Deviation of ergodic averages for substitution dynamical systems with eigenvalues of modulus 1 (Q2921109)

The topic of the article is the investigation of the asymptotic behaviour of ergodic sums for substitution dynamical systems in the non-hyperbolic case. Let \(\sigma\) be a primitive substitution on a finite alphabet~\({\mathcal{A}}\) with matrix~\({M_\sigma}\) having an eigenvalue~\({\theta}\) of modulus 1 and let \({\Gamma=(\gamma(a), a\in\mathcal{A})^t}\) be a right-eigenvector associated to \(\theta\). Let \({X_\sigma}\) be the subshift defined from \(\sigma\) (\({x\in X_\sigma}\) iff any subword of \(x\) is a subword of iteration \({\sigma^N(a)}\) for some \({N\in\mathbb{N}}\) and \({a\in\mathcal{A}}\)).NEWLINENEWLINEThe first main result states that there exists a constant~\({C=C(\sigma, \gamma)}\) such that, for any \({x\in X_\sigma}\) NEWLINE\[NEWLINE \liminf_{n\to\infty}|\gamma(x_0)+\gamma(x_1)+\ldots+\gamma(x_n)|<C,\;\;\liminf_{n\to\infty}|\gamma(x_{-n})+\ldots+\gamma(x_{-1})+\gamma(x_0)|<C. NEWLINE\]NEWLINE The proof is based on a prefix-suffix decomposition for \({x\in X_\sigma}\). As a corollary of this result, the authors prove the existence of infinitely many self-similar interval exchange transformations \(T\) on more than 4 intervals such that every affine interval exchange transformation that is semi-conjugate to~\(T\) is also conjugate to \(T.\)NEWLINENEWLINEThe second result states that if additionally \(\sigma\) has constant length~\(d\) (i.e., the word \(\sigma(a)\) has lenght \(d\) for all \({a\in\mathcal{A}}\)), the eigenvalue \({\theta=1}\) and \(t\in[0,1]\) has eventually a periodic expansion~\({\{\tau_n\}_{n\in\mathbb{N}}}\) in basis \(d\), \({t=\sum\tau_nd^n}\), then NEWLINE\[NEWLINE \frac{1}{\sqrt{n}}S_{\lfloor d^nt\rfloor}(\gamma)\to\nu, NEWLINE\]NEWLINE where \(\nu\) is a probability measure with some generalized density.NEWLINENEWLINEThe last main result states that there exist infinitely many primitive substitutions~\(\sigma\) of constant length~\(d\) with eigenvalue 1 and eigenvector~\(\gamma\) such that, for Lebesgue almost all \({t\in[0,1]}\), the sequence NEWLINE\[NEWLINE \frac{1}{\sqrt{V_n}}S_{\lfloor d^nt\rfloor}(\gamma) NEWLINE\]NEWLINE converges in distribution to the standard normal law and the variance~\(V_n\) satisfies the inequality NEWLINE\[NEWLINE \alpha_1n^{4/5}\leq V_n\leq\alpha_2n. NEWLINE\]NEWLINE The martingale approach for the CLT due to \textit{S. Sethuraman} and \textit{S. R. S. Varadhan} [Electron. J. Probab. 10, Paper No. 36, 1221--1235 (2005; Zbl 1111.60057)] is used in the proof of the last result.