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February  2021, 41(2): 507-523. doi: 10.3934/dcds.2020266

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

 Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902, USA

Received  October 2019 Revised  April 2020 Published  February 2021 Early access  July 2020

Fund Project: 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.

Citation: Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266
##### References:
 [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.

show all references

##### References:
 [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.
Shadows of $p$ on $\partial X$, from basepoints $x \in X$ (left) and $\xi \in \partial X$ (right)
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