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<p>Inspired by Pansiot’s work on substitutions, we prove a similar theorem for automorphisms of a free group <i>F</i> of finite rank: if a right-infinite word <i>X</i> represents an attracting fixed point of an automorphism of <i>F</i>, the subword complexity of <i>X</i> is equivalent to <i>n, n</i> log log <i>n, n</i> log <i>n</i>, or <i>n</i><sup>2</sup>. The proof uses combinatorial arguments analogous to Pansiot’s as well as train tracks. We also define the recurrence complexity of <i>X</i>, and we apply it to laminations. In particular, we show that attracting laminations have complexity equivalent to <i>n, n</i> log log <i>n, n</i> log <i>n</i>, or <i>n</i><sup>2</sup> (to <i>n</i> if the automorphism is fully irreducible).</p>

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A Pansiot-type subword complexity theorem for automorphisms of free groups

  • 期刊论文

  • 发表日期：

    2025年6月29日

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  • Arnaud Hilion,
  • Gilbert Levitt

摘要

Inspired by Pansiot’s work on substitutions, we prove a similar theorem for automorphisms of a free group F of finite rank: if a right-infinite word X represents an attracting fixed point of an automorphism of F, the subword complexity of X is equivalent to n, n log log n, n log n, or n2. The proof uses combinatorial arguments analogous to Pansiot’s as well as train tracks. We also define the recurrence complexity of X, and we apply it to laminations. In particular, we show that attracting laminations have complexity equivalent to n, n log log n, n log n, or n2 (to n if the automorphism is fully irreducible).