Perturbation of the Lyapunov spectra of periodic orbits (Q2901905)

Let \(f\) be a diffeomorphism of a \(d\)-dimensional compact manifold and let \(p\) be a periodic point of \(f\).NEWLINENEWLINEDenote by \(\lambda_1\leq\dots\leq\lambda_d\) the Lyapunov exponents of \(p\) and associate to the trajectory \(\gamma\) of \(p\) its Lyapunov graph \(\sigma(\gamma)=(\sigma_0,\sigma_1,\dots,\sigma_d)\), where \(\sigma_0=0\) and NEWLINE\[NEWLINE \sigma_i=\sum_{j=1}^i \lambda_j,\quad i>0. NEWLINE\]NEWLINE The authors prove several very deep and important results concerning the behavior of Lyapunov graphs for sequences of periodic points and for perturbations of diffeomorphisms. One of them reads as follows.NEWLINENEWLINEDenote by \(S_d\subset\{0\}\times{\mathbb R}^d\) the set of all convex graphs. Let \(\gamma_n=\text{orb}(p_n)\) be a sequence of periodic orbits of \(f\) whose periods tend to infinity and such that the sequence \(\gamma_n\) converges in the Hausdorff topology to a compact set \(\Lambda\) that has no dominated splitting. Then for any given \(\varepsilon\) there is an \(N\) such that if \(n\geq N\) and \(\sigma\in S_d\) is a convex graph with (i) \(\sigma_d=\sigma_d(\gamma_n)\) and (ii) \(\sigma_i\geq \sigma_i(\gamma_n)\) for \(i\in\{1,\dots,d-1\}\), then there exists a perturbation \(g\) of \(f\) belonging to the \(C^1\)-neighborhood of \(f\) of radius \(\varepsilon\) such that (i) \(g\) coincides with \(f\) outside an arbitrary neighborhood of \(\gamma_n\), (ii) \(\gamma_n\) is a periodic orbit of \(g\), and (iii) \(\sigma(\gamma_n,g)=\sigma\).NEWLINENEWLINEThe authors also consider the case of a sequence of periodic orbits that converges to a compact set having a dominated splitting and the case of a sequence of diffeomorphisms \(f_n\) that converges to \(f\) in the \(C^1\) topology and such that \(f_n\) have periodic orbits \(\gamma_n\) whose \(f_n\)-invariant probabilities \(\mu_n\) associated to \(\gamma_n\) converge in the weak-star topology to an \(f\)-invariant measure \(\mu\).