| Abstract: |
| We investigate geometric properties of graphs of Takagi and Weierstrass type functions, represented by series based on smooth functions. They are H\older continuous, and can be embedded into smooth dynamical systems, where their graphs emerge as pullback attractors. It turns out that occupation measures and Sinai-Bowen-Ruelle (SBR) measures on their stable manifolds are dual by ``time'' reversal.
Hence absolute continuity of the SBR measure is seen to be dual to the existence of local time.
The link between the rough curves considered and smooth dynamical systems can be generalized in various ways. For instance, Gaussian randomizations of Takagi curves just reproduce the trajectories of Brownian motion. Applications to regularization of singular ODE by rough signals are envisaged. This is joint work with O. Pamen (U Liverpool and AIMS Ghana) and F. Proske (U Oslo). |
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