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On the structure of sequences with minimal maximal pattern complexity
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems / Volume 46 / Issue 8 / August 2026
- Published online by Cambridge University Press:
- 17 April 2026, pp. 2131-2148
- Print publication:
- August 2026
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- Article
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In 2002, Kamae and Zamboni [Ergod. Th. & Dynam. Sys. 22(4) (2002), 1191–1199] introduced maximal pattern complexity and determined that any aperiodic sequence must have maximal pattern complexity at least $2k$
. In 2006, Kamae and Rao [European J. Combin. 27(1) (2006), 125–137] examined the maximal pattern complexity of sequences over larger alphabets and showed that sequences which have maximal pattern complexity less than $\ell k$
, for $\ell $
the size of the alphabet, must have some periodic structure. In this paper, we investigate the structure of sequences of low maximal pattern complexity over $\ell $
letters, where $\liminf _{k \to \infty } p_{\alpha }^*(k) - 3k = -\infty $
. In addition, we show that the minimal maximal pattern complexity of an aperiodic sequence which uses all $\ell $
letters is $p_{\alpha }^*(k) = 2k + \ell -2$
and give an exact structure for aperiodic sequences with this maximal pattern complexity.
