Thermodynamic Formalism on Generalized Symbolic Spaces Research - PowerPoint PPT Presentation

Thermodynamic Formalism on Generalized Symbolic Spaces Research Project jointly with R. Exel (UFSC, Brazil), R. Frausino (USP), Thiago Raszeja (USP) Rodrigo Bissacot - (USP), Brazil Partially supported by CNPq and FAPESP Thermodynamic Formalism: Ergodic Theory and Validated Numerics CIRM - 2019 Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Outline Main references 1 Generalized shift spaces 2 Countable Markov Shifts and Exel-Laca Algebras X A - a candidate to replace Σ A Renewal Shift Thermodynamic Formalism 3 (Conformal and eigen) measures on X A in (and out) of Σ A 4 New (?) Type of Phase Transition. (with E. Beltr´ an and E. Endo) 5 Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Main references - Video of Ruy Exel’s talk at the youtube channel of ICM 2018. For those who want to see more algebraic aspects of the results: groupoids, equivalence relations, C ∗ -algebras... Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Main references - Video of Ruy Exel’s talk at the youtube channel of ICM 2018. For those who want to see more algebraic aspects of the results: groupoids, equivalence relations, C ∗ -algebras... Preprints online: - Conformal Measures on Generalized Renault-Deaconu Groupoids . [RB, R. Exel, T. Raszeja, R. Frausino] - Quasi-invariant measures for generalized approximately proper equivalence relations . [RB, R. Exel, T. Raszeja, R. Frausino] Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Main references 1 Generalized shift spaces 2 Countable Markov Shifts and Exel-Laca Algebras X A - a candidate to replace Σ A Renewal Shift Thermodynamic Formalism 3 (Conformal and eigen) measures on X A in (and out) of Σ A 4 New (?) Type of Phase Transition. (with E. Beltr´ an and E. Endo) 5 Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Countable Markov Shifts - Alphabet N . Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Countable Markov Shifts - Alphabet N . - An irreducible transition matrix A ( A ( i , j ) ∈ { 0 , 1 } ). Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Countable Markov Shifts - Alphabet N . - An irreducible transition matrix A ( A ( i , j ) ∈ { 0 , 1 } ). - The Countable Markov shift Σ A , in general, is not locally compact. Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Countable Markov Shifts - Alphabet N . - An irreducible transition matrix A ( A ( i , j ) ∈ { 0 , 1 } ). - The Countable Markov shift Σ A , in general, is not locally compact. Generalized = Locally compact version of Σ A , denoted by X A : Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Countable Markov Shifts - Alphabet N . - An irreducible transition matrix A ( A ( i , j ) ∈ { 0 , 1 } ). - The Countable Markov shift Σ A , in general, is not locally compact. Generalized = Locally compact version of Σ A , denoted by X A : X A locally compact space. (in many cases compact) Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Countable Markov Shifts - Alphabet N . - An irreducible transition matrix A ( A ( i , j ) ∈ { 0 , 1 } ). - The Countable Markov shift Σ A , in general, is not locally compact. Generalized = Locally compact version of Σ A , denoted by X A : X A locally compact space. (in many cases compact) Σ A is dense in X A . Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Countable Markov Shifts - Alphabet N . - An irreducible transition matrix A ( A ( i , j ) ∈ { 0 , 1 } ). - The Countable Markov shift Σ A , in general, is not locally compact. Generalized = Locally compact version of Σ A , denoted by X A : X A locally compact space. (in many cases compact) Σ A is dense in X A . Y A = X A \ Σ A is a set of finite words of the shift, it is also dense in X A . (empty words are possible) When Σ A is locally compact, then Σ A = X A . Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Renewal shift 1 2 3 4 5 6 7 Figure: The Renewal shift Σ A Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

The generalized renewal shift X A X A = Σ A ∪ Y A Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

The generalized renewal shift X A X A = Σ A ∪ Y A Y A = { finite words ending in 1 } ∪ { ξ 0 } , where ξ 0 is the empty word. Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Main reference: - R. Exel and M. Laca, Cuntz-Krieger algebras for infinite matrices . J. Reine Angew. Math., 512 , 119-172, (1999). Partial isometries satisfiyng: ( EL 1) S ∗ i S i and S ∗ j S j commute for every i , j ∈ N ; ( EL 2) S ∗ i S j = 0 whenever i � = j ; ( EL 3) ( S ∗ i S i ) S j = A ( i , j ) S j for all i , j ∈ N ; ( EL 4) for every pair X , Y of finite subsets of N such that the quantity � � A ( X , Y , j ) := A ( x , j ) (1 − A ( y , j )) , j ∈ N x ∈ X y ∈ Y is non-zero only for a finite number of j ’s, we have �   ��  � � S ∗ (1 − S ∗  = A ( X , Y , j ) S j S ∗ x S x y S y ) j . x ∈ X y ∈ Y j ∈ N Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Exel-Laca algebras For each s ∈ N , consider the following operators on B ( ℓ 2 (Σ A )), � � δ sx if A ( s , x 0 ) = 1 , δ σ ( x ) if x ∈ [ s ] , T ∗ T s ( δ x ) = with s ( δ x ) = 0 otherwise; 0 otherwise , where { δ x } x ∈ Σ A is the canonical basis. Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Exel-Laca algebras For each s ∈ N , consider the following operators on B ( ℓ 2 (Σ A )), � � δ sx if A ( s , x 0 ) = 1 , δ σ ( x ) if x ∈ [ s ] , T ∗ T s ( δ x ) = with s ( δ x ) = 0 otherwise; 0 otherwise , where { δ x } x ∈ Σ A is the canonical basis. Definition (Exel-Laca algebra) The Exel-Laca algebra O A is the subalgebra of � O A which is the unital C ∗ -algebra generated by the partial isometries T s , s ∈ N . There exists a collection of projections indexed by the free group generated by N : e g := T g T ∗ g ∈ F N reduced word . g , These elements commute each other. Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

Main references 1 Generalized shift spaces 2 Countable Markov Shifts and Exel-Laca Algebras X A - a candidate to replace Σ A Renewal Shift Thermodynamic Formalism 3 (Conformal and eigen) measures on X A in (and out) of Σ A 4 New (?) Type of Phase Transition. (with E. Beltr´ an and E. Endo) 5 Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

The set X A Consider D A := C ∗ ( { e g : g ∈ F N } ) the commutative C ∗ -subalgebra of O A generated by these projections. Definition Given an irreducible transition matrix A on the alphabet N , define the set X A := spec D A Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31

The set X A Consider D A := C ∗ ( { e g : g ∈ F N } ) the commutative C ∗ -subalgebra of O A generated by these projections. Definition Given an irreducible transition matrix A on the alphabet N , define the set X A := spec D A Characters: nonnull linear functionals such that ϕ ( a . b ) = ϕ ( a ) .ϕ ( b ) On the weak ∗ topology it is well known that X A is at least locally compact and in many cases compact. Thermodynamic Formalism: Ergodic Theory and Rodrigo Bissacot - (USP), Brazil (University of S˜ Thermodynamic Formalism on Generalized Symbolic Spaces ao Paulo) / 31