Droubay, Justin and Pirillo 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 $d$-interval exchange transformations give us other examples on larger alphabets. In this paper we look for homomorphisms of the free monoid, which allow to construct new rich words from already known rich words. In particular we study two types of morphisms: Arnoux-Rauzy morphisms and morphisms from Class $P_{ret}$. These morphisms contain Sturmian morphisms as a subclass. We show that Arnoux-Rauzy morphisms preserve the set of all rich words. We also characterize $P_{ret}$ morphisms which preserve richness on binary alphabet.

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关于保留回文丰富度的态射

Droubay、Justin 和 Pirillo 认为每个长度为 $n$ 的单词最多包含 $n+1$ 个不同的回文。有限的“丰富词”是具有最大数量的回文因子的词。回文丰富度的定义自然可以延伸到无限词。Sturmian 词和 Rote 互补对称序列形成两类丰富的二进制词，而 Episturmian 词和编码 $d$-间隔交换变换的词为我们提供了更大字母表的其他示例。在本文中，我们寻找自由幺半群的同态，它允许从已知的丰富词构建新的丰富词。我们特别研究了两种类型的态射：Arnoux-Rauzy 态射和来自 $P_​​{ret}$ 类的态射。这些态射包含 Sturmian 态射作为子类。我们证明 Arnoux-Rauzy 态射保留了所有丰富词的集合。