This paper gives the pointwise Holder (or multifractal) spectrum of continuous functions on the interval $[0,1]$ whose graph is the attractor of an iterated function system consisting of $r\geq 2$ affine maps on $\mathbb{R}^2$. These functions satisfy a functional equation of the form $\phi(a_k x+b_k)=c_k x+d_k\phi(x)+e_k$, for $k=1,2,\dots,r$ and $x\in[0,1]$. They include the Takagi function, the Riesz-Nagy singular functions, Okamoto's functions, and many other well-known examples. It is shown that the multifractal spectrum of $\phi$ is given by the multifractal formalism when $|d_k|\geq |a_k|$ for at least one $k$, but the multifractal formalism may fail otherwise, depending on the relationship between the shear parameters $c_k$ and the other parameters. In the special case when $a_k>0$ for every $k$, an exact expression is derived for the pointwise Holder exponent at any point. These results extend recent work by the author [Adv. Math. 328 (2018), 1-39] and S. Dubuc [Expo. Math. 36 (2018), 119-142].

中文翻译：

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区间上一般自仿射函数的逐点 Hölder 谱

本文给出了区间 $[0,1]$ 上连续函数的逐点 Holder（或多重分形）谱，其图是迭代函数系统的吸引子，该系统由 $\mathbb{R 上的 $r\geq 2$ 仿射图组成}^2$。这些函数满足形式为 $\phi(a_k x+b_k)=c_k x+d_k\phi(x)+e_k$ 的函数方程，对于 $k=1,2,\dots,r$ 和 $x\in [0,1]$。它们包括 Takagi 函数、Riesz-Nagy 奇异函数、Okamoto 函数和许多其他著名的例子。表明当 $|d_k|\geq |a_k|$ 至少有一个 $k$ 时，$\phi$ 的多重分形谱是由多重分形形式给出的，否则多重分形形式可能会失败，这取决于它们之间的关系剪切参数 $c_k$ 和其他参数。在每$k$$a_k>0$的特殊情况下，为任意点的逐点 Holder 指数导出精确表达式。这些结果扩展了作者最近的工作 [Adv. 数学。328 (2018), 1-39] 和 S. Dubuc [Expo. 数学。36 (2018), 119-142]。