Abstract
Given a Lipschitz function \(f:\{1,\ldots ,d\}^\mathbb {N} \rightarrow \mathbb {R}\), for each \(\beta >0\) we denote by \(\mu _\beta \) the equilibrium measure of \(\beta f\) and by \(h_\beta \) the main eigenfunction of the Ruelle Operator \(L_{\beta f}\). Assuming that \(\{\mu _{\beta }\}_{\beta >0}\) satisfy a large deviation principle, we prove the existence of the uniform limit \(V= \lim _{\beta \rightarrow +\infty }\frac{1}{\beta }\log (h_{\beta })\). Furthermore, the expression of the deviation function is determined by its values at the points of the union of the supports of maximizing measures. We study a class of potentials having two ergodic maximizing measures and prove that a L.D.P. is satisfied. The deviation function is explicitly exhibited and does not coincide with the one that appears in the paper by Baraviera-Lopes-Thieullen which considers the case of potentials having a unique maximizing measure.
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Notes
we write \(\lim _{n,\beta \rightarrow +\infty } a_{\beta ,n}= a\), with \(a \in \mathbb {R}\), if for any \(\epsilon >0\) there exists \(L>0\) such that \(n,\beta >L \Rightarrow |a_{\beta ,n} -a|<\epsilon \). In the Eq. (2), even though \(R_{+}^{\infty }\) can assume the value \(+\infty \), we have that \(\inf _{z\in k} R_{+}^{\infty }(z)\) is finite because for any point p that belongs to the support of the maximizing measure we have \(R_{+}^{\infty }(p)=0\) and the set \(\cup _{n\ge 1} \sigma ^{-n}\{p\}\) is dense.
we use the following notations
$$\begin{aligned} 0^2=00, \, 0^3=000,\ldots ,\,0^\infty = (0000 \cdots ), \,\, 1^2=11,\,1^3=111,\ldots , \,1^\infty =(1111 \cdots ). \end{aligned}$$
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Mengue, J.K. Large Deviations for Equilibrium Measures and Selection of Subaction. Bull Braz Math Soc, New Series 49, 17–42 (2018). https://doi.org/10.1007/s00574-017-0044-x
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DOI: https://doi.org/10.1007/s00574-017-0044-x