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Looking for the Bernoulli shift. (English) Zbl 0803.58013
Consider a Hamiltonian system \(-J \dot x = A x + R_ x(t,x)\), where \(J\) is the canonical symplectic matrix, \(A\) is Hermitian, and \(JA\) is a constant matrix with nonzero real parts of all its eigenvalues. Assume that \(R\) is 1-periodic in \(t\), and is strictly convex \(\forall t\), and that for some \(\alpha > 0\), \(0 < k_ 1 < k_ 2 < +\infty\), we have \[ k_ 1 | x|^ \alpha \leq R(t,x) \leq k_ 2| x|^ \alpha. \] Suppose that the set of nonzero critical points of the dual action functional associated with the system is at most countable below the level \(c^ 1> c\), where \(c\) is the mountain pass level. Then there exists a homoclinic orbit \(x\) such that, for any \(\varepsilon > 0\) and any \(\overline{p} = (p^ 1,\dots ,p^ m) \in \mathbb{Z}^ m\) satisfying \[ \forall i : (p^{i + 1} - p^ i) \geq K(\varepsilon),\quad\text{a const. independent of }m, \] there is a homoclinic orbit \(y_{\overline{p}}\) with \[ \biggl\| y_{\overline{p}} - \sum^ m_{i = 1} x( \cdot - p^ i) \biggr\|_ \infty \leq \varepsilon. \] As a consequence, the flow of the system has a positive topological entropy.
The main result is obtained by constructing multibump homoclinic solutions via variational methods.

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E30 Variational principles in infinite-dimensional spaces
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
54C70 Entropy in general topology
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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