# What is the definition of ideal boundary?

In many papers about dynamical system, I found the word " ideal boundary". T don't know what is the definition of ideal boundary.

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For a Hadamard space $X$ there are two kinds of ideal boundaries, the set $Bd(X)$ of horofunctions up to additive constants, and the set $X(\infty)$ of equivalence classes of rays. These two are homeomorphic by the correspondence:

$$\gamma \text{ (a ray)}\to \text{the Busemann function of } \gamma$$ $$h \text{ (a horofunction)}\to \text{the gradient of } h$$

Here you can find lecture notes by Ballman, with chapter two treating the boundary at infinity via Buseman functions and via rays. You will also find definitions in this paper and this paper.

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You could also look at Bridson & Haefliger's book Metric spaces of non-positive curvature. –  HJRW May 31 '10 at 5:29