ANNALES DE

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ANNALES DE

L’INSTITUT FOURIER

Université Grenoble Alpes

LesAnnales de l’institut Fouriersont membres du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org

François Ledrappier & Lin Shu

Corrigendum to “Differentiating the stochastic entropy in negatively curved spaces under conformal changes”

Tome 71, n^o1 (2021), p. 89-96.

<http://aif.centre-mersenne.org/item/AIF_2021__71_1_89_0>

Cet article est mis à disposition selon les termes de la licence Creative Commons attribution – pas de modification 3.0 France.

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CORRIGENDUM TO “DIFFERENTIATING THE STOCHASTIC ENTROPY IN NEGATIVELY CURVED

SPACES UNDER CONFORMAL CHANGES”

Annales de l’institut Fourier , vol. 67 (2017), n°3, 1115–1183

by François LEDRAPPIER & Lin SHU

Abstract. —We correct an error in [5, Proposition 2.16], and explain the sub- sequent changes in the proof of the main results.

Résumé. —Nous corrigeons une erreur dans [5, Proposition 2.16] et nous dé- taillons les modifications à apporter aux démonstrations des principaux résultats.

There is an error in our paper [5, Proposition 2.16], in the expression of the entropy of a foliated diffusion due to our dismissing the non-symmetry of the diffusions. The main results on the regularities of the entropy (Propo- sition 3.2 and Theorem 5.1 (ii)) and the structure of their proofs are not affected. But in several places, the Martin kernels we used should be replaced by the Martin kernels of the formal adjoint operator and the details of the first step of the proof of Theorem 5.1 (ii) should be corrected.

1. Setting

In this section, we briefly recall the settings and the notations from [5].

We locate the error and introduce the new notations we use.

Let (M, g) be anm-dimensional closed Riemannian manifold with neg- ative curvature, whereg isC³, (M ,f eg) the universal cover andGthe fundamental group ofM that acts freely onMfin such a way that M can be identified with a space of orbits of this action. We choose once for all a connected fundamental domainM0 for the action ofGonM .f

Keywords:formal adjoint, Martin kernel, stochastic entropy.

2020Mathematics Subject Classification:37D40, 58J65.

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90 François LEDRAPPIER & Lin SHU

The unit tangent bundles of M and Mf, denoted SM and SMf, can be identified with M0×∂Mf and Mf×∂Mf, respectively, where ∂Mf is the geometric boundary ofMf. For eachv= (x, ξ)∈SMf, itsstable manifold with respect to the geodesic flow onSMf, denoted W^s(v), is the collection of initial vectorswof geodesics asymptotic toξand can be identified with Mf×{ξ}. The collection ofW^s(v) defines thestable foliation(lamination)W onSM, whose leaves are discrete quotients ofMfand are naturally endowed with the Riemannian metric induced fromeg. A differential operatorL on (the smooth functions on) SM with continuous coefficients is said to be subordinate to the stable foliationW, if for every smooth function F on SM the value ofL(F) at v∈SM only depends on the restriction of F to the leaf ofW that containsv. Given such an operatorL, a Borel measure monSM is calledL-harmonicif it satisfies, for every smooth function f onSM,

Z

L(f)dm= 0.

In this note, we are interested in the case L = ∆ +Y, where Y is a section of the tangent bundle ofW overSM of classC_s^k,α for somek>1 and α ∈ [0,1) such that Y^∗, the dual of Y in the cotangent bundle to W over SM satisfies dY^∗ = 0 leafwisely and such that the Pressure of the function−hX, Yiis positive (the sign convention for the argument of the Pressure function in [2] is opposite to the usual one), whereX is the geodesic spray and can be considered as a section of the tangent bundle ofW overSM. Such aLhas a uniqueL-harmonic probability measurem ([2, Theorem A]).

ExtendLto aG-equivariant operator onSMf=Mf×∂Mfwhich we shall denote with the same symbol. The operatorLdefines a Markovian family of probabilities on Ωe₊, the space of continuous paths of ωe : [0,+∞) → SMf, equipped with the smallest σ-algebra A for which the projections Rt: ωe7→ω(t) are measurable. Indeed, fore v= (x, ξ)∈SMf, let Lv denote the laminated operator ofLonW^s(v). It can be regarded as an operator on Mfwith corresponding heat kernel functions pv(t, y, z), t∈R+, y, z ∈Mf. Define

p(t,(x, ξ), d(y, η)) =p_v(t, x, y)dVol(y)δ_ξ(η),

where δξ(·) is the Dirac function at ξ. Then the diffusion process on W^s(v) with infinitesimal operator Lv is given by a Markovian family {Pw}_w∈M×{ξ}˜ , where for everyt >0 and every Borel setA⊂Mf×∂Mfwe have

Pw({eω: ω(t)e ∈A}) = Z

A

p(t,w, d(y, η)).

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The groupGacts naturally and discretely on the space Ωe+ with quotient the space Ω+ of continuous paths in SM, and this action commutes with the shift. Therefore, the measureeP=R

Pv dm(v) one Ωe+ is the extension of a shift invariant ergodic probability measureP on Ω+.Elements in Ω+

will be denoted by ω, which can be identified with its lift in Ωe+ starting from M₀. By [1], for eachv ∈SM ,f there exists the corresponding Green function Gv(·,·) on Mf×Mf. We define the Green function G(·,·) on SMf×SMfas being

G((y, η),(z, ζ)) :=G(y,η)(y, z)δη(ζ), for (y, η),(z, ζ)∈SM ,f whereδ_η is the Dirac function at η. Moreover, for x, y∈Mfandη ∈∂M ,f the following limit exists and defines theMartin kernel kv(x, y, η):

kv(x, y, η) = lim

z→η

G(x,ξ)(x, z) G_(x,ξ)(y, z).

By [2], Corollary B7, (x, ξ)7→ ∇ylogk_(x,ξ)(x, y, ξ)|y=xis a Hölder continuous function.

Denote by me the G-invariant measure which extends m on Mf×∂Mf. The(stochastic) entropy h_L of L (introduced by Kaimanovich ([3, 4])) is given by

hL:= lim

t→+∞−1 t Z

M₀×∂M˜

(logp(t,(x, ξ),(y, η)))p(t,(x, ξ), d(y, η))dm(x, ξ).e We have the following limit expressions forh_L:

Proposition 1.1 ([5, Proposition 2.4]). — ForP-a.e. paths ω∈Ω+, hL= lim

t→+∞−1

tlogp(t, ω(0), ω(t))

= lim

t→+∞−1

tlogG(ω(0), ω(t)).

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Our attempt in [5, Proposition 2.16], is to replace G in (1.1) by the Martin kernel and use it to find an integral expression. In its computation, we replaced at some point ^G_G^v^(y,z)

v(y,x), for y far away in Mf, by kv(x, z, ξ) whereas that ratio is close to the Martin kernel k^∗_v(x, z, ξ) of the formal adjointL^∗_v, which we define as follows. Forv∈SM ,f letL^∗_v be the formal adjointofLv, i.e., the operator such that for allϕ, ψ∈C_c^∞(Mf),

Z

M˜

ϕL^∗_vψ dVol˜g = Z

M˜

(Lvϕ)ψ dVolg˜.

The operatorL^∗ subordinate toW given byL^∗ψ= ∆ψ−Div(ψY) admits L^∗_v as its laminated operator on W^s(v). Observe that the operator L^∗

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is also weakly coercive and that its Green function G^∗(x, y) is given by G^∗_v(x, y) =Gv(y, x). Again, forv= (x, ξ)∈SMf, y∈Mfandη ∈∂M ,f the following limit exists and defines theMartin kernel k_v^∗(x, y, η):

(1.2) k^∗_v(x, y, η) = lim

z→η

G^∗_(x,ξ)(x, z)

G^∗_(x,ξ)(y, z) = lim

z→η

G_(x,ξ)(z, x) G_(x,ξ)(z, y).

By [2], Corollary B7, the function (x, ξ) 7→ ∇ylogk^∗_(x,ξ)(x, y, ξ)|y=x is Hölder continuous. Moreover, if we writeY =∇logfv for some C² func- tionfv, we haveL^∗_vψ=fvLv (fv)⁻¹ψ

. Thus,ψ 7→ (fv)⁻¹ψ

is a one- to-one correspondence between L^∗_v-harmonic functions and Lv-harmonic functions, so that, for allv= (x, ξ)∈SMf,y∈Mf, η∈∂Mf,

(1.3) k^∗_v(x, y, η) = f_v(y)

fv(x)k_v(x, y, η).

In [2, Appendix B], Hamenstädt studied the operatorL^∗_v and its Martin kernel k_v^∗(x, z, ξ) and she showed that they enjoy many properties of the operator L and its Martin kernel kv(x, z, ξ). In particular, our argument for Proposition 3.2 still holds after correcting k_v by k_v^∗. Finally, to get Theorem 5.1 (ii), we apply the above to operators which come from formally self-adjoint operators, it is not surprising that the final formula is the same as the one obtained when overlooking the lack of symmetry of the intermediate operator.

2. Stochastic entropy for laminated diffusions

With the above notations, the following replaces Proposition 2.16 in [5]:

Proposition 2.1. — LetL = ∆ +Y be such that Y^∗, the dual of Y in the cotangent bundle to the stable foliation overSM, satisfies dY^∗= 0 leafwisely andpr(−hX, Yi)>0. Then,

h_L=− Z

M0×∂M˜

(∆ +Y)y

logk_(x,ξ)^∗ (y, ξ) y=x

dm(x, ξ)e (2.1)

= Z

M0×∂M˜

k∇ylogk_(x,ξ)(y, ξ)|y=xk²−DivY − kYk²

dm(x, ξ).e (2.2)

Proof. — The proof of (2.1) copies the proof of Proposition 2.16 in [5], up to the point where one uses (1.2). For P-a.e. path ω ∈ Ω₊, we still denoteω its projection toMfand writev:=ω(0). Whent goes to infinity, we see that

lim sup

t→+∞

|logGv(x, ω(t))−logk^∗_v(ω(t), ξ)|<+∞.

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Indeed, letz_tbe the point on the geodesic rayγ_ω(t),ξ closest tox. Then, as t→+∞,

G_v(x, ω(t))G_v(z_t, ω(t))Gv(y, ω(t)) Gv(y, zt)

for ally on the geodesic going fromω(t) toξwithd(y, ω(t))>d(y, zt) + 1, wheremeans up to some multiplicative constant independent of t. The first comes from Harnack inequality using the fact that sup_td(x, zt) is finiteP-almost everywhere. (ForP-a.e. ω ∈ Ω₊, η = lim_t→+∞ω(t) differs fromξ andd(x, zt), ast→+∞, tends to the distance betweenxand the geodesic asymptotic toξandηin opposite directions.) The secondcomes from Ancona’s inequality ([1]). Replace Gv(y, ω(t))/Gv(y, zt) by its limit asy →ξ, which isk_(z^∗

t,ξ)(ω(t), ξ) by (1.2), which is itself k_(x,ξ)^∗ (ω(t), ξ) by Harnack inequality again.

Since the L-diffusion has leafwise infinitesimal generator ∆ +Y and is ergodic, we obtain

h_L= lim

t→+∞−1 t

Z t

0

∂

∂s(logk_(x,ξ)^∗ (ω(s), ξ))ds

=− lim

t→+∞

1 t

Z t

0

(∆ +Y)y

logk^∗_(x,ξ)(y, ξ)

_y=ω(s)ds

=− Z

M0×∂M˜

(∆ +Y)y

logk_(x,ξ)^∗ (y, ξ)

_y=xdm(x, ξ).e

For proving (2.2), one starts by using (1.3) to write h_L = −

Z

M0×∂M˜

(∆ +Y)y logk_(x,ξ)(y, ξ) + logf_(x,ξ)(y) _y=xdm(x, ξ).e Since the Martin kernelk_(x,ξ)(·, ξ) satisfies (∆ +Y)y(k_(x,ξ)(y, ξ))|y=x= 0,

(∆ +Y)y logk_(x,ξ)(y, ξ)

|y=x=−k∇y logk_(x,ξ)(y, ξ)

|y=xk². On the other hand, sinceY =∇ylogf_(x,ξ)|y=x, we have

(∆ +Y)_y logf_(x,ξ)(y)

|_y=x= DivY +kYk².

Relation (2.2) follows.

For the proof of Proposition 3.2 in [5], the places that are affected by Proposition 2.16 and should be corrected are as follows: theu₁,Z¹_t andσ₁ of page 1141 have to be replaced byu^∗₁,Z^1,∗_t andσ₁^∗fork^∗. For Lemma 3.11, the changes are only formal, replacing k⁰_v(ω(s), ξ) by kv^0,∗(ω(s), ξ) on page 1160 and in the estimate ofI3.

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3. The entropy part regularity

Letλ∈(−1,1)7→g^λ=e^2ϕ^λgbe aC³curve in the space ofC³negatively curved metrics withg⁰=gand constant volume. Denoteeg^λtheG-invariant extension ofg^λ toMf. Its geometric boundary forMfcan be identified with

∂Mfofeg. For eachg^λ, let ∆^λdenote the laminated Laplacian for its stable foliation (and also the lift to{Mf×{ξ}}_ξ∈∂M˜),m^λthe unique ∆^λ-harmonic probability measure withG-invariant liftme^λ toMf×∂Mf, h_λ the entropy of ∆^λ and k^λ_v the leafwise Martin kernels for ∆^λ. Note that the formal adjoint of ∆^λ is itself and hence has the same Martin kernel functions as

∆^λ. By [3] (see also (2.1)), hλ=−

Z

M0×∂M˜

∆^λ_y

logk_(x,ξ)^λ (y, ξ)

_y=xdme^λ(x, ξ).

LetLb^λ:= ∆ +Z^λ with Z^λ= (m−2)∇ϕ^λ◦φ,where φ:SM7→M is the projection. Then

(3.1) ∆^λ=e^−2ϕ^λ^◦φLb^λ.

Leafwisely,Z^λis the dual of the closed form (m−2)dϕ^λ◦φ.Moreover, the pressure of the function−hX, Z⁰i= 0 is positive. Therefore, there exists δ >0 such that for|λ|< δ, the pressure of the function −hX, Z^λiis still positive. For suchλ, we denote mb^λ for the G-invariant extension to SMf of the harmonic probability measure corresponding to Lb^λ with respect to metricg. We denotebk_v^λ the Martin kernels forLb^λ. LetLb^λ,∗ be the formal adjoint ofLb^λ and its Martin kernel functions are denoted by bk^λ,∗v . Letbhλ

be the entropy forLb^λ. We have by Proposition 2.1 that bhλ=−

Z

M0×∂M˜

Lb^λ_y

logbk^λ,∗_(x,ξ)(y, ξ)

_y=xdmb^λ(x, ξ).

The proof of Theorem 5.1 (ii) in [5] concerning the differential formula of hλ inλwas divided into two parts, where the first part is to show

(I)_h:= lim

λ→0

1

λ(h_λ−bh_λ) = 0

and the other is to analyze the differential ofbh_λinλ. The proof of (I)_h= 0 is affected by the use of the wrong formula of [5, Proposition 2.16]. We correct it as follows.

Claim 3.1. — We have(I)_h= 0.

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Proof. — Forλwith |λ|< δ, setVλ:=R

M₀×∂M˜e^−2ϕ^λ^◦φdme^λ. We show bhλ =V_λ⁻¹hλ. Since the volume of gλ is constant, limλ→01

λ(Vλ−1) = 0, and the claim follows.

Using (3.1), we obtain that Lb^λ,∗α= 0 iff ∆^λ e^−(m−2)ϕ^λ^◦φα

= 0. Con- sequently, we have

bk^λ,∗_v =e^(m−2)ϕ^λ^◦φk^λ_v.

Usingdmb^λ=V_λ⁻¹e^−2ϕ^λ^◦φdme^λ and (3.1), we compute that Vλbhλ=−

Z

M0×∂M˜

∆^λ_y

logbk^λ,∗_(x,ξ)(y, ξ)

_y=x dme^λ(x, ξ)

=− Z

M0×∂M˜

∆^λ

log(k_(x,ξ)^λ (y, ξ)

_y=xdme^λ(x, ξ)

−(m−2) Z

M₀×∂M˜

∆^λ(ϕ^λ◦φ)dme^λ

=− Z

M₀×∂M˜

∆^λ

log(k_(x,ξ)^λ (y, ξ)

_y=xdme^λ(x, ξ) =hλ, where the third equality holds sincem^λ is ∆^λ harmonic.

For the computation of the derivative, we have to replaceu₁byu^∗₁. Since in our case, Lb₀ = ∆ is formally self-adjoint, u^∗₁ = u₁ and our previous computation is unchanged.

BIBLIOGRAPHY

[1] A. Ancona, “Negatively curved manifolds, elliptic operators, and the Martin boundary”,Ann. Math.125(1987), no. 3, p. 495-536.

[2] U. Hamenstädt, “Harmonic measures for compact negatively curved manifolds”, Acta Math.178(1997), no. 1, p. 39-107.

[3] V. A. Ka˘imanovich, “Brownian motion and harmonic functions on covering manifolds. An entropic approach”,Dokl. Akad. Nauk SSSR 288(1986), no. 5, p. 1045- 1049.

[4] ——— , “Brownian motion on foliations: entropy, invariant measures, mixing”, Funkts. Anal. Prilozh. 22 (1988), no. 4, p. 82-83, Russian, translation in Funct.

Anal. Appl.22(1988), no. 4, p. 326-328.

[5] F. Ledrappier&L. Shu, “Differentiating the stochastic entropy for compact negatively curved spaces under conformal changes”,Ann. Inst. Fourier 67(2017), no. 3, p. 1115-1183.

Manuscrit reçu le 31 janvier 2019, accepté le 3 août 2020.

François LEDRAPPIER

Sorbonne Université, UMR 8001, LPSM Boîte Courrier 158

4, Place Jussieu

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96 François LEDRAPPIER & Lin SHU 75252 PARIS cedex 05 (France) fledrapp@nd.edu

Lin SHU

LMAM, School of Mathematical Sciences Peking University

Beijing 100871 (China) lshu@math.pku.edu.cn