AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Overlapping Iterated Function Systems from the Perspective of Metric Number Theory
About this Title
Simon Baker
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 287, Number 1428
ISBNs: 978-1-4704-6440-0 (print); 978-1-4704-7544-4 (online)
DOI: https://doi.org/10.1090/memo/1428
Published electronically: June 28, 2023
Keywords: Overlapping iterated function systems,
Khintchine’s theorem,
self-similar measures
Table of Contents
Chapters
- 1. Introduction
- 2. Statement of results
- 3. Preliminary results
- 4. Applications of Proposition 3.1
- 5. A specific family of IFSs
- 6. Proof of Theorem 2.15
- 7. Proof of Theorem 2.16
- 8. Applications of the mass transference principle
- 9. Examples
- 10. Final discussion and open problems
- Acknowledgments
Abstract
In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems that our results apply to include those arising from Bernoulli convolutions, the $\{0,1,3\}$ problem, and affine contractions with varying translation parameter. As a by-product of our analysis we obtain new proofs of some well known results due to Solomyak on the absolute continuity of Bernoulli convolutions, and when the attractor in the $\{0,1,3\}$ problem has positive Lebesgue measure.
For each $t\in [0,1]$ we let $\Phi _t$ be the iterated function system given by \begin{equation*} \Phi _{t}≔\Big \{\phi _1(x)=\frac {x}{2},\phi _2(x)=\frac {x+1}{2},\phi _3(x)=\frac {x+t}{2},\phi _{4}(x)=\frac {x+1+t}{2}\Big \}. \end{equation*} We prove that either $\Phi _t$ contains an exact overlap, or we observe Khintchine like behaviour. Our analysis shows that by studying the metric properties of limsup sets, we can distinguish between the overlapping behaviour of iterated function systems in a way that is not available to us by simply studying properties of self-similar measures.
Last of all, we introduce a property of an iterated function system that we call being consistently separated with respect to a measure. We prove that this property implies that the pushforward of the measure is absolutely continuous. We include several explicit examples of consistently separated iterated function systems.
- J. C. Alexander and J. A. Yorke, Fat baker’s transformations, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 1–23. MR 758890, DOI 10.1017/S0143385700002236
- 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
- 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 mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. of Math. (2) 164 (2006), no. 3, 971–992. MR 2259250, DOI 10.4007/annals.2006.164.971
- 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
- Yann Bugeaud, Approximation by algebraic numbers, Cambridge Tracts in Mathematics, vol. 160, Cambridge University Press, Cambridge, 2004. MR 2136100, DOI 10.1017/CBO9780511542886
- J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957. MR 87708
- R. J. Duffin and A. C. Schaeffer, Khintchine’s problem in metric Diophantine approximation, Duke Math. J. 8 (1941), 243–255. MR 4859
- G. A. Edgar, Fractal dimension of self-affine sets: some examples, Rend. Circ. Mat. Palermo (2) Suppl. 28 (1992), 341–358. Measure theory (Oberwolfach, 1990). MR 1183060
- Paul Erdös, On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61 (1939), 974–976. MR 311, DOI 10.2307/2371641
- Paul Erdös, On the smoothness properties of a family of Bernoulli convolutions, Amer. J. Math. 62 (1940), 180–186. MR 858, DOI 10.2307/2371446
- 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, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 339–350. MR 923687, DOI 10.1017/S0305004100064926
- Adriano M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962), 409–432. MR 137961, DOI 10.1090/S0002-9947-1962-0137961-5
- Adriano M. Garsia, Entropy and singularity of infinite convolutions, Pacific J. Math. 13 (1963), 1159–1169. MR 156945
- Miguel de Guzmán, Differentiation of integrals in $R^{n}$, Lecture Notes in Mathematics, Vol. 481, Springer-Verlag, Berlin-New York, 1975. With appendices by Antonio Córdoba, and Robert Fefferman, and two by Roberto Moriyón. MR 457661
- Glyn Harman, Metric number theory, London Mathematical Society Monographs. New Series, vol. 18, The Clarendon Press, Oxford University Press, New York, 1998. MR 1672558
- 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
- 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
- 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
- Børge Jessen and Aurel Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), no. 1, 48–88. MR 1501802, DOI 10.1090/S0002-9947-1935-1501802-5
- 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
- Mike Keane, Meir Smorodinsky, and Boris Solomyak, On the morphology of $\gamma$-expansions with deleted digits, Trans. Amer. Math. Soc. 347 (1995), no. 3, 955–966. MR 1290723, DOI 10.1090/S0002-9947-1995-1290723-X
- Tom Kempton, Counting $\beta$-expansions and the absolute continuity of Bernoulli convolutions, Monatsh. Math. 171 (2013), no. 2, 189–203. MR 3077931, DOI 10.1007/s00605-013-0512-3
- 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
- Simon Kochen and Charles Stone, A note on the Borel-Cantelli lemma, Illinois J. Math. 8 (1964), 248–251. MR 161355
- 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
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- R. Daniel Mauldin and Károly Simon, The equivalence of some Bernoulli convolutions to Lebesgue measure, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2733–2736. MR 1458276, DOI 10.1090/S0002-9939-98-04460-8
- A. Nagel, E. M. Stein, and S. Wainger, Differentiation in lacunary directions, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 3, 1060–1062. MR 466470, DOI 10.1073/pnas.75.3.1060
- Yuval Peres, MichałRams, Károly Simon, and Boris Solomyak, Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets, Proc. Amer. Math. Soc. 129 (2001), no. 9, 2689–2699. MR 1838793, DOI 10.1090/S0002-9939-01-05969-X
- Yuval Peres and Wilhelm Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102 (2000), no. 2, 193–251. MR 1749437, DOI 10.1215/S0012-7094-00-10222-0
- Yuval Peres, Wilhelm Schlag, and Boris Solomyak, Sixty years of Bernoulli convolutions, Fractal geometry and stochastics, II (Greifswald/Koserow, 1998) Progr. Probab., vol. 46, Birkhäuser, Basel, 2000, pp. 39–65. MR 1785620
- 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
- Yuval Peres and Boris Solomyak, Self-similar measures and intersections of Cantor sets, Trans. Amer. Math. Soc. 350 (1998), no. 10, 4065–4087. MR 1491873, DOI 10.1090/S0002-9947-98-02292-2
- 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
- MichałRams, Generic behavior of iterated function systems with overlaps, Pacific J. Math. 218 (2005), no. 1, 173–186. MR 2224595, DOI 10.2140/pjm.2005.218.173
- S. Rigot, Differentiation of measures in metric spaces, CIME-CIRM Course on New Trends on Analysis and Geometry in Metric Spaces, Lecture Notes in Mathematics, Fondazione CIME Foundation Subseries (to appear).
- David Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 99–107. MR 684247, DOI 10.1017/s0143385700009603
- Santiago Saglietti, Pablo Shmerkin, and Boris Solomyak, Absolute continuity of non-homogeneous self-similar measures, Adv. Math. 335 (2018), 60–110. MR 3836658, DOI 10.1016/j.aim.2018.06.015
- 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
- Pablo Shmerkin, On the exceptional set for absolute continuity of Bernoulli convolutions, Geom. Funct. Anal. 24 (2014), no. 3, 946–958. MR 3213835, DOI 10.1007/s00039-014-0285-4
- Pablo Shmerkin, Overlapping self-affine sets, Indiana Univ. Math. J. 55 (2006), no. 4, 1291–1331. MR 2269414, DOI 10.1512/iumj.2006.55.2718
- Pablo Shmerkin and Boris Solomyak, Absolute continuity of self-similar measures, their projections and convolutions, Trans. Amer. Math. Soc. 368 (2016), no. 7, 5125–5151. MR 3456174, DOI 10.1090/tran6696
- Pablo Shmerkin and Boris Solomyak, Zeros of $\{-1,0,1\}$ power series and connectedness loci for self-affine sets, Experiment. Math. 15 (2006), no. 4, 499–511. MR 2293600
- Károly Simon and Boris Solomyak, On the dimension of self-similar sets, Fractals 10 (2002), no. 1, 59–65. MR 1894903, DOI 10.1142/S0218348X02000963
- Károly Simon and Lajos Vágó, Singularity versus exact overlaps for self-similar measures, Proc. Amer. Math. Soc. 147 (2019), no. 5, 1971–1986. MR 3937675, DOI 10.1090/proc/14042
- Boris Solomyak, Measure and dimension for some fractal families, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, 531–546. MR 1636589, DOI 10.1017/S0305004198002680
- 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
- V. G. Sprindžuk, Metric theory of Diophantine approximation V. H. Winston & Sons, 1979.
- Péter P. Varjú, Absolute continuity of Bernoulli convolutions for algebraic parameters, J. Amer. Math. Soc. 32 (2019), no. 2, 351–397. MR 3904156, DOI 10.1090/jams/916
- 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