## Irregular sets for ratios of Birkhoff averages are residual.(English)Zbl 1309.37018

Let $$\sigma :X\rightarrow X$$ be a subshift. Given two continuous functions $$\varphi, \psi :X\rightarrow \mathbb{R}$$ with $$\psi >0$$. Consider the open interval $\mathcal{L}_{\varphi, \psi }=\left(\inf_{\mu}\frac{\int_{X}\varphi d\mu}{\int_{X}\psi d\mu},\sup_{\mu}\frac{\int_{X}\varphi d\mu}{\int_{X}\psi d\mu}\right),$ where the infimum and supremum are taken over all $$\sigma$$-invariant probability measures on $$X$$. Let $$I\subset \mathcal{L}_{\varphi,\psi}$$ and given the irregular set $$X_{\varphi, \psi,I }=\{\omega \in X: A_{\varphi,\psi}(\omega)=I \},$$ where $$A_{\varphi,\psi}(\omega)$$ is the set of accumulation points of the sequence $S_{\varphi, \psi}(\omega,n)=\frac{\sum_{i=0}^{n-1}\varphi(\sigma^i(\omega) }{\sum_{i=0}^{n-1}\psi(\sigma^i(\omega)}.$ The authors prove the following main result:
Let $$\sigma |X$$ be a subshift with the weak specification property and let $$\varphi, \psi :X\rightarrow \mathbb{R}$$ be continuous functions with $$\inf \psi >0.$$ Given a closed interval $$I\subset \mathcal{L}_{\varphi, \psi }$$ that is not a singleton, then $$X_{\varphi, \psi,I }$$ is either empty or residual.
Also, the authors give an application of their result to the pointwise dimension of a Gibbs measure on a repeller of a conformal map.

### MSC:

 37B10 Symbolic dynamics 37A30 Ergodic theorems, spectral theory, Markov operators
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