Factors of Gibbs measures on subshifts What is a Gibbs measure? - PowerPoint PPT Presentation

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction Factors of Gibbs measures on subshifts What is a Gibbs measure? Two-ish definitions Equivalence DLR theorems Sophie MacDonald Preservation of Gibbsianness UBC Mathematics Introduction Results Proof ideas West Coast Dynamics Seminar, May 2020 Closing 1 / 23

Factors of Gibbs measures on Acknowledgments subshifts Sophie MacDonald Thanks to Lior and Jayadev for inviting me to speak. Introduction What is a Gibbs measure? I am grateful to be supervised by Lior, Brian Marcus, and Two-ish definitions Equivalence Omer Angel. DLR theorems Preservation of Gibbsianness All work joint with Lu´ ısa Borsato, PhD student at Introduction Results Universidade de S˜ ao Paulo, visiting at UBC for the year, Proof ideas supported by grants 2018/21067-0 and 2019/08349-9, S˜ ao Closing Paulo Research Foundation (FAPESP). We are very grateful to Brian for his generous support and supervision, and to Tom Meyerovitch for his generous advice throughout this work. 2 / 23

Factors of Gibbs measures on Goals for this presentation subshifts Sophie MacDonald Introduction In this talk, I hope to communicate to you: What is a Gibbs measure? Two-ish definitions Equivalence • Roughly two definitions of a Gibbs measure on a subshift DLR theorems Preservation of and why they are equivalent Gibbsianness Introduction • A property defining a class of factor maps that preserve Results Gibbsianness, and some elements of the proof Proof ideas Closing • A Lanford-Ruelle theorem for irreducible sofic shifts on Z On Thursday, we can go into more detail, as interest dictates 3 / 23

Factors of Gibbs measures on Subshifts on groups subshifts Sophie MacDonald • Finite (discrete) alphabet A , countable group G Introduction • Product topology on full shift A G (compact metrizable) What is a Gibbs measure? Two-ish definitions Equivalence DLR theorems Preservation of Gibbsianness Introduction Results Proof ideas Closing 4 / 23

Factors of Gibbs measures on Subshifts on groups subshifts Sophie MacDonald • Finite (discrete) alphabet A , countable group G Introduction • Product topology on full shift A G (compact metrizable) What is a Gibbs measure? • Shift action of G on A G via ( x · g ) h = x gh Two-ish definitions Equivalence DLR theorems Preservation of Gibbsianness Introduction Results Proof ideas Closing 4 / 23

Factors of Gibbs measures on Subshifts on groups subshifts Sophie MacDonald • Finite (discrete) alphabet A , countable group G Introduction • Product topology on full shift A G (compact metrizable) What is a Gibbs measure? • Shift action of G on A G via ( x · g ) h = x gh Two-ish definitions Equivalence ◦ When G = Z , ( σ n x ) 0 = x n DLR theorems Preservation of Gibbsianness Introduction Results Proof ideas Closing 4 / 23

Factors of Gibbs measures on Subshifts on groups subshifts Sophie MacDonald • Finite (discrete) alphabet A , countable group G Introduction • Product topology on full shift A G (compact metrizable) What is a Gibbs measure? • Shift action of G on A G via ( x · g ) h = x gh Two-ish definitions Equivalence ◦ When G = Z , ( σ n x ) 0 = x n DLR theorems • A subshift is a closed, shift-invariant set X ⊆ A G Preservation of Gibbsianness Introduction Results Proof ideas Closing 4 / 23

Factors of Gibbs measures on Subshifts on groups subshifts Sophie MacDonald • Finite (discrete) alphabet A , countable group G Introduction • Product topology on full shift A G (compact metrizable) What is a Gibbs measure? • Shift action of G on A G via ( x · g ) h = x gh Two-ish definitions Equivalence ◦ When G = Z , ( σ n x ) 0 = x n DLR theorems • A subshift is a closed, shift-invariant set X ⊆ A G Preservation of Gibbsianness Introduction • Shift of finite type (SFT): subshift obtained by Results forbidding finitely many finite patterns from A G Proof ideas Closing 4 / 23

Factors of Gibbs measures on Subshifts on groups subshifts Sophie MacDonald • Finite (discrete) alphabet A , countable group G Introduction • Product topology on full shift A G (compact metrizable) What is a Gibbs measure? • Shift action of G on A G via ( x · g ) h = x gh Two-ish definitions Equivalence ◦ When G = Z , ( σ n x ) 0 = x n DLR theorems • A subshift is a closed, shift-invariant set X ⊆ A G Preservation of Gibbsianness Introduction • Shift of finite type (SFT): subshift obtained by Results forbidding finitely many finite patterns from A G Proof ideas Closing • Sliding block code: π : X → Y with π ( x · g ) = π ( x ) · g 4 / 23

Factors of Gibbs measures on Subshifts on groups subshifts Sophie MacDonald • Finite (discrete) alphabet A , countable group G Introduction • Product topology on full shift A G (compact metrizable) What is a Gibbs measure? • Shift action of G on A G via ( x · g ) h = x gh Two-ish definitions Equivalence ◦ When G = Z , ( σ n x ) 0 = x n DLR theorems • A subshift is a closed, shift-invariant set X ⊆ A G Preservation of Gibbsianness Introduction • Shift of finite type (SFT): subshift obtained by Results forbidding finitely many finite patterns from A G Proof ideas Closing • Sliding block code: π : X → Y with π ( x · g ) = π ( x ) · g ◦ Mostly care about π surjective (hence notation π ), called a factor map 4 / 23

Factors of Gibbs measures on Subshifts on groups subshifts Sophie MacDonald • Finite (discrete) alphabet A , countable group G Introduction • Product topology on full shift A G (compact metrizable) What is a Gibbs measure? • Shift action of G on A G via ( x · g ) h = x gh Two-ish definitions Equivalence ◦ When G = Z , ( σ n x ) 0 = x n DLR theorems • A subshift is a closed, shift-invariant set X ⊆ A G Preservation of Gibbsianness Introduction • Shift of finite type (SFT): subshift obtained by Results forbidding finitely many finite patterns from A G Proof ideas Closing • Sliding block code: π : X → Y with π ( x · g ) = π ( x ) · g ◦ Mostly care about π surjective (hence notation π ), called a factor map • Sofic shift: factor of an SFT 4 / 23

Factors of Gibbs measures on Subshifts on groups subshifts Sophie MacDonald • Finite (discrete) alphabet A , countable group G Introduction • Product topology on full shift A G (compact metrizable) What is a Gibbs measure? • Shift action of G on A G via ( x · g ) h = x gh Two-ish definitions Equivalence ◦ When G = Z , ( σ n x ) 0 = x n DLR theorems • A subshift is a closed, shift-invariant set X ⊆ A G Preservation of Gibbsianness Introduction • Shift of finite type (SFT): subshift obtained by Results forbidding finitely many finite patterns from A G Proof ideas Closing • Sliding block code: π : X → Y with π ( x · g ) = π ( x ) · g ◦ Mostly care about π surjective (hence notation π ), called a factor map • Sofic shift: factor of an SFT • All measures G -invariant Borel probability measures 4 / 23

Factors of Gibbs measures on Finite thermodynamics subshifts Sophie MacDonald Take a finite set { 1 , . . . , N } (e.g. patterns on Λ ⋐ G ) Introduction with “energy function” u ∈ R N and probability vector p What is a Gibbs measure? Two-ish definitions Equivalence DLR theorems Preservation of Gibbsianness Introduction Results Proof ideas Closing 5 / 23

Factors of Gibbs measures on Finite thermodynamics subshifts Sophie MacDonald Take a finite set { 1 , . . . , N } (e.g. patterns on Λ ⋐ G ) Introduction with “energy function” u ∈ R N and probability vector p What is a Gibbs measure? The free energy (volume derivative is called pressure) Two-ish definitions Equivalence DLR theorems N N � � Preservation of − p i log p i − p i u i Gibbsianness Introduction i =1 i =1 Results � �� � Proof ideas entropy H ( p ) Closing 5 / 23

Factors of Gibbs measures on Finite thermodynamics subshifts Sophie MacDonald Take a finite set { 1 , . . . , N } (e.g. patterns on Λ ⋐ G ) Introduction with “energy function” u ∈ R N and probability vector p What is a Gibbs measure? The free energy (volume derivative is called pressure) Two-ish definitions Equivalence DLR theorems N N � � Preservation of − p i log p i − p i u i Gibbsianness Introduction i =1 i =1 Results � �� � Proof ideas entropy H ( p ) Closing is uniquely maximized by the Gibbs distribution, p i = Z − 1 exp( − u i ) 5 / 23

Factors of Gibbs measures on Finite thermodynamics subshifts Sophie MacDonald Take a finite set { 1 , . . . , N } (e.g. patterns on Λ ⋐ G ) Introduction with “energy function” u ∈ R N and probability vector p What is a Gibbs measure? The free energy (volume derivative is called pressure) Two-ish definitions Equivalence DLR theorems N N � � Preservation of − p i log p i − p i u i Gibbsianness Introduction i =1 i =1 Results � �� � Proof ideas entropy H ( p ) Closing is uniquely maximized by the Gibbs distribution, p i = Z − 1 exp( − u i ) What about infinite volume? 5 / 23

Factors of Gibbs measures on Interactions subshifts Sophie MacDonald Define on every finite set Λ ⋐ G an interaction Introduction Φ Λ : X → R where Φ Λ ( x ) depends only on x Λ . What is a Gibbs measure? Two-ish definitions Equivalence DLR theorems Preservation of Gibbsianness Introduction Results Proof ideas Closing 6 / 23