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We study infinite binary words that contain few distinct palindromes. In particular, we classify such words according to their critical exponents. This extends results by Fici and Zamboni TCS 2013. Interestingly, the words with 18 and 20 palindromes happen to be morphic images of the fixed point of the morphism 001, 121, 20.
Dvořáková et al. (Thu,) studied this question.