On stochastic stability of expanding circle maps with neutral fixed points

Abstract

It is well known that the Manneville–Pomeau map with a parabolic fixed point of the form is stochastically stable for α ≥ 1 and the limiting measure is the Dirac measure at the fixed point. In this paper, we show that if α ∈ (0, 1), then it is also stochastically stable. Indeed, the stationary measure of the random map converges strongly to the absolutely continuous invariant measure for the deterministic system as the noise tends to zero.

Keywords:

• Manneville--Pomeau map
• stochastic stability
• intermittency

2000 Mathematics Subject Classification:

• Primary: 37E05
• Secondary: 37D25
• 37C40
• 37C75
• 37H99