1,194 research outputs found
Repetitions in infinite palindrome-rich words
Rich words are characterized by containing the maximum possible number of
distinct palindromes. Several characteristic properties of rich words have been
studied; yet the analysis of repetitions in rich words still involves some
interesting open problems. We address lower bounds on the repetition threshold
of infinite rich words over 2 and 3-letter alphabets, and construct a candidate
infinite rich word over the alphabet with a small critical
exponent of . This represents the first progress on an open
problem of Vesti from 2017.Comment: 12 page
Undecidability and Finite Automata
Using a novel rewriting problem, we show that several natural decision
problems about finite automata are undecidable (i.e., recursively unsolvable).
In contrast, we also prove three related problems are decidable. We apply one
result to prove the undecidability of a related problem about k-automatic sets
of rational numbers
Periodic 2-graphs arising from subshifts
Higher-rank graphs were introduced by Kumjian and Pask to provide models for
higher-rank Cuntz-Krieger algebras. In a previous paper, we constructed
2-graphs whose path spaces are rank-two subshifts of finite type, and showed
that this construction yields aperiodic 2-graphs whose -algebras are
simple and are not ordinary graph algebras. Here we show that the construction
also gives a family of periodic 2-graphs which we call \emph{domino graphs}. We
investigate the combinatorial structure of domino graphs, finding interesting
points of contact with the existing combinatorial literature, and prove a
structure theorem for the -algebras of domino graphs.Comment: 17 page
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