[Paper Review] Big Birkhoff sums in $d$-decaying Gauss like iterated function systems

Rams, Michal, Lingmin Liao+1 more|arXiv (Cornell University)|May 7, 2019

Mathematical Dynamics and FractalsMathematics1 references20 citations

TL;DR

This paper investigates the Hausdorff dimension of level sets for Birkhoff sums in $d$-decaying Gauss-like infinite iterated function systems, generalizing results from continued fraction expansions. It establishes that for potential functions growing faster than polynomially, the dimension of the set of points with a given increasing rate of Birkhoff sums is $1/d$, extending prior results on the Gauss map and confirming a phase transition at $\alpha = 1/2$ for exponential-type growth rates.

ABSTRACT

The increasing rate of the Birkhoff sums in the infinite iterated function systems with polynomial decay of the derivative (for example the Gauss map) is studied. For different unbounded potential functions, the Hausdorff dimensions of the sets of points whose Birkhoff sums share the same increasing rate are obtained.

Motivation & Objective

• To generalize multifractal analysis of Birkhoff sums from the classical Gauss map to a broad class of $d$-decaying Gauss-like iterated function systems.
• To determine the Hausdorff dimension of level sets where Birkhoff sums grow at a prescribed rate, especially for unbounded potential functions.
• To resolve the gap in dimension estimates for exponential-type growth rates $\Phi(n) = \exp(n^\alpha)$ with $\alpha \in [1/2, 1)$, confirming $\dim_H E(\Phi) = 1/2$.

Proposed method

• The authors define a $d$-decaying Gauss-like iterated function system satisfying specific geometric and derivative decay conditions, generalizing the Gauss map.
• They use symbolic dynamics and symbolic coding to represent points in the limit set via infinite sequences in $\mathbb{N}^\mathbb{N}$, with basic intervals corresponding to cylinder sets.
• A key technique involves constructing nested sets of cylinder sets based on block frequencies of symbols, using estimates on the number of admissible blocks with given sum constraints.
• The proof employs a mass distribution argument and estimates on the $s$-dimensional Hausdorff measure using a modified generating function $G(m,n,a,\varepsilon,s)$, adapted to the decay rate $d$.
• For upper bounds, they apply Lemma 2.4 and Lemma 2.5 to show that the $s$-measure vanishes for $s > 1/d$, implying $\dim_H \leq 1/d$. For lower bounds, they construct a Cantor-type subset with positive $s$-measure for $s < 1/d$, using inverse image constructions.

Experimental results

Research questions

• RQ1 What is the Hausdorff dimension of the set of points for which the Birkhoff sum of a general potential function grows at a prescribed rate in a $d$-decaying Gauss-like IFS?
• RQ2 How does the dimension depend on the growth rate of the normalizing function $\Phi(n)$, especially for $\Phi(n) = \exp(n^\alpha)$ with $\alpha \in [1/2, 1)$?
• RQ3 Does the dimension jump from 1 to $1/2$ at $\alpha = 1/2$, and what happens for $\alpha > 1/2$?
• RQ4 Can the results for the Gauss map be generalized to a broader class of infinite iterated function systems with polynomial decay of derivatives?

Key findings

• For $d$-decaying Gauss-like IFS, the Hausdorff dimension of the set $E_\varphi(\Phi)$ where Birkhoff sums grow like $\Phi(n)$ is $1/d$ when $\Phi(n)$ grows faster than polynomially.
• For $\Phi(n) = \exp(n^\alpha)$ with $\alpha \in [1/2, 1)$, the dimension is $1/2$, confirming a phase transition at $\alpha = 1/2$.
• When $\Phi(n) = \exp(\beta^n)$ with $\beta > 1$, the dimension is $1/(\beta + 1)$, extending previous results.
• The dimension is $1/d$ for general potential functions $\varphi$ satisfying certain growth and regularity conditions, under the $d$-decay assumption.
• The upper bound $\dim_H \leq 1/d$ is established by showing the $s$-dimensional Hausdorff measure vanishes for all $s > 1/d$, using estimates on cylinder set measures.
• The lower bound $\dim_H \geq 1/d$ is obtained by constructing a subset with positive $s$-measure for $s < 1/d$, using inverse image constructions and block frequency control.