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Transactions of the American Mathematical Society

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On self-affine measures with equal Hausdorff and Lyapunov dimensions


Author: Ariel Rapaport
Journal: Trans. Amer. Math. Soc. 370 (2018), 4759-4783
MSC (2010): Primary 37C45; Secondary 28A80
DOI: https://doi.org/10.1090/tran/7099
Published electronically: January 18, 2018
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Abstract: Let $ \mu $ be a self-affine measure on $ \mathbb{R}^{d}$ associated to a self-affine IFS $ \{\varphi _{\lambda }(x)=A_{\lambda }x+v_{\lambda }\}_{\lambda \in \Lambda }$ and a probability vector $ p=(p_{\lambda })_{\lambda }>0$. Assume the strong separation condition holds. Let $ \gamma _{1}\ge \cdots \ge \gamma _{d}$ and $ D$ be the Lyapunov exponents and dimension corresponding to $ \{A_{\lambda }\}_{\lambda \in \Lambda }$ and $ p^{\mathbb{N}}$, and let $ \mathbf {G}$ be the group generated by $ \{A_{\lambda }\}_{\lambda \in \Lambda }$. We show that if $ \gamma _{m+1}>\gamma _{m}=\cdots =\gamma _{d}$, if $ \mathbf {G}$ acts irreducibly on the vector space of alternating $ m$-forms, and if the Furstenberg measure $ \mu _{F}$ satisfies $ \dim _{H}\mu _{F}+D>(m+1)(d-m)$, then $ \mu $ is exact dimensional with $ \dim \mu =D$.


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Additional Information

Ariel Rapaport
Affiliation: The Hebrew University of Jerusalem, Givat Ram Campus, Jerusalem
Email: ariel.rapaport@mail.huji.ac.il

DOI: https://doi.org/10.1090/tran/7099
Keywords: Self-affine measures, Furstenberg measure, random matrices.
Received by editor(s): November 27, 2015
Received by editor(s) in revised form: September 26, 2016
Published electronically: January 18, 2018
Additional Notes: The author was supported by ERC grant 306494
Article copyright: © Copyright 2018 American Mathematical Society

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