Dimension of ergodic measures projected onto self-similar sets with overlaps

Ergodic lifts and overlap numbers

We study skew product lifts and overlap numbers for equilibrium measures μ_ψof Hölder continuous potentials ψon such lifts. We find computable formulas and estimates for the overlap numbers in several concrete significant cases of systems with overlaps. In particular we obtain iterated systems which are asymptotically irrational-to-1 and absolutely continuous on their limit sets. Then we look into the general structure of the Rokhlin conditional measures of μ_ψwith respect to different fiber partitions associated to the lift Φ, and find relations between them. Moreover we prove an estimate on …

Families of infinite parabolic IFS with overlaps: the approximating method

This work is devoted to the study of families of infinite parabolic iterated function systems (PIFS) on a closed interval parametrized by vectors in $\mathbb{R}^d$ with overlaps. We show that the Hausdorff dimension and absolute continuity of ergodic projections through a family of infinite PIFS are decided \emph{a.e.} by the growth rate of the entropy and Lyapunov exponents of the families of truncated PIFS with respect to the concentrating measures, under transversality of the family essentially. We also give an estimation on the upper bound of the Hausdorff dimension of parameters where the…

The Ergodic Theorem for a new kind of attractor of a GIFS

In 1987, J. H. Elton, has proved the first fundamental result in convergence of IFS, the Elton's Ergodic Theorem. In this work we prove the natural extension of this theorem to the projected Hutchinson measure $μ_α$ associated to a GIFSpdp $\mathcal{S}=\left(X, (ϕ_j:X^{m} \to X)_{j=0,1, ..., n-1}, (p_j)_{j=0,1, ..., n-1}\right),$ in a compact metric space $(X,d)$. More precisely, the average along of the trajectories $x_{n}(a)$ of the GIFS, starting in any initial points $x_0, ..., x_{m-1} \in X$ satisfies, for any $f \in C(X , \mathbb{R})$, $$\lim_{N\to +\infty} \frac{1}{N}\sum_{n=0 }^{N-1} f…

扩张有理半群投影测度的精确维数

Exact dimensionality of projected measures for expanding rational semigroups

研究扩张有理半群的维数理论，通过符号空间与黎曼球面的斜积，证明了投影条件测度的Ledrappier-Young型公式及其精确维数性质。

Physical Measures for Certain Partially Hyperbolic Attractors on 3-Manifolds

In this work, we study ergodic properties of certain partially hyperbolic attractors whose central direction has a neutral behavior, the main feature is a condition of transversality between unstable leaves when projected by the stable holonomy. We prove that partial hyperbolic attractors satisfying conditions of transversality between unstable leaves via the stable holonomy, neutrality in the central direction and regularity of the stable foliation admits a finite number of physical measures, coinciding with the ergodic u-Gibbs States, whose union of the basins has full Lebesgue measure. More…

二维环面$\n\mathbb{T}^2$上投影无维数下降的遍历$\times p$-不变测度

Ergodic $\times p$-invariant measures on $\mathbb{T}^2$ with no dimension dropping projections

本研究构造了二维环面上维数为$s$的遍历$\times p$-不变测度，其所有直线投影均保持维数且平面管质量满足$O(w^s)$界，同时证明正维数拟Bernoulli $T_p$-不变测度必存在维数下降投影。