Earthquakes and Thurston's boundary for the Teichmüller space of the universal hyperbolic solenoid

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A measured laminations on the universal hyperbolic solenoid § is, by our definition, a leafwise measured lamination with appropriate continuity for the transverse variations. An earthquakes on theuniversal hyperbolic solenoid § is uniquely determined by a measured lamination on §; it is a leafwise earthquake with the leafwise earthquake measure equal to the leafwise measured lamination. Leafwise earthquakes fit together to produce a new hyperbolic metric on § which is transversely continuous and we show that any two hyperbolic metrics on § are connected by an earthquake. We also establish the space of projective measured lamination PML(§) as a natural Thurston-type boundary to the Teichmüller space T(§) of the universal hyperbolic solenoid §. The (baseleaf preserving) mapping class group MCGBLP(§) acts continuously on the closure T(§)∪ PML(§) of T(§). Moreover, the set of transversely locally constant measured laminations on § is dense in ML(§).