On the generalized dimensions of physical measures of chaotic flows

TL;DR

The paper establishes a general dimension-transformation principle for physical measures of chaotic flows that are constructed as suspensions over Poincaré maps: for all $q\neq 1$, $D_q(\mu)=D_q(\mu_R)+1$, and $d_{\mu}(x)=d_{\mu_R}(\xi)+1$, with $\xi=\pi_1\circ\Theta^{-1}(x)$; if $\mu_R$ is exact-dimensional, the same shift holds for $D_1$. This result is first proved in the bounded roof function case and then extended to general roofs via truncation arguments. The framework is then applied to two canonical systems: (i) a Rössler-like flow whose $D_q$ spectrum is computed explicitly in terms of the singularity parameter $\alpha$, producing a sharp formula for $D_q(\hat{\mu})$ that aligns with numerical estimates; (ii) Lorenz-like flows, for which the existence and exact-dimensionality of the base measure yield $D_1(\mu)=D_1(\mu_R)+1$ and a universal lower bound $D_q^{-}(\mu)\ge 2$, providing a concrete benchmark for the spectrum in singular-hyperbolic attractors.

Abstract

We prove that if $μ$ is the physical measure of a $C^2$ flow in $\mathbb{R}^d, d \geq 3,$ diffeomorphically conjugated to a suspension flow based on a Poincaré application $R$ with physical measure $μ_{R}$, then $D_{q}(μ)=D_{q}(μ_{R})+1$, where $D_{q}$ denotes the generalized dimension of order $q \neq1$. We also show that a similar result holds for the local dimensions of $μ$ and, under the additional hypothesis of exact-dimensionality of $μ_{R}$, that our result extends to the case $q=1$. We apply these results to estimate the $D_{q}$ spectrum associated with Rössler systems and turn our attention to Lorenz-like flows, proving the existence of their information dimension and giving a lower bound for their generalized dimensions.

Figures (4)

• Figure 1: Up: Attractor of the velocity field $\tilde{f}_c$. The box $B$ is represented in blue. Down: Attractor of the flow generated by $\bar{f}_c$. The Poincaré section $\Sigma$ is represented in blue. In both cases, we took with $a=b=0.1$ and $c=18$
• Figure 2: Left: Graphical representation of the unimodal map $\mathbf{T}$, associated with the Rössler flow of parameters $a=b=0.1$, $c=18$ (the axes are inverted). Right: 1-D cross-section of the attractor with the Poincaré section $\Sigma$ for different discretization steps $h$.
• Figure 3: Schwartzian derivative of $\mathbf{T}$ for the parameters $a=b=0.1$ and $c=18$.
• Figure 4: Representation of the $D_q$ spectrum of the measure $\hat{\mu}$, as given by formula (\ref{['dqrr']}) for $\alpha=1/2$.

Theorems & Definitions (18)

• Definition 1
• Theorem 2
• Proposition 3
• Lemma 4
• Lemma 5
• Proposition 6
• Theorem 7
• Remark 8
• Remark 9
• Proposition 10