  We prove that the existence of a $1$-Lipschitz retraction (a contraction)
from a space $X$ onto its subspace $A$ implies the persistence diagram of $A$
embeds into the persistence diagram of $X$. As a tool we introduce tight
injections of persistence modules as maps inducing the said embeddings. We show
contractions always exist onto shortest loops in metric graphs and conjecture
on existence of contractions in planar metric graphs onto all loops of a
shortest homology basis.
  Of primary interest are contractions onto loops in geodesic spaces. These act
as ideal circular coordinates. Furthermore, as the Theorem of Adamaszek and
Adams describes the pattern of persistence diagram of $S^1$, a contraction $X
\to S^1$ implies the same pattern appears in persistence diagram of $X$.
