Assouad谱和准Assouad维数：两个谱的故事
The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra
摘要：
我们认为Assouad频谱，介绍了弗雷泽和余，随着一个自然的变种，我们称之为“上Assouad频谱”。这些光谱被设计成在上盒计数和Assouad维度之间插值。已知Assouad谱在其域的左手侧接近上盒计数维度，但不一定在右侧接近Assouad维度。在这里，我们表明，它必然会接近\emph{quasi-Assouad dimension}在右手边的域。我们进一步表明，上Assouad频谱可以表示的Assouad频谱，从而激发使用的定义Fraser-Yu。我们还提供了一个大家庭的例子，展示了新的现象有关的阿苏德光谱的形式。例如，我们证明了它可以是严格凹的，表现出任何阶的相变，并且不需要分段可微。
英文摘要：
We consider the Assouad spectrum, introduced by Fraser and Yu, along with a natural variant that we call the `upper Assouad spectrum'. These spectra are designed to interpolate between the upper box-counting and Assouad dimensions. It is known that the Assouad spectrum approaches the upper box-counting dimension at the left hand side of its domain, but does not necessarily approach the Assouad dimension on the right. Here we show that it necessarily approaches the \emph{quasi-Assouad dimension} at the right hand side of its domain. We further show that the upper Assouad spectrum can be expressed in terms of the Assouad spectrum, thus motivating the definition used by Fraser-Yu. We also provide a large family of examples demonstrating new phenomena relating to the form of the Assouad spectrum. For example, we prove that it can be strictly concave, exhibit phase transitions of any order, and need not be piecewise differentiable.
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