point \(x\) is given by \(\mu_{x,t}(A)= \mu({\mathrm{e}}^{-t}A+x)/\mu(B(x,{\mathrm{e}}^{-t}))\) for a measurable set \(A\subset B(0,1)\), where \(B(a,r)\) denotes the closed metric ball at \(a\) of radius \(r\). Tangent measures are defined to be the weak*-accumulation points of the scenery; these reflect the local structure of \(\mu\). There are Radon measures with the property that every non-zero Radon measure arises as a tangent measure at almost every point. Here this unmanageable complexity is constrained in a natural way for dynamics, by proving that self-similar measures satisfying a separation property have a regularity property called uniformly scaling. This is done using ideas from ergodic theory alongside a geometric analysis arising from the separation property.


37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations
37A05 Dynamical aspects of measure-preserving transformations
28A80 Fractals
28D05 Measure-preserving transformations
28D10 One-parameter continuous families of measure-preserving transformations
Full Text: DOI arXiv


[1] Algom, A., Rodriguez Hertz, F. and Wang, Z.. Pointwise normality and Fourier decay for self-conformal measures. Preprint, 2021, arXiv:2012.06529. · Zbl 1484.42010
[2] Dayan, Y., Ganguly, A. and Weiss, B.. Random walks on tori and normal numbers in self-similar sets. Preprint, 2020, arXiv:2002.00455.
[3] Erdös, P.. On a family of symmetric Bernoulli convolutions. Amer. J. Math.61 (1939), 974-976. · JFM 65.1308.01
[4] Falconer, K.. Fractal Geometry: Mathematical Foundations and Applications. Wiley, Hoboken, NJ, 2013. · Zbl 0689.28003
[5] Feng, D.-J.. Gibbs properties of self-conformal measures and the multifractal formalism. Ergod. Th. & Dynam. Sys.27(3) (2007), 787-812. · Zbl 1126.28003
[6] Feng, D.-J.. Uniformly scaling property of self-similar measures with the finite type condition. Unpublished manuscript.
[7] Feng, D.-J. and Lau, K.-S.. Multifractal formalism for self-similar measures with weak separation condition. J. Math. Pures Appl. (9)92(4) (2009), 407-428. · Zbl 1184.28009
[8] Ferguson, A., Fraser, J. M. and Sahlsten, T.. Scaling scenery of \(\left(\times m,\times n\right)\) invariant measures. Adv. Math.268 (2015), 564-602. · Zbl 1302.28029
[9] Fraser, J. M., Henderson, A. M., Olson, E. J. and Robinson, J. C.. On the Assouad dimension of self-similar sets with overlaps. Adv. Math.273 (2015), 188-214. · Zbl 1317.28014
[10] Fraser, J. and Pollicott, M.. Uniform scaling limits for ergodic measures. J. Fractal Geom.4(1) (2017), 1-19. · Zbl 1380.37017
[11] Furstenberg, H.. Ergodic fractal measures and dimension conservation. Ergod. Th. & Dynam. Sys.28(2) (2008), 405-422. · Zbl 1154.37322
[12] Garsia, A. M.. Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc.102 (1962), 409-432. · Zbl 0103.36502
[13] Gavish, M.. Measures with uniform scaling scenery. Ergod. Th. & Dynam. Sys.31(1) (2011), 33-48. · Zbl 1225.37012
[14] Hare, K., Hare, K. and Rutar, A.. When the weak separation condition implies the generalized finite type condition. Proc. Amer. Math. Soc.149(4) (2020), 1555-1568. · Zbl 1466.28010
[15] Hochman, M.. Dynamics on fractals and fractal distributions. Preprint, 2013, arXiv:1008.3731.
[16] Hochman, M.. On self-similar sets with overlaps and inverse theorems for entropy. Ann. of Math. (2)180(2) (2014), 773-822. · Zbl 1337.28015
[17] Hochman, M.. Dimension theory of self-similar sets and measures. Proc. Int. Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited Lectures. Eds. Sirakov, B., Ney De Souza, P. and Viana, M.. World Scientific Publishing, Hackensack, NJ, 2018, pp. 1949-1972. · Zbl 1447.28007
[18] Hochman, M. and Shmerkin, P.. Local entropy averages and projections of fractal measures. Ann. of Math. (2)175(3) (2012), 1001-1059. · Zbl 1251.28008
[19] Hochman, M. and Shmerkin, P.. Equidistribution from fractal measures. Invent. Math.202(1) (2015), 427-479. · Zbl 1409.11054
[20] Kac, M.. On the notion of recurrence in discrete stochastic processes. Bull. Amer. Math. Soc.53 (1947), 1002-1010. · Zbl 0032.41802
[21] Käenmäki, A. and Rossi, E.. Weak separation condition, Assouad dimension, and Furstenberg homogeneity. Ann. Acad. Sci. Fenn. Math.41(1) (2016), 465-490. · Zbl 1342.28016
[22] Kakutani, S.. Random ergodic theorems and Markoff processes with a stable distribution. Proc. 2nd Berkeley Symp. on Mathematical Statistics and Probability. Ed. Neyman, J.. University of California Press, Berkeley, 1951, pp. 247-261. · Zbl 0044.33901
[23] Kim, H. J.. Skew product action. Int. J. Contemp. Math. Sci.1(5-8) (2006), 205-211. · Zbl 1175.37007
[24] Lau, K.-S. and Ngai, S.-M.. Multifractal measures and a weak separation condition. Adv. Math.141(1) (1999), 45-96. · Zbl 0929.28007
[25] Lau, K.-S. and Ngai, S.-M.. A generalized finite type condition for iterated function systems. Adv. Math.208(2) (2007), 647-671. · Zbl 1113.28006
[26] Lau, K.-S. and Wang, X.-Y.. Iterated function systems with a weak separation condition. Studia Math.161(3) (2004), 249-268. · Zbl 1062.28009
[27] O’Neil, T.. A measure with a large set of tangent measures. Proc. Amer. Math. Soc.123(7) (1995), 2217-2220. · Zbl 0827.28002
[28] Shmerkin, P.. On Furstenberg’s intersection conjecture, self-similar measures, and the \({L}^q\) norms of convolutions. Ann. of Math. (2)189(2) (2019), 319-391. · Zbl 1426.11079
[29] Varjú, P. P.. On the dimension of Bernoulli convolutions for all transcendental parameters. Ann. of Math. (2)189(3) (2019), 1001-1011. · Zbl 1426.28024
[30] Zerner, M. P. W.. Weak separation properties for self-similar sets. Proc. Amer. Math. Soc.124(11) (1996), 3529-3539. · Zbl 0874.54025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.