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

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

Information

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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