A Large Deviation Principle for Gibbs States on countable Markov shifts at zero temperature Rodrigo Bissacot - IME USP Jointly with Edgardo P´ erez (IME-USP, Brazil) and Jairo K. Mengue (UFRGS, Brazil) Workshop on Functional Analysis and Dynamical Systems Florian´ opolis - Brasil - 24 February 2015 supported by FAPESP 1/32
Motivation Common belief (among people working in countable Markov shifts) ”The thermodynamic formalism for topologically mixing countable Markov shifts with BIP property is similar to that of subshifts of finite type defined on finite alphabets.” 2/32
Setting: X = Σ A ( S ) is a Markov subshift of a one-dimensional lattice: Σ A ( S ) ⊆ S N (or S Z ). where S = {− 1 , +1 } , { 1 , 2 , ..., k } or N [our case]. The potential f : X → R always more then continuous (Lipschitz, H¨ older, summable variation, Walters). Shift map σ : S N → S N ; σ ( x ) = σ ( x 0 , x 1 , x 2 , ... ) = ( x 1 , x 2 , x 3 , ... ). 3/32
Setting: X = Σ A ( S ) is a Markov subshift of a one-dimensional lattice: Σ A ( S ) ⊆ S N (or S Z ). where S = {− 1 , +1 } , { 1 , 2 , ..., k } or N [our case]. The potential f : X → R always more then continuous (Lipschitz, H¨ older, summable variation, Walters). Shift map σ : S N → S N ; σ ( x ) = σ ( x 0 , x 1 , x 2 , ... ) = ( x 1 , x 2 , x 3 , ... ). 3/32
Setting: X = Σ A ( S ) is a Markov subshift of a one-dimensional lattice: Σ A ( S ) ⊆ S N (or S Z ). where S = {− 1 , +1 } , { 1 , 2 , ..., k } or N [our case]. The potential f : X → R always more then continuous (Lipschitz, H¨ older, summable variation, Walters). Shift map σ : S N → S N ; σ ( x ) = σ ( x 0 , x 1 , x 2 , ... ) = ( x 1 , x 2 , x 3 , ... ). 3/32
Infinite matrix A : N × N → { 0 , 1 } Σ A ( N ) := { x ∈ N N : A ( x i , x i +1 ) = 1 , ∀ i ≥ 0 } . cylinders: Fix a ∈ N . [ a ] = { x = ( a , x 1 , x 2 , ... ) ∈ Σ A ( N ) } [ a 0 a 1 ... a n − 1 ] = { x = ( a 0 , a 1 , ..., a n − 1 , x n , x n +1 , ... ) ∈ Σ A ( N ) } 4/32
Infinite matrix A : N × N → { 0 , 1 } Σ A ( N ) := { x ∈ N N : A ( x i , x i +1 ) = 1 , ∀ i ≥ 0 } . cylinders: Fix a ∈ N . [ a ] = { x = ( a , x 1 , x 2 , ... ) ∈ Σ A ( N ) } [ a 0 a 1 ... a n − 1 ] = { x = ( a 0 , a 1 , ..., a n − 1 , x n , x n +1 , ... ) ∈ Σ A ( N ) } 4/32
Infinite matrix A : N × N → { 0 , 1 } Σ A ( N ) := { x ∈ N N : A ( x i , x i +1 ) = 1 , ∀ i ≥ 0 } . cylinders: Fix a ∈ N . [ a ] = { x = ( a , x 1 , x 2 , ... ) ∈ Σ A ( N ) } [ a 0 a 1 ... a n − 1 ] = { x = ( a 0 , a 1 , ..., a n − 1 , x n , x n +1 , ... ) ∈ Σ A ( N ) } 4/32
The matrix A is finitely primitive , when there exist a finite subset F ⊆ N and an integer K 0 ≥ 0 such that, for any pair of symbols i , j ∈ N such that [ i ] � = ∅ and [ j ] � = ∅ , one can find ℓ 1 , ℓ 2 , . . . , ℓ K 0 ∈ F satisfying A ( i , ℓ 1 ) A ( ℓ 1 , ℓ 2 ) · · · A ( ℓ K 0 , j ) = 1 . 5/32
The matrix (or the shift) A has the big image property (BIP) , when there exist a finite subset F ⊆ N such that, for any symbol with [ i ] � = ∅ , there exist ℓ i and r i in F such that A ( ℓ i , i ) = A ( i , r i ) = 1. In topologically mixing Markov shifts: A has the BIP property ⇐ ⇒ A is finitely primitive. 6/32
The matrix (or the shift) A has the big image property (BIP) , when there exist a finite subset F ⊆ N such that, for any symbol with [ i ] � = ∅ , there exist ℓ i and r i in F such that A ( ℓ i , i ) = A ( i , r i ) = 1. In topologically mixing Markov shifts: A has the BIP property ⇐ ⇒ A is finitely primitive. 6/32
Beyond the Finite Primitive case Renewal shifts Example: Let A = ( a ij ) i , j ∈ N be the transition matrix such that there exists an increasing sequence of naturals ( d i ) i ∈ N for which a 00 = a i +1 , i = a 1 , d i = 1 , ∀ i ∈ N and the others coefficients are zero. 7/32
A topologically mixing Markov shift without the BIP property Renewal Shift d 1 d 2 0 2 4 5 Figura : Example of Renewal shift.
f : Σ A ( N ) → R has summable variation when ∞ � Var( f ) := Var k ( f ) < ∞ . k =1 where [ f ( x ) − f ( y )] ∀ k ≥ 1 [ r ∈ (0 , 1)] Var k ( f ) := sup x , y ∈ Σ A ( N ) d ( x , y ) ≤ rk f is locally H¨ older continuous when there exist H f > 0 such that Var k ( f ) ≤ H f r k , ∀ k ≥ 1. f can be unbounded! 9/32
f : Σ A ( N ) → R has summable variation when ∞ � Var( f ) := Var k ( f ) < ∞ . k =1 where [ f ( x ) − f ( y )] ∀ k ≥ 1 [ r ∈ (0 , 1)] Var k ( f ) := sup x , y ∈ Σ A ( N ) d ( x , y ) ≤ rk f is locally H¨ older continuous when there exist H f > 0 such that Var k ( f ) ≤ H f r k , ∀ k ≥ 1. f can be unbounded! 9/32
Thermodynamic Formalism Let f : Σ A ( N ) → R with summable variation. For each β > 0 we define: Pressure � n − 1 � 1 � � β f ( σ i x ) P ( β f ) = lim n log exp 1 [ a ] ( x ) , n →∞ σ n ( x )= x i =0 The definition doesn’t depend of the symbol a because we always assume that Σ A ( N ) is topologically mixing. 10/32
For each β the equilibrium measures µ β (for a potential β f ) are the probability measures satisfying the variational principle : P ( β f ): � � � � � P ( β f ) = sup h ( m ) + β fdm ; β fdm > −∞ = h ( µ β )+ β fd µ β m ∈ M σ where h ( m ) is the Kolmogorov-Sinai entropy of the measure m . M σ = invariant (for σ ) probability measures defined over the borel sets of Σ A ( N ) 11/32
(Sarig, Mauldin-Urbanski) Theorem (Ruelle-Perron-Frobenius) Let Σ A ( N ) be a finitely primitive shift, f a potential with summable variation and P ( f ) < ∞ . Then, for any β > 0 , if λ β = e P ( β f ) there exists a probability measure ν β finite and positive in cylinders and a continuos function h β > 0 such that L ∗ β f ν β = λ β ν β , L β f h β = λ β h β and µ β = h β d ν β is a equilibirum measure for β f . Let f : Σ A ( N ) → R be a function. The Ruelle-Perron-Frobenius operator ig given by: L f : C b (Σ A ( N )) → C b (Σ A ( N )) e f ( ix ) g ( ix ). ( L f g )( x ) = � i ∈ N 12/32
(Sarig, Mauldin-Urbanski) Theorem (Ruelle-Perron-Frobenius) Let Σ A ( N ) be a finitely primitive shift, f a potential with summable variation and P ( f ) < ∞ . Then, for any β > 0 , if λ β = e P ( β f ) there exists a probability measure ν β finite and positive in cylinders and a continuos function h β > 0 such that L ∗ β f ν β = λ β ν β , L β f h β = λ β h β and µ β = h β d ν β is a equilibirum measure for β f . Let f : Σ A ( N ) → R be a function. The Ruelle-Perron-Frobenius operator ig given by: L f : C b (Σ A ( N )) → C b (Σ A ( N )) e f ( ix ) g ( ix ). ( L f g )( x ) = � i ∈ N 12/32
(Sarig, Mauldin-Urbanski) Theorem (Ruelle-Perron-Frobenius) Let Σ A ( N ) be a finitely primitive shift, f a potential with summable variation and P ( f ) < ∞ . Then, for any β > 0 , if λ β = e P ( β f ) there exists a probability measure ν β finite and positive in cylinders and a continuos function h β > 0 such that L ∗ β f ν β = λ β ν β , L β f h β = λ β h β and µ β = h β d ν β is a equilibirum measure for β f . Let f : Σ A ( N ) → R be a function. The Ruelle-Perron-Frobenius operator ig given by: L f : C b (Σ A ( N )) → C b (Σ A ( N )) e f ( ix ) g ( ix ). ( L f g )( x ) = � i ∈ N 12/32
In the previous theorem: P ( β f ) is real analytic in the parameter β . For each β , there exist only one equilibrium measure. h β is uniformly bounded away from zero and from infinity. µ β is a Gibbs measure. Exactly as in the shifts with a finite number of symbols. 13/32
In the previous theorem: P ( β f ) is real analytic in the parameter β . For each β , there exist only one equilibrium measure. h β is uniformly bounded away from zero and from infinity. µ β is a Gibbs measure. Exactly as in the shifts with a finite number of symbols. 13/32
In the previous theorem: P ( β f ) is real analytic in the parameter β . For each β , there exist only one equilibrium measure. h β is uniformly bounded away from zero and from infinity. µ β is a Gibbs measure. Exactly as in the shifts with a finite number of symbols. 13/32
In the previous theorem: P ( β f ) is real analytic in the parameter β . For each β , there exist only one equilibrium measure. h β is uniformly bounded away from zero and from infinity. µ β is a Gibbs measure. Exactly as in the shifts with a finite number of symbols. 13/32
Definition (Bowen definition) An invariant measure µ is called of Gibbs Measure for the potential f : Σ A ( N ) → R when there exist constants C 1 , C 2 > 0 such that µ [ x 0 . . . x n − 1 ] C 1 ≤ exp( S n f ( x ) − nP ( f )) ≤ C 2 x ∈ [ x 0 . . . x n − 1 ] . Spoiler! See the talk of Artur Lopes about equivalent definitions for Gibbsianity. 14/32
Definition (Bowen definition) An invariant measure µ is called of Gibbs Measure for the potential f : Σ A ( N ) → R when there exist constants C 1 , C 2 > 0 such that µ [ x 0 . . . x n − 1 ] C 1 ≤ exp( S n f ( x ) − nP ( f )) ≤ C 2 x ∈ [ x 0 . . . x n − 1 ] . Spoiler! See the talk of Artur Lopes about equivalent definitions for Gibbsianity. 14/32
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