In this talk, we discuss about the smooth random dynamical systems on surfaces. Based on the measure rigidity work by Aaron Brown and Federico Rodriguez Hertz, we know that a stationary measure has SRB property unless there is a certain obstruction. Here, SRB property implies that, morally, the measure is 'absolutely continuous' along 1 dimensional piece.
In this talk, we consider about a different mechanism to get ‘measure rigidity’ which promote SRB to absolute continuity using ‘transversality' motivated by Tsujii’s works on partially hyperbolic endomorphisms and Bernoulli convolution.
The talk is based on a joint work in progress with Aaron Brown, Davi Obata, and Yuping Ruan.