Thermodynamic formalism of rational maps Juan Rivera-Letelier U. of - PowerPoint PPT Presentation

Thermodynamic formalism of rational maps Juan Rivera-Letelier U. of Rochester Makarov fest Saas-Fee, Switzerland March  -  , 

Integral means and geometric pressure  Integral means spectrum;  Quadratic Julia sets;  Geometric pressure function.

Integral means spectrum φ : D → C : Univalent, φ ( z ) =  z + b  z + b  z  + ··· . ��  π � � � � t d θ � � � φ ′ ( re i θ ) log  β φ ( t ) := limsup . | log (  − r ) | r →  − Integral means spectrum.

Integral means spectrum φ : D → C : Univalent, φ ( z ) =  z + b  z + b  z  + ··· . ��  π � � � � t d θ � � � φ ′ ( re i θ ) log  β φ ( t ) := limsup . | log (  − r ) | r →  − Integral means spectrum. B ( t ) := sup β φ ( t ) . φ Universal spectrum. For | t | <  , B ( t ) = t  Conjeture:  . B (  ) = Littlewood ’s constant; ⇒ Hölder domains and Brenan ’s conjectures.

Littlewood ’s constant z + b  z + b  z  + ··· . φ : D → C : Univalent, φ ( z ) =  logLength ( φ ( { z ∈ D : | z | = r } )) β φ (  ) = limsup . | log (  − r ) | r →  − Length = Euclidean length in C . Theorem ( Littlewood ,  ; Carleson – Jones ,  ) z + b  z + b  z  + ··· , For every φ ( z ) =  | b n | � n B (  ) . Moreover, B (  ) is the least constant with this property. B (  ) <  .  , Hedenmalm – Shimorin ,  . B (  ) >  .  , Beliaev – Smirnov ,  ;

Littlewood ’s constant Figure : Equipotentials of φ ( z ) =  z + z , for r =  −    ,  −    ,  −    , and  −    .

Littlewood ’s constant Figure : Extremal functions must have a fractal nature

Quadratic Julia sets For c ∈ C : f c : C → C f c ( z ) := z  + c �→ z � � z  ∈ C : ( f n K c := c ( z  )) n ≥  is bounded Filled Julia set of f c ; = complement of the attracting basin of infinity. J c := ∂ K c Julia set of f c .

Quadratic Julia sets Figure : Quadratic Julia set; from Tomoki Kawahira ’s gallery.

Quadratic Julia sets Figure : Another quadratic Julia set, from Arnaud Chéritat ’s gallery.

The spectrum as a pressure function c ∈ C : Such that J c is connected; φ c : D → C : Conformal representation of C \ K c , φ c ( z ) =  z + b  z + b  z  + ··· . The universal spectrum can be computed with Julia sets of arbitrary degree ( Binder , Jones , Makarov , Smirnov ).

The spectrum as a pressure function c ∈ C : Such that J c is connected; φ c : D → C : Conformal representation of C \ K c , φ c ( z ) =  z + b  z + b  z  + ··· . The universal spectrum can be computed with Julia sets of arbitrary degree ( Binder , Jones , Makarov , Smirnov ). � � P c ( t ) := β φ c ( t ) − t +  log  ; Geometric pressure function of f c . �  | Df n c ( z ) | − t ; = lim n log n →∞ z ∈ f − n c ( z  ) = spectral radius of the transfer operator.

Multifractal analysis ρ c : Harmonic measure of J c = Maximal entropy measure of f c . D c ( α ) := HD ( { z ∈ J c : ρ c ( B ( z , r )) ∼ r α } ) . Local dimension spectrum; Frequently D c is analytic (!!!). Theorem ( Sinaï , Ruelle , Bowen ,  ’s) ⇒ f c uniformly hyperbolic D c and P c are analytic and � � t + α P c ( t ) D c ( α ) = inf . log  t ∈ R ∼ Legendre transform; Morally: P c is analytic ⇔ D c is analytic.

Classification of phase transitions  Basic properties of the geometric pressure function;  Negative spectrum;  Phase transitions are of freezing type;  Positive spectrum tricothomy;  Phase transitions at infinity.

Geometric pressure function Variational Principle � � � P c ( t ) = sup h µ − t log | Df c | d µ . µ invariant probability on J c h µ = measure-theoretic entropy. Definition • Equilibrium state for the potential − t log | Df c | : = A measure µ realizing the supremum. • Phase transition : = A parameter at which P c is not analytic. Comparison with statistical mechanics.

Geometric pressure function � � � P c ( t ) = sup h µ − t log | Df c | d µ . µ invariant probability on J c • P c is convex, Lipschitz , and non-increasing; • P c (  ) = log  topological entropy of f c ;

Geometric pressure function � � � P c ( t ) = sup h µ − t log | Df c | d µ . µ invariant probability on J c • P c is convex, Lipschitz , and non-increasing; • P c (  ) = log  topological entropy of f c ; • P c ( t ) ≥ max {− t χ inf ( c ) , − t χ sup ( c ) } , where P c ( t ) χ sup ( c ) := lim ; − t t → + ∞ = Supremum of Lyapunov exponents. P c ( t ) χ inf ( c ) := lim . − t t →−∞ = Infimum of Lyapunov exponents.

Geometric pressure function � � � P c ( t ) = sup h µ − t log | Df c | d µ . µ invariant probability on J c • P c is convex, Lipschitz , and non-increasing; • P c (  ) = log  topological entropy of f c ; • P c ( t ) ≥ max {− t χ inf ( c ) , − t χ sup ( c ) } , where P c ( t ) χ sup ( c ) := lim ; − t t → + ∞ = Supremum of Lyapunov exponents. P c ( t ) χ inf ( c ) := lim . − t t →−∞ = Infimum of Lyapunov exponents. Theorem (Generalized Bowen formula, Przytycki ,  ) inf { t ∈ R : P c ( t ) =  } = HD hyp ( J c ) .

Negative spectrum Mechanism: Gap in the Lyapunov spectrum . ⇔ there is a finite set Σ such that f ( Σ ) = Σ , f −  ( Σ ) \ Σ ⊂ Crit ( f ) .

Negative spectrum Mechanism: Gap in the Lyapunov spectrum . ⇔ there is a finite set Σ such that f ( Σ ) = Σ , f −  ( Σ ) \ Σ ⊂ Crit ( f ) . These phase transitions are removable. Makarov – Smirnov ,  .

Phase transitions are of freezing type

Phase transitions are of freezing type Theorem ( Pryzycki – RL ,  ) P c ( t  ) > max {− t  χ inf ( c ) , − t  χ sup ( c ) } P c is analytic at t = t  . ⇒

Positive spectrum tricothomy  m log | Df m χ crit ( c ) := liminf c ( c ) | . m → + ∞  χ crit ( c ) <  ⇔ f c is uniformly hyperbolic; Levin – Przytycki – Shen ,  .  χ crit ( c ) =  ⇔ Phase transition at the first zero of P c ; ⇔ χ inf ( c ) =  Przytycki – RL – Smirnov (  ), “High-temperature phase transition” Mechanism: Lack of expansion .

Positive spectrum tricothomy  m log | Df m χ crit ( c ) := liminf c ( c ) | . m → + ∞  χ crit ( c ) <  ⇔ f c is uniformly hyperbolic; Levin – Przytycki – Shen ,  .  χ crit ( c ) =  ⇔ Phase transition at the first zero of P c ; ⇔ χ inf ( c ) =  Przytycki – RL – Smirnov (  ), “High-temperature phase transition” Mechanism: Lack of expansion .  χ crit ( c ) >  ⇔ f c is Collet – Eckmann Non-uniformly hyperbolic in a strong sense; Any phase transition in this case must be at “low-temperature”: After the first zero of the geometric pressure function.

Positive spectrum tricothomy Theorem ( Coronel – RL ,  ) There is c ∈ R such that χ crit ( c ) >  and such that f c has a phase transition at some t ∗ > HD hyp ( J c ) . Moreover, c can be chosen so that the critical point of f c is non-recurrent. Examples show the phase transition can be of first order, or of “infinite order”; Inspired conformal Cantor of Makarov and Smirnov (  ). Mechanism: Irregularity of the critical orbit .

Phase transitions at infinity Theorem ( Coronel – RL ,  (hopefully ...)) There is a quadratic-like map f such that: • For every t >  there is a unique equilibrium state ρ t for − t log | Df | ; • lim t → + ∞ ρ t does not exists. Theorem (Sensitive dependence of equilibria) There is a quadratic-like map f such that, for every sequence ( t ( ℓ )) ℓ ≥  going to infinity, there is � f arbitrarily close to f such that ρ t of � • For every t >  there is a unique equilibrium state � f for − t log | D � f | ; • lim ℓ → + ∞ � ρ t ( ℓ ) does not exists.