Growth of monomial algebras, simple rings and free subalgebras

We construct finitely generated simple algebras with prescribed growth types, which can be arbitrarily taken from a large variety of (super-polynomial) growth types. This (partially) answers a question raised by the author in a recent paper. Our construction goes through a construction of finitely generated just-infinite, primitive monomial algebras with prescribed growth type, from which we construct uniformly recurrent infinite words with subword complexity having the same growth type. We also discuss the connection between entropy of algebras and their homomorphic images, as well as the degrees of their generators of free subalgebras.