# On the invariant Cantor sets of period doubling type of infinitely renormalizable area-preserving maps

## TL;DR

This paper proves that invariant Cantor sets of period doubling type in infinitely renormalizable area-preserving maps are contained in Lipschitz curves but cannot be smooth, extending prior dissipative map results to conservative systems.

## Contribution

It establishes the Lipschitz regularity of invariant Cantor sets in area-preserving maps and proves they are not contained in smooth curves, extending previous dissipative map results.

## Key findings

- Invariant Cantor sets are contained in Lipschitz curves.
- Such sets are never contained in smooth curves.
- The methods adapt dissipative techniques to conservative maps.

## Abstract

In this paper we show that the invariant Cantor set of period doubling type of any infinitely renormalizable area-preserving map in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point is always contained in a Lipschitz curve but never contained in a smooth curve. This extends previous results by de Carvalho, Lyubich and Martens about strongly dissipative maps of the plane close to unimodal maps to the area-preserving setting. The method used for constructing the Lipschitz curve is very similar to the method used in the dissipative case but proving the nonexistence of smooth curves requires new techniques.