Exponentially small splitting of separatrices near a period-doubling bifurcation in area-preserving maps

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We consider the conservative Hénon family at the period-doubling bifurcation of its fixed point and demonstrate that the separatrices of the fixed saddle point nearing the bifurcation split exponentially: given that λ+ is the smaller of the eigenvalues of the saddle point, the angle between the separatrices along the homoclinic orbit satisfies α= O(e-π² |λ+|)+ O(e-2 (1-κ) π² |λ+|), for any positive κ<1.