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Bochi-Katok-Rodriguez Hertz proposed in [BKH21] a program on the flexibilityof Lyapunov exponents for conservative Anosov diffeomorphisms, and obtainedpartial results in this direction. For conservative Anosov diffeomorphisms withstrong hyperbolic properties we establish extended flexibility results fortheir Lyapunov exponents. We give examples of Anosov diffeomorphisms with thestrong unstable exponent larger than the strong unstable exponent of the linearpart. We also give examples of derived from Anosov diffeomorphisms with themetric entropy entropy larger than the entropy of the linear part.  These results rely on a new type of deformation which goes beyond theprevious Shub-Wilkinson and Baraviera-Bonatti techniques for conservativesystems having some invariant directions. In order to estimate the Lyapunovexponents even after breaking the invariant bundles, we obtain an abstractresult (Theorem 3.1) which gives bounds on exponents of some specific cocyclesand which can be applied in various other settings. We also include variousinteresting comments in the appendices: our examples are $\mathcal{C}^2$robust, the Lyapunov exponents are continuous (with respect to the map) evenafter breaking the invariant bundles, a similar construction can be obtainedfor the case of multiple eigenvalues.

Extended flexibility of Lyapunov exponents for Anosov diffeomorphisms