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On self-affine measures with equal Hausdorff and Lyapunov dimensions.(English)Zbl 1386.37021

Summary: Let \( \mu \) be a self-affine measure on \( \mathbb{R}^{d}\) associated to a self-affine IFS \( \{\phi _{\lambda }(x)=A_{\lambda }x+v_{\lambda }\}_{\lambda \in \Lambda }\) and a probability vector \( p=(p_{\lambda })_{\lambda }>0\). Assume the strong separation condition holds. Let \( \gamma _{1}\geq \cdots \geq \gamma _{d}\) 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 \( \dim \mu =D\).

MSC:

37C45 Dimension theory of smooth dynamical systems
28A80 Fractals

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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