Group Extended Markov Systems, Amenability, and the Perron-Frobenius Operator

We characterise amenability of a countable group in terms of the spectral radius of the Perron-Frobenius operator associated to a group extension of a countable Markov shift and a H\"older continuous potential. This extends a result of Day for random walks and recent work of Stadlbauer for dynamical systems. Moreover, we show that, if the potential satisfies a symmetry condition with respect to the group extension, then the logarithm of the spectral radius of the Perron-Frobenius operator is given by the Gurevic pressure of the potential.

Theorem 1.1 provides an extension of a result of Day for random walks on groups ([Day64, Theorem 1]). The Perron-Frobenius operator corresponds to the convolution operator acting on ℓ 2 (G) in Day's setting.
Stadlbauer ([Sta13]) gave a criterion for amenability in terms of the Gurevič pressure, which is more in the spirit of Kesten's characterisation of amenability in terms of the growth rate of return probabilities (see [Kes59b,Kes59a]). More precisely, Stadlbauer proved the following two implications. Here, P (ϕ • π 1 , σ ⋊ Ψ) refers to the Gurevič pressure of ϕ • π 1 with respect to σ ⋊ Ψ.
Comparing Stadlbauer's results with Theorem 1.1, we note that the spectral radius of the Perron-Frobenius operator can be used to characterise amenability for an arbitrary Hölder continuous potential. In contrast to this, the criterion in terms of the Gurevič pressure involves a certain symmetry condition on the Markov shift and the potential under consideration (see [Sta13,p.455] for the definition of a symmetric group extension and a weakly symmetric potential). For an amenable group, it may happen that P (ϕ • π 1 , σ ⋊ Ψ) < P (ϕ, σ ), if ϕ is not weakly symmetric. In fact, the gap between these pressures can be arbitrarily large, as the following example illustrates.
For a Markov shift with a finite alphabet, it is well-known that the Gurevič pressure of a Hölder continuous potential coincides with the logarithm of the spectral radius of the Perron-Frobenius operator. However, if the Markov shift is constructed over an infinite alphabet, then the Gurevič pressure is less or equal to the logarithm of the spectral radius. The Gurevič pressure describes the growth of iterates of the Perron-Frobenius operator on functions supported on a cylindrical set, whereas for the spectral radius, we have to consider functions supported on the whole space (see Lemma 3.2). To relate the spectral radius of the Perron-Frobenius operator to the Gurevič pressure, we introduce the following notions of symmetry, generalizing those given in [Jae14, Definition 3.10].
Definition 1.3. Let (Σ × G, σ ⋊ Ψ) be a group extended Markov system and ϕ : Σ → R. Let α ≥ 1. We say that ϕ is asymptotically α-symmetric with respect to Ψ, if there exist n 0 ∈ N and sequences (c n ) ∈ (R + ) N and (N n ) ∈ N N with the property that lim n (c n ) 1/(2n) = α, lim n n −1 N n = 0 and such that, for each g ∈ G and for all n ≥ n 0 , we have We say that ϕ is compactly asymptotically α-symmetric with respect to Ψ if there exists a sequence (Σ k ) k∈N of topologically mixing subshifts of Σ with finite alphabet I k ⊂ I, such that k∈N I k = I, the set Ψ Σ * k is a subgroup of G, and ϕ |Σ k is asymptotically α-symmetric with respect to Ψ |I * k . If α is not specified, then we will tacitly assume that α = 1.
Remark 1.4. If ϕ is (compactly) asymptotically α-symmetric with respect to Ψ, then so is ϕ + log h − log h • σ + P, for each P ∈ R and for each function h : Σ → R + , which is bounded away from zero and infinity.
The following proposition generalises [Jae14, Corollary 3.17], where a locally constant potential ϕ on a finite state Markov shift was considered (see Remark 1.8 for details). The proposition relates Theorem 1.1 to Stadlbauer's results and is also of independent interest, since it relates the Gurevič pressure of a not necessarily recurrent potential ([Sar01]) on Σ × G to the spectrum of the Perron-Frobenius operator.
If ϕ is not assumed to be Hölder continuous, or if Σ does not satisfy the b.i.p. property as in (2), then a Gibbs measure for ϕ does not exist. To obtain an inequality as in the previous corollary, the existence of an approximating sequence of Hölder continuous potentials is sufficient. More precisely, suppose there exists a sequence of subgroups (G j ) of G, group extended Markov systems (Σ j × G j , σ j ⋊ Ψ j ) and potentials ϕ j : Σ j → R, such that ϕ j is asymptotically α j -symmetric with respect to Ψ j and the assumptions of Theorem 1.1 hold for each j ∈ N. If moreover lim j P (ϕ j , σ j ) ≥ P (ϕ, σ ), lim j P (ϕ j • π 1 , σ j ⋊ Ψ j ) ≤ P (ϕ • π 1 , σ ⋊ Ψ) and lim j α j ≤ α, then we have P (ϕ • π 1 , σ ⋊ Ψ) ≥ P (ϕ, σ ) − log α, provided that G is amenable. Using this approach, Stadlbauer derived the assertion in (2).
We make use of this approach to give a similar result for compactly asymptotically α-symmetric potentials of medium variation. Note that P (ϕ, σ ) is allowed to be infinite in the following corollary.
Acknowledgement. The author thanks Manuel Stadlbauer and Sara Munday for valuable comments on an ealier version of this paper. The author thanks the referee for his/her helpful comments on the paper.
If Σ is the Markov shift with alphabet I whose incidence matrix consists entirely of 1s, then we have that Σ = I N and Σ n = I n for all n ∈ N. Then we set I * := Σ * . For ω, τ ∈ I * we denote by ωτ ∈ I * the concatenation of ω and τ, which is defined by ωτ := ω 1 , . . . , ω |ω| , τ 1 , . . . , τ |τ| for ω, τ ∈ I * . Note that I * forms a semigroup with respect to the concatenation operation. The semigroup I * is the free semigroup over the set I and satisfies the following universal property: For each semigroup S and for every map u : I → S, there exists a unique semigroup homomorphism u : I * → S such that u (i) = u (i), for all i ∈ I.
We equip I N with the product topology of the discrete topology on I. The Markov shift Σ ⊂ I N is equipped with the subspace topology. A countable basis of this topology on Σ is given by the cylindrical sets {[ω] : ω ∈ Σ * }. We will make use of the following metric generating the topology on Σ. For β > 0, we define the metric d β on Σ given by A function ϕ : Σ → R is also called a potential. For n ∈ N, we use S n ϕ : Σ → R to denote the ergodic sum of ϕ with respect to σ , in other words, S n ϕ : We say that ϕ is Hölder continuous if there exists β > 0 such that ϕ is β -Hölder continuous. The function ϕ is of medium variation, if ϕ is continuous and if there exists a sequence (D n ) ∈ R N with lim n (D n ) 1/n = 1 such that, for all n ∈ N, ω ∈ Σ n and x, y ∈ [ω], we have e S n ϕ(x)−S n ϕ(y) ≤ D n .
We need the following topological mixing properties for Markov shifts.
Definition 2.1. Let Σ be a Markov shift with alphabet I ⊂ N.
• Σ is topologically mixing if, for all i, j ∈ I, there exists n 0 ∈ N with the property that, for all n ≥ n 0 , there exists ω ∈ Σ n such that iω j ∈ Σ * . The Gurevič pressure of ϕ with respect to σ is, for each a ∈ I, given by The next definition goes back to the work of Ruelle and Bowen (cf. [Rue69], [Bow75]).
Definition 2.4. Let Σ be a Markov shift and let ϕ : Σ → R be Hölder continuous with P (ϕ, σ ) < ∞. We say that a Borel probability measure µ is a Gibbs measure for ϕ if there exists a constant C µ ≥ 1 such that

PROOFS
Let us first state the necessary definitions and notations which are needed for the proofs of our main results.
If Σ is a topologically mixing Markov shift with the b.i.p. property, and if ϕ : Σ → R is Hölder continuous with P (ϕ, σ ) < ∞, then there exists a unique σ -invariant Gibbs measure µ ϕ for ϕ by Theorem 2.5. For p ∈ N ∪ {∞} and ψ ∈ L p Σ, B (Σ) , µ ϕ , we denote by ψ p the L p -norm of ψ, where B (Σ) is the Borel sigma algebra of Σ. For a group extended Markov system (Σ × G, σ ⋊ Ψ), Stadlbauer ([Sta13]) introduced the Banach space H p , |·| p , given by Denote by ½ Ω the indicator function of a set Ω ⊂ Σ × G. Let H c be the closed subspace of H ∞ generated by ½ Σ×{g} : g ∈ G , which is isomorphic to the Hilbert space ℓ 2 (G). We use (·, ·) to denote the inner product on H c , given by ( For closed subspaces V 1 , V 2 ⊂ H ∞ and a bounded linear operator T : V 1 → V 2 , the operator norm of T is given by T : where C = C µ ϕ ≥ 1 denotes the constant of the Gibbs measure µ ϕ given by (2.1). Clearly, the linear operators A and T n are positive and bounded with A = 1 and T n ≤ A L n ϕ•π 1 = L n ϕ•π 1 , for each n ∈ N. To state the next lemma, let us recall the definition of Λ n ([Sta13, Proposition 5.2]), which is given by Lemma 3.2. Under the assumptions of Theorem 1.1, suppose that L ϕ (½) = ½. Then we have the following.
Proof. Let n ∈ N. The assertion in (1) follows because T n is positive and |T n ( f )| ∞ = L n ϕ•π 1 ( f ) 1 , for each f ∈ H + c . The first inequality in (2) holds since A = 1. To prove the second inequality we show that Hence, (3.2) follows. To prove L n ϕ•π 1 ≤ C T n , let f ∈ H + c . By the Gibbs property (2.1) of µ ϕ , there exists C ≥ 1 such that, for each g ∈ G and x 0 ∈ Σ, and finishes the proof of (2).
To prove the converse implication, suppose that G is amenable. It follows from a well-known result of Day ([Day64, Theorem 1(d)]) that T n = 1, for each n ∈ N. To prove this, observe that by (3.4), we have that T n is the right convolution operator with respect to the probability density on G, given by µ ϕ {x ∈ Σ : Ψ (x 1 , . . . , x n ) = g} , for each g ∈ G. Finally, we conclude that ρ L ϕ•π 1 = 1 by Lemma 3.2 (3). The proof is complete.
The proof is complete.
Lemma 3.3. Let Σ be a Markov shift and let ϕ : Σ → R be of medium variation. Suppose that ϕ is asymptotically α-symmetric with respect to Ψ, for some α ≥ 1. Then there exists a sequence of Hölder continuous functions ϕ j : Σ → R and (D j ) ∈ R N , D j ≥ 1, j ∈ N, such that ϕ j is asymptotically α(D j ) 1/(2 j)symmetric, lim j (D j ) 1/(2 j) = 1 and lim j ϕ j = ϕ, where the convergence of (ϕ j ) is uniformly on compact subsets of Σ.
Proof. Define ϕ j (x) := inf ϕ (y) : y ∈ [x 1 , . . . , x j ] , for each x ∈ Σ and j ∈ N. Since ϕ is of medium variation, there exists a sequence (D j ) ∈ R N , D j ≥ 1, such that, for all j, n ∈ N, ω ∈ Σ n and x ∈ [ω], where ⌊u⌋ denotes the largest integer not greater than u. Since ϕ is asymptotically α-symmetric, there exist n 0 ∈ N and sequences (c n ) ∈ R N and (N n ) ∈ N N with the property that lim n (c n ) 1/(2n) = α, lim n n −1 N n = 0 and such that (1.1) holds, for each g ∈ G and for all n ≥ n 0 . Let j ∈ N. By (1.1) and (3.9) we have for each n ∈ N with n > max {N n , n 0 } and g ∈ G, Since lim n c n max n−N n ≤l≤n+N n (D j ) ⌊ l j ⌋ sup k< j D k 1/(2n) = α(D j ) 1/(2 j) , we have that ϕ j is asymptotically α(D j ) 1/(2 j) -symmetric. By continuity of ϕ j and ϕ, and using that ϕ j ≤ ϕ j+1 , for each j ∈ N, we have that lim j ϕ j = ϕ locally uniformly by Dini's Theorem.