Analogues of Khintchine’s theorem for random attractors

Baker, Simon; Troscheit, Sascha

doi:10.1090/tran/8537

Analogues of Khintchine’s theorem for random attractors
HTML articles powered by AMS MathViewer

by Simon Baker and Sascha Troscheit PDF

Trans. Amer. Math. Soc. 375 (2022), 1411-1441 Request permission

Abstract:

In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to Khintchine from Diophantine approximation to hold almost surely for stochastically self-similar and self-affine random iterated function systems.

References

D. Allen and B. Bárány, Diophantine approximation on fractals: Hausdorff measures of shrinking targets on self-conformal sets, Mathematika, to appear.
Simon Baker, An analogue of Khintchine’s theorem for self-conformal sets, Math. Proc. Cambridge Philos. Soc. 167 (2019), no. 3, 567–597. MR 4015651, DOI 10.1017/s030500411800049x
Simon Baker, Approximation properties of $\beta$-expansions, Acta Arith. 168 (2015), no. 3, 269–287. MR 3342324, DOI 10.4064/aa168-3-4
Simon Baker, Approximation properties of $\beta$-expansions II, Ergodic Theory Dynam. Systems 38 (2018), no. 5, 1627–1641. MR 3819995, DOI 10.1017/etds.2016.108
S. Baker, Overlapping iterated function systems from the perspective of metric number theory, Mem. Amer. Math. Soc. (To appear).
Balázs Bárány, Michael Hochman, and Ariel Rapaport, Hausdorff dimension of planar self-affine sets and measures, Invent. Math. 216 (2019), no. 3, 601–659. MR 3955707, DOI 10.1007/s00222-018-00849-y
Itai Benjamini and Boris Solomyak, Spacings and pair correlations for finite Bernoulli convolutions, Nonlinearity 22 (2009), no. 2, 381–393. MR 2475552, DOI 10.1088/0951-7715/22/2/008
Victor Beresnevich, Detta Dickinson, and Sanju Velani, Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc. 179 (2006), no. 846, x+91. MR 2184760, DOI 10.1090/memo/0846
Victor Beresnevich and Sanju Velani, A note on zero-one laws in metrical Diophantine approximation, Acta Arith. 133 (2008), no. 4, 363–374. MR 2457266, DOI 10.4064/aa133-4-5
Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Second revised edition, Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin, 2008. With a preface by David Ruelle; Edited by Jean-René Chazottes. MR 2423393
R. J. Duffin and A. C. Schaeffer, Khintchine’s problem in metric Diophantine approximation, Duke Math. J. 8 (1941), 243–255. MR 4859
Kenneth Falconer, Fractal geometry, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014. Mathematical foundations and applications. MR 3236784
Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
K. J. Falconer, Random fractals, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 3, 559–582. MR 857731, DOI 10.1017/S0305004100066299
K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 339–350. MR 923687, DOI 10.1017/S0305004100064926
Siegfried Graf, Statistically self-similar fractals, Probab. Theory Related Fields 74 (1987), no. 3, 357–392. MR 873885, DOI 10.1007/BF00699096
Michael Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2) 180 (2014), no. 2, 773–822. MR 3224722, DOI 10.4007/annals.2014.180.2.7
M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy in $R^d$, Mem. Amer. Math. Soc. (To appear).
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
Esa Järvenpää, Maarit Järvenpää, Meng Wu, and Wen Wu, Random affine code tree fractals: Hausdorff and affinity dimensions and pressure, Math. Proc. Cambridge Philos. Soc. 162 (2017), no. 2, 367–382. MR 3604920, DOI 10.1017/S0305004116000694
A. Khintchine, Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann. 92 (1924), no. 1-2, 115–125 (German). MR 1512207, DOI 10.1007/BF01448437
Thomas Jordan, Mark Pollicott, and Károly Simon, Hausdorff dimension for randomly perturbed self affine attractors, Comm. Math. Phys. 270 (2007), no. 2, 519–544. MR 2276454, DOI 10.1007/s00220-006-0161-7
Henna Koivusalo, Dimension of uniformly random self-similar fractals, Real Anal. Exchange 39 (2013/14), no. 1, 73–90. MR 3261900
Dimitris Koukoulopoulos and James Maynard, On the Duffin-Schaeffer conjecture, Ann. of Math. (2) 192 (2020), no. 1, 251–307. MR 4125453, DOI 10.4007/annals.2020.192.1.5
Jason Levesley, Cem Salp, and Sanju L. Velani, On a problem of K. Mahler: Diophantine approximation and Cantor sets, Math. Ann. 338 (2007), no. 1, 97–118. MR 2295506, DOI 10.1007/s00208-006-0069-8
Lars Olsen, Random geometrically graph directed self-similar multifractals, Pitman Research Notes in Mathematics Series, vol. 307, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994. MR 1297123, DOI 10.2307/4351476
Yuval Peres, Károly Simon, and Boris Solomyak, Absolute continuity for random iterated function systems with overlaps, J. London Math. Soc. (2) 74 (2006), no. 3, 739–756. MR 2286443, DOI 10.1112/S0024610706023258
Yuval Peres and Boris Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Res. Lett. 3 (1996), no. 2, 231–239. MR 1386842, DOI 10.4310/MRL.1996.v3.n2.a8
Tomas Persson and Henry W. J. Reeve, A Frostman-type lemma for sets with large intersections, and an application to diophantine approximation, Proc. Edinb. Math. Soc. (2) 58 (2015), no. 2, 521–542. MR 3341452, DOI 10.1017/S0013091514000066
Tomas Persson and Henry W. J. Reeve, On the Diophantine properties of $\lambda$-expansions, Mathematika 59 (2013), no. 1, 65–86. MR 3028172, DOI 10.1112/S0025579312001076
Mark Pollicott and Károly Simon, The Hausdorff dimension of $\lambda$-expansions with deleted digits, Trans. Amer. Math. Soc. 347 (1995), no. 3, 967–983. MR 1290729, DOI 10.1090/S0002-9947-1995-1290729-0
A. Rapaport, Proof of the exact overlaps conjecture for systems with algebraic contractions, Ann. Sci. Éc. Norm. Supér, to appear.
A. Rapaport and P. Varju, Self-similar measures associated to a homogeneous system of three maps, arXiv::2010.01022.
Pablo Shmerkin and Ville Suomala, Spatially independent martingales, intersections, and applications, Mem. Amer. Math. Soc. 251 (2018), no. 1195, v+102. MR 3756896, DOI 10.1090/memo/1195
Pablo Shmerkin, On Furstenberg’s intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions, Ann. of Math. (2) 189 (2019), no. 2, 319–391. MR 3919361, DOI 10.4007/annals.2019.189.2.1
Boris Solomyak, On the random series $\sum \pm \lambda ^n$ (an Erdős problem), Ann. of Math. (2) 142 (1995), no. 3, 611–625. MR 1356783, DOI 10.2307/2118556
Sascha Troscheit, On the dimensions of attractors of random self-similar graph directed iterated function systems, J. Fractal Geom. 4 (2017), no. 3, 257–303. MR 3732642, DOI 10.4171/JFG/51
Péter P. Varjú, On the dimension of Bernoulli convolutions for all transcendental parameters, Ann. of Math. (2) 189 (2019), no. 3, 1001–1011. MR 3961088, DOI 10.4007/annals.2019.189.3.9