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Register a new userA Dichotomy for the Weierstrass-type functions
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For a real analytic periodic function $phi:mathbb{R}to mathbb{R}$, an integer $bge 2$ and $lambdain (1/b,1)$, we prove the following dichotomy for the Weierstrass-type function $W(x)=sumlimits_{nge 0}{{lambda}^nphi(b^nx)}$: Either $W(x)$ is real analytic, or the Hausdorff dimension of its graph is equal to $2+log_blambda$. Furthermore, given $b$ and $phi$, the former alternative only happens for finitely many $lambda$ unless $phi$ is constant.
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Let $f$ be a $C^{2+epsilon}$ expanding map of the circle and $v$ be a $C^{1+epsilon}$ real function of the circle. Consider the twisted cohomological equation $v(x) = alpha (f(x)) - Df(x) alpha (x)$ which has a unique bounded solution $alpha$. We prove that $alpha$ is either $C^{1+epsilon}$ or nowhere differentiable, and if $alpha$ is nowhere differentiable then the Newton quotients of $alpha$, after an appropriated normalization, converges in distribution to the normal distribution, with respect to the unique absolutely continuous invariant probability of $f$.
We show that the graph of the classical Weierstrass function $sum_{n=0}^infty lambda^n cos (2pi b^n x)$ has Hausdorff dimension $2+loglambda/log b$, for every integer $bge 2$ and every $lambdain (1/b,1)$. Replacing $cos(2pi x)$ by a general non-constant $C^2$ periodic function, we obtain the same result under a further assumption that $lambda b$ is close to $1$.
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 the graphs of the functions are identified as their global attractors. They possess stable manifolds hosting Sinai-Bowen-Ruelle (SBR) measures. We systematically exploit a telescoping property of associated measures to give an alternative proof of the absolute continuity of the SBR measure for large enough $gamma$ with square-integrable density. Telescoping allows a macroscopic argument using the transversality of the flow related to the mapping describing the stable manifold. The smoothness of the SBR measure can be used to compute the Hausdorff dimension of the graphs of the original Weierstrass functions and investigate their local times.
For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. Moreover, we introduce the notion of weak kinematical similarity and prove a reducibility result by the spectral theorem.
In this paper we study several stronger forms of sensitivity for continuous surjective selfmaps on compact metric spaces and relations between them. The main result of the paper states that a minimal system is either multi-sensitive or an almost one-to-one extension of its maximal equicontinuous factor, which is an analog of the Auslander-Yorke dichotomy theorem. For minimal dynamical systems, we also show that all notions of thick sensitivity, multi-sensitivity and thickly syndetical sensitivity are equivalent, and all of them are much stronger than sensitivity.
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