Advanced Search
Article Contents
Article Contents

# The unique measure of maximal entropy for a compact rank one locally CAT(0) space

The author would like to thank three anonymous referees, who all made helpful suggestions to improve the paper. The author was partially supported by NSF RTG 1045119

• Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. We prove the Bowen-Margulis measure on the space of geodesics is the unique measure of maximal entropy for the geodesic flow, which has topological entropy equal to the critical exponent of the Poincaré series.

Mathematics Subject Classification: Primary:37D40, 37B40;Secondary:28D20.

 Citation:

• Figure 1.  Shadows of $p$ on $\partial X$, from basepoints $x \in X$ (left) and $\xi \in \partial X$ (right)

•  [1] W. Ballmann, Lectures on Spaces of Nonpositive Curvature, vol. 25 of DMV Seminar, Birkhäuser Verlag, Basel, 1995, With an appendix by Misha Brin. doi: 10.1007/978-3-0348-9240-7. [2] R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X. [3] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, vol. 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9. [4] R. Engelking, Theory of Dimensions Finite and Infinite, vol. 10 of Sigma Series in Pure Mathematics, Heldermann Verlag, Lemgo, 1995. [5] B. Kleiner, The local structure of length spaces with curvature bounded above, Math. Z., 231 (1999), 409-456.  doi: 10.1007/PL00004738. [6] G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds, Ann. of Math. (2), 148 (1998), 291-314.  doi: 10.2307/120995. [7] A. Lytchak and K. Nagano, Geodesically complete spaces with an upper curvature bound, Geom. Funct. Anal., 29 (2019), 295-342.  doi: 10.1007/s00039-019-00483-7. [8] A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567-573.  doi: 10.2307/1971239. [9] R. Ricks, Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one ${\rm{CAT}}$(0) spaces, Ergodic Theory Dynam. Systems, 37 (2017), 939-970.  doi: 10.1017/etds.2015.78. [10] E. L. Swenson, A cut point theorem for ${\rm{CAT}}$(0) groups, J. Differential Geom., 53 (1999), 327-358.  doi: 10.4310/jdg/1214425538. [11] P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1982.

Figures(1)

## Article Metrics

HTML views(506) PDF downloads(236) Cited by(0)

## Other Articles By Authors

• on this site
• on Google Scholar

### Catalog

/

DownLoad:  Full-Size Img  PowerPoint