Lectures on dynamics, fractal geometry, and metric number theory

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October 2014, 8(3&4): 437-497. doi: 10.3934/jmd.2014.8.437

Lectures on dynamics, fractal geometry, and metric number theory

Michael Hochman ^1,

1.	Einstein Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel

Received December 2013 Revised August 2014 Published April 2015

Abstract
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These notes are based on lectures delivered in the summer school ``Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory'', held in Będlewo, Poland, in the summer of 2011. The course is an exposition of Furstenberg's conjectures on ``transversality'' of the maps $x\rightarrow ax $mod1 and $x\mapsto bx$mod1 for multiplicatively independent integers $a,b$, and of the associated problems on intersections and sums of invariant sets for these maps. The first part of the course is a short introduction to fractal geometry. The second part develops the theory of Furstenberg's CP-chains and local entropy averages, ending in proofs of the sumset problem and of the known case of the intersections conjecture.

Keywords: transversality of semigroups., Hausdorff dimension, Marstrand slice theorem, CP-process, entropy dimension, Marstrand projection theorem.

Mathematics Subject Classification: Primary: 28A80; Secondary: 37F99, 37A9.

Citation: 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

References:

[1]	R. Broderick, Y. Bugeaud, L. Fishman, D. Kleinbock and B. Weiss, Schmidt's game, fractals, and numbers normal to no base, Math. Res. Lett., 17 (2010), 307-321. doi: 10.4310/MRL.2010.v17.n2.a10.
[2]	J. W. S. Cassels, On a problem of Steinhaus about normal numbers, Colloq. Math., 7 (1959), 95-101.
[3]	T. M. Cover and J. A. Thomas, Elements of Information Theory, Second edition, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006.
[4]	P. Erdős, Some unconventional problems in number theory, Math. Mag., 52 (1979), 67-70. doi: 10.2307/2689842.
[5]	K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.
[6]	H. Furstenberg, Intersections of Cantor sets and transversality of semigroups, in Problems in Analysis (Sympos. Salomon Bochner, Princeton Univ., Princeton, N.J., 1969), Princeton Univ. Press, Princeton, N.J., 1970, 41-59.
[7]	H. Furstenberg, Ergodic fractal measures and dimension conservation, Ergodic Theory Dynam. Systems, 28 (2008), 405-422. doi: 10.1017/S0143385708000084.
[8]	M. Hochman, Dynamics on fractals and fractal distributions, preprint, 2010.
[9]	M. Hochman and P. Shmerkin, Local entropy and dimension of projections, to appear in Annals of Mathematics, 2009.
[10]	M. Hochman and P. Shmerkin, Local entropy averages and projections of fractal measures, Ann. of Math. (2), 175 (2012), 1001-1059. doi: 10.4007/annals.2012.175.3.1.
[11]	B. Host, Nombres normaux, entropie, translations, Israel J. Math., 91 (1995), 419-428. doi: 10.1007/BF02761660.
[12]	B. R. Hunt and V. Yu. Kaloshin, How projections affect the dimension spectrum of fractal measures, Nonlinearity, 10 (1997), 1031-1046. doi: 10.1088/0951-7715/10/5/002.
[13]	J. C. Lagarias, Ternary expansions of powers of 2, J. Lond. Math. Soc. (2), 79 (2009), 562-588. doi: 10.1112/jlms/jdn080.
[14]	R. Lyons, Strong laws of large numbers for weakly correlated random variables, Michigan Math. J., 35 (1988), 353-359. doi: 10.1307/mmj/1029003816.
[15]	P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 44, Cambridge, 1995. doi: 10.1017/CBO9780511623813.
[16]	Y. Peres and P. Shmerkin, Resonance between Cantor sets, Ergodic Theory Dynam. Systems, 29 (2009), 201-221. doi: 10.1017/S0143385708000369.
[17]	D. Preiss, Geometry of measures in $R^n$: Distribution, rectifiability, and densities, Ann. of Math. (2), 125 (1987), 537-643. doi: 10.2307/1971410.
[18]	W. M. Schmidt, On normal numbers, Pacific J. Math., 10 (1960), 661-672. doi: 10.2140/pjm.1960.10.661.
[19]	T. Wolff, Recent work connected with the Kakeya problem, in Prospects in Mathematics (Princeton, NJ, 1996), Amer. Math. Soc., Providence, RI, 1999, 129-162.

show all references

References:

[1]	R. Broderick, Y. Bugeaud, L. Fishman, D. Kleinbock and B. Weiss, Schmidt's game, fractals, and numbers normal to no base, Math. Res. Lett., 17 (2010), 307-321. doi: 10.4310/MRL.2010.v17.n2.a10.
[2]	J. W. S. Cassels, On a problem of Steinhaus about normal numbers, Colloq. Math., 7 (1959), 95-101.
[3]	T. M. Cover and J. A. Thomas, Elements of Information Theory, Second edition, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006.
[4]	P. Erdős, Some unconventional problems in number theory, Math. Mag., 52 (1979), 67-70. doi: 10.2307/2689842.
[5]	K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.
[6]	H. Furstenberg, Intersections of Cantor sets and transversality of semigroups, in Problems in Analysis (Sympos. Salomon Bochner, Princeton Univ., Princeton, N.J., 1969), Princeton Univ. Press, Princeton, N.J., 1970, 41-59.
[7]	H. Furstenberg, Ergodic fractal measures and dimension conservation, Ergodic Theory Dynam. Systems, 28 (2008), 405-422. doi: 10.1017/S0143385708000084.
[8]	M. Hochman, Dynamics on fractals and fractal distributions, preprint, 2010.
[9]	M. Hochman and P. Shmerkin, Local entropy and dimension of projections, to appear in Annals of Mathematics, 2009.
[10]	M. Hochman and P. Shmerkin, Local entropy averages and projections of fractal measures, Ann. of Math. (2), 175 (2012), 1001-1059. doi: 10.4007/annals.2012.175.3.1.
[11]	B. Host, Nombres normaux, entropie, translations, Israel J. Math., 91 (1995), 419-428. doi: 10.1007/BF02761660.
[12]	B. R. Hunt and V. Yu. Kaloshin, How projections affect the dimension spectrum of fractal measures, Nonlinearity, 10 (1997), 1031-1046. doi: 10.1088/0951-7715/10/5/002.
[13]	J. C. Lagarias, Ternary expansions of powers of 2, J. Lond. Math. Soc. (2), 79 (2009), 562-588. doi: 10.1112/jlms/jdn080.
[14]	R. Lyons, Strong laws of large numbers for weakly correlated random variables, Michigan Math. J., 35 (1988), 353-359. doi: 10.1307/mmj/1029003816.
[15]	P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 44, Cambridge, 1995. doi: 10.1017/CBO9780511623813.
[16]	Y. Peres and P. Shmerkin, Resonance between Cantor sets, Ergodic Theory Dynam. Systems, 29 (2009), 201-221. doi: 10.1017/S0143385708000369.
[17]	D. Preiss, Geometry of measures in $R^n$: Distribution, rectifiability, and densities, Ann. of Math. (2), 125 (1987), 537-643. doi: 10.2307/1971410.
[18]	W. M. Schmidt, On normal numbers, Pacific J. Math., 10 (1960), 661-672. doi: 10.2140/pjm.1960.10.661.
[19]	T. Wolff, Recent work connected with the Kakeya problem, in Prospects in Mathematics (Princeton, NJ, 1996), Amer. Math. Soc., Providence, RI, 1999, 129-162.

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