We prove that every $C^1$ diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub’s entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive.

For n ≥ 1, given an n-dimensional locally (n-1)-connected compact space X and a finite Borel measure μ without atoms at isolated points, we prove that for a generic (in the uniform metric) continuous map f:X → X, the set of points which are chain recurrent under f has μ-measure zero. The same is true for n = 0 (skipping the local connectedness assumption).

The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties. The work is motivated by two applications: an existence result for the problem of maximizing the rate of...

Let $L:TM\to ℝ$ be a Tonelli Lagrangian function (with $M$ compact and connected and $dimM\ge 2$). The tiered Aubry set (resp. Mañé set) ${𝒜}^T\left(L\right)$ (resp. ${𝒩}^T\left(L\right)$) is the union of the Aubry sets (resp. Mañé sets) $𝒜(L+\lambda )$ (resp. $𝒩(L+\lambda )$) for $\lambda$ closed 1-form. Then1.the set ${𝒩}^T\left(L\right)$ is closed, connected and if $dim{H}^1\left(M\right)\ge 2$, its intersection with any energy level is connected and chain transitive;2.for $L$ generic in the Mañé sense, the sets $\overline{{𝒜}^T\left(L\right)}$ and $\overline{{𝒩}^T\left(L\right)}$ have no interior;3.if the interior of $\overline{{𝒜}^T\left(L\right)}$ is non empty, it contains a dense subset of periodic points.We then give an example...