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.
We introduce two parametrized families of piecewise affine maps on $[0,1]^2$ and $[0,1]^3$, as generalizations of the heterochaos baker maps which were introduced and investigated in [Y. Saiki, H. Takahasi, J. A. Yorke, Nonlinearity, 34 (2021), 5744--5761] as minimal models of the unstable dimension variability in multidimensional dynamical systems. We show that natural coding spaces of these maps coincide with the Dyck system that has come from the theory of languages. Based on this coincidence, we start to develop a complementary analysis on their invariant measures. As a first attempt, we s…
Let $μ$ be a self-affine measure on $\mathbb{R}^{d}$ associated to an affine IFS $Φ$ and a positive probability vector $p$. Suppose that the maps in $Φ$ do not have a common fixed point, and that standard irreducibility and proximality assumptions are satisfied by their linear parts. We show that $\dimμ$ is equal to the Lyapunov dimension $\dim_{L}(Φ,p)$ whenever $d=3$ and $Φ$ satisfies the strong separation condition (or the milder strong open set condition). This follows from a general criteria ensuring $\dimμ=\min\{d,\dim_{L}(Φ,p)\}$, from which earlier results in the planar case also follo…
We study continuous countably (strictly) monotone maps defined on a tame graph, i.e., a special Peano continuum for which the set containing branchpoints and endpoints has a countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a constant slope map $g$ of a countably affine tame graph. In particular, we show that in the case of a Markov map $f$ that corresponds to recurrent transition matrix, the condition is satis…
We study equilibrium measures (Käenmäki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier-Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of the important coordinate projection of the measure. In particular, we do this by showing that the Käenmäki measure is equal to the sum …
