论文标题
关于与重叠的自相似措施的多重形式主义
On multifractal formalism for self-similar measures with overlaps
论文作者
论文摘要
令$μ$为IFS $φ= \ {ϕ_i \} _ {i = 1}^\ ell $在$ \ mathbb r^d $（$ d \ ge 1 $）上产生的自相似度量。当$φ$是定期的（请参见定义〜1.1）时，我们给出了$ l^q $ -spectrum $τ_μ（q）$ $ $ $ $ $ $ $ $ $ [0,1] $的明确公式，并表明$τ_μ$在$（0,1] $的$（0,1）$ $ $ $的情况下， [τ_μ'（1），τ_μ'（0+）$。在$ l^q $ -spectrum上，自相似措施。
Let $μ$ be a self-similar measure generated by an IFS $Φ=\{ϕ_i\}_{i=1}^\ell$ of similarities on $\mathbb R^d$ ($d\ge 1$). When $Φ$ is dimensional regular (see Definition~1.1), we give an explicit formula for the $L^q$-spectrum $τ_μ(q)$ of $μ$ over $[0,1]$, and show that $τ_μ$ is differentiable over $(0,1]$ and the multifractal formalism holds for $μ$ at any $α\in [τ_μ'(1),τ_μ'(0+)]$. We also verify the validity of the multifractal formalism of $μ$ over $[τ_μ'(\infty),τ_μ'(0+)]$ for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds. For one of them, the proof appeals to a recent result of Shmerkin on the $L^q$-spectrum of self-similar measures.
