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We consider the pseudo‐Anosov elements of the mapping class group of a surface of genus   g   that fix a rank   k   subgroup of the first homology of the surface. We show that the smallest entropy among these is comparable to    (  k  +  1  )  /  g   . This interpolates between results of Penner and of Farb and the second and third authors, who treated the cases of    k  =  0    and    k  =  2  g   , respectively, and answers a question of Ellenberg. We also show that the number of conjugacy classes of pseudo‐Anosov mapping classes as above grows (as a function of   g  ) like a polynomial of degree   k  .

Pseudo‐Anosov stretch factors and homology of mapping tori