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I would like, if possible, a simple example of a hyperbolic set that is not locally maximal.

What kind of dynamic phenomenon should occur for the appearance of hyperbolic set that is not maximal.

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up vote 5 down vote accepted

Consider the Smale horseshoe, call it $\Lambda$, with the corresponding map $f\colon \Lambda\to\Lambda$. Then $\Lambda$ is a locally maximal hyperbolic set. Moreover, $(\Lambda,f)$ is topologically conjugate to $(\Sigma,\sigma)$, where $\Sigma = \{0,1\}^\mathbb{Z}$ is the full shift on two symbols. Call the topological conjugacy between the two $\pi\colon \Sigma\to \Lambda$.

Let $X\subset \Sigma$ be any closed $\sigma$-invariant subset. Then $\pi(X)\subset \Lambda$ is a hyperbolic set but is not necessarily locally maximal. Indeed, $\pi(X)$ is locally maximal if and only if $X$ is a subshift of finite type.

As an extreme example of what may happen, you may take $X$ to be some minimal subshift that is not periodic, then $\pi(X)$ is a hyperbolic set that does not contain any periodic orbits. Similarly you can construct hyperbolic sets on which the map is uniquely ergodic, or has more or less any kind of behaviour you wish -- at least, anything you can realise with a shift space, which is quite a lot (see for example the book by Denker, Grillenberger, and Sigmund). The point of local maximality is that you ensure the types of behaviour that are typically associated with hyperbolic dynamics.

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See TODD FISHER's thesis: Hyperbolic sets that are not locally maximal (, where he constructed some new examples of hyperbolic sets which are not contained in locally maximal hyperbolic sets.

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