Perturbation techniques for dissipative \(C^1\)-diffeomorphisms have been developed, and a survey is contained in the paper [S. Crovisier, Perturbation de la dynamique de difféomorphismes en topologie \(C^1\). Paris: Société Mathématique de France (SMF) (2013; Zbl 1300.37001)].
As the authors say, the aim of this paper is “to systematically extend the perturbation tools to the conservative settings, trying as much as possible to follow the local approach of [J. Bochi and C. Bonatti, Proc. Lond. Math. Soc. (3) 105, No. 1, 1–48 (2012; Zbl 1268.37027)].” The topics discussed in the paper are:

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Franks’ lemma, linearization and preservation of homoclinic connections;

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perturbation of the spectrum of periodic orbits: achieving simplicity, realness or equal modulus for the stable and for the unstable eigenvalues;

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further perturbations of the tangent dynamics above periodc orbits: making the angle between stable and unstable spaces arbitrarily small;

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birth of homoclinic tangencies for a hyperbolic periodic orbit without strong dominated splitting.