Phase Transitions of the Multifractal Spectrum

Multifractal analysis of Lyapunov exponent for the backward continued fraction map

In this note we study the multifractal spectrum of Lyapunov exponents for interval maps with infinitely many branches and a parabolic fixed point. It turns out that, in strong contrast with the hyperbolic case, the domain of the spectrum is unbounded and points of non-differentiability might exist. Moreover, the spectrum is not concave. We establish conditions that ensure the existence of inflection points. We also study the thermodynamic formalism for such maps. We prove that the pressure function is real analytic in a certain interval and then it becomes equal to zero.

Multifractal analysis of weak Gibbs measures for non-uniformly expanding C^1 maps

We consider the local dimension spectrum of a weak Gibbs measure on a C^1 non-uniformly hyperbolic system of Manneville- Pomeau type. We present the spectrum in three ways: using invariant measures, uniformly hyperbolic ergodic measures and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems.

Gatzouras--Lalley 测度的多重分形分析

Multifractal analysis of Gatzouras--Lalley measures

本文研究 Gatzouras--Lalley 型自仿射测度的多重分形分析，证明 Olsen 谱的变分公式，并指出多重分形形式在可微点成立，但一般可能失败且存在跳跃间断和多个相变。

Multifractal analysis for quotients of Birkhoff sums for countable Markov maps

This paper is devoted to study multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show that the spectrum varies analytically in parts of its domain. We apply our results to show that the Birkhoff spectrum for the Manneville-Pomeau map can be discontinuous, showing the remarkable differences with the uniformly hyperbolic setting. We also obtain results describing the Birkhoff spectrum of suspension flows. Examples involving continued fractions …

Multifractal analysis for multimodal maps

Given a multimodal interval map $f:I \to I$ and a Hölder potential $ϕ:I \to \mathbb{R}$, we study the dimension spectrum for equilibrium states of $ϕ$. The main tool here is inducing schemes, used to overcome the presence of critical points. The key issue is to show that enough points are `seen' by a class of inducing schemes. We also compute the Lyapunov spectrum. We obtain the strongest results when $f$ is a Collet-Eckmann map, but our analysis also holds for maps satisfying much weaker growth conditions.

From thermodynamic and spectral phase transitions to multifractal analysis

It is known that all uniformly expanding or hyperbolic dynamics have no phase transition with respect to Hölder continuous potentials. In \cite{BC21}, is proved that for all transitive $C^{1+α}-$local diffeomorphism $f$ on the circle, that is neither a uniformly expanding map nor invertible, has a unique thermodynamic phase transition with respect to the geometric potential, in other words, the topological pressure function $\mathbb{R} \ni t \mapsto P_{top}(f,-t\log|Df|)$ is analytic except at a point $t_{0} \in (0 , 1]$. Also it is proved spectral phase transitions, in other words, the transf…

The Multifractal Spectra of V-Statistics

Let $(X, T)$ be a topological dynamical system and let $Φ: X^r \to \mathbb{R}$ be a continuous function on the product space $X^r= X\times ... \times X$ ($r\ge 1$). We are interested in the limit of V-statistics taking $Φ$ as kernel: [\lim_{n\to \infty} n^{-r}\sum_{1\le i_1, ..., i_r\le n} Φ(T^{i_1}x, ..., T^{i_r} x).] The multifractal spectrum of topological entropy of the above limit is expressed by a variational principle when the system satisfies the specification property. Unlike the classical case ($r=1$) where the spectrum is an analytic function when $Φ$ is Hölder continuous, the spect…

源自随机替换的测度的多重分形分析

Multifractal analysis of measures arising from random substitutions

本文研究源自随机替换的频率测度的正则性，推导了自然测度类的 $L^q$ 谱闭式公式并证明多重分形形式成立，同时引入膨胀词 $L^q$ 谱概念，为多重分形分析提供了新测度类。

Large Deviation Multifractal Analysis of a Process Modeling TCP CUBIC

Multifractal characteristics of the Internet traffic have been discovered and discussed in several research papers so far. However, the origin of this phenomenon is still not fully understood. It has been proven that the congestion control mechanism of the Internet transport protocol, i.e., the mechanism of TCP Reno can generate multifractal traffic properties. Nonetheless, TCP Reno does not exist in today's network any longer, surprisingly traffic multifractality has still been observed. In this paper we give the theoretical proof that TCP CUBIC, which is the default TCP version in the Linux …

Typical Borel measures on $[0,1]d$ satisfy a multifractal formalism

In this article, we prove that in the Baire category sense, measures supported by the unit cube of $\R^d$ typically satisfy a multifractal formalism. To achieve this, we compute explicitly the multifractal spectrum of such typical measures $μ$. This spectrum appears to be linear with slope 1, starting from 0 at exponent 0, ending at dimension $d$ at exponent $d$, and it indeed coincides with the Legendre transform of the $L^q$-spectrum associated with typical measures $μ$.

Partition zeta functions, multifractal spectra, and tapestries of complex dimensions

For a Borel measure and a sequence of partitions on the unit interval, we define a multifractal spectrum based on coarse Holder regularity. Specifically, the coarse Holder regularity values attained by a given measure and with respect to a sequence of partitions generate a sequence of lengths (or rather, scales) which in turn define certain Dirichlet series, called the partition zeta functions. The abscissae of convergence of these functions define a multifractal spectrum whose concave envelope is the (geometric) Hausdorff multifractal spectrum which follows from a certain type of Moran constr…

高维多标度分析用于双曲面上的尖点旋绕过程

Higher-dimensional multifractal analysis for the cusp winding process on hyperbolic surfaces

本文研究了双曲面上多尖点旋绕过程的多标度分析，建立了其Hausdorff维度谱的条件变分原理，并证明了该维度谱的解析性。

On multifractal formalism for self-similar measures with overlaps

Let $μ$ be a self-similar measure generated by an IFS $Φ=\{ϕ_i\}_{i=1}^\ell$ of similarities on $\mathbb R^d$ ($d\ge 1$). When $Φ$ is dimensional regular (see Definition~1.1), we give an explicit formula for the $L^q$-spectrum $τ_μ(q)$ of $μ$ over $[0,1]$, and show that $τ_μ$ is differentiable over $(0,1]$ and the multifractal formalism holds for $μ$ at any $α\in [τ_μ'(1),τ_μ'(0+)]$. We also verify the validity of the multifractal formalism of $μ$ over $[τ_μ'(\infty),τ_μ'(0+)]$ for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds.…