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A classical construction due to Newhouse creates horseshoes from hyperbolicperiodic orbits with large period and weak domination through local$C^1$-perturbations. Our main theorem shows that, when one works in the $C^1$topology, the entropy of such horseshoes can be made arbitrarily close to anupper bound deriving from Ruelle's inequality, i.e., the sum of the positiveLyapunov exponents (or the same for the inverse diffeomorphism, whichever issmaller). Adapting classical techniques, we use perturbations that are localand can be chosen to preserve volume or symplectic form or a homoclinicconnection.  This optimal entropy creation yields a number of consequences for$C^1$-generic diffeomorphisms, especially in the absence of a dominatedsplitting. For instance, in the conservative settings, we find formulas for thetopological entropy, deduce that the topological entropy is continuous but notlocally constant at the generic diffeomorphism and we prove that these genericdiffeomorphisms have no measure of maximum entropy. In the dissipative setting,we show the locally generic existence of infinitely many homoclinic classeswith entropy bounded away from zero.

Entropy of $C^1$ diffeomorphisms without a dominated splitting