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Entropy conjecture for continuous maps of nilmanifolds

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Abstract

In 1974 Michael Shub asked the following question [29]: When is the topological entropy of a continuous mapping of a compact manifold into itself is estimated from below by the logarithm of the spectral radius of the linear mapping induced in the cohomologies with real coefficients? This estimate has been called the Entropy Conjecture (EC). In 1977 the second author and Michał Misiurewicz proved [23] that EC holds for all continuous mappings of tori. Here we prove EC for all continuous mappings of compact nilmanifolds. Also generalizations for maps of some solvmanifolds and another proof via Lefschetz and Nielsen numbers, under the assumption the map is not homotopic to a fixed points free map, are provided.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań, ul. Umultowska 87, 61-614, Poznań, Poland

    Wacław Marzantowicz

  2. Istitute of Mathematics of the Polish Academy of Science, ul. Śniadeckich 8, 00-950, Warszawa, Poland

    Feliks Przytycki

Authors
  1. Wacław Marzantowicz
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  2. Feliks Przytycki
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Correspondence to Wacław Marzantowicz.

Additional information

Research supported by KBN grant no. 2P03A 04522.

Research supported by Foundation for Polish Science and by KBN grant no. 2P03A 03425.

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Marzantowicz, W., Przytycki, F. Entropy conjecture for continuous maps of nilmanifolds. Isr. J. Math. 165, 349–379 (2008). https://doi.org/10.1007/s11856-008-1015-0

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  • DOI: https://doi.org/10.1007/s11856-008-1015-0

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