Lq-spectra of measures on planar non-conformal attractors

Kenneth John Falconer; Jonathan Fraser; Lawrence David Lee

doi:10.1017/etds.2020.106

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

Kenneth John Falconer, Jonathan Fraser, Lawrence David Lee^*

^*Corresponding author for this work

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

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+α 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.

Original language	English
Pages (from-to)	3288-3306
Number of pages	19
Journal	Ergodic Theory and Dynamical Systems
Volume	41
Issue number	11
Early online date	26 Oct 2020
DOIs	https://doi.org/10.1017/etds.2020.106
Publication status	Published - Nov 2021

Keywords

Lq-spectrum
Generalised q-dimensions
Non-conformal attractor
Modified singular value function
Self-affine measure

Access to Document

10.1017/etds.2020.106

2005.09361Submitted manuscript, 613 KB
NonlinearLq_revised
Copyright © The Author(s), 2020. Published by Cambridge University Press. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1017/etds.2020.106.
Accepted author manuscript, 568 KBLicence: CC BY-NC-ND

Cite this

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title = "Lq-spectra of measures on planar non-conformal attractors",

abstract = "We study the Lq-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 C1+α 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 Lq-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-spectrum, Generalised q-dimensions, Non-conformal attractor, Modified singular value function, Self-affine measure",

author = "Falconer, \{Kenneth John\} and Jonathan Fraser and Lee, \{Lawrence David\}",

note = "Funding: KJF and JMF were supported by an EPSRC Standard Grant (EP/R015104/1). JMF was also supported by a Leverhulme Trust Research Project Grant (RPG-2019-034). LDL was supported by an EPSRC Doctoral Training Grant.",

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language = "English",

volume = "41",

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AU - Falconer, Kenneth John

AU - Fraser, Jonathan

AU - Lee, Lawrence David

N1 - Funding: KJF and JMF were supported by an EPSRC Standard Grant (EP/R015104/1). JMF was also supported by a Leverhulme Trust Research Project Grant (RPG-2019-034). LDL was supported by an EPSRC Doctoral Training Grant.

PY - 2021/11

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N2 - We study the Lq-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 C1+α 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 Lq-spectrum of Bernoulli measures supported on such sets by using an appropriately defined analogue of the singular value function and an appropriate pressure function.

AB - We study the Lq-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 C1+α 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 Lq-spectrum of Bernoulli measures supported on such sets by using an appropriately defined analogue of the singular value function and an appropriate pressure function.

KW - Lq-spectrum

KW - Generalised q-dimensions

KW - Non-conformal attractor

KW - Modified singular value function

KW - Self-affine measure

UR - https://arxiv.org/pdf/2005.09361.pdf

UR - https://www.doi.org/10.1017/etds.2020.106

UR - https://www.scopus.com/pages/publications/85094956585

U2 - 10.1017/etds.2020.106

DO - 10.1017/etds.2020.106

M3 - Article

SN - 0143-3857

VL - 41

SP - 3288

EP - 3306

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 11

ER -

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

Abstract

Keywords

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New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Multifractal measures : from self-affine to nonlinear

Cite this

Lq-spectra of measures on planar non-conformal attractors

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Projects

New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Student theses

Multifractal measures : from self-affine to nonlinear

Cite this

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