Multifractal and multi-multifractal of one-dimensional expanding Markov maps (Q1819208)

Let \(\mu\) be an invariant Gibbs measure for a \(C^{1+\delta}\) one-dimensional expanding Markov map. The standard method to compute the singularity dimension spectrum or \textit{multifractal} spectrum of \(\mu\), i.e., \[ f_{\mu}(\alpha):=\left\{x\in \text{ supp} \mu: \alpha_{\mu}(x):= \lim_{r\rightarrow 0} {\log\mu(B(x,r))\over \log r} =\alpha \right\},\quad \alpha\geq 0, \] is to construct a one-parameter continuum of Gibbs measures \(\{\mu_{\beta}\}_{\beta\in[-\infty,+\infty]}\) such that \(\alpha_{\mu_{\beta}}(x)=f_{\mu}(\alpha)\) \(\mu_{\beta}\)-a.e., where \(\alpha=-\tau'(\beta)\) and \(\tau(\beta)\) is chosen to be the unique number that satisfies a suitable pressure equation involving the potential for \(\mu\). This paper proposes to consider the two-parameter continuum of Gibbs measures \(\{\mu_{\beta,\kappa}\}\), \((\beta,\kappa)\in \bar{\mathbb R}\times \bar{\mathbb R}\), where \(\{\mu_{\beta,\kappa}\}_{\kappa}\) is the family of Gibbs measures associated with \(\mu_{\beta}\) as the family \(\{\mu_{\beta}\}_{\beta}\) it is with \(\mu\); and then to compute the multifractal spectra of the measures \(\mu_{\beta}\). This is called the first-stage multi-multifractal spectrum of \(\mu\). This computation iterated in the obvious way defines the \(n\)-stage multi-multifractal spectrum of \(\mu\). Also, well-known results on the dimension, entropy and Lyapunov singularity spectra of one-dimensional expanding maps are proved in the paper using large deviation methods. As an application, the extremal measures \(\mu_{\pm_ \infty}\) are shown to have degenerate spectra (i.e., \(f_{\mu_{\pm \infty}}(\alpha)\) are characteristic functions of a point).