Uniqueness of conformal measures and local mixing for Anosov groups

Publication date: 24 November 2021

Abstract: In the late seventies, Sullivan showed that for a convex cocompact subgroup

G a m m a

of

o p e r a t o r n a m e {S O}^{c} i r c (n, 1)

with critical exponent

d e l t a > 0

, any

G a m m a

-conformal measure on

p a r t i a l m a t h b b H^{n}

of dimension

d e l t a

is necessarily supported on the limit set

L a m b d a

and that the conformal measure of dimension

d e l t a

exists uniquely. We prove an analogue of this theorem for any Zariski dense Anosov subgroup of a connected semisimple real algebraic group

G

of rank at most

3

. We also obtain the local mixing for generalized BMS measures on

including Haar measures.

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