Dynamic Metric Spaces

Question

Fellow Mathers,

Is there a name for, and is there established theory on, metric spaces that evolve over time?

What I mean by an evolving metric space is one in which the distances between the objects in your set can change through time.

Thank you much to anyone who can point me in the right direction.

Answer 1

I'm glad to find your question. One of my research interests is to study dynamic metric spaces from the perspective of metric geometry and persistent homology. In my papers, you can find

1. a certain definition of dynamic metric spaces,

2. metrics on dynamic metric spaces, and

3. some invariants of dynamic metric spaces.

Here are the links:

• Spatiotemporal Persistent Homology for Dynamic Metric Spaces (arXiv version)
• Stable Signatures for Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence

Hope this helps you. FYI, some other results will appear in my thesis.

Answer 2

It depends exactly on what sort of dynamics you are after. In metric space theory the Gromov-Hausdorff metric is a way to endow a collection of metric spaces with a metric. You can then speak of convergence of metric spaces. You can view a sequence of metric spaces as a metric space evolving through time in this setting. You can even consider a path of metric spaces for continuous time.