By operating Denjoy like surgery on a piecewise linear map, we constructed a family of$C^1$maps$f_\alpha \ (1<\alpha<3 )$admitting the following properties:
1)$f_\alpha$admits a hyperbolic repelling Cantor set$\mathcal{A}_\alpha$with positive Lebesgue measure, and$\mathcal{A}_\alpha$is also a wild attractor of$f_{\alpha}$;
2) The attractor$\mathcal{A}_\alpha$is accessible: the difference set$\mathbb{B}(A_\alpha)\backslash A_\alpha$between the basin of attraction$\mathbb{B}(A_\alpha)$and$A_\alpha$has positive Lebesgue measure;
3) The family is structurally stable:$f_{\alpha}$is topologically conjugate to$f_{\alpha'}$for all$1<\alpha,\ \alpha'<3$.
The surgery involves blowing up the discontinuity and its preimages set into open intervals. The$C^1$smoothness of$f_{\alpha}$is ensured by the prescribed lengths of glued intervals and the maps defined on the glued intervals.
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