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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

Russell Ricks

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

Received  October 2019 Revised  April 2020 Published  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.

Keywords: Entropy, nonpositive curvature, geodesic flows.
Mathematics Subject Classification: Primary:37D40, 37B40;Secondary:28D20.
Citation: Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[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.  Google Scholar

[4]

R. Engelking, Theory of Dimensions Finite and Infinite, vol. 10 of Sigma Series in Pure Mathematics, Heldermann Verlag, Lemgo, 1995.  Google Scholar

[5]

B. Kleiner, The local structure of length spaces with curvature bounded above, Math. Z., 231 (1999), 409-456.  doi: 10.1007/PL00004738.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567-573.  doi: 10.2307/1971239.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1982.  Google Scholar

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.  Google Scholar

[2]

R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[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.  Google Scholar

[4]

R. Engelking, Theory of Dimensions Finite and Infinite, vol. 10 of Sigma Series in Pure Mathematics, Heldermann Verlag, Lemgo, 1995.  Google Scholar

[5]

B. Kleiner, The local structure of length spaces with curvature bounded above, Math. Z., 231 (1999), 409-456.  doi: 10.1007/PL00004738.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567-573.  doi: 10.2307/1971239.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1982.  Google Scholar

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