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We investigate Takagi-type functions with roughness parameter $gamma$ that are Holder continuous with coefficient $H=frac{loggamma}{log eh}.$ Analytical access is provided by an embedding into a dynamical system related to the baker transform where the graphs of the functions are identified as their global attractors. They possess stable manifolds hosting Sinai-Bowen-Ruelle (SBR) measures. We show that the SBR measure is absolutely continuous for large enough $gamma$. Dually, where duality is related to time reversal, we prove that for large enough $gamma$ a version of the Takagi-type curve centered around fibers of the associated stable manifold possesses a square integrable local time.
We investigate Weierstrass functions with roughness parameter $gamma$ that are Holder continuous with coefficient $H={loggamma}/{log frac12}.$ Analytical access is provided by an embedding into a dynamical system related to the baker transform where
Let $(X,mathcal{B},mu)$ be a standard probability space. We give new fundamental results determining solutions to the coboundary equation: begin{eqnarray*} f = g - g circ T end{eqnarray*} where $f in L^p$ and $T$ is ergodic invertible measure preserv
Certain dynamical models of competition have a unique invariant hypersurface to whichevery nonzero tractory is asymptotic, having simple geometry and topology.
Let $DeltasubsetneqV$ be a proper subset of the vertices $V$ of the defining graph of an aperiodic shift of finite type $(Sigma_{A}^{+},S)$. Let $Delta_{n}$ be the union of cylinders in $Sigma_{A}^{+}$ corresponding to the points $x$ for which the fi
We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the arithmetic sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, ErdH