The Berkovich projective line is an analytic space over a non-Archimedean field. It can also be constructed as an inverse limit of finite rooted trees. 

We find how to associate C*-algebras generated by partial isometries to the Berkovich line. This allows us to construct several spectral triples on this space. 

Finally, we show that invariant measures, such as the Patterson-Sullivan measure, can be obtained as certain KMS-states of the crossed product algebra with a subgroup of PGL2(Cp). 

This is a joint work with Masoud Khalkhali.