• Previous Article
    Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards
  • JMD Home
  • This Issue
  • Next Article
    Rigidity of Julia sets for Hénon type maps
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

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.
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

[1]

Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591

[2]

Nikolai Dokuchaev. Dimension reduction and Mutual Fund Theorem in maximin setting for bond market. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1039-1053. doi: 10.3934/dcdsb.2011.16.1039

[3]

Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023

[4]

Colin J. Cotter, Michael John Priestley Cullen. Particle relabelling symmetries and Noether's theorem for vertical slice models. Journal of Geometric Mechanics, 2019, 11 (2) : 139-151. doi: 10.3934/jgm.2019007

[5]

Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018

[6]

Qiang Li. A kind of generalized transversality theorem for $C^r$ mapping with parameter. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1043-1050. doi: 10.3934/dcdss.2017055

[7]

Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405

[8]

Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503

[9]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[10]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457

[11]

Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293

[12]

Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098

[13]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235

[14]

Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125

[15]

Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015

[16]

Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417

[17]

Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020

[18]

Cristina Lizana, Leonardo Mora. Lower bounds for the Hausdorff dimension of the geometric Lorenz attractor: The homoclinic case. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 699-709. doi: 10.3934/dcds.2008.22.699

[19]

Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993

[20]

Aline Cerqueira, Carlos Matheus, Carlos Gustavo Moreira. Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. Journal of Modern Dynamics, 2018, 12: 151-174. doi: 10.3934/jmd.2018006

2018 Impact Factor: 0.295

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]