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Talk V:Stochastic Properties and Multifractal AnalysisCincinnati (2009)Volker MAYERUniversité de Lille IU.M.R. CNRS 8524http://math.univ-lille1.fr/ mayer/joint work with Mariusz UrbańskiVolker MAYER (USTL)Stochastic Properties and MFA1 / 16

Introduction: Zdunik’s smooth or fractal theoremConsider pc (z) z 2 c and the associated Julia set Jc .Jc is connected iff c M the Mandelbrot set.Volker MAYER (USTL)Stochastic Properties and MFA2 / 16

Introduction: Zdunik’s smooth or fractal theoremConsider pc (z) z 2 c and the associated Julia set Jc .Jc is connected iff c M the Mandelbrot set.There are several dynamically important measures for the dynamical system (pc , Jc ).Volker MAYER (USTL)Stochastic Properties and MFA2 / 16

Introduction: Zdunik’s smooth or fractal theoremConsider pc (z) z 2 c and the associated Julia set Jc .Jc is connected iff c M the Mandelbrot set.There are several dynamically important measures for the dynamical system (pc , Jc ).A. Zdunik (in her 1990 Inv. Math paper) comparedthe Hausdorff measure HM δcwith the harmonic measure µc(in fact the measure of maximal entropy).Volker MAYER (USTL)Stochastic Properties and MFA2 / 16

Introduction: Zdunik’s smooth or fractal theoremConsider pc (z) z 2 c and the associated Julia set Jc .Jc is connected iff c M the Mandelbrot set.There are several dynamically important measures for the dynamical system (pc , Jc ).A. Zdunik (in her 1990 Inv. Math paper) comparedthe Hausdorff measure HM δcwith the harmonic measure µc(in fact the measure of maximal entropy).Theorem (Zdunik(Version for polynomials))δc Hdim(µc ) excepted if pc is a power map z 7 z d or a Chebychev polynomial.Volker MAYER (USTL)Stochastic Properties and MFA2 / 16

Introduction: Zdunik’s smooth or fractal theoremConsider pc (z) z 2 c and the associated Julia set Jc .Jc is connected iff c M the Mandelbrot set.There are several dynamically important measures for the dynamical system (pc , Jc ).A. Zdunik (in her 1990 Inv. Math paper) comparedthe Hausdorff measure HM δcwith the harmonic measure µc(in fact the measure of maximal entropy).Theorem (Zdunik(Version for polynomials))δc Hdim(µc ) excepted if pc is a power map z 7 z d or a Chebychev polynomial.Theorem (Makarov)If Jc is connected (hence A simply connected) then Hdim(µc ) 1.Volker MAYER (USTL)Stochastic Properties and MFA2 / 16

Introduction: Zdunik’s smooth or fractal theoremVolker MAYER (USTL)Stochastic Properties and MFA3 / 16

Introduction: Zdunik’s smooth or fractal theoremVolker MAYER (USTL)Stochastic Properties and MFA4 / 16

Introduction: Zdunik’s smooth or fractal theoremVolker MAYER (USTL)Stochastic Properties and MFA5 / 16

Introduction: Zdunik’s smooth or fractal theoremReason for the dichotomy in Zdunik’s result:Her work is based on stochastic limit theorems (Central Limit Theorem, Law of theIterated Logarithm) andVolker MAYER (USTL)Stochastic Properties and MFA6 / 16

Introduction: Zdunik’s smooth or fractal theoremReason for the dichotomy in Zdunik’s result:Her work is based on stochastic limit theorems (Central Limit Theorem, Law of theIterated Logarithm) andsmooth cases asymptotic variance σ 2 0.Volker MAYER (USTL)Stochastic Properties and MFA6 / 16

The action of Lt on Hölder spacesSetting.f : C Ĉ hyperbolic meromorphic of finite order s.t. f ′ (z) z α1 f (z) α2 .γ f (z) dz ′′Derivative wrt dσ γ dz z α2 : f (z) σ : f (z) γ(z) .Volker MAYER (USTL)Stochastic Properties and MFA7 / 16

The action of Lt on Hölder spacesSetting.f : C Ĉ hyperbolic meromorphic of finite order s.t. f ′ (z) z α1 f (z) α2 .γ f (z) dz ′′Derivative wrt dσ γ dz z α2 : f (z) σ : f (z) γ(z) .The transfer operator associated to the potential φt t log f ′ σ is given byXXLt g (w ) g (z)e φt g (z) f ′ (z) tσ .f (z) wVolker MAYER (USTL)(1)f (z) wStochastic Properties and MFA7 / 16

The action of Lt on Hölder spacesSetting.f : C Ĉ hyperbolic meromorphic of finite order s.t. f ′ (z) z α1 f (z) α2 .γ f (z) dz ′′Derivative wrt dσ γ dz z α2 : f (z) σ : f (z) γ(z) .The transfer operator associated to the potential φt t log f ′ σ is given byXXLt g (w ) g (z)e φt g (z) f ′ (z) tσ .f (z) w(1)f (z) wRecall that1 L̃t e P(t) Lt is the normalized operatorVolker MAYER (USTL)Stochastic Properties and MFA7 / 16

The action of Lt on Hölder spacesSetting.f : C Ĉ hyperbolic meromorphic of finite order s.t. f ′ (z) z α1 f (z) α2 .γ f (z) dz ′′Derivative wrt dσ γ dz z α2 : f (z) σ : f (z) γ(z) .The transfer operator associated to the potential φt t log f ′ σ is given byXXLt g (w ) g (z)e φt g (z) f ′ (z) tσ .f (z) w(1)f (z) wRecall that1 L̃t e P(t) Lt is the normalized operator2 L̃t fixes the density ρt dµt /dmt (hence 1 is eigenvalue).Volker MAYER (USTL)Stochastic Properties and MFA7 / 16

The action of Lt on Hölder spacesSetting.f : C Ĉ hyperbolic meromorphic of finite order s.t. f ′ (z) z α1 f (z) α2 .γ f (z) dz ′′Derivative wrt dσ γ dz z α2 : f (z) σ : f (z) γ(z) .The transfer operator associated to the potential φt t log f ′ σ is given byXXLt g (w ) g (z)e φt g (z) f ′ (z) tσ .f (z) w(1)f (z) wRecall that1 L̃t e P(t) Lt is the normalized operator2 L̃t fixes the density ρt dµt /dmt (hence 1 is eigenvalue).Now we have to1 let Lt act on Hölder functions. Given h : J C, letf h(y ) h(x) vβ (h) supforallx,y Jwith0 y x δ,f y x βbe the β-variation of the function h. Consider the space of β-Hölder or simplyHölder continuous functionsHβ (Jf ) {h : Jf C : h β }Volker MAYER (USTL)with normStochastic Properties and MFA h β vβ (h) h .7 / 16

The action of Lt on Hölder spacesSetting.f : C Ĉ hyperbolic meromorphic of finite order s.t. f ′ (z) z α1 f (z) α2 .γ f (z) dz ′′Derivative wrt dσ γ dz z α2 : f (z) σ : f (z) γ(z) .The transfer operator associated to the potential φt t log f ′ σ is given byXXLt g (w ) g (z)e φt g (z) f ′ (z) tσ .f (z) w(1)f (z) wRecall that1 L̃t e P(t) Lt is the normalized operator2 L̃t fixes the density ρt dµt /dmt (hence 1 is eigenvalue).Now we have to1 let Lt act on Hölder functions. Given h : J C, letf h(y ) h(x) vβ (h) supforallx,y Jwith0 y x δ,f y x βbe the β-variation of the function h. Consider the space of β-Hölder or simplyHölder continuous functionsHβ (Jf ) {h : Jf C : h β }2with norm h β vβ (h) h .Potential of the form φ t log f ′ σ h, h Hβ (Jf ).Denote associated normalized operator by L̃φ .Volker MAYER (USTL)Stochastic Properties and MFA7 / 16

The action of Lt on Hölder spacesTwo Norm inequality and Spectral GapThe key ingredient for all what follows is the two norm inequality:LemmaIf φ t log f ′ σ h, h Hβ (Jf ) then there exists a constant c1 0 such that L̃nφ g β 1 g β c1 g 2for all n 1 large enough and every g Hβ . In particular, L̃φ (Hβ ) Hβ .Volker MAYER (USTL)Stochastic Properties and MFA8 / 16

The action of Lt on Hölder spacesTwo Norm inequality and Spectral GapThe key ingredient for all what follows is the two norm inequality:LemmaIf φ t log f ′ σ h, h Hβ (Jf ) then there exists a constant c1 0 such that L̃nφ g β 1 g β c1 g 2for all n 1 large enough and every g Hβ . In particular, L̃φ (Hβ ) Hβ .Modulo some details,this allows to apply Ionescu-Tulcea and Marinescu in order to get:Theorem- Spectral Gap: The number 1 is a simple isolated eigenvalue of L̃φ : Hβ Hβ andthe rest of the spectrum is contained in D(0, r ) for some 0 r 1.- Decomposition: We have the decomposition L̃φ Q1 S where QR1 : Hβ Cρφ isa projector on the eigenspace Cρφ (given by the formula Q1 (g ) ( g dmφ )ρφ ),Q1 S S Q1 0 and S n β C ξ nfor some constant C 0, some constant ξ (0, 1) and all n 1.Volker MAYER (USTL)Stochastic Properties and MFA8 / 16

The action of Lt on Hölder spacesTheorem- Spectral Gap: The number 1 is a simple isolated eigenvalue of L̃φ : Hβ Hβ andthe rest of the spectrum is contained in D(0, r ) for some 0 r 1.- Decomposition: We have the decomposition L̃φ Q1 S where QR1 : Hβ Cρφ isa projector on the eigenspace Cρφ (given by the formula Q1 (g ) ( g dmφ )ρφ ),Q1 S S Q1 0 and S n β C ξ nfor some constant C 0, some constant ξ (0, 1) and all n 1.RemarkL̃φ : Cb (Jf ) Cb (Jf ) does not have a spectral gap!Volker MAYER (USTL)Stochastic Properties and MFA9 / 16

The action of Lt on Hölder spacesTheorem- Spectral Gap: The number 1 is a simple isolated eigenvalue of L̃φ : Hβ Hβ andthe rest of the spectrum is contained in D(0, r ) for some 0 r 1.- Decomposition: We have the decomposition L̃φ Q1 S where QR1 : Hβ Cρφ isa projector on the eigenspace Cρφ (given by the formula Q1 (g ) ( g dmφ )ρφ ),Q1 S S Q1 0 and S n β C ξ nfor some constant C 0, some constant ξ (0, 1) and all n 1.RemarkL̃φ : Cb (Jf ) Cb (Jf ) does not have a spectral gap!Corollary (Mixing)With the previous notations we have, for every n 1, that L̃nφ Q1 S n and that Z L̃nφ (g ) g dmφ ρφ kS n (g )kβ C ξ n kg kβ , g Hβ .βHence,L̃nφ (g ) RVolker MAYER (USTL) g dmφ ρφ exponentially when n .Stochastic Properties and MFA9 / 16

Stochastic properties: Decay of correlations and CLTThis topic concerns the asymptotic behavior of Sn ψ Pn 1n 0ψ f k for appropriate ψ.Since the equilibrium state µφ is ergodic it follows from Birkhoffs Ergodic Theorem thatZ1Sn ψ(z) ψ dµφ for µφ a.e. z Jf .nRIn the centered case ( ψ dµφ 0), that we consider from now on, n1 Sn ψ 0 µφ –a.e.The random variables Xk ψ f k have same distribution (µφ is invariant).Volker MAYER (USTL)Stochastic Properties and MFA10 / 16

Stochastic properties: Decay of correlations and CLTThis topic concerns the asymptotic behavior of Sn ψ Pn 1n 0ψ f k for appropriate ψ.Since the equilibrium state µφ is ergodic it follows from Birkhoffs Ergodic Theorem thatZ1Sn ψ(z) ψ dµφ for µφ a.e. z Jf .nRIn the centered case ( ψ dµφ 0), that we consider from now on, n1 Sn ψ 0 µφ –a.e.The random variables Xk ψ f k have same distribution (µφ is invariant).The classical Central Limit Theorem (CLT) says that 1n Sn ψ converges in distribution toa Gaussian random variable N (0, σ 2 ) if σ 2 0 and if the variables Xk are independent.Volker MAYER (USTL)Stochastic Properties and MFA10 / 16

Stochastic properties: Decay of correlations and CLTThis topic concerns the asymptotic behavior of Sn ψ Pn 1n 0ψ f k for appropriate ψ.Since the equilibrium state µφ is ergodic it follows from Birkhoffs Ergodic Theorem thatZ1Sn ψ(z) ψ dµφ for µφ a.e. z Jf .nRIn the centered case ( ψ dµφ 0), that we consider from now on, n1 Sn ψ 0 µφ –a.e.The random variables Xk ψ f k have same distribution (µφ is invariant).The classical Central Limit Theorem (CLT) says that 1n Sn ψ converges in distribution toa Gaussian random variable N (0, σ 2 ) if σ 2 0 and if the variables Xk are independent.The later is however not the case and the defect of independence of ψ and U n ψ ψ f nis measured by the correlationsRRRCn (ψ1 , ψ2 ) : ψ1 · (ψ2 f n ) dµφ ψ1 dµφ ψ2 dµφ R (ψ1 E ψ1 ) (ψ2 E ψ2 ) f n dµφ ,Rwhere we put E ψ ψ dµφ .RIn the centered case E ψ1 E ψ2 0, then Cn (ψ1 , ψ2 ) ψ1 ψ2 f n dµφ .Volker MAYER (USTL)Stochastic Properties and MFA10 / 16

Stochastic properties: Decay of correlations and CLTDecay of CorrelationsTheoremFor all (centered) ψ1 Oβ”mainly Hölder”,ψ2 L1 (mφ ) Cn (ψ1 , ψ2 ) ξ n kL̃φ (ψ1 ρφ )kβ kψ2 kL1 (mφ ) ,where ξ (0, 1) comes from the spectral gap Theorem.Volker MAYER (USTL)Stochastic Properties and MFA11 / 16