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

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