The construction of a spectral cocycle from the case of one-dimensional substitution flows [A. I. Bufetov and B. Solomyak. A spectral cocycle for substitution systems and translation flows. J. Anal. Math.141(1) (2020), 165–205] is extended to the setting of pseudo-self-similar tilings in ${\mathbb R}^d$ , allowing expanding similarities with rotations. The pointwise upper Lyapunov exponent of this cocycle is used to bound the local dimension of spectral measures of deformed tilings. The deformations are considered, following the work of Treviño [Quantitative weak mixing for random substitution tilings. Israel J. Math., to appear], in the simpler, non-random setting. We review some of the results of Treviño in this special case and illustrate them on concrete examples.

中文翻译：

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用于替代平铺的光谱循环

从一维替代流的情况构建谱余环[AI Bufetov 和 B. Solomyak。用于替代系统和平移流的谱共循环。J.肛门。数学。141(1) (2020), 165–205]扩展到伪自相似平铺的设置 ${\mathbb R}^d$ ，允许通过旋转扩大相似性。该余循环的逐点上李雅普诺夫指数用于限制变形平铺的谱测量的局部维数。根据 Trevino 的工作[随机替换平铺的定量弱混合。以色列 J.马斯。，出现]，在更简单的非随机设置中。我们回顾了特雷维诺在这个特殊情况下的一些结果，并用具体的例子来说明它们。