Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-19T05:03:15.709Z Has data issue: false hasContentIssue false
  • English
  • Français

Random affine code tree fractals: Hausdorff and affinity dimensions and pressure

Published online by Cambridge University Press:  20 July 2016

ESA JÄRVENPÄÄ ,
MAARIT JÄRVENPÄÄ ,
MENG WU  and
WEN WU
ESA JÄRVENPÄÄ
Affiliation:
Mathematics, P.O. Box 3000, 90014 University of Oulu, Finland. e-mails: [email protected]; [email protected]; [email protected]
MAARIT JÄRVENPÄÄ
Affiliation:
Mathematics, P.O. Box 3000, 90014 University of Oulu, Finland. e-mails: [email protected]; [email protected]; [email protected]
MENG WU
Affiliation:
Mathematics, P.O. Box 3000, 90014 University of Oulu, Finland. e-mails: [email protected]; [email protected]; [email protected]
WEN WU*
Affiliation:
School of Mathematics, South China University of Technology, Guangzhou 510641, P. R. China. Mathematics, P.O. Box 3000, 90014 University of Oulu, Finland. e-mail: [email protected]
*
†Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that for random affine code tree fractals the affinity dimension is almost surely equal to the unique zero of the pressure function. As a consequence, we show that the Hausdorff, packing and box counting dimensions of such systems are equal to the zero of the pressure. In particular, we do not presume the validity of the Falconer–Sloan condition or any other additional assumptions which have been essential in all the previously known results.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 2 , March 2017 , pp. 367 - 382
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bárány, B. On the Hausdorff dimension of a family of self-similar sets with complicated overlaps. Fund. Math. 206 (2009), 49–59.Google Scholar
[2] Barnsley, M., Hutchinson, J. E. and Stenflo, Ö. A fractal valued random iteration algorithm and fractal hierarchy. Fractals 13 (2005), 111–146.CrossRefGoogle Scholar
[3] Barnsley, M., Hutchinson, J. E. and Stenflo, Ö. V-variable fractals: fractals with partial self similarity. Adv. Math. 218 (2008), 2051–2088.CrossRefGoogle Scholar
[4] Barnsley, M., Hutchinson, J. E. and Stenflo, Ö. V-variable fractals: dimension results. Forum Math. 24 (2012), 445–470.Google Scholar
[5] Barral, J. and Feng, D.-J. Multifractal formalism for almost all self-affine measures. Comm. Math. Phys. 318 (2013), 473–504.Google Scholar
[6] Bedford, T. Crinkly curves, Markov partitions and dimension. Ph.D. Thesis. University of Warwick (1984).Google Scholar
[7] Dajani, K., Jiang, K. and Kempton, T. Self-affine sets with positive Lebesgue measure. Indag. Math. (N.S.) 25 (2014), no. 4, 774–784.CrossRefGoogle Scholar
[8] Falconer, K. J. The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Phil. Soc. 103 (1988), 339–350.Google Scholar
[9] Falconer, K. J. Dimensions of self-affine sets: a survey. Further developments in fractals and related fields. Trends Math. (Birkhäuser/Springer, New York, 2013), 115–134.Google Scholar
[10] Falconer, K. J. and Miao, J. Dimensions of self-affine fractals and multifractals generated by upper triangular matrices. Fractals 15 (2007), 289–299.Google Scholar
[11] Falconer, K. J. and Miao, J. Exceptional sets for self-affine fractals. Math. Proc. Camb. Phil. Soc. 145 (2008), 669–684.Google Scholar
[12] Falconer, K. J. and Miao, J. Random subsets of self-affine fractals. Mathematika 56 (2010), 61–76.Google Scholar
[13] Falconer, K. J. and Sloan, A. Continuity of subadditive pressure for self-affine sets. Real Anal. Exchange 34 (2009), 413–427.CrossRefGoogle Scholar
[14] Feng, D.-J. Lyapunov exponents for products of matrices and multifractal analysis, Part II: General matrices. Israel J. Math. 170 (2009), 355–394.Google Scholar
[15] Feng, D.-J. and Shmerkin, P. Non-conformal repellers and the continuity of pressure for matrix cocycles. Geom. Funct. Anal. 24 (2014), 1101–1128.Google Scholar
[16] Feng, D.-J. and Wang, Y. A class of self-affine sets and self-affine measures. J. Fourier Anal. Appl. 11 (2005), 107–124.Google Scholar
[17] Ferguson, A., Jordan, T. and Rams, M. Dimension of self-affine sets with holes. Ann. Acad. Sci. Fenn. Math. 40 (2015), 63–88.Google Scholar
[18] Ferguson, A., Jordan, T. and Shmerkin, P. The Hausdorff dimension of the projections of self-affine carpets. Fund. Math. 209 (2010), no. 3, 193–213.Google Scholar
[19] Fraser, J. Dimension theory and fractal constructions based on self-affine carpets. Ph.D. Thesis. University of St. Andrews (2013).Google Scholar
[20] Fraser, J. Assouad type dimensions and homogeneity of fractals. Trans. Amer. Math. Soc. 366 (2014), no. 12, 6687–6733.Google Scholar
[21] Fraser, J. and Shmerkin, P. On the dimensions of a family of overlapping self-affine carpets. Ergodic Theory Dynam. Systems, to appear.Google Scholar
[22] Jordan, T., Pollicott, M. and Simon, K. Hausdorff dimension for randomly perturbed self affine attractors. Comm. Math. Phys. 270 (2007), 519–544.Google Scholar
[23] Jordan, T. and Rams, M. Packing spectra for Bernoulli measures supported on Bedford–McMullen carpets. Fund. Math. 229 (2015), no. 2, 171–196.CrossRefGoogle Scholar
[24] Järvenpää, E., Järvenpää, M., Käenmäki, A., Koivusalo, H., Stenflo, Ö. and Suomala, V. Dimensions of random affine code tree fractals. Ergodic Theory Dynam. Systems. 34 (2014), 854–875.Google Scholar
[25] Järvenpää, E., Järvenpää, M., Li, B. and Stenflo, Ö. Random affine code tree fractals and Falconer–Sloan condition. Ergodic Theory Dynam. Systems. 36 (2016), 1516–1533.Google Scholar
[26] Mauldin, R. D. and Urbański, M. Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets. Cambridge Tracts in Mathematics 148 (Cambridge University Press, Cambridge, 2003).Google Scholar
[27] McMullen, C. The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96 (1984), 1–9.Google Scholar
[28] Przytycki, F. and Urbański, M. On the Hausdorff dimension of some fractal sets. Studia Math. 93 (1989), 155–186.Google Scholar
[29] Rossi, E. Local dimensions of measures on infinitely generated self-affine sets. J. Math. Anal. Appl. 413 (2014), no. 2, 1030–1039.Google Scholar
[30] Simon, K. The dimension theory of almost self-affine sets and measures. in Fractals, wavelets, and their applications. Springer Proc. Math. Stat. 92. (Springer, Cham, 2014), 103–127.Google Scholar
[31] Solomyak, B. Measure and dimensions for some fractal families. Math. Proc. Camb. Phil. Soc. 124 (1998), 531–546.Google Scholar