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Monotonicity of entropy for real multimodal maps


Authors: Henk Bruin and Sebastian van Strien
Journal: J. Amer. Math. Soc. 28 (2015), 1-61
MSC (2010): Primary 37E05; Secondary 37B40
DOI: https://doi.org/10.1090/S0894-0347-2014-00795-5
Published electronically: June 23, 2014
MathSciNet review: 3264762
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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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Additional Information

Henk Bruin
Affiliation: Faculty of Mathematics, University of Vienna, Oskar Morgenstern Platz 1, A-1090 Vienna, Austria
Email: henk.bruin@univie.ac.at

Sebastian van Strien
Affiliation: Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
Email: s.van-strien@imperial.ac.uk

DOI: https://doi.org/10.1090/S0894-0347-2014-00795-5
Received by editor(s): May 20, 2009
Received by editor(s) in revised form: June 12, 2010, September 13, 2012, and October 3, 2013
Published electronically: June 23, 2014
Additional Notes: The first author was supported by EPSRC [Grants GR/S91147/01 and EP/F037112/1].
The second author was supported by a Royal Society Leverhulme Trust Senior Research Fellowship, a Visitor’s Travel grant from the Netherlands Organisation for Scientific Research (NWO) and the Marie Curie grant MRTN-CT-2006-035651 (CODY)
Article copyright: © Copyright 2014 American Mathematical Society