ISRAEL JOURNAL OF MATHEMATICS 165 (2008), 349–379 DOI: 10.1007/s11856-008-1015-0 ENTROPY CONJECTURE FOR CONTINUOUS MAPS OF NILMANIFOLDS BY Wac law Marzantowicz ∗ Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Pozna´ n, ul. Umultowska 87 61-614 Pozna´ n, Poland e-mail: marzan@amu.edu.pl AND Feliks Przytycki ∗∗ Institute of Mathematics of the Polish Academy of Science, ul. ´ Sniadeckich 8, 00-950 Warszawa, Poland e-mail: feliksp@impan.gov.pl ABSTRACT In 1974 Michael Shub asked the following question [29] : When is the topo- logical 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 Micha l Misiurewicz proved [23] that EC holds for all continuous map- pings of tori. Here we prove EC for all continuous mappings of compact nilmanifolds. Also generalizations for maps of some solvmanifolds and an- other proof via Lefschetz and Nielsen numbers, under the assumption the map is not homotopic to a fixed points free map, are provided. * Research supported by KBN grant no. 2P03A 04522. ** Research supported by Foundation for Polish Science and by KBN grant no. 2P03A 03425. Received February 9, 2006 and in revised form June 1, 2006 349