A note on two approaches to the thermodynamic formalism (Q977922)

This paper considers and compares two approaches to a thermodynamic-formalism called an inducing scheme. An inducing scheme provides a means of using symbolic dynamics to study equilibrium states of non-uniformly hyperbolic maps. The setting is as follows: a compact metric space \(X\), a continuous map \(f: X\to X\), and a continuous potential function \(\varphi: X\to\mathbb{R}\). The connection between this original system and symbolic dynamics is provided by the inducing scheme, a concept that generalizes the idea of the first return map. On a subset \(W\subset X\) (called the base of the inducing scheme) the induced map \(F: W\to W\) sends \(x\) to \(f^{\tau(x)}\), where \(\tau(x)\) is the inducing times; \(\tau(x)\) is a return time to \(W\), but generally not the first return time. With suitable choice of \(W\) and \(\tau\) the induced map \(F\) can -- in many cases -- be made equivalent to the full shift on a countable alphabet whose symbols correspond to subsets of \(W\) on which \(\tau(x)\) is constant. Using this correspondence one obtains a correspondence between potentials on \(X\) and induced potentials on symbol space, and between measures on \(X\) and lifted measures on symbol space. However, the correspondence is incomplete since there can be measures on \(X\) that do not correspond to a measure on symbol space. In previous work Pesin and Senti, working on the symbol space side, give conditions on the induced potential that imply existence of an equiibrium state. Working with the original system, Bruin and Todd use inducing schemes to study a class of one-dimensional maps more general than those considered by Pesin and Senti. They establish existence of an equilibrium state for a class of potentials that satisfy the so-called bounded range condition. The author of this paper considers the relationships between the Pesin-Senti and Bruin-Todd results. He proves that: (1) for the inducing schemes that Bruin and Todd consider, their conditions imply Pesin and Senti's; (2) for any inducing scheme satisfying a liftability condition and a growth rate estimate, the bounded range conditions alone implies Pesin and Senti's conditions and; (3) for a broad range of inducing schemes there is a potential function that does not have bounded range but whose induced potential satisfies Pesin and Senti's conditions.