Monotonicity of entropy for real multimodal maps

Bruin, Henk; van Strien, Sebastian

doi:10.1090/S0894-0347-2014-00795-5

Monotonicity of entropy for real multimodal maps
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by Henk Bruin and Sebastian van Strien

J. Amer. Math. Soc. 28 (2015), 1-61

DOI: https://doi.org/10.1090/S0894-0347-2014-00795-5

Published electronically: June 23, 2014

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Abstract:

In 1992, Milnor posed the Monotonicity Conjecture that within a family of real multimodal polynomial interval maps with only real critical points, the isentropes, i.e., the sets of parameters for which the topological entropy is constant, are connected. This conjecture was already proved in the mid-1980s for quadratic maps by a number of different methods, see A. Douady (1993, 1995), A. Douady and J.H. Hubbard (1984, 1985), W. de Melo and S. van Strein (1993), J. Milnor and W. Thurston (1986, 1988), and M. Tsujii (2000). In 2000, Milnor and Tresser, provided a proof for the case of cubic maps. In this paper we will prove the general case of this 20 year old conjecture.

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