HOROSPHERICAL INVARIANT MEASURES AND A RANK DICHOTOMY FOR ANOSOV GROUPS

Or Landesberg; Minju Lee; Elon Lindenstrauss; Hee Oh

doi:10.3934/jmd.2023009

HOROSPHERICAL INVARIANT MEASURES AND A RANK DICHOTOMY FOR ANOSOV GROUPS

Or Landesberg, Minju Lee, Elon Lindenstrauss, Hee Oh

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Let G =^∏r i=1 G_i be a product of simple real algebraic groups of rank one and Γ an Anosov subgroup of G with respect to a minimal parabolic subgroup. For each v in the interior of a positive Weyl chamber, let Rv ⊂Γ\G denote the Borel subset of all points with recurrent exp(R₊v)-orbits. For a maximal horospherical subgroup N of G, we show that the N-action on Rv is uniquely ergodic if r = rank(G) ≤ 3 and v belongs to the interior of the limit cone of Γ, and that there exists no N-invariant Radon measure on Rv otherwise.

Original language	American English
Pages (from-to)	331-362
Number of pages	32
Journal	Journal of Modern Dynamics
Volume	19
DOIs	https://doi.org/10.3934/jmd.2023009
State	Published - 2023

Bibliographical note

Funding Information:
Received September 15, 2021; revised October 31, 2022. 2020 Mathematics Subject Classification: Primary: 37A17, 37A40, 22E40; Secondary: 22F30. Key words and phrases: Horospherical flow, Anosov subgroups, infinite measure rigidity. OL: Partially supported by ISF-Moked grant 2095/19. EL: Partially supported by ERC 2020 grant no. 833423. HO: Partially supported by the NSF grant 0003086.

Publisher Copyright:
© 2023, American Institute of Mathematical Sciences. All rights reserved.

Keywords

Anosov subgroups
Horospherical flow
infinite measure rigidity

Access to Document

10.3934/jmd.2023009

Cite this

@article{50076e0635bb4a70bd0f74d2371fdde1,

title = "HOROSPHERICAL INVARIANT MEASURES AND A RANK DICHOTOMY FOR ANOSOV GROUPS",

abstract = "Let G =∏r i=1 Gi be a product of simple real algebraic groups of rank one and Γ an Anosov subgroup of G with respect to a minimal parabolic subgroup. For each v in the interior of a positive Weyl chamber, let Rv ⊂Γ\G denote the Borel subset of all points with recurrent exp(R+v)-orbits. For a maximal horospherical subgroup N of G, we show that the N-action on Rv is uniquely ergodic if r = rank(G) ≤ 3 and v belongs to the interior of the limit cone of Γ, and that there exists no N-invariant Radon measure on Rv otherwise.",

keywords = "Anosov subgroups, Horospherical flow, infinite measure rigidity",

author = "Or Landesberg and Minju Lee and Elon Lindenstrauss and Hee Oh",

note = "Funding Information: Received September 15, 2021; revised October 31, 2022. 2020 Mathematics Subject Classification: Primary: 37A17, 37A40, 22E40; Secondary: 22F30. Key words and phrases: Horospherical flow, Anosov subgroups, infinite measure rigidity. OL: Partially supported by ISF-Moked grant 2095/19. EL: Partially supported by ERC 2020 grant no. 833423. HO: Partially supported by the NSF grant 0003086. Publisher Copyright: {\textcopyright} 2023, American Institute of Mathematical Sciences. All rights reserved.",

year = "2023",

doi = "10.3934/jmd.2023009",

language = "American English",

volume = "19",

pages = "331--362",

journal = "Journal of Modern Dynamics",

issn = "1930-5311",

publisher = "American Institute of Mathematical Sciences",

}

TY - JOUR

T1 - HOROSPHERICAL INVARIANT MEASURES AND A RANK DICHOTOMY FOR ANOSOV GROUPS

AU - Landesberg, Or

AU - Lee, Minju

AU - Lindenstrauss, Elon

AU - Oh, Hee

N1 - Funding Information: Received September 15, 2021; revised October 31, 2022. 2020 Mathematics Subject Classification: Primary: 37A17, 37A40, 22E40; Secondary: 22F30. Key words and phrases: Horospherical flow, Anosov subgroups, infinite measure rigidity. OL: Partially supported by ISF-Moked grant 2095/19. EL: Partially supported by ERC 2020 grant no. 833423. HO: Partially supported by the NSF grant 0003086. Publisher Copyright: © 2023, American Institute of Mathematical Sciences. All rights reserved.

PY - 2023

Y1 - 2023

N2 - Let G =∏r i=1 Gi be a product of simple real algebraic groups of rank one and Γ an Anosov subgroup of G with respect to a minimal parabolic subgroup. For each v in the interior of a positive Weyl chamber, let Rv ⊂Γ\G denote the Borel subset of all points with recurrent exp(R+v)-orbits. For a maximal horospherical subgroup N of G, we show that the N-action on Rv is uniquely ergodic if r = rank(G) ≤ 3 and v belongs to the interior of the limit cone of Γ, and that there exists no N-invariant Radon measure on Rv otherwise.

AB - Let G =∏r i=1 Gi be a product of simple real algebraic groups of rank one and Γ an Anosov subgroup of G with respect to a minimal parabolic subgroup. For each v in the interior of a positive Weyl chamber, let Rv ⊂Γ\G denote the Borel subset of all points with recurrent exp(R+v)-orbits. For a maximal horospherical subgroup N of G, we show that the N-action on Rv is uniquely ergodic if r = rank(G) ≤ 3 and v belongs to the interior of the limit cone of Γ, and that there exists no N-invariant Radon measure on Rv otherwise.

KW - Anosov subgroups

KW - Horospherical flow

KW - infinite measure rigidity

UR - http://www.scopus.com/inward/record.url?scp=85159948354&partnerID=8YFLogxK

U2 - 10.3934/jmd.2023009

DO - 10.3934/jmd.2023009

M3 - Article

AN - SCOPUS:85159948354

SN - 1930-5311

VL - 19

SP - 331

EP - 362

JO - Journal of Modern Dynamics

JF - Journal of Modern Dynamics

ER -

HOROSPHERICAL INVARIANT MEASURES AND A RANK DICHOTOMY FOR ANOSOV GROUPS

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this