Ledrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. In this paper, we show that a self-affine measure on the plane satisfies the Ledrappier-Young formula if the corresponding iterated function system (IFS) satisfies the strong separation condition and the linear parts satisfy the dominated splitting condition. We give sufficient conditions, inspired by Ledrappier and by Falconer and Kempton, that the dimensions of such a self-affine measure is equal to the Lyapunov dimension. We show some applications, namely, we give another proof for Hueter-Lalley's theorem and we consider self-affine measures and sets generated by lower triangular matrices.
In this paper, we solve the long standing open problem on exact dimensionality of self-affine measures on the plane. We show that every self-affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system. In higher dimensions, under certain assumptions, we prove that self-affine and quasi self-affine measures are exact dimensional. In both cases, the measures satisfy the Ledrappier-Young formula.
Assuming strong irreducibility and proximality, we prove that the Furstenberg measure, corresponding to a finitely supported measure on the general linear group of a finite dimensional real vector space, is exact dimensional. We also establish a Ledrappier-Young type formula for its dimension. The general strategy of the proof is based on the argument given by Feng for the exact dimensionality of self-affine measures.
Under a suitable bunching condition, we establish that stable holonomies inside center-stable manifolds for $C^{1+β}$ diffeomorphisms are uniformly bi-Lipschitz and in fact $C^{1+\text{Hölder}}$. This verifies that the Pugh-Shub theory for ergodicity holds for suitably center-bunched, $C^{1+β}$, essentially accessible, partially hyperbolic diffeomorphism and verifies that the Ledrappier-Young entropy formulas hold for $C^{1+β}$ diffeomorphisms of compact manifolds.
$M$ is a Riemannian, boundaryless, and compact manifold with $\dim M\geq 2$, and $f$ is a $C^{1+β}$ ($β>0$) diffeomorphism of $M$. $φ$ is a Hölder continuous potential on $M$. We construct an invariant and absolutely continuous family of measures (with transformation relations defined by $φ$), which sit on local unstable leaves. We present two main applications. First, given an ergodic homoclinic class $H_χ(p)$, we prove that $φ$ admits a local equilibrium state on $H_χ(p)$ if and only if $φ$ is "recurrent on $H_χ(p)$" (a condition tested by counting periodic points), and one of the leaf measu…
We study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show that these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier-Young formula.
In this paper we study the dimension theory of planar self-affine sets satisfying dominated splitting in the linear parts and strong separation condition. The main results of this paper is the existence of dimension maximizing Gibbs measures (Käenmäki measures). To prove this phenomena, we show that the Ledrappier-Young formula holds for Gibbs measures and we introduce a transversality type condition for the strong-stable directions on the projective space.
Let $W_{λ,b}(x)=\sum_{n=0}^\inftyλ^n g(b^n x)$ where $b\geqslant2$ is an integer and $g(u)=\cos(2πu)$ (classical Weierstrass function). Building on work by Ledrappier (1992), Baránsky, Bárány and Romanowska (2013) and Tsujii (2001), we provide an elementary proof that the Hausdorff dimension of $W_{λ,b}$ equals $2+\frac{\logλ}{\log b}$ for all $λ\in(λ_b,1)$ with a suitable $λ_b<1$. This reproduces results by Baránsky, Bárány and Romanowska without using the dimension theory for hyperbolic measures of Ledrappier and Young (1985,1988), which is replaced by a simple telescoping argument together …
An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension of the self-affine set is the affinity dimension for Lebesgue almost every translation vectors. Similar statement was proven by Jordan, Pollicott, and Simon in 2007 for the dimension of self-affine measures. In this article, we have an orthogonal approach. We introduce a class of self-affine systems in which, given translation vectors, we get the same results…
Let $\{S_i\}_{i\in Λ}$ be a finite contracting affine iterated function system (IFS) on ${\Bbb R}^d$. Let $(Σ,σ)$ denote the two-sided full shift over the alphabet $Λ$, and $π:Σ\to {\Bbb R}^d$ be the coding map associated with the IFS. We prove that the projection of an ergodic $σ$-invariant measure on $Σ$ under $π$ is always exact dimensional, and its Hausdorff dimension satisfies a Ledrappier-Young type formula. Furthermore, the result extends to average contracting affine IFSs. This completes several previous results and answers a folklore open question in the community of fractals. Some ap…
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 …
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 …
This paper presents a general procedure based on using the method of types to calculate the box dimension of sets. The approach unifies and simplifies multiple box counting arguments. In particular, we use it to generalize the formula for the box dimension of self-affine carpets of Gatzouras-Lalley and of Barański type to their higher dimensional sponge analogues. In addition to a closed form, we also obtain a variational formula which resembles the Ledrappier-Young formula for Hausdorff dimension.
We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge $K\subset \mathbb{R}^k$ ($k\ge 2$) obtained by using some percolation process in $[0,1]^k$. To do so, we first exhibit a Ledrappier-Young type formula for the Hausdorff dimensions of statistically self-affine measures supported on $K$. This formula presents a new feature compared to its deterministic or random dynamical version. Then, we establish a variational principle expressing $\dim_H K$ as the supremum of the Hausdorff dimensions of statistically self-affine measures supported on $K$, and show th…
This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teresópolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov exponents and metric entropy -- as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for aff…
