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Let $f$ be a $C^1$-diffeomorphism and $\mu$ be a hyperbolic ergodic$f$-invariant Borel probability measure with positive measure-theoreticentropy. Assume that the Oseledec splitting $$T_xM=E_1(x) \oplus\cdots\oplusE_s(x) \oplus E_{s+1}(x) \oplus\cdots\oplus E_l(x) $$ is dominated on theOseledec basin $\Gamma$. We give extensions of Katok's Horseshoes construction.Moreover there is a dominated splitting corresponding to Oseledec subspace onhorseshoes.

The approximation of Lyapunov exponents by horseshoes for $C^1$-diffeomorphisms with dominated splitting