In this paper, we show a new relation between phase transition in Statistical Mechanics and the dimension of space harmonic functions (SHF) for transfer operator. This is accomplished by extending classical Ruelle-Perron-Frobenius theory to realm low regular potentials defined on either finite or infinite (uncountable) alphabets. We also give an example potential having where Perron-Frobenius e...

In this paper, we study the thermodynamic phase transition in one dimensional systems with short range interaction. We begin by reviewing some famous non-existence result, such as Landau’s argument and van Hove’s theorem. And we point out that the PerronFrobenius theorem is generally used to determine whether there is phase transition in such kind of system. Then several examples that Perron-Fr...

In this paper, we attempt to answer the three questions about the invariant probability distribution for stochastic matrices: (1) does every stochastic matrix have an invariant probability distribution?; (2) is the invariant probability distribution unique?; and (3) when can we conclude that the power of a stochastic matrix converges? To answer these questions, we present the Perron-Frobenius T...

Using Walters’ version of the Ruelle-Perron-Frobenius Theorem we show the existence and uniqueness of KMS states for a certain one-parameter group of automorphisms on a C*-algebra associated to a positively expansive map on a compact metric space.

Poles of the Ihara zeta function associated with a finite graph are described by graph-theoretic quantities. Elementary proofs based on the notions of oriented line graphs, Perron-Frobenius operators, and discrete Laplacians are provided for Bass’s theorem on the determinant expression of the zeta function and Hashimoto’s theorems on the pole at u = 1.

we give a new proof of the well known wedderburn's little theorem (1905) that a finite‎ ‎division ring is commutative‎. ‎we apply the concept of frobenius kernel in frobenius representation theorem in finite group‎ ‎theory to build a proof‎.

Using the Perron-Frobenius operator we establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic. We apply the result to asymptotically periodic transformations and give an extensive specific example using the tent map.

We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly useful in generalizing certain areas where the spectral theory of matrices has traditionally played an important role. For illustration, we will discuss a m...