The persistence of the Chekanov-Eliashberg algebra

We apply the barcodes of persistent homology theory to the Chekanov-Eliashberg algebra of a Legendrian submanifold to deduce displacement energy bounds for arbitrary Legendrians. We do not require the full Chekanov-Eliashberg algebra to admit an augmentation as we linearize the algebra only below a certain action level. As an application we show that it is not possible to $C^0$-approximate a stabilized Legendrian by a Legendrian that admits an augmentation.