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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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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
Published electronically: June 23, 2014


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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Bibliographic Information
  • Henk Bruin
  • Affiliation: Faculty of Mathematics, University of Vienna, Oskar Morgenstern Platz 1, A-1090 Vienna, Austria
  • MR Author ID: 329851
  • Email:
  • Sebastian van Strien
  • Affiliation: Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
  • Email:
  • 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)
  • © Copyright 2014 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 28 (2015), 1-61
  • MSC (2010): Primary 37E05; Secondary 37B40
  • DOI:
  • MathSciNet review: 3264762