Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-14T17:08:46.643Z Has data issue: false hasContentIssue false
  • English
  • Français

Pointwise Assouad dimension for measures

Part of: Classical measure theory Set functions and measures on spaces with additional structure

Published online by Cambridge University Press:  21 December 2022

Roope Anttila Open the ORCID record for Roope Anttila [Opens in a new window]
Roope Anttila*
Affiliation:
Research Unit of Mathematical Sciences, University of Oulu, P.O.Box 8000, Oulu, FI-90014, Finland roope.anttila@oulu.fi
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and study its properties in relation to the global Assouad dimension. We show that, in general, the value of the pointwise Assouad dimension may differ from the global counterpart, but in many classical cases, the pointwise Assouad dimension exhibits similar exact dimensionality properties as the classical local dimension, namely it equals the global Assouad dimension almost everywhere. We also prove an explicit formula for the Assouad dimension of certain invariant measures with place-dependent probabilities supported on self-conformal sets.

Keywords

pointwise doubling measurepointwise Assouad dimensionquasi-Bernoulli measureself-conformal setplace-dependent probabilities

MSC classification

Primary: 28A80: Fractals
Secondary: 28C15: Set functions and measures on topological spaces (regularity of measures, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 6 , December 2023 , pp. 2053 - 2078
Check for updates
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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

Assouad, P.. Espaces métriques, plongements, facteurs. Thèse de doctorat dÉtat, Univ. Paris XI, 91405 Orsay, 1977.Google Scholar
Barnsley, M. F., Demko, S. G., Elton, J. H. and Geronimo, J. S.. Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities. Ann. Inst. Henri Poincare 24 (1988), 367–394.Google Scholar
Björn, A., Björn, J. and Lehrbäck, J.. Sharp capacity estimates for annuli in weighted $\mathbb {R}^n$ and in metric spaces. Math. Z. 286 (2017), 1173–1215.CrossRefGoogle Scholar
Bowen, R.. Equilibrium states and the ergodic theory of Anosov diffeomorphisms (Berlin: Springer-Verlag, 1975).CrossRefGoogle Scholar
Bárány, B., Hochman, M. and Rapaport, A.. Hausdorff dimension of planar self-affine sets and measures. Invent. Math. 216 (2019), 601–659.CrossRefGoogle Scholar
Falconer, K.. Techniques in fractal geometry (Chichester: John Wiley & Sons Ltd, 1996).Google Scholar
Falconer, K. J., Fraser, J. M. and Käenmäki, A.. Minkowski dimension for measures. Preprint, 2020.Google Scholar
Fan, A. H. and Lau, K.-L.. Iterated function system and Ruelle operator. J. Math. Anal. Appl. 231 (1999), 319–344.CrossRefGoogle Scholar
Fraser, J. M.. Assouad dimension and fractal geometry. Cambridge Tracts in Mathematics (Cambridge: Cambridge University Press, 2020).CrossRefGoogle Scholar
Fraser, J. M. and Howroyd, D. C.. On the upper regularity dimensions of measures. Indiana Univ. Math. J. 69 (2020), 685–712.CrossRefGoogle Scholar
Heurteaux, Y.. Estimations de la dimension inférieure et de la dimension supérieure des mesures. Ann. Inst. Henri Poincare 34 (1998), 309–338.CrossRefGoogle Scholar
Hochman, M.. On self-similar sets with overlaps and inverse theorems for entropy. Ann. Math. 180 (2014), 773–782.CrossRefGoogle Scholar
Hochman, M. and Rapaport, A.. Hausdorff dimension of planar self-affine sets and measures with overlaps. J. Eur. Math. Soc. 24 (2021), 2361–2441.CrossRefGoogle Scholar
Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
Keith, S. and Laakso, T.. Conformal Assouad dimension and modulus. Geom. Funct. Anal. 14 (2004), 1278–1321.CrossRefGoogle Scholar
Käenmäki, A., Lehrbäck, J. and Vuorinen, M.. Dimensions, Whitney covers, and tubular neighborhoods. Indiana Univ. Math. J. 62 (2013), 1861–1889.CrossRefGoogle Scholar
Kigami, J.. Analysis on fractals (Cambridge: Cambridge University Press, 2001).CrossRefGoogle Scholar
Le Donne, E. and Rajala, T.. Assouad dimension, Nagata dimension, and uniformly close metric tangents. Indiana Univ. Math. J. 64 (2015), 21–54.CrossRefGoogle Scholar
Luukkainen, J. and Saksman, E.. Every complete doubling metric space carries a doubling measure. Proc. Am. Math. Soc. 126 (1998), 531–534.CrossRefGoogle Scholar
Mackay, J. M. and Tyson, J. T.. Conformal dimension: theory and application, University Lecture Series (USA: American Mathematical Society, 2010).CrossRefGoogle Scholar
Mauldin, R. D. and Urbański, M.. Dimensions and measures in infinite iterated function systems. Proc. London Math. Soc. 73 (1996), 105–173.CrossRefGoogle Scholar
Olsen, L.. Self-affine multifractal Sierpinski sponges in $R^d$. Pac. J. Math. 183 (1998), 143–199.CrossRefGoogle Scholar
Troscheit, S.. On quasisymmetric embeddings of the Brownian map and continuum trees. Probab. Theory Relat. Fields 179 (2021), 1023–1046.CrossRefGoogle Scholar
Vol'berg, A. L. and Konyagin, S. V.. There is a homogeneous measure on any compact subset in $\mathbb {R}^n$. Sov. Math. Dokl. 30 (1984), 453–456. (Russian).Google Scholar
Yung, P.-L.. Doubling properties of self-similar measures. Indiana Univ. Math. J. 56 (2007), 965–990.CrossRefGoogle Scholar