On self-affine measures with equal Hausdorff and Lyapunov dimensions
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- by Ariel Rapaport PDF
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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 $\dim \mu =D$.References
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Additional Information
- Ariel Rapaport
- Affiliation: The Hebrew University of Jerusalem, Givat Ram Campus, Jerusalem
- MR Author ID: 1088705
- Email: ariel.rapaport@mail.huji.ac.il
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 4759-4783
- MSC (2010): Primary 37C45; Secondary 28A80
- DOI: https://doi.org/10.1090/tran/7099
- MathSciNet review: 3812095