Naive entropy of dynamical systems
- Creators
- Burton, Peter
Abstract
We study an invariant of dynamical systems called naive entropy, which is defined for both measurable and topological actions of any countable group. We focus on nonamenable groups, in which case the invariant is two-valued, with every system having naive entropy either zero or infinity. Bowen has conjectured that when the acting group is sofic, zero naive entropy implies sofic entropy at most zero for both types of systems. We prove the topological version of this conjecture by showing that for every action of a sofic group by homeomorphisms of a compact metric space, zero naive entropy implies sofic entropy at most zero. This result and the simple definition of naive entropy allow us to show that the generic action of a free group on the Cantor set has sofic entropy at most zero. We observe that a distal Γ-system has zero naive entropy in both senses, if Γ has an element of infinite order. We also show that the naive entropy of a topological system is greater than or equal to the naive measure entropy of the same system with respect to any invariant measure.
Additional Information
© 2017 Hebrew University of Jerusalem. Received July 27, 2015 and in revised form March 26, 2016. We thank Alexander Kechris for introducing us to this topic, and Lewis Bowen for allowing us to read his preprint [7]. We also thank the anonymous referee for numerous helpful comments. This research was partially supported by NSF grant DMS-0968710.Attached Files
Submitted - 1503.06360.pdf
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Additional details
- Eprint ID
- 77986
- Resolver ID
- CaltechAUTHORS:20170607-072116987
- DMS-0968710
- NSF
- Created
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2017-06-07Created from EPrint's datestamp field
- Updated
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2021-11-15Created from EPrint's last_modified field