The persistence of a relative Rabinowitz–Floer complex

We give a quantitative refinement of the invariance of the Legendrian contact homology algebra in general contact manifolds.We show that in this general case, the Lagrangian cobordism trace of a Legendrian isotopy defines a DGA stable tame isomorphism, which is similar to a bifurcation invariance proof for a contactization contact manifold.We use this result to construct a relative version of the Rabinowitz-Floer complex defined for Legendrians that also satisfies a quantitative invariance, and study its persistent homology barcodes.We apply these barcodes to prove several results, including: displacement energy bounds for Legendrian submanifolds in terms of the oscillatory norms of the contact Hamiltonians; a proof of Rosen and Zhang's nondegeneracy conjecture for the Shelukhin-Chekanov-Hofer metric on Legendrian submanifolds; and the nondisplaceability of the standard Legendrian real-projective space inside the contact real-projective space.53D10, 53D42 Ãfor n 1;as well as for the high-dimensional "lens spaces" given as the quotients S 2nC1 =Z k for a subgroup Z k S 1 .We will here consider the case RP 2nC1 D S 2nC1 =Z 2 ; see Proposition 6.4 for the relevant index computation in the case of RP 2nC1 .The computations for the sphere and lens spaces are analogous.