Dimension and measures on sub-self-affine sets. (English) Zbl 1206.28013
The authors study the dimension of a typical so-called sub-self-affine set which extends the self-affine set from the classical case studied by J. Hutchinson, K. J. Falconer and others. In Theorem 5.2 the same result for a typical sub-self-affine set is proved and further, that such a set carries an invariant measure of full Hausdorff dimension as obtained by Falconer for sub-self-similar sets satisfying the open set condition. The proof of Theorem 5.2 is based on the existence of an equilibrium measure (Section 3). The authors study in section 4 some properties of the topological pressure (Theorem 4.4).
The paper is well written and the subject is a very current one. It answers a question of Falconer.
The paper is well written and the subject is a very current one. It answers a question of Falconer.
Reviewer: Nicolae-Adrian Secelean (Sibiu)
MSC:
| 28A80 | Fractals |
| 28A78 | Hausdorff and packing measures |
| 37C45 | Dimension theory of smooth dynamical systems |
References:
| [1] | Bandt C.: Self-similar sets. I. Topological Markov chains and mixed self-similar sets. Math. Nachr. 142, 107–123 (1989) · Zbl 0707.28004 · doi:10.1002/mana.19891420107 |
| [2] | Barański K.: Hausdorff dimension of the limit sets of some planar geometric constructions. Adv. Math. 210(1), 215–245 (2007) · Zbl 1116.28008 · doi:10.1016/j.aim.2006.06.005 |
| [3] | Bauer H.: Measure and integration theory. de Gruyter Studies in Mathematics, vol. 26. Walter de Gruyter & Co., Berlin (2001) · Zbl 0985.28001 |
| [4] | Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture notes in mathematics, vol. 470 (1975) · Zbl 0308.28010 |
| [5] | Cao Y.-L., Feng D.-J., Huang W.: The thermodynamic formalism for sub-additive potentials. Discrete Contin. Dyn. Syst. 20(3), 639–657 (2008) · Zbl 1140.37319 |
| [6] | Cholewa P.W.: Remarks on the stability of functional equations. Aequationes Math. 27(1), 76–86 (1984) · Zbl 0549.39006 · doi:10.1007/BF02192660 |
| [7] | Falconer K.J.: The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103(2), 339–350 (1988) · Zbl 0642.28005 · doi:10.1017/S0305004100064926 |
| [8] | Falconer K.J.: The dimension of self-affine fractals II. Math. Proc. Cambridge Philos. Soc. 111(1), 169–179 (1992) · Zbl 0797.28004 · doi:10.1017/S0305004100075253 |
| [9] | Falconer K.J.: Bounded distortion and dimension for nonconformal repellers. Math. Proc. Cambridge Philos. Soc. 115(2), 315–334 (1994) · Zbl 0808.58028 · doi:10.1017/S030500410007211X |
| [10] | Falconer K.J.: Sub-self-similar sets. Trans. Am. Math. Soc. 347(8), 3121–3129 (1995) · Zbl 0844.28005 · doi:10.2307/2154776 |
| [11] | Falconer K.J.: Techniques in Fractal Geometry. Wiley, England (1997) · Zbl 0869.28003 |
| [12] | Falconer K.J., Miao J.: Dimensions of self-affine fractals and multifractals generated by upper- triangular matrices. Fractals 15(3), 289–299 (2007) · Zbl 1137.28302 · doi:10.1142/S0218348X07003587 |
| [13] | Feng, D.-J., Käenmäki, A.: Equilibrium states for the pressure function for products of matrices (2009, in preparation) |
| [14] | Feng D.-J., Lau K.-S.: The pressure function for products of non-negative matrices. Math. Res. Lett. 9(2), 363–378 (2002) · Zbl 1116.37302 |
| [15] | Heurteaux Y.: Estimations de la dimension inférieure et de la dimension supérieure des mesures. Ann. Inst. H. Poincaré Probab. Stat. 34(3), 309–338 (1998) · Zbl 0903.28005 · doi:10.1016/S0246-0203(98)80014-9 |
| [16] | Hueter I., Lalley S.P.: Falconer’s formula for the Hausdorff dimension of a self-affine set in \({\mathbb{R}^{2}}\) . Ergod. Theory Dyn. Syst. 15(1), 77–97 (1995) · Zbl 0867.28006 · doi:10.1017/S0143385700008257 |
| [17] | Hutchinson J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981) · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055 |
| [18] | Jordan T., Pollicott M., Simon K.: Hausdorff dimension for randomly perturbed self-affine attractors. Comm. Math. Phys. 270(2), 519–544 (2007) · Zbl 1119.28004 · doi:10.1007/s00220-006-0161-7 |
| [19] | Käenmäki A.: On natural invariant measures on generalised iterated function systems. Ann. Acad. Sci. Fenn. Math. 29(2), 419–458 (2004) · Zbl 1078.37014 |
| [20] | Käenmäki A., Shmerkin P.: Overlapping self-affine sets of Kakeya type. Ergod. Theory Dyn. Syst. 29(3), 941–965 (2009) · Zbl 1173.28004 · doi:10.1017/S0143385708080474 |
| [21] | Käenmäki A., Vilppolainen M.: Separation conditions on controlled Moran constructions. Fund. Math. 200(1), 69–100 (2008) · Zbl 1148.28007 · doi:10.4064/fm200-1-2 |
| [22] | Mattila P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge University Press, Cambridge (1995) · Zbl 0819.28004 |
| [23] | Phelps R.R.: Lectures on Choquet’s Theorem. D. Van Nostrand Co., Inc., N.J.-Toronto, Ont.-London (1966) · Zbl 0135.36203 |
| [24] | Rams M.: Measures of maximal dimension for linear horseshoes. Real Anal. Exchange 31(1), 55–62 (2005/2006) · Zbl 1096.37013 |
| [25] | Rockafellar R.T.: Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton (1970) · Zbl 0193.18401 |
| [26] | Steele J.M.: Kingman’s subadditive ergodic theorem. Ann. Inst. H. Poincaré Probab. Stat. 25(1), 93–98 (1989) · Zbl 0669.60039 |
| [27] | Stein E.M., Shakarchi R.: Real analysis. Measure theory, integration, and Hilbert spaces. Princeton Lectures in Analysis, III. Princeton University Press, Princeton (2005) · Zbl 1081.28001 |
| [28] | Temam R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988) · Zbl 0662.35001 |
| [29] | Walters P.: An Introduction to Ergodic Theory. Springer, New York (1982) · Zbl 0475.28009 |
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