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Coarse entropy of metric spaces
William Geller1, Michał Misiurewicz1, Damian Sawicki2
1Department of Mathematical Sciences, Indiana University Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202 USA.
Geometriae Dedicata
|October 14, 2024
Summary
Coarse entropy, a new measure for large-scale dynamics, is invariant for isometries and spaces. This coarse invariant is either zero or infinity, offering new insights beyond volume growth, especially for bounded geometry and quasi-geodesic spaces.
Area of Science:
- Metric spaces
- Dynamical systems
- Geometric topology
Background:
- Coarse geometry analyzes metric spaces at large scales.
- Coarse entropy is a novel tool for studying dynamics from a coarse perspective.
Purpose of the Study:
- To establish coarse entropy as a coarse invariant.
- To investigate the properties and applications of coarse entropy.
Main Methods:
- Proving invariance of coarse entropy under isometries.
- Analyzing the dichotomy of coarse entropy values (zero or infinity).
- Characterizing this dichotomy for specific classes of spaces.
Main Results:
- All isometries of a metric space share the same coarse entropy.
- Coarse entropy is a coarse invariant of the space.
- Coarse entropy is always zero or infinity, distinct from volume growth in general.
- The dichotomy is characterized for spaces with bounded geometry and quasi-geodesic spaces.
Conclusions:
- Coarse entropy provides a new invariant for metric spaces.
- It offers obstructions for coarse embeddings, even when volume growth does not.
- This invariant is particularly useful for spaces with bounded geometry and quasi-geodesic spaces.
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