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Periodic points and measures for a class of skew-products. (English) Zbl 1487.37038

Summary: We consider the \(C^1\)-open set \(\mathcal{V}\) of partially hyperbolic diffeomorphisms on the space \(\mathbb{T}^2\times\mathbb{T}^2\) whose non-wandering set is not stable, introduced by M. Shub in [Topological transitive diffeomorphisms in \(\mathbb{T}^4\), Proceedings of the Symposium on Differential Equations and Dynamical Systems, 39–40 (1971), Berlin Heidelberg: Springer, Berlin Heidelberg]. Firstly, we show that the non-wandering set of each diffeormorphism in \(\mathcal{V}\) is a limit of horseshoes in the sense of entropy. Afterwards, we establish the existence of a \(C^2\)-open set \(\mathcal{U}\) of \(C^2\)-diffeomorphisms in \(\mathcal{V}\) and of a \(C^2\)-residual subset \(\operatorname{Re}\) of \(\mathcal{U}\) such that any diffeomorphism in \(\operatorname{Re}\) has equal topological and periodic entropies, is asymptotic per-expansive, has a sub-exponential growth rate of the periodic orbits and admits a principal strongly faithful symbolic extension with embedding. Besides, such a diffeomorphism has a unique probability measure with maximal entropy describing the distribution of periodic orbits. Under an additional assumption, we prove that the skew-products in \(\mathcal{U}\) preserve a unique ergodic SRB measure, which is physical, whose basin has full Lebesgue measure and which coincides with the measure with maximal entropy.

MSC:

37D30 Partially hyperbolic systems and dominated splittings
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37C55 Periodic and quasi-periodic flows and diffeomorphisms
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
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