arXiv subjects

Atsuya Otani

Publications and source records attributed to Atsuya Otani.

4 recordsLinked to original sources

arXiv2017

Fractal dimensions of graph of Weierstrass-type function and local Hölder exponent spectra

We study several fractal properties of the Weierstrass-type function \[ W(x)=\sum_{n=0} ^\infty λ(x) λ(τx) \cdots λ(τ^{n-1}x)\, g(τ^n x), \] where $τ:[0,1)\to[0,1)$ is a cookie cutter map with possibly fractal repeller, and $λ$ and $g$ are functions with proper regularity. In the first part, we determine the box dimension of the graph of $W$ and Hausdorff dimension of its randomised version. In the second part, the Housdorff spectrum of the local Hölder exponent is characterised in terms of thermodynamic formalisms. Furthermore, in the randomised case, a novel formula for the lifted Hausdorff spectrum on the graph is provided.

math.DS↗

arXiv2016

Chaotically driven sigmoidal maps

We consider skew product dynamical systems $f:Θ\times\mathbb{R}\toΘ\times\mathbb{R}, f(θ,y)=(Tθ,f_θ(y))$ with a (generalized) baker transformation $T$ at the base and uniformly bounded increasing $C^3$ fibre maps $f_θ$ with negative Schwarzian derivative. Under a partial hyperbolicity assumption that ensures the existence of strong stable fibres for $f$ we prove that the presence of these fibres restricts considerably the possible structures of invariant measures - both topologically and measure theoretically, and that this finally allows to provide a "thermodynamic formula" for the Hausdorff dimension of set of those base points over which the dynamics are synchronized, i.e. over which the global attractor consists of just one point.

math.DS↗

arXiv2015

An entropy formula for a non-self-affine measure with application to Weierstrass-type functions

Let $ τ: [0,1] \rightarrow [0,1] $ be a piecewise expanding map with full branches. Given $ λ: [0,1] \rightarrow (0,1) $ and $ g : [0,1] \rightarrow \mathbb{R} $ satisfying $ τ' λ> 1 $, we study the Weierstrass-type function \[ \sum _{n=0} ^\infty λ^n (x) \, g (τ^n (x)), \] where $ λ^n (x) := λ(x) λ(τ(x)) \cdots λ(τ^{n-1} (x)) $. Under certain conditions, Bedford proved that the box counting dimension of its graph is given as the unique zero of the topological pressure function \[ s \mapsto P ((1-s) \log τ' + \log λ) . \] We give a sufficient condition under which the Hausdorff dimension also coincides with this value. We adopt a dynamical system theoretic approach which was originally used to investigate special cases including the classical Weierstrass functions. For this purpose we prove a new Ledrappier-Young entropy formula, which is a conditional version of Pesin's formula, for non-invertible dynamical systems. Our formula holds for all lifted Gibbs measures on the graph of the above function, which are generally not self-affine.

math.DS↗

arXiv2012

Bifurcation and Hausdorff dimension in families of chaotically driven maps with multiplicative forcing

We study bifurcations of invariant graphs in skew product dynamical systems driven by hyperbolic surface maps T like Anosov surface diffeomorphisms or baker maps and with one-dimensional concave fibre maps under multiplicative forcing when the forcing is scaled by a parameter r=e^{-t}. For a range of parameters two invariant graphs (a trivial and a non-trivial one) coexist, and we use thermodynamic formalism to characterize the parameter dependence of the Hausdorff and packing dimension of the set of points where both graphs coincide. As a corollary we characterize the parameter dependence of the dimension of the global attractor A_t: Hausdorff and packing dimension have a common value dim(A_t), and there is a critical parameter t_c determined by the SRB measure of T^{-1} such that dim(A_t)=3 for t < t_c and t --> dim(A_t) is strictly decreasing for t_c < t < t_{max}.

math.DS↗