Statistical properties for holomorphic endomorphisms of morphisms - PowerPoint PPT Presentation

Statistical properties for endo- Statistical properties for holomorphic endomorphisms of morphisms F. Bianchi, projective spaces T.-C. Dinh Introduction Invariant measures Tien-Cuong Dinh (joint with Fabrizio Bianchi) Statistical properties Good Banach spaces Conference in Several Complex Variables, 18-21 August, 2020 organised by Shiferaw Berhanu and Ming Xiao

Outline Statistical properties for endo- morphisms F. Bianchi, 1 Introduction T.-C. Dinh Introduction Invariant measures 2 Invariant probability measures Statistical properties Good Banach spaces 3 Statistical properties 4 Good Banach spaces

Endomorphisms, Fatou set and Julia set Statistical properties for endo- morphisms Let P k be the complex projective space of dimension k . F. Bianchi, T.-C. Dinh Let f : P k → P k be a holomorphic endomorphism of degree d � 2. Introduction The iterate of order n of f is of degree d n and given by Invariant measures f n := f ◦ · · · ◦ f Statistical ( n times). properties If V ⊂ P k is a subvariety of dimension p then (counting multiplicity) Good Banach spaces deg f n ( V ) = d np deg V deg f − n ( V ) = d n ( k − p ) deg V . and In particular, for p = 0 and V = { z } is a point # f n ( z ) = 1 # f − n ( z ) = d nk . and Generic polynomial maps f : C k → C k of degree d can be extended to a holomorphic endomorphism f : P k → P k .

Endomorphisms, Fatou set and Julia set Statistical properties for endo- morphisms F. Bianchi, In dimension k = 1, a famous example is f : C (or P 1 ) → C (or P 1 ) T.-C. Dinh f ( z ) = z 2 + c Introduction ( c is a constant) Invariant f 4 ( z ) = ((( z 2 + c ) 2 + c ) 2 + c ) 2 + c . measures Statistical The phase space P 1 is divided into two disjoint completely invariant properties Good sets: the Fatou set (open) and the Julia set (compact) Banach spaces P 1 = 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 .

Julia sets Statistical properties for endo- morphisms F. Bianchi, T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces Figure: Julia set of z 2 + c with c = 0.687 + 0.312 i . Source : mcgoodwin.net.

Julia sets Statistical properties for endo- morphisms F. Bianchi, T.-C. Dinh Introduction Invariant measures Statistical properties Good Banach spaces Figure: Julia set of z 2 + c with c = 0.285 + 0.01 i . Source : WikiMedia.

Any dimension Statistical properties for endo- In dimension k , we have several Julia sets morphisms P k = J 0 ⊃ J 1 ⊃ · · · ⊃ J k . F. Bianchi, T.-C. Dinh Introduction We only consider the small Julia set J k where the dynamics is most Invariant chaotic. measures For simplicity, we assume f is generic in a sense close to ”the absence Statistical properties of periodic critical point”: sub-exponential growth of multiplicity Good Banach multiplicity ( f n , z ) � A n for all z ∈ P k , A > 1 and n big. spaces There is no way to describe all orbits ( z , f ( z ) , f 2 ( z ) , . . . ) of f on J k . Probabilistic point of view (for canonical invariant probability measureson J k ) Study the sequence of random variables u , u ◦ f , u ◦ f 2 , . . . where u : P k → R is an observable (function of suitable regularity).

Expected outcomes Statistical properties for endo- morphisms Remarks F. Bianchi, T.-C. Dinh With respect to invariant probability measures, the random variables u ◦ f n are identically distributed: for all n , m and interval [ a , b ] Introduction Invariant measures measure of { u ◦ f n ∈ [ a , b ] } = measure of { u ◦ f m ∈ [ a , b ] } . Statistical properties These random variables are clearly not independent (i.d. but not i.i.d.). Good Banach spaces We expect that the dependence (correlation) between u ◦ f n and u ◦ f m is weak when | n − m | is large. We expect that properties of i.i.d. random variables still hold for the sequence u ◦ f n : 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.

Outline Statistical properties for endo- morphisms F. Bianchi, 1 Introduction T.-C. Dinh Introduction Invariant measures 2 Invariant probability measures Statistical properties Good Banach spaces 3 Statistical properties 4 Good Banach spaces

Backward orbits of points Statistical If z ∈ P k , then # f − n ( z ) = d kn (counting multiplicity). properties for endo- morphisms If δ z is the Dirac mass at z , then F. Bianchi, � T.-C. Dinh d − kn ( f n ) ∗ ( δ z ) = d − kn δ w Introduction w ∈ f − n ( z ) Invariant measures is a probability measure. Statistical properties Good Banach spaces Figure: Backward orbit of a point.

Equidistribution of points: case without weight Statistical properties for endo- Theorem (Fornaess-Sibony, Briend-Duval, D.-Sibony) morphisms There is an invariant probability measure µ with support J k such that F. Bianchi, T.-C. Dinh n → ∞ d − kn ( f n ) ∗ ( δ z ) = µ for every z ∈ P k . lim Introduction Invariant measures Moreover, the convergence is exponentially fast. Statistical properties Remarks Good Banach spaces To see the speed of convergence we consider H¨ older 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 J k could be a Cantor set or equal to P k but not pluripolar.

Case with weight Statistical properties Consider a weight φ : P k → R and the measures for endo- morphisms � F. Bianchi, e φ ( w )+ ··· + φ ( f n − 1 ( w )) δ w µ φ , z , n := (with accumulated weight). T.-C. Dinh w ∈ f − n ( z ) Introduction Invariant measures Statistical properties Good Banach spaces Assume max φ − min φ < log d and a weak regularity: for some p > 2 ∀ x , y ∈ P k : | φ ( x ) − φ ( y ) | � ( 1 + | log dist ( x , y ) | ) − p . Example: any H¨ older continuous function satisfies the last condition.

Case with weight Statistical properties for endo- morphisms Theorem (Urbanski-Zdunik, Bianchi-D.) F. Bianchi, T.-C. Dinh There are an invariant probability measure µ φ with support J k , a number λ > 0 and a continuous function ρ : P k → R + such that for m φ := ρ − 1 µ φ Introduction Invariant measures n → ∞ λ − n µ φ , z , n = ρ ( z ) m φ for every z ∈ P k . lim Statistical properties Moreover, if φ satisfies a suitable regularity (e.g. H¨ older continuity), then Good Banach the convergence is exponentially fast. spaces Remarks The points in f − n ( z ) , with weights, are equidistributed with respect to m φ when n → ∞ . The measure µ φ maximises the pressure involving φ (similar to the entropy).

Duality and Perron-Frobenius operator: case without weight Statistical properties for endo- Let g : P k → R be a test continuous function. Then morphisms F. Bianchi, � � � � T.-C. Dinh � ( f n ) ∗ ( δ z ) , g � = δ w , g = g ( w ) . Introduction w ∈ f − n ( z ) w ∈ f − n ( z ) Invariant measures Define � ( f n ) ∗ ( g )( z ) := Statistical g ( w ) . properties w ∈ f − n ( z ) Good Banach We have spaces � ( f n ) ∗ ( δ z ) , g � = ( f n ) ∗ ( g )( z ) = � δ z , ( f n ) ∗ g � . So ( f n ) ∗ acting on functions is dual to ( f n ) ∗ acting on measures. In this setting, ( f n ) ∗ is the Perron-Frobenius operator. Equidistribution: convergence of d − kn ( f n ) ∗ ( δ z ) convergence of d − kn ( f n ) ∗ ( g ) . ⇐ ⇒ Notice that ( f n ) ∗ = ( f ∗ ) ◦ · · · ◦ ( f ∗ ) and ( f n ) ∗ = ( f ∗ ) ◦ · · · ◦ ( f ∗ ) .

Duality and Perron-Frobenius operator: case with weight Statistical properties for endo- morphisms F. Bianchi, T.-C. Dinh The operators acting on measures Introduction � Invariant e φ ( w )+ ··· + φ ( f n − 1 ( w )) δ w . δ z − → measures w ∈ f − n ( z ) Statistical properties The Perron-Frobenius operators acting on functions Good Banach � spaces e φ ( w )+ ··· + φ ( f n − 1 ( w )) g ( w ) . → L n L n g − φ ( g ) with φ ( g )( z ) := w ∈ f − n ( z ) ⇒ convergence of λ − n L n Equidistribution ⇐ φ ( g ) for suitable λ > 0. Notice that L n φ = L φ ◦ · · · ◦ L φ .

Linear dynamics: ideal situation (to be applied for L := L φ ) Statistical properties Consider a linear continuous operator L : E → E on a Banach space E . for endo- morphisms Assume that λ > 0 is an isolated eigenvalue of multiplicity 1. So there F. Bianchi, is a vector v ∈ E \ { 0 } such that L ( v ) = λv . T.-C. Dinh Assume that the spectrum of L is contained in Introduction Invariant measures D ( 0, r ) ∪ { λ } for some r < λ (spectral gap). Statistical properties Then λ − n L n ( g ) → c g v exponentially fast for some constant c g (modulo v , we have λ − n L n → 0 exponentially fast because r < λ ). Good Banach spaces Figure: Spectral gap and linear dynamics. Stability (for small t ∈ C ): similar properties for small perturbations L t : E → E with λ t ≃ λ , r t ≃ r and v t ≃ v .