Thermodynamic Formalism for Generalized Markov Shifts on Infinitely - PowerPoint PPT Presentation

Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States Thiago Raszeja Institute of Mathematics and Statistics - University of S˜ ao Paulo (USP) October, 2020 Joint work with R. Bissacot, R. Exel and R. Frausino. Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 1 / 49

Markov shifts: Definition Markov shifts: let I be a countable set (alphabet) and A be a { 0 , 1 } -matrix indexed by I . Let Σ A := { x ∈ I N 0 : A ( x i , x i +1 ) = 1 , i ∈ N 0 } , and consider the shift map σ : Σ A → Σ A given by σ ( x 0 x 1 x 2 · · · ) = x 1 x 2 x 3 · · · . The (one-sided) Markov shift space is the pair (Σ A , σ ), where Σ A is endowed with the product topology. Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 2 / 49

Markov shifts: Examples Full shift: A ( i , j ) = 1 for every i , j ∈ I Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 3 / 49

Markov shifts: Examples Full shift: A ( i , j ) = 1 for every i , j ∈ I Renewal shift: I = N , A (1 , n ) = A ( n + 1 , n ) = 1 for every n ∈ N and A ( i , j ) = 0 in the rest of the matrix entries. · · · 1 2 3 4   1 1 1 1 1 · · ·   · · · 1 0 0 0 0     0 1 0 0 0 · · ·   A =   · · · 0 0 1 0 0     0 0 0 1 0 · · ·   . . . . . ... . . . . . . . . . . Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 3 / 49

Markov shifts: Topology and Transitivity Admissible words: w ∈ I N , for some N ∈ N such that N = 1 or N > 1 and it is allowed by the matrix, i.e., for all i = 0 , 1 , ..., N − 2 A ( w i , w i +1 ) = 1 , Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 4 / 49

Markov shifts: Topology and Transitivity Admissible words: w ∈ I N , for some N ∈ N such that N = 1 or N > 1 and it is allowed by the matrix, i.e., for all i = 0 , 1 , ..., N − 2 A ( w i , w i +1 ) = 1 , Topology: product topology of discrete topology. Equivalently: the topology is generated by the cylinder sets [ w ] = { x ∈ Σ A : x i = w i , i = 0 , 1 , ..., | w | − 1 } , where w is admissible; the topology is induced by the metric d ( x , y ) = 2 − inf { p ∈ N 0 : x p � = y p } . Obs: σ is a local homeomorphism. Transitivity: for every two letters there is a admissible path linking them. (standing hypothesis) Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 4 / 49

Markov shifts: Topology and Transitivity Topological facts: | I | < ∞ ⇐ ⇒ Σ A is compact; | I | = ∞ , A row-finite ⇐ ⇒ Σ A is locally compact and non-compact; | I | = ∞ , A not row-finite ⇐ ⇒ Σ A is not even locally compact; Losing local compactness = losing topological weaponry to attack problems. Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 5 / 49

Markov shifts: Topology and Transitivity Topological facts: | I | < ∞ ⇐ ⇒ Σ A is compact; | I | = ∞ , A row-finite ⇐ ⇒ Σ A is locally compact and non-compact; | I | = ∞ , A not row-finite ⇐ ⇒ Σ A is not even locally compact; Losing local compactness = losing topological weaponry to attack problems. Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 5 / 49

Cuntz-Krieger algebras Consider a family { S i } n i =1 of partial isometries which satisfies the relations n n � � S j S ∗ S ∗ A ( i , j ) S j S ∗ j = 1 and i S i = j . j =1 j =1 Definition The Cuntz-Krieger algebra O A is the universal algebra generated by a family of partial isometries { S i } n i =1 which satisfies the relations above. Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 6 / 49

Cuntz-Krieger algebras The Cuntz-Krieger algebras, encodes the Marov shift space in its algebraic structure. For example: for each ω = ω 0 · · · ω k − 1 , S ω := S ω 0 · · · S ω k − 1 � = 0 ⇐ ⇒ ω is admissible; the projections S ω S ∗ ω pairwise commutes and hence they generate a commutative C ∗ -sub-algebra D A ⊆ O A s.t. D A ≃ C (Σ A ) , with Sf = ( f ◦ σ ) S , f ∈ C (Σ A ) , where S := n − 1 / 2 � n i =1 S i . Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 7 / 49

Cuntz-Krieger algebras The Cuntz-Krieger algebras, encodes the Markov shift space in its algebraic structure. For example: for each ω = ω 0 · · · ω k − 1 , S ω := S ω 0 · · · S ω k − 1 � = 0 ⇐ ⇒ ω is admissible; the projections S ω S ∗ ω pairwise commutes and hence they generate a commutative C ∗ -sub-algebra D A ⊆ O A s.t. D A ≃ C (Σ A ) . Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 8 / 49

Cuntz-Krieger algebras The Cuntz-Krieger algebras, encodes the Markov shift space in its algebraic structure. For example: for each ω = ω 0 · · · ω k − 1 , S ω := S ω 0 · · · S ω k − 1 � = 0 ⇐ ⇒ ω is admissible; the projections S ω S ∗ ω pairwise commutes and hence they generate a commutative C ∗ -sub-algebra D A ⊆ O A s.t. D A ≃ C (Σ A ) . This encoding holds for compact Σ A (finite) alphabet. Is it possible to extend this for countable alphabets? Answer: Yes! Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 8 / 49

Historical Remarks 1977: Cuntz-Algebras (full shift matrix, includes the non-compact case); 1980: Cuntz-Krieger Algebras (compact case, more general matrices); 1997: Alex Kumjian, David Pask, Iain Raeburn and Jean Renault (locally compact case, groupoid C ∗ -algebras); 1999: Exel-Laca algebras (non locally compact case); 2000: J. Renault (groupoid C ∗ -algebra approach to Exel-Laca algebras). Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 9 / 49

Exel-Laca algebras - Countable Alphabet case Consider a countably infinite transition matrix A and the universal unital C ∗ -algebra � O A generated by a family of partial isometries { S j : j ∈ N } which satisfies the relations below: ( 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 � � (1 − A ( y , j )) , j ∈ N A ( X , Y , j ) := A ( x , j ) 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 Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 10 / 49

Exel-Laca algebras - Countable Alphabet case Definition The Exel-Laca algebra O A is the C ∗ -subalgebra of � O A generated the a family of partial isometries { S i } i ∈ N . Some observations: codim O A ≤ 1; it is in fact a generalization from the finite case (it recovers the CK conditions and it is necessarily unital); from now, the alphabet is N ; we will impose A transitive, which is the basis of the theory of Markov shifts; the word codification of Σ A is also valid in the sense S ω := S ω 0 · · · S ω k − 1 � = 0 ⇐ ⇒ ω is admissible . Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 11 / 49

Exel-Laca algebras - a very convenient faithful representation For each s ∈ N , consider the following operators on B ( ℓ 2 (Σ A )), � � δ σ ( x ) if x ∈ [ s ] , δ sx if A ( s , x 0 ) = 1 , T ∗ T s ( δ x ) = and s ( δ x ) = 0 otherwise; 0 otherwise , where { δ x } x ∈ Σ A is the canonical basis. Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 12 / 49

Exel-Laca algebras - a very convenient faithful representation For each s ∈ N , consider the following operators on B ( ℓ 2 (Σ A )), � � δ σ ( x ) if x ∈ [ s ] , δ sx if A ( s , x 0 ) = 1 , T ∗ T s ( δ x ) = and s ( δ x ) = 0 otherwise; 0 otherwise , where { δ x } x ∈ Σ A is the canonical basis. Theorem O A ≃ C ∗ ( { T s : s ∈ N } ) and O A ≃ C ∗ ( { T s : s ∈ N } ∪ { 1 } ) . Transitivity = ⇒ no terminal circuits = ⇒ faithful representation. Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 12 / 49

Exel-Laca algebras - a very convenient faithful representation In particular, we define the projections Q i := T ∗ i T i , i ∈ N given by � δ ω if ω ∈ σ ([ s ]); Q s ( δ ω ) = 0 otherwise. Theorem � O A is isomorphic to � �� � � T ∗ span T α Q i β : F finite ; α, β finite admissible words . i ∈ F Moreover, O A is isomorphic to � �� � � β : F finite ; α, β finite admissible words , T ∗ span T α Q i . ( α, F , β ) � = ( ∅ , ∅ , ∅ ) i ∈ F Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 13 / 49