Abstract:

Consider, for any $n in mathbb{N}$, the set $X_n$ of all n-periodic patterns with positive topological entropy and the set $Y_nsubset X_n$ of all n-periodic patterns with no division. Let $lambda_n$ be the unique real root of the polynomial $x^n − 2x − 1$ in $(1, infty)$. We construct explicitly an n-periodic pattern $Q_n$ with no division whose entropy is $log(lambda_n)$. For periods which are powers of primes we prove that this entropy is minimum in the set $Y_n$. We also conjecture that the entropy of $Q_n$ is also minimum in the set $X_n$, for any $n$.