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Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards
Lectures on dynamics, fractal geometry, and metric number theory
1. | Einstein Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel |
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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