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.


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