Matching collapse and expansion across Matter Trapping surfaces in inhomogeneous $\Lambda$CDM models

In the present work we examine the MTS, for the restriction to spherical dust plus $\Lambda$, proving that it actually is a characteristic surface of the Cauchy problem (generated by its characteristic curves), which opens the possibility for infinite solutions. This translate as the MTS being a boundary between arbitrarily independent solutions, reminiscent of the Birkhoff theorem effects. This property is illustrated with combinations of 3 examples containing MTSs and $\Lambda$ ($\Lambda$CDM, Schwarzschild-de\,Sitter, Lema\^itre-Tolman-Bondi-de\,Sitter: LTBdS -- i.e. the inhomogeneous, spherically symmetric $\Lambda$CDM). The LTBdS model presents a static, stable MTS for the first time.