Statistical properties for holomorphic endomorphisms of morphisms - - PowerPoint PPT Presentation

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Statistical properties for holomorphic endomorphisms of projective spaces

Tien-Cuong Dinh (joint with Fabrizio Bianchi)

Conference in Several Complex Variables, 18-21 August, 2020

• rganised by Shiferaw Berhanu and Ming Xiao

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Outline

1 Introduction 2 Invariant probability measures 3 Statistical properties 4 Good Banach spaces

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Endomorphisms, Fatou set and Julia set

Let Pk be the complex projective space of dimension k. Let f : Pk → Pk be a holomorphic endomorphism of degree d 2. The iterate of order n of f is of degree dn and given by fn := f ◦ · · · ◦ f (n times). If V ⊂ Pk is a subvariety of dimension p then (counting multiplicity) deg fn(V) = dnp deg V and deg f−n(V) = dn(k−p) deg V. In particular, for p = 0 and V = {z} is a point #fn(z) = 1 and #f−n(z) = dnk. Generic polynomial maps f : Ck → Ck of degree d can be extended to a holomorphic endomorphism f : Pk → Pk.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Endomorphisms, Fatou set and Julia set

In dimension k = 1, a famous example is f : C (or P1) → C (or P1) f(z) = z2 + c (c is a constant) f4(z) = (((z2 + c)2 + c)2 + c)2 + c. The phase space P1 is divided into two disjoint completely invariant sets: the Fatou set (open) and the Julia set (compact) P1 = F ⊔ J f−1(F) = f(F) = F f−1(J) = f(J) = J. The dynamics in F is tame and stable, and the dynamics in J is chaotic. Problem (a main problem) Understand the dynamics (namely, the orbits of points) in J.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Julia sets

Figure: Julia set of z2 + c with c = 0.687 + 0.312i. Source : mcgoodwin.net.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Julia sets

Figure: Julia set of z2 + c with c = 0.285 + 0.01i. Source : WikiMedia.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Any dimension

In dimension k, we have several Julia sets Pk = J0 ⊃ J1 ⊃ · · · ⊃ Jk. We only consider the small Julia set Jk where the dynamics is most chaotic. For simplicity, we assume f is generic in a sense close to ”the absence

• f periodic critical point”: sub-exponential growth of multiplicity

multiplicity(fn, z) An for all z ∈ Pk, A > 1 and n big. There is no way to describe all orbits (z, f(z), f2(z), . . .) of f on Jk. Probabilistic point of view (for canonical invariant probability measuresonJk) Study the sequence of random variables u, u ◦ f, u ◦ f2, . . . where u : Pk → R is an observable (function of suitable regularity).

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Expected outcomes

Remarks With respect to invariant probability measures, the random variables u ◦ fn are identically distributed: for all n, m and interval [a, b] measure of {u ◦ fn ∈ [a, b]} = measure of {u ◦ fm ∈ [a, b]}. These random variables are clearly not independent (i.d. but not i.i.d.). We expect that the dependence (correlation) between u ◦ fn and u ◦ fm is weak when |n − m| is large. We expect that properties of i.i.d. random variables still hold for the sequence u ◦ fn: the law of large numbers (ergodicity, mixing, K-mixing, exponential mixing), central limit theorem, Berry-Esseen theorem, local central limit theorem, almost sure central limit theorem, large deviation principle, law of iterated logarithms, almost sure invariance principle.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Outline

1 Introduction 2 Invariant probability measures 3 Statistical properties 4 Good Banach spaces

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Backward orbits of points

If z ∈ Pk, then #f−n(z) = dkn (counting multiplicity). If δz is the Dirac mass at z, then d−kn(fn)∗(δz) = d−kn

• w∈f−n(z)

δw is a probability measure.

Figure: Backward orbit of a point.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Equidistribution of points: case without weight

Theorem (Fornaess-Sibony, Briend-Duval, D.-Sibony) There is an invariant probability measure µ with support Jk such that lim

n→∞ d−kn(fn)∗(δz) = µ

for every z ∈ Pk. Moreover, the convergence is exponentially fast. Remarks To see the speed of convergence we consider H¨

• lder continuous test

functions. The measure µ satisfies several interesting properties. In particular, it is the measure of maximal entropy k log d. Recall that we assume that f is generic. Otherwise, the statement is more complicated. The small Julia set Jk could be a Cantor set or equal to Pk but not pluripolar.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Case with weight

Consider a weight φ : Pk → R and the measures µφ,z,n :=

• w∈f−n(z)

eφ(w)+···+φ(fn−1(w))δw (with accumulated weight). Assume max φ − min φ < log d and a weak regularity: for some p > 2 ∀x, y ∈ Pk : |φ(x) − φ(y)| (1 + | log dist(x, y)|)−p. Example: any H¨

• lder continuous function satisfies the last condition.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Case with weight

Theorem (Urbanski-Zdunik, Bianchi-D.) There are an invariant probability measure µφ with support Jk, a number λ > 0 and a continuous function ρ : Pk → R+ such that for mφ := ρ−1µφ lim

n→∞ λ−nµφ,z,n = ρ(z)mφ

for every z ∈ Pk. Moreover, if φ satisfies a suitable regularity (e.g. H¨

• lder continuity), then

the convergence is exponentially fast. Remarks The points in f−n(z), with weights, are equidistributed with respect to mφ when n → ∞. The measure µφ maximises the pressure involving φ (similar to the entropy).

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Duality and Perron-Frobenius operator: case without weight

Let g : Pk → R be a test continuous function. Then (fn)∗(δz), g =

• w∈f−n(z)

δw, g

• =
• w∈f−n(z)

g(w). Define (fn)∗(g)(z) :=

• w∈f−n(z)

g(w). We have (fn)∗(δz), g = (fn)∗(g)(z) = δz, (fn)∗g. So (fn)∗ acting on functions is dual to (fn)∗ acting on measures. In this setting, (fn)∗ is the Perron-Frobenius operator. Equidistribution: convergence of d−kn(fn)∗(δz) ⇐ ⇒ convergence of d−kn(fn)∗(g). Notice that (fn)∗ = (f∗) ◦ · · · ◦ (f∗) and (fn)∗ = (f∗) ◦ · · · ◦ (f∗).

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Duality and Perron-Frobenius operator: case with weight

The operators acting on measures δz − →

• w∈f−n(z)

eφ(w)+···+φ(fn−1(w))δw. The Perron-Frobenius operators acting on functions g − → Ln

φ(g)

with Ln

φ(g)(z) :=

• w∈f−n(z)

eφ(w)+···+φ(fn−1(w))g(w). Equidistribution ⇐ ⇒ convergence of λ−nLn

φ(g) for suitable λ > 0.

Notice that Ln

φ = Lφ ◦ · · · ◦ Lφ.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Linear dynamics: ideal situation (to be applied for L := Lφ)

Consider a linear continuous operator L : E → E on a Banach space E. Assume that λ > 0 is an isolated eigenvalue of multiplicity 1. So there is a vector v ∈ E \ {0} such that L(v) = λv. Assume that the spectrum of L is contained in D(0, r) ∪ {λ} for some r < λ (spectral gap). Then λ−nLn(g) → cgv exponentially fast for some constant cg (modulo v, we have λ−nLn → 0 exponentially fast because r < λ).

Figure: Spectral gap and linear dynamics.

Stability (for small t ∈ C): similar properties for small perturbations Lt : E → E with λt ≃ λ, rt ≃ r and vt ≃ v.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Outline

1 Introduction 2 Invariant probability measures 3 Statistical properties 4 Good Banach spaces

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Statistical properties: a rough strategy

Problem Study the identically distributed random variables u, u ◦ f, u ◦ f2 . . . in the probability space (Pk, µφ). Find a good Banach space E and consider the perturbations Lφ+tu of Lφ with t ∈ C small. We have for g ∈ E Ln

φ+tug(z) =

• w∈f−n(z)

eφ(w)+···+φ(fn−1(w))+t(u(w)+···+u(fn−1(w)))g(w) Using Taylor’s expansion in t and its coefficients gives statistical properties of u (standard arguments but need to solve some technical problems, Nagaev, Guivarc’h, Liverani, Gou¨ ezel...). Problem To find a good Banach space E, i.e. to find a good norm for Lφ and its small perturbations Lφ+tu. In particular, we need Lφ+tu : E → E.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Spectral gap and consequences

Theorem (Bianchi-D., for every 0 < γ 1, new even in 1D) There is a Banach space (E, · ) such that

1 · C0 · · Cγ (H¨

• lder norm);

2 Assume that φ < ∞ (e.g. φCγ < ∞). Then Lφ : E → E has an

isolated maximal eigenvalue λ of multiplicity 1 and a spectral gap.

3 If u < ∞, then Lφ+tu, with |t| small enough, satisfies a similar

property. Corollary Assume that φ < ∞ and u < ∞. Then the sequence u, u ◦ f, u ◦ f2 . . . satisfies strong ergodic theorems (mixing, K-mixing, exponential mixing, mixing of all order), the central limit theorem, Berry-Esseen theorem, local central limit theorem, almost sure central limit theorem, large deviation principle, almost sure invariance principle and law of iterated logarithms. Remark (related results) Fornaess-Sibony, D-Nguyen-Sibony, Dupont, Szostakiewicz-Urbanski-Zdunik. In 1D: Ruelle, Smirnov, Makarov, Denker, Przytycki, Urbanski, Haydn ...

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Ergodic theorem or Law of Large Numbers

Corollary (there are stronger versions such as mixing, K-mixing...) For µφ-almost every point z we have lim

n→∞

1 n

• u(z) + u(f(z)) + · · · + u(fn−1(z))
• =
• udµφ (mean value)
• r equivalently, by duality

lim

n→∞

1 n

• δz + δf(z) + · · · + δfn−1(z)
• = µφ.

So the orbit of z is asymptotically equidistributed with respect to µφ. By adding to u a suitable constant, we can assume for simplicity that

• udµφ = 0.

So we have for µφ-almost every point z we have lim

n→∞

1 n

• u(z) + u(f(z)) + · · · + u(fn−1(z))
• = 0.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Central Limit Theorem

Corollary (there are more precise versions with convergence rate) There is a number σ 0 such that we have the following convergence in law 1 √n

• u(z) + u(f(z)) + · · · + u(fn−1(z))
• −

→ the Gaussian N(0, σ2). Moreover, σ = 0 ⇐ ⇒ N(0, σ2) = δ0 ⇐ ⇒ u is a coboundary, that is u = v ◦ f − v for some continuous v on the small Julia set.

Figure: Gaussian distributions. Source: internet.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Law of iterated logarithms

Corollary For µφ-almost every z, we have lim sup

n→∞

1 √2n log log n

• u(z) + u(f(z)) + · · · + u(fn−1(z))
• = 1

and lim inf

n→∞

1 √2n log log n

• u(z) + u(f(z)) + · · · + u(fn−1(z))
• = −1.

Figure: Law of iterated logarithms.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Large Deviation Principle

Corollary (Large Deviation Theorem, exponential concentration) For every ǫ > 0, there is a constant c(ǫ) > 0 such that µφ

• z :
• 1

n

• u(z) + u(f(z)) + · · · + u(fn−1(z))
• > ǫ
• e−c(ǫ)n.

This is a consequence of the following. Corollary (Large Deviation Principle) There is a strictly convex function c(ǫ) defined in a neighbourhood of 0 ∈ R such that c(0) = 0 and lim

n→∞

1 n log µφ

• z :

1 n

• u(z) + u(f(z)) + · · · + u(fn−1(z))
• > ǫ
• = −c(ǫ).

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Outline

1 Introduction 2 Invariant probability measures 3 Statistical properties 4 Good Banach spaces

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Main difficulties: we want Lφ : E → E, Lφ is a perturbation of f∗

Smooth functions are good for perturbations but not invariant by Perron-Frobenius operators. Example (in dimension k = 1 for f(z) = z2) f∗(u)(z) =

• w∈f−1(z)

u(w) = u(√z) + u(−√z). So u γ-H¨

• lder
• =

⇒ f∗(u) γ-H¨

• lder (only 1

2γ-H¨

• lder).

Functions spanned by psh ones are good for f∗ but not for perturbations. More generally, complex analysis objects like psh functions, positive closed currents are rigid for perturbations. Recall that we also need a spectral gap. We spent more than 1 year to find good norms and there are more than 10 candidates which are not good.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Construction of norms and semi-norms (almost norms)

glogp := sup

x,y |g(x) − g(y)|(1 + | log dist(x, y)|)p

(exp weaker than H¨

• lder).

g∗ := inf T + + T − with T ± positive closed (1, 1)-currents such that ddcg = T + − T −. gp ≃ g∗ + glogp. For (1, 1)-current R and 0 < α < 1 Rp,α := min

• c : ∃S such that |R| c

∞

• n=0

αnd−(k−1)n(fn)∗(S)

• with S positive closed (1, 1)-current of mass 1 and potential uSp 1.

gp,α := i∂g ∧ ∂gp,α dynamical Sobolev norm. gp,α,γ := inf

• c 0: ∀ 0 < ǫ 1 ∃ g(1)

ǫ , g(2) ǫ :

g = g(1)

ǫ

+ g(2)

ǫ ,

• g(1)

ǫ

• p,α c(1/ǫ)1/γ,
• g(2)

ǫ

• ∞ cǫ
• (cost of regularization)

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Technical problems and a key point

Recall that Ln

φ(g)(z) :=

• w∈f−n(z)

eφ(w)+···+φ(fn−1(w))g(w). We need to evaluate i∂Ln

φ(g) ∧ ∂Ln φ(g)

with the above norms in order to get a spectral gap. There are other technical problems. Lemma If u is psh and g is such that i∂g ∧ ∂g ddcu then the modulus of continuity of g is controlled by the one of u. We can avoid local analysis by using global positive closed currents and quasi-psh functions.

Statistical properties for endo- morphisms

• F. Bianchi,

T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces

Thank you and stay safe !

Figure: Barbers and Social Distancing. Source : internet.