Topological Entropy for Power-Law Unimodal Maps

Abstract

In this paper we prove that the monotonicity of kneading sequences and topological entropy, a fundamental structural property of the quadratic family, extends to the class of power-law unimodal maps $f_a(x)=a-|x|^r$ for arbitrary critical exponent $r>1$. This generalization is nontrivial: the absence of polynomial structure and the presence of non-integer criticality preclude the direct use of classical arguments. Our approach adapts and extends the Milnor-Thurston framework by introducing a Thurston-type operator associated with the critical orbit and establishing a determinant identity that relates its linearization to the parameter derivative of the orbit. The main difficulty proving positivity of this determinant in the absence of algebraic structure - is resolved via a contraction argument on an associated Torelli space endowed with the Teichm\"uller metric, extending Thurston's pullback construction beyond the polynomial setting, that is to critical powers $r=2^\nu/k$, $\nu\geq 1$, $k$ odd, and finally use continuity in $r$. As a consequence, we show that the kneading sequence varies monotonically with the parameter, and hence that the topological entropy is an increasing function of $a$. Our results show that the combinatorial organization of parameter space familiar from the quadratic family persists for unimodal maps with arbitrary power-law criticality, indicating that monotonicity of entropy is a robust phenomenon beyond polynomial dynamics.