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We derive a sufficient condition for topological horseshoe and uniformhyperbolicity of a 4-dimensional symplectic map, which is introduced bycoupling the two 2-dimensional H\'enon maps via linear terms. The coupledH\'enon map thus constructed can be viewed as a simple map modeling thehorseshoe in higher dimensions. We show that there are two different types ofhorseshoes, each of which is realized around different anti-integrable limitsin the parameter regime.

Coupled H\'enon Map, Part I: Topological Horseshoes and Uniform Hyperbolicity