Rigidity for infinitely renormalizable area-preserving maps

TL;DR

This paper investigates the rigidity properties of period doubling Cantor sets in area-preserving maps, confirming the Jacobian Rigidity Conjecture in the area-preserving case and extending understanding of smooth conjugacy in dynamical systems.

Contribution

It proves that for area-preserving maps, the period doubling Cantor sets are smoothly conjugated, supporting the Jacobian Rigidity Conjecture in this specific case.

Findings

Period doubling Cantor sets of area-preserving maps are smoothly conjugated.

Supports the Jacobian Rigidity Conjecture for the area-preserving case.

Extends rigidity results to the universality class of the Eckmann-Koch-Wittwer fixed point.

Abstract

The period doubling Cantor sets of strongly dissipative Henon-like maps with different average Jacobian are not smoothly conjugated. The Jacobian Rigidity Conjecture says that the period doubling Cantor sets of two-dimensional Henon-like maps with the same average Jacobian are smoothly conjugated. This conjecture is true for average Jacobian zero, e.g. the one-dimensional case. The other extreme case is when the maps preserve area, e.g. the average Jacobian is one. Indeed, the period doubling Cantor set of area-preserving maps in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point are smoothly conjugated.