AI 中文总结
本文研究紧致字母表上原始替换的唯一遍历性，证明即使对自然数一点紧化，原始性也不充分，并给出强幂收敛与拟紧性等价的充要条件。
AI 中文摘要
我们考虑在紧致Hausdorff字母表上由连续替换生成的一般化子位移。尽管原始性仍蕴含由替换生成的子位移的极小性，但Durand、Ormes和Petite通过构造Cantor字母表的反例表明，与有限情形相反，原始性不再蕴含唯一遍历性。这里我们证明，即使对于自然数的一点紧化这一可谓最简单的例子，原始性对于唯一遍历性，甚至自然长度函数的存在性，仍然是不充分的。在之前与Mañibo和Rust的工作中，我们证明了对于不可约替换，唯一遍历性和自然长度函数的存在性可由重整化替换算子\(T\)的强幂收敛推出；我们还发展了有时能确认此性质的充分判据。我们证明此结果的一个部分逆命题：对于承认自然长度函数的不可约替换（例如，所有不可约常长度替换），唯一遍历性蕴含\(T\)的强幂收敛。然后我们考虑仅含有限多个聚点的字母表情形，展示如何从相关的有限替换（由聚点的替换行为决定）导出\(T\)的本质谱半径的上界（通常还有精确公式）。这有时可用于证明\(T\)的拟紧性，从而证明原始替换的唯一遍历性。对于至少含一个孤立点的字母表的原始替换，我们证明\(T\)的强幂收敛与拟紧性是等价的，并且事实上，这些性质等价于替换迭代在所有字母表中的种子上的词长均匀增长。
英文摘要
We consider generalised subshifts generated by continuous substitution on compact Hausdorff alphabets. Although primitivity still implies minimality of the subshift generated by the substitution, Durand, Ormes and Petite showed, in contrast to the finite case, that primitivity no longer implies unique ergodicity, by constructing counter-examples with Cantor alphabet. Here we show that, even for the arguably simplest case of the one-point compactification of the natural numbers, primitivity is still insufficient for unique ergodicity, or even the existence of a natural length function. In previous work with Mañibo and Rust we showed that, for irreducible substitutions, unique ergodicity and existence of a natural length function follow from strong power convergence of the renormalised substitution operator \(T\); sufficient criteria were also developed that can sometimes confirm this property. We show a partial converse to this: for irreducible substitutions admitting a natural length function (for instance, all irreducible constant length substitutions), unique ergodicity implies strong power convergence of \(T\). We then consider the case of alphabets with only finitely many accumulation points, showing how upper bounds (and usually an exact formula) for the essential spectral radius of \(T\) can be derived from associated finite substitutions, determined by the behaviour of substitution of the accumulation points. This may sometimes be used to show quasi-compactness of \(T\), and thus unique ergodicity for primitive substitutions. For primitive substitutions of alphabets with at least one isolated point, we show that strong power convergence and quasi-compactness of \(T\) are equivalent and, in fact, that these properties are equivalent to iteration of substitution growing words in length uniformly across all seeds in the alphabet.
Comments28 pages
