Renormalization for a Class of Dynamical Systems: some Local and Global Properties

TL;DR

This paper investigates the period doubling renormalization operator for certain dynamical systems with laminar regimes, establishing the existence of explicit fixed points and exploring global properties and rigidity results.

Contribution

It introduces explicit fixed points for the renormalization operator in dynamical systems with laminar regimes and proves global uniqueness and rigidity results.

Findings

Existence of explicit attracting fixed points for the renormalization operator.

Identification of the log derivative of the Manneville-Pomeau map as a fixed point.

Proved global rigidity and uniqueness of the renormalization operator in shift spaces.

Abstract

We study the period doubling renormalization operator for dynamics which present two coupled laminar regimes with two weakly expanding fixed points. We focus our analysis on the potential point of view, meaning we want to solve $$V=\mathcal{R} (V):=V\circ f\circ h+V \circ h,$$ where $f$ and $h$ are naturally defined. Under certain hypothesis we show the existence of a explicit ``attracting'' fixed point $V^*$ for $\mathcal{R} $. We call $\mathcal{R}$ the renormalization operator which acts on potentials $V$. The log of the derivative of the main branch of the Manneville-Pomeau map appears as a special ``attracting'' fixed point for the local doubling period renormalization operator. We also consider an analogous definition for the one-sided 2-full shift $\S$ (and also for the two-sided shift) and we obtain a similar result. Then, we consider global properties and we prove two rigidity results. Up to some weak assumptions, we get the uniqueness for the renormalization operator in the shift. In the last section we show (via a certain continuous fraction expansion) a natural relation of the two settings: shift acting on the Bernoulli space $\{0,1\}^\mathbb{N}$ and Manneville-Pomeau-like map acting on an interval.