Consider the Fibonacci semigroup $<L,RLRR=RLL>$ with a Bernoulli walk $P(R)=p, P(L)=1p$. Is the entropy $H(p)$ an unimodal function with maximum at p=0.5? Is this true for all finitely generated semigroups (with some symmetries)? For the free semigroup $<L,R>$ (the Shift space) it is well known and easy to prove that $H(p)=(plog(p)+(1p)log(1p))$.
