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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.12238 |
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| _version_ | 1866916005414961152 |
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| author | Benedicks, Michael Rodrigues, Ana |
| author_facet | Benedicks, Michael Rodrigues, Ana |
| contents | 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 Teichmüller metric, extending Thurston's pullback construction beyond the polynomial setting, that is to critical powers $r=2^ν/k$, $ν\geq 1$, $k$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12238 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topological Entropy for Power-Law Unimodal Maps Benedicks, Michael Rodrigues, Ana Dynamical Systems 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 Teichmüller metric, extending Thurston's pullback construction beyond the polynomial setting, that is to critical powers $r=2^ν/k$, $ν\geq 1$, $k$ 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. |
| title | Topological Entropy for Power-Law Unimodal Maps |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2605.12238 |