Local Equivalence of Metrics for Multiparameter Persistence Modules

An ideal invariant for multiparameter persistence would be discriminative, computable and stable. In this work we analyse the discriminative power of a stable, computable invariant of multiparameter persistence modules: the fibered bar code. The fibered bar code is equivalent to the rank invariant and encodes the bar codes of the 1-parameter submodules of a multiparameter module. This invariant is well known to be globally incomplete. However in this work we show that the fibered bar code is locally complete for finitely presented modules by showing a local equivalence of metrics between the interleaving distance (which is complete on finitely-presented modules) and the matching distance on fibered bar codes. More precisely, we show that: for a finitely-presented multiparameter module $M$ there is a neighbourhood of $M$, in the interleaving distance $d_I$, for which the matching distance, $d_0$, satisfies the following bi-Lipschitz inequalities $\frac{1}{34}d_I(M,N) \leq d_0(M,N) \leq d_I(M,N)$ for all $N$ in this neighbourhood about $M$. As a consequence no other module in this neighbourhood has the same fibered bar code as $M$.