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R. Ruviaro

Publications and source records attributed to R. Ruviaro.

4 recordsLinked to original sources
arXiv2020

On the Dimension of the Space of Harmonic Functions on Transitive Shift Spaces

In this paper, we show a new relation between phase transition in one-dimensional Statistical Mechanics and the multiplicity of the dimension of the space of harmonic functions for an extension of the classical transfer operator. We accomplish this by extending the classical Ruelle-Perron-Frobenius theory to the realm of low regular potentials. This is done by establishing finer properties of the associated conformal measures and thoroughly developing a method to obtain information on the maximal eigenspace of a suitably constructed family of Markov Processes. Our results are valid in the setting of finite and infinite alphabets. Several new applications are given to illustrate the theory. For example, we determine the support of a large class of equilibrium states associated with low regular potentials, including ones allowing phase transition. Additionally, we prove a version of the Functional Central Limit Theorem for equilibrium states. A remarkable aspect of this result is that it does not require the spectral gap property of the associated transfer operator. It is valid for long-range spins systems that might not be positively correlated and for non-local observables.

math.DS↗
arXiv2020

The Double Transpose of the Ruelle Operator

In this paper we study the double transpose of the $L^1(X,\mathscr{B}(X),ν)$-extensions of the Ruelle transfer operator $\mathscr{L}_{f}$ associated to a general real continuous potential $f\in C(X)$, where $X=E^{\mathbb{N}}$, the alphabet $E$ is any compact metric space and $ν$ is a maximal eigenmeasure. For this operator, denoted by $\mathbb{L}^{**}_{f}$, we prove the existence of some non-negative eigenfunction, in the Banach lattice sense, associated to $ρ(\mathscr{L}_{f})$, the spectral radius of the Ruelle operator acting on $C(X)$. As an application, we obtain a sufficient condition ensuring that the natural extension of the Ruelle operator to $L^1(X,\mathscr{B}(X),ν)$ has an eigenfunction associated to $ρ(\mathscr{L}_{f})$. These eigenfunctions agree with the usual maximal eigenfunctions, when the potential $f$ belongs to the Hölder, Walters or Bowen class. We also construct solutions to the classical and generalized variational problem, using the eigenvector constructed here.

math.DS↗
arXiv2019

Mini-Max Algorithm via Pohozaev Manifold

A new algorithm for solving non-homogeneous asymptotically linear and superlinear problems is proposed. The ground state solution of the problem, which in general is obtained as a mini-max of the associated functional, is obtained as the minimum of the functional constrained to the Pohozaev manifold instead. Examples are given of the use of this method for finding numerical radially symmetric positive solutions depending on various parameters.

math.AP↗