Monotonicity of Topological Entropy for One-Parameter Families of Unimodal Mappings

Abstract

For a special class of one-parameter families of unimodal mappings of the form f _t(x): [0, 1] → [0, 1], f _t = atx/(x + t), 0 ≤ x ≤ 1/2, we establish that, for t ε [0, 1/(a − 2)], a > 2, the topological entropy h(f _t) is a function monotonically increasing in the parameter. We prove that there exists a class of one-parameter families of unimodal mappings f _t that contains the family indicated above and establish conditions under which the topological entropy h(f _t) is a function monotonically increasing in the parameter.