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$L^{q}$-spectra of measures on planar non-conformal attractors

Part of: Basic linear algebra Classical measure theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  26 October 2020

KENNETH J. FALCONER ,
JONATHAN M. FRASER  and
LAWRENCE D. LEE
KENNETH J. FALCONER
Affiliation:
School of Mathematics & Statistics, University of St Andrews, St Andrews, KY16 9SS, UK (e-mail: kjf@st-andrews.ac.uk, jmf32@st-andrews.ac.uk)
JONATHAN M. FRASER
Affiliation:
School of Mathematics & Statistics, University of St Andrews, St Andrews, KY16 9SS, UK (e-mail: kjf@st-andrews.ac.uk, jmf32@st-andrews.ac.uk)
LAWRENCE D. LEE*
Affiliation:
School of Mathematics & Statistics, University of St Andrews, St Andrews, KY16 9SS, UK (e-mail: kjf@st-andrews.ac.uk, jmf32@st-andrews.ac.uk)
*
e-mail: ldl@st-andrews.ac.uk
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Abstract

We study the $L^{q}$ -spectrum of measures in the plane generated by certain nonlinear maps. In particular, we consider attractors of iterated function systems consisting of maps whose components are $C^{1+\alpha }$ and for which the Jacobian is a lower triangular matrix at every point subject to a natural domination condition on the entries. We calculate the $L^{q}$ -spectrum of Bernoulli measures supported on such sets by using an appropriately defined analogue of the singular value function and an appropriate pressure function.

Keywords

Lq-spectrumgeneralized q-dimensionsnon-conformal attractormodified singular value functionself-affine measure

MSC classification

Primary: 28A80: Fractals 37C45: Dimension theory of dynamical systems
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3288 - 3306
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Bárány, B.. Subadditive pressure for IFS with triangular maps. Bull. Pol. Acad. Sci. Math. 57 (2009), 263–278.10.4064/ba57-3-8CrossRefGoogle Scholar
Barral, J. and Feng, D.-J.. Multifractal formalism for almost all self-affine measures. Comm. Math. Phys. 318 (2013), 473–504.10.1007/s00220-013-1676-3CrossRefGoogle Scholar
Cao, Y., Pesin, Y. and Zhao, Y.. Dimension estimates for non-conformal repellers and continuity of sub-additive topological pressure. Geom. Funct. Anal. 29 (2019), 1325–1368.10.1007/s00039-019-00510-7CrossRefGoogle Scholar
Falconer, K. J.. Bounded distortion and dimension for non-conformal repellers. Math. Proc. Cambridge Philos. Soc. 115 (1994), 315–334.10.1017/S030500410007211XCrossRefGoogle Scholar
Falconer, K. J.. Techniques in Fractal Geometry. Wiley, Chichester, 1997.Google Scholar
Falconer, K. J.. Generalised dimensions of measures on self-affine sets. Nonlinearity 12 (1999), 877–891.10.1088/0951-7715/12/4/308CrossRefGoogle Scholar
Falconer, K. J.. Generalised dimensions of measures on almost self-affine sets. Nonlinearity 23 (2010), 1047–1069.10.1088/0951-7715/23/5/002CrossRefGoogle Scholar
Falconer, K. J.. Fractal Geometry: Mathematical Foundations and Applications, 3rd edn. Wiley, Hoboken, NJ, 2014.Google Scholar
Falconer, K. J. and Miao, J.. Dimensions of self-affine fractals and multifractals generated by upper-triangular matrices. Fractals 15 (2007), 289–299.10.1142/S0218348X07003587CrossRefGoogle Scholar
Feng, D.-J. and Simon, K.. Dimension estimates for ${C}^1$ iterated function systems and repellers. Part I. Preprint, 2020, arXiv:2007.15320.10.1017/etds.2021.92CrossRefGoogle Scholar
Feng, D.-J. and Wang, Y.. A class of self-affine sets and self-affine measures. J. Fourier Anal. Appl. 11 (2005), 107–124.10.1007/s00041-004-4031-4CrossRefGoogle Scholar
Fraser, J. M.. On the ${L}^q$ -spectrum of planar self-affine measures. Trans. Amer. Math. Soc. 368 (2016), 5579–5620.10.1090/tran/6523CrossRefGoogle Scholar
Hu, H.. Dimensions of invariant sets of expanding maps. Comm. Math. Phys. 176 (1996), 307–320.10.1007/BF02099551CrossRefGoogle Scholar
Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.10.1512/iumj.1981.30.30055CrossRefGoogle Scholar
Kolossváry, I. and Simon, K.. Triangular Gatzouras-Lalley-type planar carpets with overlaps. Nonlinearity 32 (2019), 3294–3341.10.1088/1361-6544/ab1757CrossRefGoogle Scholar
Manning, A. and Simon, K.. Subadditive pressure for triangular maps. Nonlinearity 20 (2007), 133–149.10.1088/0951-7715/20/1/009CrossRefGoogle Scholar
Ngai, S.-M.. A dimension result arising from the ${L}^q$ -spectrum of a measure. Proc. Amer. Math. Soc. 125 (1997), 2943–2951.10.1090/S0002-9939-97-03974-9CrossRefGoogle Scholar
Olsen, L.. A multifractal formalism. Adv. Math. 116 (1995), 82–196.10.1006/aima.1995.1066CrossRefGoogle Scholar
Peres, Y. and Solomyak, B.. Existence of ${L}^q$ -dimensions and entropy dimension for self-conformal measures. Indiana Univ. Math. J. 49 (2000), 1603–1621.10.1512/iumj.2000.49.1851CrossRefGoogle Scholar