In this paper, we study the irregular multifractal spectrum of the Gauss map, by investigating the upper and lower Birkhoff spectra. For potentials satisfying a weakly tempered distortion property, we establish that the Hausdorff dimensions of the classical, upper, and lower Birkhoff level sets are identical. Furthermore, we derive an exact formula for the dimension of the intersection of the upper and lower Birkhoff level sets, proving that this dimension equals the minimum of the two individual dimensions. These findings broaden the multifractal theory previously developed for conformal repellers and finite branch expanding systems to the infinite branch context of the Gauss map. As applications, we furnish complete multifractal descriptions of the upper and lower Lyapunov spectra of the Gauss map, as well as the upper and lower arithmetic-mean spectra of continued fractions.

Given a Fuchsian group $\Gamma $, we introduce escape rate spectra associated with $\Gamma $ to investigate the transient behavior of the geodesic flow on the hyperbolic surface $\mathbb {D}/\Gamma $. Our definition is motivated by Bishop’s linear escape set. In the special case when $\mathbb {D}/\Gamma $ is a hyperbolic $\mathbb {Z}$-covering of a surface uniformized by a generalized Schottky group, we completely determine the escape rate spectra by the convex conjugate of a generalized Poincaré exponent.

We study the sets of points where a Lévy function and a translated Lévy function share a given couple of Hölder exponents, and we investigate how their Hausdorff dimensions depend on the translation parameter.

In this article, we calculate the Birkhoff spectrum in terms of the Hausdorff dimension of level sets for Birkhoff averages of continuous potentials for a certain family of diagonally affine iterated function systems. Also, we study Besicovitch–Eggleston sets for finite generalized Lüroth series number systems with redundancy. The redundancy refers to the fact that each number $x \in [0,1]$ has uncountably many expansions in the system. We determine the Hausdorff dimension of digit frequency sets for such expansions along fibres.

We perform a multifractal analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. Our main result provides a formula for the Hausdorff dimension of level sets of prescribed growth rates in terms of a generalized Poincaré exponent of the Fuchsian group. We employ symbolic dynamics developed by Bowen and Series, ergodic theory and thermodynamic formalism to prove the analyticity of the dimension spectrum.

The purpose of this study is two-fold. First, the Hausdorff dimension formula of the multidimensional multiplicative subshift (MMS) in $\mathbb {N}^d$ is presented. This extends the earlier work of Kenyon et al [Hausdorff dimension for fractals invariant under multiplicative integers. Ergod. Th. & Dynam. Sys. 32(5) (2012), 1567–1584] from $\mathbb {N}$ to $\mathbb {N}^d$. In addition, the preceding work of the Minkowski dimension of the MMS in $\mathbb {N}^d$ is applied to show that their Hausdorff dimension is strictly less than the Minkowski dimension. Second, the same technique allows us to investigate the multifractal analysis of multiple ergodic average in $\mathbb {N}^d$. Precisely, we extend the result of Fan et al, [Multifractal analysis of some multiple ergodic averages. Adv. Math. 295 (2016), 271–333] of the multifractal analysis of multiple ergodic average from $\mathbb {N}$ to $\mathbb {N}^d$.

We examine whether the differences in the legal origins of countries (Common Law versus Civil Law) can explain the variations in the price efficiencies of the stock markets of different countries. Based on multifractal detrended fluctuation analysis of the daily stock indices of 34 countries over 21 years, we find that the stock price indices in Common Law origin countries show greater price efficiency than the stock price indices in Civil Law countries. These results provide additional evidence that the legal origins of countries affect their economic activities and outcomes.

For any self-similar measure $\mu $ in $\mathbb {R}$, we show that the distribution of $\mu $ is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the iterated function system of similarities (IFS). This generalizes the net interval construction of Feng from the equicontractive finite-type case. When the measure satisfies the weak separation condition, we prove that this directed graph has a unique attractor. This allows us to verify the multifractal formalism for restrictions of $\mu $ to certain compact subsets of $\mathbb {R}$, determined by the directed graph. When the measure satisfies the generalized finite-type condition with respect to an open interval, the directed graph is finite and we prove that if the multifractal formalism fails at some $q\in \mathbb {R}$, there must be a cycle with no vertices in the attractor. As a direct application, we verify the complete multifractal formalism for an uncountable family of IFSs with exact overlaps and without logarithmically commensurable contraction ratios.

We study the thermodynamic formalism of a $C^{\infty }$ non-uniformly hyperbolic diffeomorphism on the 2-torus, known as the Katok map. We prove for a Hölder continuous potential with one additional condition, or geometric $t$ -potential $\unicode[STIX]{x1D711}_{t}$ with $t<1$ , the equilibrium state exists and is unique. We derive the level-2 large deviation principle for the equilibrium state of $\unicode[STIX]{x1D711}_{t}$ . We study the multifractal spectra of the Katok map for the entropy and dimension of level sets of Lyapunov exponents.

Multifractal analysis is known as a useful tool in signal analysis. However, themethods are often used without methodological validation. In this study, wepresent multidimensional models in order to validate multifractal analysismethods.