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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
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: let I be a countable set (alphabet) and A be a {0, 1}-matrix indexed by I. Let ΣA := {x ∈ I N0 : A(xi, xi+1) = 1, i ∈ N0}, and consider the shift map σ : ΣA → ΣA given by σ(x0x1x2 · · · ) = x1x2x3 · · · . The (one-sided) Markov shift space is the pair (ΣA, σ), where ΣA is endowed with the product topology.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 2 / 49
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
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 · · · A = 1 1 1 1 1 · · · 1 · · · 1 · · · 1 · · · 1 · · · . . . . . . . . . . . . . . . ...
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 3 / 49
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., A(wi, wi+1) = 1, for all i = 0, 1, ..., N − 2
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 4 / 49
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., A(wi, wi+1) = 1, for all i = 0, 1, ..., N − 2 Topology: product topology of discrete topology. Equivalently: the topology is generated by the cylinder sets [w] = {x ∈ ΣA : xi = wi, i = 0, 1, ..., |w| − 1}, where w is admissible; the topology is induced by the metric d(x, y) = 2− inf{p∈N0:xp=yp}. 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
Topological facts: |I| < ∞ ⇐ ⇒ ΣA is compact; |I| = ∞, A row-finite ⇐ ⇒ ΣA is locally compact and non-compact; |I| = ∞, A not row-finite ⇐ ⇒ ΣA 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
Topological facts: |I| < ∞ ⇐ ⇒ ΣA is compact; |I| = ∞, A row-finite ⇐ ⇒ ΣA is locally compact and non-compact; |I| = ∞, A not row-finite ⇐ ⇒ ΣA 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
Consider a family {Si}n
i=1 of partial isometries which satisfies the relations n
SjS∗
j = 1
and S∗
i Si = n
A(i, j)SjS∗
j .
Definition
The Cuntz-Krieger algebra OA is the universal algebra generated by a family of partial isometries {Si}n
i=1 which satisfies the relations above.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 6 / 49
The Cuntz-Krieger algebras, encodes the Marov shift space in its algebraic
for each ω = ω0 · · · ωk−1, Sω := Sω0 · · · Sωk−1 = 0 ⇐ ⇒ ω is admissible; the projections SωS∗
ω pairwise commutes and hence they generate a
commutative C ∗-sub-algebra DA ⊆ OA s.t. DA ≃ C(ΣA), with Sf = (f ◦ σ)S, f ∈ C(ΣA), where S := n−1/2 n
i=1 Si.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 7 / 49
The Cuntz-Krieger algebras, encodes the Markov shift space in its algebraic structure. For example: for each ω = ω0 · · · ωk−1, Sω := Sω0 · · · Sωk−1 = 0 ⇐ ⇒ ω is admissible; the projections SωS∗
ω pairwise commutes and hence they generate a
commutative C ∗-sub-algebra DA ⊆ OA s.t. DA ≃ C(ΣA).
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 8 / 49
The Cuntz-Krieger algebras, encodes the Markov shift space in its algebraic structure. For example: for each ω = ω0 · · · ωk−1, Sω := Sω0 · · · Sωk−1 = 0 ⇐ ⇒ ω is admissible; the projections SωS∗
ω pairwise commutes and hence they generate a
commutative C ∗-sub-algebra DA ⊆ OA s.t. DA ≃ C(ΣA). This encoding holds for compact ΣA (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
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
Consider a countably infinite transition matrix A and the universal unital C ∗-algebra OA generated by a family of partial isometries {Sj : j ∈ N} which satisfies the relations below: (EL1) S∗
i Si and S∗ j Sj commute for every i, j ∈ N;
(EL2) S∗
i Sj = 0 whenever i = j;
(EL3) (S∗
i Si)Sj = A(i, j)Sj for all i, j ∈ N;
(EL4) 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 is non-zero only for a finite number of j’s, we have
S∗
x Sx
y∈Y
(1 − S∗
y Sy)
=
A(X, Y , j)SjS∗
j .
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 10 / 49
Definition
The Exel-Laca algebra OA is the C∗-subalgebra of OA generated the a family of partial isometries {Si}i∈N. Some observations: codim OA ≤ 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 ΣA is also valid in the sense Sω := Sω0 · · · Sωk−1 = 0 ⇐ ⇒ ω is admissible.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 11 / 49
For each s ∈ N, consider the following operators on B(ℓ2(ΣA)), Ts(δx) =
0 otherwise; and T ∗
s (δx) =
0 otherwise, where {δx}x∈ΣA is the canonical basis.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 12 / 49
For each s ∈ N, consider the following operators on B(ℓ2(ΣA)), Ts(δx) =
0 otherwise; and T ∗
s (δx) =
0 otherwise, where {δx}x∈ΣA is the canonical basis.
Theorem
OA ≃ C ∗({Ts : s ∈ N}) and OA ≃ C ∗({Ts : 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
In particular, we define the projections Qi := T ∗
i Ti, i ∈ N given by
Qs(δω) =
0 otherwise.
Theorem
span
Qi
β : F finite; α, β finite admissible words
Moreover, OA is isomorphic to span
Qi
β : F finite; α, β finite admissible words,
(α, F, β) = (∅, ∅, ∅)
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 13 / 49
Definition
Let DA be the commutative unital C ∗-subalgebra of OA given by
QiT ∗
α : F finite; α finite word
DA := span
QiT ∗
α : F finite; α finite word; (α, F) = (∅, ∅)
Obs: codim DA ≤ 1.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 14 / 49
Consider the free group F = FN and the map T : F → OA, s → Ts, s−1 → Ts−1 := T ∗
s .
Also, for any word g in F, take its reduced form g = x1...xn and define that T realizes the mapping g → Tg := Tx1 · · · Txn, and that Te = 1.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 15 / 49
Consider the projections eg := TgT ∗
g ,
g ∈ F reduced word.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 16 / 49
Consider the projections eg := TgT ∗
g ,
g ∈ F reduced word.
Theorem (R. Exel, M. Laca (1999))
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 16 / 49
Consider the projections eg := TgT ∗
g ,
g ∈ F reduced word.
Theorem (R. Exel, M. Laca (1999))
Definition (Generalized Markov shift space)
Given an irreducible transition matrix A on the alphabet N, the generalized Markov shift spaces are the sets XA := spec DA and
DA, both endowed by the weak∗ topology.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 16 / 49
Some remarks: XA is always locally compact, and XA is always compact. when XA = XA, then XA is the Alexandrov’s compactification of XA, the extra point is ϕ0(eg) :=
if g = e; 0,
a sufficient condition for XA = XA is A having a full row of 1’s: Qj = T ∗
j Tj = 1 =
⇒ OA unital; DA and DA are C∗-subalgebras of diagonal operators of B(ℓ2(ΣA)).
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 17 / 49
Some relations between ΣA and XA:
the evaluation map);
topological embedding). (ϕ(eg))g∈F ∈ {0, 1}F.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 18 / 49
More remarks: ΣA is a dense subset of XA; if ΣA is locally compact, then XA = ΣA; YA := Σc
A, when it is not empty, is also dense in XA.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 19 / 49
Consider the compact set Ωτ
A =
ξ ∈ {0, 1}F : ξe = 1, ξ connected, if ξω = 1, then there exists at most one y ∈ N s.t. ξωy = 1, if ξω = ξωy = 1, then for all x ∈ N (ξωx−1 = 1 ⇔ A(x, y) = 1) Cayley Tree: vertices = elements of F filled according the rules above.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 20 / 49
g a ga
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 21 / 49
g a ga 1 2 x y ω ωy A(x, y) = 1 4 5 x y ω ωy A(x, y) = 0
Figure 1: The black dots represents that the configuration ξ is filled.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 21 / 49
Positive words: (admissible) elements of F+ plus elements of ΣA. Given ω positive, define ω := {e, ω0, ω0ω1, ω0ω1ω2, · · · } Let ξ ∈ Ωτ
A.
Stem: the positive word κ(ξ) such that {g ∈ F : ξg = 1} ∩ F+ = ω. Root: the set of edges that disembogues to the stem. Given ξ ∈ YA, Rξ := {j ∈ N : ξκ(ξ)j−1 = 1}
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 22 / 49
Positive words: (admissible) elements of F+ plus elements of ΣA. Given ω positive, define ω := {e, ω0, ω0ω1, ω0ω1ω2, · · · } Let ξ ∈ Ωτ
A.
Stem: the positive word κ(ξ) such that {g ∈ F : ξg = 1} ∩ F+ = ω. Root: the set of edges that disembogues to the stem. Given ξ ∈ YA, Rξ := {j ∈ N : ξκ(ξ)j−1 = 1} If ξ ∈ ΣA, then it is uniquely determined by κ(ξ). If ξ ∈ YA, then it is uniquely determined by (κ(ξ), Rξ).
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 22 / 49
XA = YA ⊔ ΣA.
Figure 2: Density of ΣA vs elements of YA.
ω ωj1 ωj2 ωj3 ωj4 ω ωj1 ωj2 ωj3 ωj4 ω ωj1 ωj2 ωj3 ωj4 ω ωj1 ωj2 ωj3 ωj4
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 23 / 49
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 · · ·
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 24 / 49
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 · · · full line of 1’s = ⇒ OA unital = ⇒ XA compact; 1 is the unique infinite emitter; the unique accumulation point of the sequence of the columns of A is c = 1 . . . .
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 24 / 49
e
1 1 2 1 1 2 1 1 1 2 2 2 1 1 1 1 2 3 3 3 1 4 1 5 6 1 2 2 1 4 1 1 2 3 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 1 · · · · · · · · · · · · 1 1 2 3 1 2 · · · · · · · · · · · · 1 2 · · · · · · 1 · · · · · · 3 1 1 2 4 1 2 1 3 · · · · · · 1 5 · · · · · · 1 2 1 · · · · · · 1 3 · · · · · · 1 2 4 1−1 1−12−1 1−12−11−1 1−12−11−12−1 1−12−11−12−13−1
Figure 3: The empty stem configuration of the renewal shift. F.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 25 / 49
321 e 3 2 1−1 32 3 1 2 3 1 1 1 2 2 1 1 1 1 1 3 2 2 3 ξ0 \ {e1} 3 1−1 1 4 4−1 1 5 4−1 1−1 4−1 5−1 1 1 2 6
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 26 / 49
321 e 3 2 1
− 1
32 3 1 2 3 1 1 1 2 2 1 1 1 1 1 3 2 2 3 η0 3 1
− 1
1 4 4
− 1
1 5 4
− 1
1
− 1
4
− 1
5
− 1
1 1 2 6 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 21 3−1 2 e 1 2 3 η0 1 1 1 2 3 · · · · · · · · · · · · 1 21−1 2 1 1 4 3
− 1
1
− 1
3
− 1
4
− 1
1 1 2 5 · · · · · · · · · · · ·
shift action
σ Figure 4: Shift action on the 321 stem configuration for the renewal shift.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 27 / 49
YA-family: let {ξ0,e}e∈E , the collection of all configurations on XA (or
YA(ξ0,e) :=
σ−n(ξ0,e). Renewal shift: unique YA-family.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 28 / 49
A(1, n) = A(2, 2n) = 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 5 6 · · ·
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 29 / 49
full line of 1’s = ⇒ OA unital = ⇒ XA compact; 1 and 2 are the only infinite emitters; two accumulation points of the sequence of the columns of A c1 = 1 1 . . . and c2 = 1 . . . ; two YA families.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 30 / 49
Figure 5: The two emtpty stem configurations of the pair renewal shift. The left configuration ξ0,1 satisfies Rξ0,1(e) = c1, while the right one ξ0,2 corresponds to Rξ0,2(e) = c2, where c1 and c2 are the last two column vectors
2 2 1 2 3 2
− 1
1 2 3 2 e 1 1
− 1
· · · · · ·
1
· · · · · · · · · · · ·
1 1 2 1 2
· · ·
2
· · ·
1
· · · · · · · · · · · ·
2 1 3
· · · · · · · · · · · ·
1 2 1 2 3 2 2 1 2 3 e 1 1
− 1
1
· · · · · · · · ·
1 2 1 4 1 3
· · ·
1
· · · · · ·
2
· · · · · ·
2
· · · · · ·
1
· · ·
1 2
· · ·
3
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 31 / 49
Generalized cylinder: given ω ∈ F, define Cω := {ξ ∈ XA : ξω = 1}. A typical element of the basis: F ⊔
Cw(n), w(n) positive words, F ⊆ YA.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 32 / 49
nski Transactions AMS 91’.
Definition (Conformal measure)
Let (X, F) be a measurable space, σ : U ⊆ X → X a measurable endomorphism and D : U → [0, ∞) also measurable. A set A ⊆ U is called special if A ∈ F and σA := σ ↾A: A → σ(A) is injective. A measure µ in X is said to be D-conformal in the sense of Patterson when µ(σ(A)) =
Ddµ, (1) for all special sets A.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 33 / 49
nski Transactions AMS 91’.
Definition (Conformal measure)
Let (X, F) be a measurable space, σ : U ⊆ X → X a measurable endomorphism and D : U → [0, ∞) also measurable. A set A ⊆ U is called special if A ∈ F and σA := σ ↾A: A → σ(A) is injective. A measure µ in X is said to be D-conformal in the sense of Patterson when µ(σ(A)) =
Ddµ, (1) for all special sets A. As an example, in the Markov shifts, a Borel set contained in a cylinder set is special. For us, U = Dom σ, X = XA and D = eβF, for some potential
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 33 / 49
Theorem (R. Bissacot, R. Exel, R. Frausino, T.R.)
Consider a potential F : XA \ {ξ0} → R and β > 0, we have the following: (i) If inf F > 0, for β > log 2
inf F , there exists a unique eβF-conformal
probability measure µβ that vanishes in ΣA. (ii) If 0 ≤ sup F < +∞ and β ≤ log 2
sup F , there are no eβF-conformal
probability measures that vanish in ΣA.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 34 / 49
Corollary (R. Bissacot, R. Exel, R. Frausino, T. R.)
Let F ≡ 1. Then, for the constant βc = log 2, the result follows: (1) For β > βc we have a unique eβ-conformal probability measure that vanishes on ΣA. (2) For β ≤ βc there is no eβ- conformal probability measure that vanishes on ΣA. (3) lim
β→βc µβ = µβc (weak∗ convergence) where µβc lives on ΣA and it is a
conformal measure in the classical framework.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 35 / 49
Corollary (R. Bissacot, R. Exel, R. Frausino, T. R.)
Let F ≡ 1. Then, for the constant βc = log 2, the result follows: (1) For β > βc we have a unique eβ-conformal probability measure that vanishes on ΣA. (2) For β ≤ βc there is no eβ- conformal probability measure that vanishes on ΣA. (3) lim
β→βc µβ = µβc (weak∗ convergence) where µβc lives on ΣA and it is a
conformal measure in the classical framework. µβc[α] = 2−|α|, where α is a word ending in 1.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 35 / 49
Convex combinations of conformal measures are conformal measures.
Corollary (R. Bissacot, R. Exel, R. Frausino, T. R.)
Let F : U → XA be a potential. The extremal eF-conformal probabilities living on YA are precisely the the ones living on each YA-family.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 36 / 49
Theorem (R. Bissacot, R. Exel, R. Frausino, T. R.)
For the pair renewal shift and constant potential F = 1, there exists a critical value βc = log(1 + √ 2) s.t. (i) for β > βc there exist two extremal eβ-conformal probabilities living
(ii) for β = βc there exists a unique eβ-conformal probability µβ living on ΣA; (iii) for β < βc, there are no eβ-conformal probabilities; (iv) for β > βc, let Φβ be the set of eβ-conformal probabilities, then dH(µβc, Φβ) → 0 as β → βc.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 37 / 49
Theorem (R. Bissacot, R. Exel, R. Frausino, T. R.)
For the pair renewal shift and constant potential F = 1, there exists a critical value βc = log(1 + √ 2) s.t. (i) for β > βc there exist two extremal eβ-conformal probabilities living
(ii) for β = βc there exists a unique eβ-conformal probability µβ living on ΣA; (iii) for β < βc, there are no eβ-conformal probabilities; (iv) for β > βc, let Φβ be the set of eβ-conformal probabilities, then dH(µβc, Φβ) → 0 as β → βc. Hausdorff distance: dH(µβc, Φβ) = max
y∈Φβ
d(µβc, y), sup
y∈Φβ
d(µβc, y)
y∈Φβ
d(µβc, y), where d is a metric compatible with the weak∗-topology.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 37 / 49
Definition (Eigenmeasure associated to the Ruelle Transformation)
Consider the Borel σ-algebra BX. A measure µ on BX is said to be an eigenmeasure with eigenvalue λ for the Ruelle transformation LβF when
LβF(f )(x)dµ(x) = λ
f (x)dµ(x), (2) for all f ∈ Cc(U). In other words, the equation (2) can be rewritten by using (??) as
eβF(y)f (y)dµ(x) = λ
f (x)dµ(x), (3) for all f ∈ Cc(U). As in the standard theory of countable Markov shifts, when a measure m satisfies the equation (3) we write L∗
βFµ = λµ.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 38 / 49
Consider ΣA (topologically mixing), a ∈ N and the partition function: Zn(f , [a]) :=
efn(x)✶[a](x). (4) The Gurevich pressure is given by PG(f ) := lim
n
1 n log Zn(f , [a]).
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 39 / 49
Consider ΣA (topologically mixing), a ∈ N and the partition function: Zn(f , [a]) :=
efn(x)✶[a](x). (4) The Gurevich pressure is given by PG(f ) := lim
n
1 n log Zn(f , [a]). Analogously for x ∈ XA we have the partition function and the pressure at x: Zn(βF, x) :=
eβFn(y) and P(βF, x) := lim sup
n→∞
1 n log Zn(βF, x).
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 39 / 49
Consider ΣA (topologically mixing), a ∈ N and the partition function: Zn(f , [a]) :=
efn(x)✶[a](x). (4) The Gurevich pressure is given by PG(f ) := lim
n
1 n log Zn(f , [a]). Analogously for x ∈ XA we have the partition function and the pressure at x: Zn(βF, x) :=
eβFn(y) and P(βF, x) := lim sup
n→∞
1 n log Zn(βF, x). Renewal shift: PG(βF) = P(βF, x) for every x ∈ XA, when F is bounded above and F(x) = g(x0) − g(x0 + a), for some g continuous.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 39 / 49
Theorem (M. Denker, M. Yuri (2015))
Let XA be compact and suppose there exists a x ∈ XA such that P(F, x) is finite, then there exists an eigenmeasure m for LF with eigenvalue eP(F,x).
Theorem (R. Bissacot, R. Exel, R. Frausino, T.R.)
Let A be the renewal shift transition matrix and XA its generalized Markov shift space. Consider the potential F : U → R given by F(x) = log(x0) − log(x0 + 1). . Then, for every β > 0, there exists a unique eigenmeasure associated to the eigenvalue λβ = ePG (βF). Moreover, there is critical value βc > 0, which is the solution for ζ(βc) = 2 such that (i) if β > βc, then the eigenmeasure lives on YA; (ii) if β ≤ βc, then the eigenmeasure lives on ΣA.
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Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 44 / 49
Take matrix A as follows: for each p prime number and n ∈ N we have A(n + 1, n) = A(1, n) = A(p, pn) = 1 and zero for the other entries of A. A = 1 1 1 1 1 1 1 1 1 · · · 1 1 1 1 · · · 1 1 1 · · · 1 · · · 1 1 · · · 1 · · · 1 1 · · · 1 · · · 1 · · · . . . . . . . . . . . . . . . . . . . . . . . . . . . ... .
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 45 / 49
XA is compact. c(p) = 1 . . . 1 (p-th coordinate) . . .
c(1) = 1 . . . . Infinite YA-families (countable).
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Theorem (R. Bissacot, R. Exel, R. Frausino, T.R.)
Consider potentials F : U → R, we have the following: (i) Suppose inf F > 0. For each p ∈ {1} ∪ {p ∈ N : p is a prime number} and β > log 3
inf F , there exists a unique eβF-conformal probability
measure µβ,p that vanishes out of YA(ξ0(p)). (ii) Suppose 0 ≤ sup F < +∞ and β ≤ log 2
sup F , there are no eβF-conformal
probability measures which vanish in ΣA.
Thiago Raszeja Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States October, 2020 47 / 49
Let the transition matrix A, which its columns are periodic sequences in {0, 1}N excluding the zero sequence. Also, we order the columns by increasing the minimal period. Set A(n, 1) = 1 1 1 1 1 1 . . . , A(n, 2) = 1 1 1 . . . and A(n, 2) = 1 1 1 . . . . For the columns with same minimal period, take any ordering.
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A is transitive (also, topologically mixing); ΣA is not locally compact; XA is not compact; there are uncountable many YA-families
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