Lectures on dynamics, fractal geometry, and metric number theory

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

Lectures on dynamics, fractal geometry, and metric number theory

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. 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.
[3]	T. M. Cover and J. A. Thomas, Elements of Information Theory,, Second edition, (2006).
[4]	P. Erdős, Some unconventional problems in number theory,, Math. Mag., 52 (1979), 67. doi: 10.2307/2689842.
[5]	K. J. Falconer, The Geometry of Fractal Sets,, Cambridge Tracts in Mathematics, (1986).
[6]	H. Furstenberg, Intersections of Cantor sets and transversality of semigroups,, in Problems in Analysis* (Sympos. Salomon Bochner*, (1969), 41.
[7]	H. Furstenberg, Ergodic fractal measures and dimension conservation,, Ergodic Theory Dynam. Systems, 28 (2008), 405. 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. doi: 10.4007/annals.2012.175.3.1.
[11]	B. Host, Nombres normaux, entropie, translations,, Israel J. Math., 91 (1995), 419. 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. 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. doi: 10.1112/jlms/jdn080.
[14]	R. Lyons, Strong laws of large numbers for weakly correlated random variables,, Michigan Math. J., 35 (1988), 353. doi: 10.1307/mmj/1029003816.
[15]	P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability,, Cambridge Studies in Advanced Mathematics, (1995). doi: 10.1017/CBO9780511623813.
[16]	Y. Peres and P. Shmerkin, Resonance between Cantor sets,, Ergodic Theory Dynam. Systems, 29 (2009), 201. doi: 10.1017/S0143385708000369.
[17]	D. Preiss, Geometry of measures in $R^n$: Distribution, rectifiability, and densities,, Ann. of Math. (2), 125 (1987), 537. doi: 10.2307/1971410.
[18]	W. M. Schmidt, On normal numbers,, Pacific J. Math., 10 (1960), 661. doi: 10.2140/pjm.1960.10.661.
[19]	T. Wolff, Recent work connected with the Kakeya problem,, in Prospects in Mathematics* (Princeton*, (1996), 129.

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. 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.
[3]	T. M. Cover and J. A. Thomas, Elements of Information Theory,, Second edition, (2006).
[4]	P. Erdős, Some unconventional problems in number theory,, Math. Mag., 52 (1979), 67. doi: 10.2307/2689842.
[5]	K. J. Falconer, The Geometry of Fractal Sets,, Cambridge Tracts in Mathematics, (1986).
[6]	H. Furstenberg, Intersections of Cantor sets and transversality of semigroups,, in Problems in Analysis* (Sympos. Salomon Bochner*, (1969), 41.
[7]	H. Furstenberg, Ergodic fractal measures and dimension conservation,, Ergodic Theory Dynam. Systems, 28 (2008), 405. 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. doi: 10.4007/annals.2012.175.3.1.
[11]	B. Host, Nombres normaux, entropie, translations,, Israel J. Math., 91 (1995), 419. 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. 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. doi: 10.1112/jlms/jdn080.
[14]	R. Lyons, Strong laws of large numbers for weakly correlated random variables,, Michigan Math. J., 35 (1988), 353. doi: 10.1307/mmj/1029003816.
[15]	P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability,, Cambridge Studies in Advanced Mathematics, (1995). doi: 10.1017/CBO9780511623813.
[16]	Y. Peres and P. Shmerkin, Resonance between Cantor sets,, Ergodic Theory Dynam. Systems, 29 (2009), 201. doi: 10.1017/S0143385708000369.
[17]	D. Preiss, Geometry of measures in $R^n$: Distribution, rectifiability, and densities,, Ann. of Math. (2), 125 (1987), 537. doi: 10.2307/1971410.
[18]	W. M. Schmidt, On normal numbers,, Pacific J. Math., 10 (1960), 661. doi: 10.2140/pjm.1960.10.661.
[19]	T. Wolff, Recent work connected with the Kakeya problem,, in Prospects in Mathematics* (Princeton*, (1996), 129.

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