In a joint work with Sylvain Crovisier and Todd Fisher, we strengthen Newhouse’s construction of horseshoes from homoclinic tangencies. We obtain an entropy arbitrarily close to Ruelle’s bound: the sum of the positive exponents (or the same for the inverse, whichever is smaller).

Our perturbations are local, preserve a pre existing volume or symplectic form and homoclinic connection, building on works of Gourmelon and Bochi-Bonatti among others.

A number of consequences follows.

For instance, C1-generic conservative diffeomorphisms without domination have no measure of maximal entropy and are Borel conjugate to a Markov shifts (up to periodic points). Theit topological entropy is nowhere locally constant and there is a generalization of the entropy formula obtained by Catalan and Tahzibi in dimension 2.

We also obtain a nonempty open set of C1-diffeomorphisms which generically have infinitely many homoclinic classes with topological entropy bounded away from zero.

You can read it all here!

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