Fractal Dimension of nonattracting chaotic sets

The fractal dimension of a nonattracting chaotic set provides information about its geometric complexity and can often be of practical use. For example in the case of a chaotic saddle on a (fractal) basin boundary between two basins of attraction where the boundary is the stable set of the chaotic saddle. Then, the fractal dimension of the saddle and of the boundary provide information about the impact of small changes to the initial conditions on the future behaviour of the system, when the system is in a state close to the boundary.
This information is highly relevant in the context of climate tipping phenomena.
Building on Edward Ott’s and David Sweet’s work from 2000, we will discuss how to rigorously construct a measure on a chaotic repellor which leads to the estimation of its fractal dimension. Further, we discuss the fractal dimension of its stable and unstable set.