Abstract: 

Because of their sensitivity to deterministic perturbations, chaotic systems are usually studied via their invariant measures. As the dynamics changes, one intuitively expects the physically meaningful invariant measures to change smoothly: this assumption has been used in the scientific literature for sixty years. However, for one-dimensional logistic maps this so-called linear response property fails, and a rigorous explanation for its success in more complex systems has been lacking. 

In this talk I will consider what happens when you add a dimension, and show that whether certain two-dimensional maps have a linear response reduces to a novel mixing property called “conditional mixing”. I will present some results suggesting that conditional mixing is likely to hold very generally.