子移位间可测饱和补偿函数的存在性及其应用
Existence of a measurable saturated compensation function between subshifts and its applications
摘要：
我们证明了存在一个有界的Borel可测饱和补偿函数的子移位之间的因子映射。作为应用，我们得到了由整数值对角矩阵给出的环面上扩张非共形映射的紧不变集的豪斯多夫维数和满豪斯多夫维数测度.这些问题在[19]中研究了在饱和补偿函数存在的条件下，符号表示为有限型移位的紧不变集。通过给出一个符号表示为无条件子移位的紧不变集的豪斯多夫维数公式，并将全维不变遍历测度刻画为可测补偿函数的常倍数的遍历平衡态，推广了上述结果.对于符号表示为有限型拓扑混合移位的紧致不变集，利用可测补偿函数研究了唯一性及其唯一不变遍历测度的性质.我们的积极成果缩小的可能性，其中一个例子有一个以上的措施，全尺寸可以找到。
英文摘要：
We show the existence of a bounded Borel measurable saturated compensation function for a factor map between subshifts. As an application, we find the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. These problems were studied in [19] for a compact invariant set whose symbolic representation is a shift of finite type under the condition of the existence of a saturated compensation function. We extend the results by presenting a formula for the Hausdorff dimension for a compact invariant set whose symbolic representation is a subshift without the condition and characterizing the invariant ergodic measures of full dimension as the ergodic equilibrium states of a constant multiple of a measurable compensation function. For a compact invariant set whose symbolic representation is a topologically mixing shift of finite type, we study uniqueness and the properties for the unique invariant ergodic measure of full dimension by using a measurable compensation function. Our positive results narrow the possibility of where an example having more than one measure of full dimension can be found.
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