Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts with Eric O. Endo (NYU-Shanghai) and Elmer R. Beltr´ an (IME-USP) Rodrigo Bissacot - (USP), Brazil Partially supported by FAPESP and CNPq XXIII Brazilian School of Probability ICMC - 2019 XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Outline Setting 1 Thermodynamic Formalism 2 Infinite DLR Measures 3 New (?) Type of Phase Transition. 4 XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Countable Markov Shifts - Alphabet N . XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Countable Markov Shifts - Alphabet N . - An irreducible transition matrix A ( A ( i , j ) ∈ { 0 , 1 } ). XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Countable Markov Shifts - Alphabet N . - An irreducible transition matrix A ( A ( i , j ) ∈ { 0 , 1 } ). Σ A = { ( x 0 , x 1 , . . . ) ∈ N N : A ( x i , x i +1 ) = 1 for every i } . XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Countable Markov Shifts - Alphabet N . - An irreducible transition matrix A ( A ( i , j ) ∈ { 0 , 1 } ). Σ A = { ( x 0 , x 1 , . . . ) ∈ N N : A ( x i , x i +1 ) = 1 for every i } . - Countable Markov shifts Σ A , in general, are not locally compact. XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Renewal shift 1 2 3 4 5 6 7 Figure: The Renewal shift Σ A XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Thermodynamic Formalism σ : Σ A → Σ A defined by σ ( x 0 , x 1 , . . . ) = ( x 1 , x 2 , . . . ). (shift map) XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Thermodynamic Formalism σ : Σ A → Σ A defined by σ ( x 0 , x 1 , . . . ) = ( x 1 , x 2 , . . . ). (shift map) φ : Σ A → R measurable. (potential) XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Thermodynamic Formalism σ : Σ A → Σ A defined by σ ( x 0 , x 1 , . . . ) = ( x 1 , x 2 , . . . ). (shift map) φ : Σ A → R measurable. (potential) n − 1 � φ ( σ i x ) for n ≥ 1. φ n ( x ) := i =0 XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Thermodynamic Formalism σ : Σ A → Σ A defined by σ ( x 0 , x 1 , . . . ) = ( x 1 , x 2 , . . . ). (shift map) φ : Σ A → R measurable. (potential) n − 1 � φ ( σ i x ) for n ≥ 1. φ n ( x ) := i =0 For n ≥ 1, the n -variation of φ is given by var n ( φ ) = sup {| φ ( x ) − φ ( y ) | : x 0 = y 0 , ..., x n − 1 = y n − 1 } φ has summable variations when � n ≥ 2 var n ( φ ) < ∞ . XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Thermodynamic Formalism σ : Σ A → Σ A defined by σ ( x 0 , x 1 , . . . ) = ( x 1 , x 2 , . . . ). (shift map) φ : Σ A → R measurable. (potential) n − 1 � φ ( σ i x ) for n ≥ 1. φ n ( x ) := i =0 For n ≥ 1, the n -variation of φ is given by var n ( φ ) = sup {| φ ( x ) − φ ( y ) | : x 0 = y 0 , ..., x n − 1 = y n − 1 } φ has summable variations when � n ≥ 2 var n ( φ ) < ∞ . var n ( φ ) ≤ constant .λ n , 0 < λ < 1 (locally H¨ older) var 1 ( φ ) = + ∞ is allowed. XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Ruelle operator φ : Σ A → R be a measurable potential. Ruelle operator : For measurable function f and x ∈ Σ A , � e φ ( y ) f ( y ) . L φ ( f )( x ) = y ∈ Σ A σ ( y )= x Let µ sigma-finite measure, λ > 0. (eigenmeasures) � � for each f ∈ L 1 ( µ ) L φ f ( x ) d µ ( x ) = λ f ( x ) d µ ( x ) , L ∗ Notation: φ ( µ ) = λµ Eigenmeasures from the Generalized Ruelle-Perron-Frobenius’ Theorem are sigma-finite but can be infinite! XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Motivation ””””” - We should consider infinite measures on statistical mechanics, people in ergodic theory already did this...”””” by Charles Pfister at CIRM, Marseille, in 2013. Classical Reference: An Introduction to Infinite Ergodic Theory, 1997 By Jon Aaronson. XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
DLR Measures on the Reversal Renewal Shift Before the infinite case... XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
DLR Measures on the Reversal Renewal Shift Before the infinite case... 1 2 3 4 5 6 7 XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
DLR Measures on the Reversal Renewal Shift Before the infinite case... 1 2 3 4 5 6 7 Take z = 12345 ... ∈ Σ A and ν = δ z . XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
DLR Measures on the Reversal Renewal Shift Before the infinite case... 1 2 3 4 5 6 7 Take z = 12345 ... ∈ Σ A and ν = δ z . ν is a DLR measure... for which potential??? Reference: Thermodynamic Formalism for Transient Potential Functions , Ofer Shwartz, CMP, 2019. XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
DLR Measures on the Reversal Renewal Shift Before the infinite case... 1 2 3 4 5 6 7 Take z = 12345 ... ∈ Σ A and ν = δ z . ν is a DLR measure... for which potential??? ν is a DLR measure for ANY potential!!!!!! Reference: Thermodynamic Formalism for Transient Potential Functions , Ofer Shwartz, CMP, 2019. XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Infinite DLR Measures Definition Let Σ A be a Markov shift, ν be a measure on the Borel sigma algebra B , and φ : Σ A → R be a measurable potential. We say that ν is φ -DLR if, for every n ≥ 1, i) the restriction of ν to the sub- σ -algebra σ − n B is sigma-finite, ii) for every cylinder [ a ] of length n , we have ( x ) = e φ n ( a σ n x ) 1 { a σ n x ∈ Σ A } 1 [ a ] | σ − n B � � ν -a.e. (1) E ν , � e φ n ( y ) σ n y = σ n x XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Infinite DLR Measures Proposition φ : Σ A → R be a measurable potential and ν such that � L φ 1 � ∞ < ∞ . If, ν ([ a ]) < ∞ for each a ∈ N . L ∗ φ ( ν ) = λν Then, ν is φ -DLR. XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Phase Transitions Buzzi-Sarig (ETDS-2003): Let Σ A be a topologically mixing Markov shift, if φ : Σ A → R is regular enough with sup φ < ∞ and P G ( φ ) < ∞ . Then there exists at most one equilibrium measure m and, when does exist, m = hd µ where h and µ are the eigenfunction and eigenmeasure associated to λ = e P G ( φ ) . XXIII Brazilian School of Probability ICMC - Rodrigo Bissacot - (USP), Brazil (University of S˜ Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts ao Paulo) / 14
Recommend
More recommend
