In this paper, we investigate the double ergodicity of strong horseshoe maps, defined as onto maps whose phase spaces act as attrac tors for their inverse iterations. We prove that such maps, when possessing the reverse bounded distortion property, are doubly ergodic with respect to the Lebesgue measure. Additionally, we establish the robustness of double ergodicity and weak mixing for a C1-perturbed doubling map on the circle, demonstrating that all maps in a C1-neighborhood, as given in Theorem B, share these properties.
