Characterization of Infinite LSP Words and Endomorphisms Preserving the LSP Property

Answering a question of G. Fici, we give an [Formula: see text]-adic characterization of the family of infinite LSP words, that is, the family of infinite words having all their left special factors as prefixes. More precisely we provide a finite set of morphisms [Formula: see text] and an automaton [Formula: see text] such that an infinite word is LSP if and only if it is [Formula: see text]-adic and one of its directive words is recognizable by [Formula: see text]. Then we characterize the endomorphisms that preserve the property of being LSP for infinite words. This allows us to prove that there exists no set [Formula: see text] of endomorphisms for which the set of infinite LSP words corresponds to the set of [Formula: see text]-adic words. This implies that an automaton is required no matter which set of morphisms is used.