[Paper Review] Dimensional characteristics of invariant measures for circle diffeomorphisms

Victoria Sadovskaya|ArXiv.org|Sep 2, 2008

Mathematical Dynamics and Fractals6 references3 citations

TL;DR

This paper constructs $C^{∞}$ circle diffeomorphisms with Liouville rotation numbers such that the unique invariant measure exhibits highly irregular dimensional behavior: its pointwise and box dimensions do not exist almost everywhere, with lower pointwise and lower box dimensions equal to any prescribed value $\beta \in [0,1]$. The construction uses a refined version of the Anosov-Katok method with explicit quantitative estimates to achieve precise control over the measure's local scaling properties.

ABSTRACT

We consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number $τ$ there exists a $C^\infty$ circle diffeomorphism with rotation number $τ$ such that the pointwise and box dimensions of its unique invariant measure do not exist. Moreover, the lower pointwise and lower box dimensions can equal any value $0\le β\le 1$.

Motivation & Objective

• To investigate the dimensional characteristics—pointwise, box, and Hausdorff—of invariant measures for circle diffeomorphisms with irrational rotation numbers.
• To resolve the behavior of these dimensions in the case of Liouville rotation numbers, which are rapidly approximable by rationals and known to produce non-uniform dynamics.
• To construct explicit $C^\infty$ diffeomorphisms with a given Liouville rotation number such that the invariant measure's pointwise and box dimensions do not exist almost everywhere.
• To show that the lower pointwise and lower box dimensions of the invariant measure can be made equal to any value $\beta \in [0,1]$.

Proposed method

• A refined version of the Anosov-Katok method is employed to construct $C^\infty$ diffeomorphisms with prescribed rotation numbers.
• The construction proceeds iteratively via a sequence of $C^\infty$ diffeomorphisms $h_n$, each approximating the final map $h$, with controlled distortion and scaling.
• The invariant measure $\mu$ is defined as the weak limit of pushforwards of Lebesgue measure under the $h_n$, ensuring ergodicity and uniqueness.
• Quantitative estimates on the scaling of intervals and the modulus of continuity of the conjugating map $h$ are used to control the pointwise dimension.
• Hölder continuity of $h$ with exponent $\beta$ is proven by induction, ensuring $\underline{d}_\mu(x) \geq \beta$ almost everywhere.
• The construction ensures $\overline{d}_\mu(x) = 1$ a.e. by controlling the decay of measure on small balls using rapidly shrinking intervals $E_n$.

Experimental results

Research questions

• RQ1 Can the pointwise dimension of an invariant measure for a $C^\infty$ circle diffeomorphism fail to exist almost everywhere, even when the system is uniquely ergodic?
• RQ2 For a Liouville rotation number, can the lower pointwise dimension of the invariant measure be made equal to any prescribed value $\beta \in [0,1]$?
• RQ3 Is it possible to construct a $C^\infty$ circle diffeomorphism with a given Liouville rotation number such that the lower and upper box dimensions of the invariant measure differ?
• RQ4 What is the relationship between the Diophantine or Liouville nature of the rotation number and the existence or non-existence of pointwise dimension for the invariant measure?
• RQ5 Can the Hausdorff and box dimensions of the invariant measure be made unequal in the Liouville case?

Key findings

• For any Liouville number $\tau$ and any $\beta \in [0,1]$, there exists a $C^\infty$ circle diffeomorphism with rotation number $\tau$ such that the lower pointwise dimension $\underline{d}_\mu(x) = \beta$ and the upper pointwise dimension $\overline{d}_\mu(x) = 1$ for $\mu$-almost every $x \in S^1$.
• The lower box dimension $\underline{\dim}_B \mu = \beta$ and the upper box dimension $\overline{\dim}_B \mu = 1$ for the invariant measure $\mu$.
• The Hausdorff dimension $\dim_H \mu = \beta$, which matches the lower box dimension.
• The pointwise dimension does not exist almost everywhere because $\underline{d}_\mu(x) = \beta < 1 = \overline{d}_\mu(x)$ for $\mu$-a.e. $x$.
• The construction explicitly realizes all possible values of $\beta \in [0,1]$ for the lower pointwise and lower box dimensions, demonstrating a full range of non-uniform dimensional behavior.
• The invariant measure is singular with respect to Lebesgue measure, as its dimension is strictly less than 1, and the measure is not equivalent to Lebesgue.