Thermodynamic Formalism for Transient Potential Functions

Abstract

We study the thermodynamic formalism of locally compact Markov shifts with transient potential functions. In particular, we show that the Ruelle operator admits positive continuous eigenfunctions and positive Radon eigenmeasures in forms of Martin kernels. These eigenmeasures can be characterized in terms of the direction of escape to infinity of their orbits, when viewed inside a suitable Martin-like compactification of the underlying shift space. We relate these results to first-order phase transitions in one-dimensional lattice gas models with infinite set of states. This work complements earlier works by Sarig [32,33] who focused on the recurrent scenario. * ofer.shwartz@weizmann.ac.il 2. Representation of eigenvectors: We extend Martin's representation theorem [23] to the context of Ruelle operator. Specifically, we construct a compactification X + of X + with boundary M = X + \ X + and construct a kernel K(f, ω|λ) (f ∈ C + c (X + ), ω ∈ X + ) s.t. every Radon measure µ with L * φ µ = λµ has the form µ(·) = M K(·, ω|λ)dν(ω) 1.1 Topological Markov shifts, Ruelle operator and transience Let S be an infinite countable set of states and let A ∈ {0, 1} S×S be a transition matrix over S. For a subset A ⊆ Z and a vector x ∈ S A , we denote by (x) i the i-th coordinate of x. The (positive) one-sided topological Markov shift (TMS) is the spacewith the transformation T : X + → X + , (T (x)) i = (x) i+1 and the metric d(x, y) = 2 − inf{i≥0:(x)i =(y)i} .(1)

The conformal measures of a normal subgroup of a cocompact Fuchsian group

“…Proof. This result can be derived from the classical Choquet theory but also follows from the unique representation on the minimal boundary (see equation (2) and also [49]) and by the linearity of the mapping in Theorem 1.1. COROLLARY 3.2.…”

“…As the theory of the eigenmeasures of the Ruelle operator is well developed (see [10,18,35,41,42,49,51]), this correspondence can be a step towards a classification of the conformal measures. In particular, in [49] the author presented the eigenmeasures of a transient Ruelle operator (see the definitions in §2.2) in terms of points on a Martin boundary. Thus, Conformal measures of normal subgroups of cocompact Fuchsian groups 3 the classification of the conformal measures directly translates to the identification of the Martin boundary.…”

“…For a t-transient potential with summable variations, the author introduced in [49] a Martin boundary which represents all eigenmeasures (and analogously all eigenfunctions) of the Ruelle operator, for the eigenvalue t. We briefly describe the construction here.…”

“…In §3 we show that for every δ ≥ δ Ŵ there is a linear one-to-one correspondence between the extremal (Ŵ, δ)-conformal measures and eigenmeasures of a suitable Ruelle operator. As the theory of the eigenmeasures of the Ruelle operator is well developed (see [10,18,35,41,42,49,51]), this correspondence can be a step towards a classification of the conformal measures. In particular, in [49] the author presented the eigenmeasures of a transient Ruelle operator (see the definitions in §2.2) in terms of points on a Martin boundary.…”

The conformal measures of a normal subgroup of a cocompact Fuchsian group

“…The theory of the eigenmeasures of the Ruelle operator is well developed, see [7,35,36,29,45,43]. In particular, in [43] the author presented the eigenmeasures of a transient Ruelle operator (see definition in Section 2.2) in terms of points on a Martin boundary. Thus, the classification of the conformal measures translates to the identification of the Martin boundary.…”