# On substitutions closed under derivation: examples

## TL;DR

This paper explores specific classes of infinite words fixed by morphisms, demonstrating that certain sets of morphisms remain closed under derivation, with examples including episturmian and period doubling morphisms.

## Contribution

It provides the first known examples of sets of morphisms closed under derivation, expanding understanding of fixed points in infinite words.

## Key findings

- Standard episturmian morphisms form a closed set under derivation.
- The period doubling morphism set is also closed under derivation.
- Derived words in these sets are again fixed by morphisms from the same set.

## Abstract

We study infinite words fixed by a morphism and their derived words. A derived word is a coding of return words to a factor. We exhibit two examples of sets of morphisms which are closed under derivation --- any derived word with respect to any factor of the fixed point is again fixed by a morphism from this set. The first example involves standard episturmian morphisms, and the second concerns the period doubling morphism.