Linear response for Sinai billiards with small holes

TL;DR

This paper analyzes how the survival probability measure in Sinai billiards changes with small boundary holes, establishing differentiability at zero hole size and calculating the derivative.

Contribution

It provides the first rigorous proof of the differentiability of the survival measure with respect to small boundary holes in Sinai billiards and computes the derivative.

Findings

Survival probability measure is differentiable at zero hole size.

The derivative of the survival measure with respect to hole size is explicitly computed.

Abstract

We show that the conditional survival probability measure for a Sinai billiard with a small hole on the boundary of the table is differentiable with respect to the size t of the hole at t = 0 and we compute the derivative.