Annealed and quenched representations of the Gauss-Rényi measure by "periodic points"

TL;DR

The paper studies random dynamical systems generated by i.i.d. compositions of the Gauss map $T_0$ and the Rényi map $T_1$, which yield random continued fractions. It develops annealed and quenched representations of the Gauss-Rényi measure $oldsymbol{ u}_p$ in terms of random cycles by embedding the system into a skew-product and proving a level-2 large deviation principle for periodic points, then transferring this LDP to the original Gauss-Rényi map via inducing and symbolic coding to a countable full shift. The main contributions include a complete annealed and quenched description of the Gauss-Rényi measure through random cycles, a detailed analysis of the induced system and distortion properties, and level-2 as well as level-1 LDPs with information about digit frequencies in the random continued fraction. The results illuminate how periodic-like structures govern the statistics of random compositions with infinitely-many branches, providing a rigorous framework for understanding random continued fractions and their invariant measures with implications for ergodic properties and digit frequencies.

Abstract

We consider independently identically distributed random compositions of the Gauss and Rényi maps that generate random continued fractions. Using methods of ergodic theory, thermodynamic formalism and large deviations, we show that weighted cycles of this random dynamical system equidistribute with respect to the Gauss-Rényi measure. We present both annealed (sample-averaged) and quenched (samplewise) results.

Figures (2)

• Figure 1: The graph of the Gauss map $T_0$ (left) and that of the Rényi map $T_1$ (right): $T_0^{-1}(0)=\{1/k\colon k\in\mathbb N\}$, $T_1^{-1}(0)=\{(k-1)/k\colon k\in\mathbb N\}$; $T_0^{-1}(1)=T_1^{-1}(1)=\emptyset$; $T_10=0$, $T_1'0=1$.
• Figure 2: The inducing domain $\widehat{\Lambda}$ associated with the inducing scheme $(\Lambda\setminus\varDelta(1),t_{\Lambda\setminus\varDelta(1)})$ is contained in $\bigcup_{k=2}^\infty\varDelta(k)$, the shaded area.

Theorems & Definitions (65)

• Theorem 1.1: quenched representation of the Gauss-Rényi measure
• Theorem 1.2: annealed representation of the Gauss-Rényi measure
• Corollary 1.3: quenched and annealed representations of the Radon-Nikodým derivative
• Theorem 1.4: annealed level-2 Large Deviation Principle
• Theorem 1.5: quenched level-2 large deviations
• Proposition 2.1
• Lemma 2.2: NT24
• proof : Proof of Proposition \ref{['random-lem']}
• Proposition 2.3
• proof : Proof of the second assertion of Proposition \ref{['CP']}