Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

TRANSITION PROBABILITY OF A DISCRETE GEODESIC FLOW ON THE STANDARD NON-UNIFORM QUOTIENT OF PGL3

  • Sanghoon Kwon (Department of Mathematical Education Catholic Kwandong University)
  • Received : 2023.07.31
  • Accepted : 2023.11.27
  • Published : 2024.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

We describe the local transition probability of a singular diagonal action on the standard non-uniform quotient of PGL3 associated to the type 1 geodesic flow. As a consequence, we deduce the property of strongly positive recurrence.

Keywords

Acknowledgement

The author would like to express his gratitude to Prof. Seonhee Lim for her encouragement of this project. This work has been supported by NRF grant (No. RS-2023-00237811).

References

  1. P. Abramenko and K. S. Brown, Buildings, Graduate Texts in Mathematics, 248, Springer, New York, 2008. https://doi.org/10.1007/978-0-387-78835-7
  2. J. S. Athreya, A. Ghosh, and A. Prasad, Ultrametric logarithm laws, II, Monatsh. Math. 167 (2012), no. 3-4, 333-356. https://doi.org/10.1007/s00605-012-0376-y
  3. D. I. Cartwright and W. M lotkowski, Harmonic analysis for groups acting on triangle buildings, J. Austral. Math. Soc. Ser. A 56 (1994), no. 3, 345-383.
  4. K. Golubev and O. Parzanchevski, Spectrum and combinatorics of two-dimensional Ramanujan complexes, Israel J. Math. 230 (2019), no. 2, 583-612. https://doi.org/10.1007/s11856-019-1828-z
  5. S. Gouezel, B. Schapira, and S. Tapie, Pressure at infinity and strong positive recurrence in negative curvature, Comment. Math. Helv. 98 (2023), no. 3, 431-508. https://doi.org/10.4171/cmh/552
  6. S. Hong and S. Kwon, Spectrum of weighted adjacency operator on a non-uniform arithmetic quotient of PGL3, Combinatorics and Number Theory 13 (2024), no. 2, 103-122. https://doi.org/10.2140/cnt.2024.13.103
  7. M.-H. Kang and W.-C. W. Li, Zeta functions of complexes arising from PGL(3), Adv. Math. 256 (2014), 46-103. https://doi.org/10.1016/j.aim.2013.12.033
  8. S. Kwon, Effective mixing and counting in Bruhat-Tits trees, Ergodic Theory Dynam. Systems 38 (2018), no. 1, 257-283. https://doi.org/10.1017/etds.2016.28
  9. S. Kwon and S. Lim, Limiting distribution of geodesics in a geometrically finite quotients of regular trees, Groups Geom. Dyn. 15 (2021), no. 1, 35-55. https://doi.org/10.4171/ggd/590
  10. S. Mozes, Actions of Cartan subgroups, Israel J. Math. 90 (1995), no. 1-3, 253-294. https://doi.org/10.1007/BF02783216
  11. F. Paulin and U. Shapira, On continued fraction expansions of quadratic irrationals in positive characteristic, Groups Geom. Dyn. 14 (2020), no. 1, 81-105. https://doi.org/10.4171/ggd/535
  12. S. Ruette, On the Vere-Jones classification and existence of maximal measures for countable topological Markov chains, Pacific J. Math. 209 (2003), no. 2, 366-380. https://doi.org/10.2140/pjm.2003.209.365
  13. R. Ruhr and O. Sarig, Effective intrinsic ergodicity for countable state Markov shifts, Israel J. Math. 251 (2022), no. 2, 679-735. https://doi.org/10.1007/s11856-022-2436-x