Assouad type dimensions and homogeneity of fractals

Fraser, Jonathan

doi:10.1090/S0002-9947-2014-06202-8

Assouad type dimensions and homogeneity of fractals
HTML articles powered by AMS MathViewer

by Jonathan M. Fraser PDF

Trans. Amer. Math. Soc. 366 (2014), 6687-6733 Request permission

Abstract:

We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural ‘dimension pair’. In particular, we compute these dimensions for certain classes of self-affine sets and quasi-self-similar sets and study their relationships with other notions of dimension, such as the Hausdorff dimension for example. We also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity.

References

Patrice Assouad, Espaces métriques, plongements, facteurs, Publications Mathématiques d’Orsay, No. 223-7769, Université Paris XI, U.E.R. Mathématique, Orsay, 1977 (French). Thèse de doctorat. MR 0644642
Patrice Assouad, Étude d’une dimension métrique liée à la possibilité de plongements dans $\textbf {R}^{n}$, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 15, A731–A734 (French, with English summary). MR 532401
Krzysztof Barański, Hausdorff dimension of the limit sets of some planar geometric constructions, Adv. Math. 210 (2007), no. 1, 215–245. MR 2298824, DOI 10.1016/j.aim.2006.06.005
T. Bedford. Crinkly curves, Markov partitions and box dimensions in self-similar sets, Ph.D dissertation, University of Warwick, 1984.
Tim Bedford, Dimension and dynamics for fractal recurrent sets, J. London Math. Soc. (2) 33 (1986), no. 1, 89–100. MR 829390, DOI 10.1112/jlms/s2-33.1.89
Per Bylund and Jaume Gudayol, On the existence of doubling measures with certain regularity properties, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3317–3327. MR 1676303, DOI 10.1090/S0002-9939-00-05405-8
K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 339–350. MR 923687, DOI 10.1017/S0305004100064926
K. J. Falconer, Dimensions and measures of quasi self-similar sets, Proc. Amer. Math. Soc. 106 (1989), no. 2, 543–554. MR 969315, DOI 10.1090/S0002-9939-1989-0969315-8
K. J. Falconer, Sub-self-similar sets, Trans. Amer. Math. Soc. 347 (1995), no. 8, 3121–3129. MR 1264809, DOI 10.1090/S0002-9947-1995-1264809-X
Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
Kenneth Falconer, Fractal geometry, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR 2118797, DOI 10.1002/0470013850
De-Jun Feng and Yang Wang, A class of self-affine sets and self-affine measures, J. Fourier Anal. Appl. 11 (2005), no. 1, 107–124. MR 2128947, DOI 10.1007/s00041-004-4031-4
Jonathan M. Fraser, On the packing dimension of box-like self-affine sets in the plane, Nonlinearity 25 (2012), no. 7, 2075–2092. MR 2947936, DOI 10.1088/0951-7715/25/7/2075
Steven P. Lalley and Dimitrios Gatzouras, Hausdorff and box dimensions of certain self-affine fractals, Indiana Univ. Math. J. 41 (1992), no. 2, 533–568. MR 1183358, DOI 10.1512/iumj.1992.41.41031
Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
M. Hochman. On self-similar sets with overlaps and inverse theorems for entropy, preprint, to appear in Annals of Math. (2013), arXiv:1212.1873v4.
J. D. Howroyd, On Hausdorff and packing dimension of product spaces, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 4, 715–727. MR 1362951, DOI 10.1017/S0305004100074545
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
A. Käenmäki, J. Lehrbäck and M. Vuorinen. Dimensions, Whitney covers, and tubular neighborhoods, to appear in Indiana Univ. Math. J., arXiv:1209.0629v2.
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
R. Kenyon and Y. Peres, Measures of full dimension on affine-invariant sets, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 307–323. MR 1389626, DOI 10.1017/S0143385700008828
A. L. Vol′berg and S. V. Konyagin, On measures with the doubling condition, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 666–675 (Russian); English transl., Math. USSR-Izv. 30 (1988), no. 3, 629–638. MR 903629
D. G. Larman, A new theory of dimension, Proc. London Math. Soc. (3) 17 (1967), 178–192. MR 203691, DOI 10.1112/plms/s3-17.1.178
D. G. Larman, On Hausdorff measure in finite-dimensional compact metric spaces, Proc. London Math. Soc. (3) 17 (1967), 193–206. MR 210874, DOI 10.1112/plms/s3-17.2.193
Jouni Luukkainen, Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, J. Korean Math. Soc. 35 (1998), no. 1, 23–76. MR 1608518
Jouni Luukkainen and Eero Saksman, Every complete doubling metric space carries a doubling measure, Proc. Amer. Math. Soc. 126 (1998), no. 2, 531–534. MR 1443161, DOI 10.1090/S0002-9939-98-04201-4
John M. Mackay, Assouad dimension of self-affine carpets, Conform. Geom. Dyn. 15 (2011), 177–187. MR 2846307, DOI 10.1090/S1088-4173-2011-00232-3
John M. Mackay and Jeremy T. Tyson, Conformal dimension, University Lecture Series, vol. 54, American Mathematical Society, Providence, RI, 2010. Theory and application. MR 2662522, DOI 10.1090/ulect/054
Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
Pertti Mattila and R. Daniel Mauldin, Measure and dimension functions: measurability and densities, Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 1, 81–100. MR 1418362, DOI 10.1017/S0305004196001089
John McLaughlin, A note on Hausdorff measures of quasi-self-similar sets, Proc. Amer. Math. Soc. 100 (1987), no. 1, 183–186. MR 883425, DOI 10.1090/S0002-9939-1987-0883425-3
Curt McMullen, The Hausdorff dimension of general Sierpiński carpets, Nagoya Math. J. 96 (1984), 1–9. MR 771063, DOI 10.1017/S0027763000021085
L. Olsen, Measurability of multifractal measure functions and multifractal dimension functions, Hiroshima Math. J. 29 (1999), no. 3, 435–458. MR 1728608, DOI 10.32917/hmj/1206124851
L. Olsen, On the Assouad dimension of graph directed Moran fractals, Fractals 19 (2011), no. 2, 221–226. MR 2803238, DOI 10.1142/S0218348X11005282
Eric Olson, Bouligand dimension and almost Lipschitz embeddings, Pacific J. Math. 202 (2002), no. 2, 459–474. MR 1888175, DOI 10.2140/pjm.2002.202.459
James C. Robinson, Dimensions, embeddings, and attractors, Cambridge Tracts in Mathematics, vol. 186, Cambridge University Press, Cambridge, 2011. MR 2767108
Andreas Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), no. 1, 111–115. MR 1191872, DOI 10.1090/S0002-9939-1994-1191872-1
Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710
Feng Xie, Yongcheng Yin, and Yeshun Sun, Uniform perfectness of self-affine sets, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3053–3057. MR 1993212, DOI 10.1090/S0002-9939-03-06976-4