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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

Michael Hochman 1,

1. 

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

Received  December 2013 Revised  August 2014 Published  April 2015

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.  Google Scholar

[2]

J. W. S. Cassels, On a problem of Steinhaus about normal numbers,, Colloq. Math., 7 (1959), 95.   Google Scholar

[3]

T. M. Cover and J. A. Thomas, Elements of Information Theory,, Second edition, (2006).   Google Scholar

[4]

P. Erdős, Some unconventional problems in number theory,, Math. Mag., 52 (1979), 67.  doi: 10.2307/2689842.  Google Scholar

[5]

K. J. Falconer, The Geometry of Fractal Sets,, Cambridge Tracts in Mathematics, (1986).   Google Scholar

[6]

H. Furstenberg, Intersections of Cantor sets and transversality of semigroups,, in Problems in Analysis (Sympos. Salomon Bochner, (1969), 41.   Google Scholar

[7]

H. Furstenberg, Ergodic fractal measures and dimension conservation,, Ergodic Theory Dynam. Systems, 28 (2008), 405.  doi: 10.1017/S0143385708000084.  Google Scholar

[8]

M. Hochman, Dynamics on fractals and fractal distributions,, preprint, (2010).   Google Scholar

[9]

M. Hochman and P. Shmerkin, Local entropy and dimension of projections,, to appear in Annals of Mathematics, (2009).   Google Scholar

[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.  Google Scholar

[11]

B. Host, Nombres normaux, entropie, translations,, Israel J. Math., 91 (1995), 419.  doi: 10.1007/BF02761660.  Google Scholar

[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.  Google Scholar

[13]

J. C. Lagarias, Ternary expansions of powers of 2,, J. Lond. Math. Soc. (2), 79 (2009), 562.  doi: 10.1112/jlms/jdn080.  Google Scholar

[14]

R. Lyons, Strong laws of large numbers for weakly correlated random variables,, Michigan Math. J., 35 (1988), 353.  doi: 10.1307/mmj/1029003816.  Google Scholar

[15]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability,, Cambridge Studies in Advanced Mathematics, (1995).  doi: 10.1017/CBO9780511623813.  Google Scholar

[16]

Y. Peres and P. Shmerkin, Resonance between Cantor sets,, Ergodic Theory Dynam. Systems, 29 (2009), 201.  doi: 10.1017/S0143385708000369.  Google Scholar

[17]

D. Preiss, Geometry of measures in $R^n$: Distribution, rectifiability, and densities,, Ann. of Math. (2), 125 (1987), 537.  doi: 10.2307/1971410.  Google Scholar

[18]

W. M. Schmidt, On normal numbers,, Pacific J. Math., 10 (1960), 661.  doi: 10.2140/pjm.1960.10.661.  Google Scholar

[19]

T. Wolff, Recent work connected with the Kakeya problem,, in Prospects in Mathematics (Princeton, (1996), 129.   Google Scholar

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.  Google Scholar

[2]

J. W. S. Cassels, On a problem of Steinhaus about normal numbers,, Colloq. Math., 7 (1959), 95.   Google Scholar

[3]

T. M. Cover and J. A. Thomas, Elements of Information Theory,, Second edition, (2006).   Google Scholar

[4]

P. Erdős, Some unconventional problems in number theory,, Math. Mag., 52 (1979), 67.  doi: 10.2307/2689842.  Google Scholar

[5]

K. J. Falconer, The Geometry of Fractal Sets,, Cambridge Tracts in Mathematics, (1986).   Google Scholar

[6]

H. Furstenberg, Intersections of Cantor sets and transversality of semigroups,, in Problems in Analysis (Sympos. Salomon Bochner, (1969), 41.   Google Scholar

[7]

H. Furstenberg, Ergodic fractal measures and dimension conservation,, Ergodic Theory Dynam. Systems, 28 (2008), 405.  doi: 10.1017/S0143385708000084.  Google Scholar

[8]

M. Hochman, Dynamics on fractals and fractal distributions,, preprint, (2010).   Google Scholar

[9]

M. Hochman and P. Shmerkin, Local entropy and dimension of projections,, to appear in Annals of Mathematics, (2009).   Google Scholar

[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.  Google Scholar

[11]

B. Host, Nombres normaux, entropie, translations,, Israel J. Math., 91 (1995), 419.  doi: 10.1007/BF02761660.  Google Scholar

[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.  Google Scholar

[13]

J. C. Lagarias, Ternary expansions of powers of 2,, J. Lond. Math. Soc. (2), 79 (2009), 562.  doi: 10.1112/jlms/jdn080.  Google Scholar

[14]

R. Lyons, Strong laws of large numbers for weakly correlated random variables,, Michigan Math. J., 35 (1988), 353.  doi: 10.1307/mmj/1029003816.  Google Scholar

[15]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability,, Cambridge Studies in Advanced Mathematics, (1995).  doi: 10.1017/CBO9780511623813.  Google Scholar

[16]

Y. Peres and P. Shmerkin, Resonance between Cantor sets,, Ergodic Theory Dynam. Systems, 29 (2009), 201.  doi: 10.1017/S0143385708000369.  Google Scholar

[17]

D. Preiss, Geometry of measures in $R^n$: Distribution, rectifiability, and densities,, Ann. of Math. (2), 125 (1987), 537.  doi: 10.2307/1971410.  Google Scholar

[18]

W. M. Schmidt, On normal numbers,, Pacific J. Math., 10 (1960), 661.  doi: 10.2140/pjm.1960.10.661.  Google Scholar

[19]

T. Wolff, Recent work connected with the Kakeya problem,, in Prospects in Mathematics (Princeton, (1996), 129.   Google Scholar

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