Multifractal formalism for Benedicks-Carleson quadratic maps (Q2925259)

The authors develop a multifractal formalism for the special family of functions known as Benedicks-Carleson maps. Multifractal Analysis is a field extensively studied especially in the last two decades. One of the main issues in this subject is to describe multifractal spectra to have an idea of a quantification of the complexity of the system. The authors consider the family of maps NEWLINE\[NEWLINE\begin{aligned} f_a : [ -1, 1] \to [ -1, 1] \\ f_a (x) = 1 - ax^2,\end{aligned}NEWLINE\]NEWLINE with \(0 < a \leq 2\). The multifractal spectrum to be analyzed is that corresponding to Birkhoff ergodic averages, i.e., the multifractal decomposition NEWLINE\[NEWLINEK_\alpha = \left\{x : \lim_{n \to \infty}\frac1n \sum^n_{i=0} \varphi (f^i (x))= \alpha \right\},NEWLINE\]NEWLINE where \(\varphi : [ -1, 1] \to \mathbb R\) is a continuous map. For the dimension spectrum, i.e., for the function \(B_\alpha = \dim_H K_\alpha\), a variational formula is obtained, like that one previously given by Takens and Verbitski for systems with specification. For the estimation of the spectrum, the authors construct fractal sets, and a family of uniformly ergodic systems.NEWLINENEWLINEThey also establish a large deviation process for the Lebesgue measure.