关于保持回文丰富性的态射
On morphisms preserving palindromic richness
AI总结：
本文研究自由幺半群中保持回文丰富性的态射，刻画了二元字母表上保持丰富性的 $P_{ret}$ 态射，并证明每个 Arnoux-Rauzy 态射共轭于 $P_{ret}$ 类中且保持丰富性。
摘要：
已知长度为 $n$ 的每个单词至多包含 $n+1$ 个不同的回文。有限丰富单词是具有最大数量回文因子的单词。回文丰富性的定义可以自然地扩展到无限单词。Sturmian 单词和 Rote 互补对称序列构成两类二元丰富单词，而 episturmian 单词和编码对称 $d$-区间交换变换的单词则在更大的字母表上给出了其他例子。在本文中，我们研究自由幺半群的态射，这些态射使我们能够从已知的丰富单词构造新的丰富单词。我们关注 $P_{ret}$ 类中的态射。该类包含在字母表上单射且满足特定回文性质的态射：对于该类中的每个态射 $φ$，存在一个回文 $w$，使得对于每个字母 $a$，$φ(a)w$ 是 $w$ 的第一个完全返回单词。我们刻画了在二元字母表上保持丰富性的 $P_{ret}$ 态射。我们还研究了作用于具有更多字母的字母表上的标记 $P_{ret}$ 态射。特别地，我们证明了每个 Arnoux-Rauzy 态射都共轭于 $P_{ret}$ 类中的一个态射，并且它保持丰富性。
英文摘要：
It is known that each word of length $n$ contains at most $n+1$ distinct palindromes. A finite rich word is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while episturmian words and words coding symmetric $d$-interval exchange transformations give us other examples on larger alphabets. In this paper we look for morphisms of the free monoid, which allow us to construct new rich words from already known rich words. We focus on morphisms in Class $P_{ret}$. This class contains morphisms injective on the alphabet and satisfying a particular palindromicity property: for every morphism $φ$ in the class there exists a palindrome $w$ such that $φ(a)w$ is a first complete return word to $w$ for each letter $a$. We characterize $P_{ret}$ morphisms which preserve richness over a binary alphabet. We also study marked $P_{ret}$ morphisms acting on alphabets with more letters. In particular we show that every Arnoux-Rauzy morphism is conjugated to a morphism in Class $P_{ret}$ and that it preserves richness.
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