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



What is the definition of ideal boundary?
For a [Hadamard space](http://en.wikipedia.org/wiki/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](http://www.math.psu.edu/petrunin/papers/akp-papers/ballmann-lect.pdf) 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](http://www.mathnet.or.kr/mathnet/kms_tex/24289.pdf) paper and [this](http://arxiv.org/abs/math/0510105) paper.


