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In this work we exhibit flexibility phenomena for some (countable) groups acting by order preserving homeomorphisms of the line. More precisely, we show that if a left orderable group admits an amalgam decomposition of the form    G  =   F  n    ∗  Z    F  mwhere    n  +  m  ⩾  3   , then every faithful action of   G   on the line by order preserving homeomorphisms can be approximated by another action (without global fixed points) that is  not  semi‐conjugated to the initial action. We deduce that    LO  (  G  )   , the space of left orders of   G  , is a Cantor set.  In the special case where    G  =   π  1    (  Σ  )is the fundamental group of a closed hyperbolic surface, we found finer techniques of perturbation. For instance, we exhibit a single representation whose conjugacy class in dense in the space of representations. This entails that the space of representations without global fixed points ofπ  1    (  Σ  )into    H  o  m  e   o  +    (  R  )is connected, and also that the natural conjugation action ofπ  1    (  Σ  )on    LO  (   π  1    (  Σ  )   )    has a dense orbit.

Orderings and flexibility of some subgroups of H o m e o + ( R )