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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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