A Proof of the Existence and Simplicity of a Maximal Eigenvalue for RuellePerronFrobenius Operators

We give a new proof of a result due to Ruelle about the existence and simplicity of a unique maximal eigenvalue for a Ruelle^Perron^Frobenius operator acting on some Ho« lder continuous function space. Mathematics Subject Classi¢cations (1991): Primary 58F23, Secondary 30C62. Key words: locally expanding, mixing, Ruelle^Perron^Frobenius operator, maximal eigenvalue. 1. Introduction Ruelle's Theorem represents remarkable progress in the study of thermodynamical formalism. The theorem concerns transfer operators with positive weights associated with certain dynamical systems. These kind of operators are called Ruelle^ Perron^Frobenius operators. Ruelle's theorem was ¢rst proved by Ruelle [8, 9] in the study of the existence and uniqueness of the Gibbs measure associated with a one-sided ¢nite-type subshift and a Ho« lder continuous potential function.The result was extended to continuous positive expansive transformations byWalters [12] by using g-measures. It has become a standard technique in thermodynamical formalism. In the original proof (see [2, 12]), the dual operator acting on the space of measures was ¢rst studied and the Schauder^Tychono¡ ¢xed point theorem was used with the dual operator and then some di¤cult analysis followed. There is another proof of Ruelle's theorem associated with a one-sided ¢nite-type subshift by Fan [3] by applying an idea from probability theory. A key part of Ruelle's theorem deals with the existence and simplicity of a unique maximal eigenvalue for a Ruelle^ Perron^Frobenius operator. A more geometric proof of this part is given by Ferrero and Schmitt [4] using the Hilbert projective metrics de¢ned by Birkho¡ [1] on convex cones in Banach spaces. They showed that Ruelle^Perron^Frobenius operator acting on a certain Ho« lder continuous function space contracts the Hilbert projective metrics of certain convex cones in this Ho« lder continuous function space. So the existence and *The author is supported in part by NSF grants and PSC-CUNY awards. Letters in Mathematical Physics 48: 211^219, 1999. 211 # 1999 Kluwer Academic Publishers. Printed in the Netherlands. simplicity of the unique maximal eigenvalue follows from the contracting ¢xed point theorem. Here we give a new proof of the existence and simplicity of a unique maximal eigenvalue for a Ruelle^Perron^Frobenius operator acting on a certain Ho« lder continuous function space without using any ¢xed-point theorems and any properties for convex cones. The existence and simplicity of a unique maximal eigenvalue for a Ruelle^Perron^Frobenius operator has many interesting applications. One of them is to Krzyzewski^Szlenk's theorem [5] (see also [6, 11]) concerning the existence and uniqueness of the smooth probability invariant measure for a C1‡a locally expanding map from a compact C2 Riemannian manifold into itself. The Letter is organized as follows. In Section 2, we de¢ne local expanding and mixing maps from a compact metric space into itself and Ruelle^Perron^Frobenius operators. In the same section, we give our proof of the main theorem due to Ruelle. Before presenting our new proof, we ¢rst prove several key lemmas (Lemmas 1 to 3). In Section 3, we provide an application (Corollary) to a result from Krzyzewski and Szlenk about smooth probability invariant measures in dynamical systems. 2. Existence and Simplicity of a Maximal Eigenvalue Let …X ; d† be a compact metric space and let B…x; r† mean the open ball centered at x with radius r > 0. Let f : X ! X be a locally expanding map, i.e., there are constants l > 1 and b > 0 such that f jB…x; b† is homeomorphic for any x in X and d… f …x†; f …x0††X ld…x; x0† for any x and x0 in X with d…x; x0† < b. For a locally expanding map, there is a constant integer N0 > 0 such that #… f ÿ1…x††WN0 for all x in X.We say that f is mixing if, for any open set U of X, there is an integer n > 0 such that f n…U† ˆ X. Henceforth, we suppose that f is a locally expanding andmixing map.Then for any x and y in X such that f …x† ˆ y, there is a neighborhood V of y such that f has the local inverse g : V ! g…V † and fg ˆ identity, g…y† ˆ x.We can further chooseV such that g is contracting, i.e., d…g…y†; g…y0††W lÿ1d…y; y0† for any y and y0 in V. Since X is compact, there is a ¢xed constant 0 < a0 < 1 such that V can be picked as B…y; a0†. LetR denote the real line and let C0…X ;R† be the space of all continuous functions f : X ! R with the supremum norm jjfjj ˆ maxx2X fjf…x†jg. Let Ca…X ;R†, for 0 < aW 1, be the space of all a-Ho« lder continuous functions f, i.e., f is in C0…X ;R† and satis¢es sup x6ˆy2X jf…x† ÿ f…y†j d…x; y† <1: We say two functions f1 Xf2 if f1…x†Xf2…x† for all x in X. Let Ca K;s…X ;R†, for 212 YUNPING JIANG