zbMATH — the first resource for mathematics

Gradual doubling property of Hutchinson orbits. (English) Zbl 1363.28005
Summary: The classical self-similar fractals can be obtained as fixed points of the iteration technique introduced by Hutchinson. The well known results of Mosco show that typically the limit fractal equipped with the invariant measure is a (normal) space of homogeneous type. But the doubling property along this iteration is generally not preserved even when the starting point, and of course the limit point, both have the doubling property. We prove that the elements of Hutchinson orbits possess the doubling property except perhaps for radii which decrease to zero as the step of the iteration grows, and in this sense, we say that the doubling property of the limit is achieved gradually. We use this result to prove the uniform upper doubling property of the orbits.

28A78 Hausdorff and packing measures
28A75 Length, area, volume, other geometric measure theory
Full Text: DOI Link
[1] H. Aimar, M. Carena, B. Iaffei: Boundedness of the Hardy-Littlewood maximal operator along the orbits of contractive similitudes. J. Geom. Anal. 23 (2013), 1832–1850. · Zbl 1279.28012
[2] H. Aimar, M. Carena, B. Iaffei: On approximation of maximal operators. Publ. Math. 77 (2010), 87–99. · Zbl 1224.42058
[3] H. Aimar, M. Carena, B. Iaffei: Discrete approximation of spaces of homogeneous type. J. Geom. Anal. 19 (2009), 1–18. · Zbl 1178.28002
[4] P. Assouad: Étude d’une dimension métrique liée à la possibilité de plongements dans \(\mathbb{R}\)n. C. R. Acad. Sci., Paris, Sér. A 288 (1979), 731–734. (In French.) · Zbl 0409.54020
[5] R. R. Coifman, M. de Guzman: Singular integrals and multipliers on homogeneous spaces. Rev. Un. Mat. Argentina 25 (1970), 137–143. · Zbl 0249.43009
[6] R. R. Coifman, G. Weiss: Non-Commutative Harmonic Analysis on Certain Homogeneous Spaces. Study of Certain Singular Integrals. Lecture Notes in Mathematics 242, Springer, Berlin, 1971. · Zbl 0224.43006
[7] K. Falconer: Techniques in Fractal Geometry. John Wiley, Chichester, 1997. · Zbl 0869.28003
[8] J. E. Hutchinson: Fractals and self similarity. Indiana Univ. Math. J. 30 (1981), 713–747. · Zbl 0598.28011
[9] T. Hytönen: A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. Publ. Mat., Barc. 54 (2010), 485–504. · Zbl 1246.30087
[10] T. Hytönen, S. Liu, D. Yang, D. Yang: Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces. Can. J. Math. 64 (2012), 892–923. · Zbl 1250.42044
[11] T. Hytönen, H. Martikainen: Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces. J. Geom. Anal. 22 (2012), 1071–1107. · Zbl 1261.42017
[12] T. Hytönen, D. Yang, D. Yang: The Hardy space H 1 on non-homogeneous metric spaces. Math. Proc. Camb. Philos. Soc. 153 (2012), 9–31. · Zbl 1250.42076
[13] B. Iaffei, L. Nitti: Riesz type potentials in the framework of quasi-metric spaces equipped with upper doubling measures. ArXiv:1309.3755 (2013).
[14] J. Kigami: Analysis on Fractals. Cambridge University Press, Cambridge, 2001. · Zbl 0998.28004
[15] J. Kigami: A harmonic calculus on the Sierpiński spaces. Japan J. Appl. Math. 6 (1989), 259–290. · Zbl 0686.31003
[16] P. A. P. Moran: Additive functions of intervals and Hausdorff measure. Proc. Camb. Philos. Soc. 42 (1946), 15–23. · Zbl 0063.04088
[17] U. Mosco: Variational fractals. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 25 (1997), 683–712. · Zbl 1016.28010
[18] R. S. Strichartz: Differential Equations on Fractals. A Tutorial. Princeton University Press, Princeton, 2006.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.