Volkova O. Topological and symbolic complexity of unimodal maps

It was proved that for some families of piecewise-linear maps (so called barn maps) the kneading invariant and topological entropy were monotone increasing. It was shown that the kneading invariant and topological entropy of some families of barn maps were non-monotone as a function of the parameter. We gave a dynamical argument why our scheme of getting non-monotonicity of kneading invariants is not possible for maps with negative Schwarzian derivative. Some special conditions for monotonicity of entropy for non-polynomial families were generalized and new classes of one-parameter families of unimodal maps were found for which topological entropy is monotone with respect to the parameter. We investigated symbolic complexity of subshifts generated by interval maps with specific combinatorial properties. The word-complexity of Fibonacci-like kneading subshifts was obtained exactly.