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A filling curve γ on a based surface  S  determines a pseudo‐Anosov homeomorphism  P (γ) of  S  via the process of ‘point‐pushing along γ’. We consider the relationship between the self‐intersection number i(γ) of γ and the dilatation λ γ  of  P (γ); our main result is that (i(γ) + 1) 1/5  ⩽ λ γ  ⩽ 9 i(γ) . We also bound the least dilatation of any pseudo‐Anosov in the point‐pushing subgroup of a closed surface and prove that this number tends to infinity with the genus. Lastly, we investigate the minimal entropy of any pseudo‐Anosov homeomorphism obtained by pushing along a curve with self‐intersection number  k  and show that, for a closed surface, this number grows like log( k ).

Dilatation versus self‐intersection number for point‐pushing pseudo‐Anosov homeomorphisms