Let $F$ be a self similar set in $R^2$. Assuming $F$ is generated by an IFS $\Phi$ with strong separation, we characterize the affine maps $g:R^2 \rightarrow R^2$ such that $g(F) \subseteq F$ (these maps are called self embeddings of $F$). If the group generated by the orthogonal parts of the similarities in $\Phi$ is infinite, we show that any such self embedding must be a similarity. If the group generated by the orthogonal parts is finite and $\Phi$ has a uniform contraction ratio r, we show that the linear part of any such self embedding is diagonalizable, and the norm of each of its eigenvalues is a rational power of $r$.

We also prove (in a joint work with Mike Hochman) a related result for strictly self affine Bedford-McMullen carpets. Namely, let $F\subseteq R^2$ be a Bedford-McMullen carpet defined by multiplicatively independent exponents. Assuming some mild conditions on $F$, we prove that any similarity self embedding of $F$ is an isometry, composed of reflections about lines parallel to the axes.

Let $A=\{A_1,\ldots,A_N\}$ be a finite set of contracting non-singular matrices and let $\Psi=\{x\mapsto A_ix+t_i\}_{i=1}^N$ be a finite iterated function system of affine transformations on the plane. In this talk, we present that if $\Psi$ satisfies the strong separation condition and the matrices $A_i$ act totally irreducibly on $\mathbb{R}^2$, then the Hausdorff dimension of the attractor of $\Psi$ is equal to the affinity dimension. Moreover, the Hausdorff dimension of every fully supported self-affine measure is equal to the Lyapunov dimension. This is a joint work with Mike Hochman and Ariel Rapaport.

We show that fractal percolation sets (AKA Mandelbrot percolation) in $\mathbb{R}^{d}$ almost surely (conditioned on being non-empty) intersect the set of badly approximable numbers (and every other hyperplane absolute winning set) with maximal Hausdorff dimension, i.e. the intersection has the same dimension as the fractal itself. We rely on properties of Galton - Watson processes to show the existence of hyperplane diffuse subsets with Hausdorff dimensions approaching the dimension of the fractal.

For a given deterministic measure we construct a random measure on the Brownian path that has expectation the given measure. For the construction we introduce the concept of weak convergence of random measures in probability. The machinery can be extended to more general sets than Brownian path.

In this talk, we investigate the local dimensions of certain fractal measures. We prove the exact dimensionality of  ergodic invariant measures for every contractive affine iterated function systems.  These measures are the push-forwards of the ergodic measures in the symbolic space under the coding map, and include all the self-affine measures.  We also establish the Ledrappier-Young type dimension formula. This completes  several  previous results.  Applications are given to the dimension of self-affine sets. Part of the results extends to average-contractive affine iterated function systems.

It was proved by Oberlin and by Falconer and Mattila  that the union of any $s$-Hausdorff-dimensional family of affine hyperplanes in $\mathbb R^n$ has Hausdorff dimension $s+n-1$ if $s\in [0,1]$, and positive Lebesgue-measure if $s > 1$. We generalize the result for the range $s \in [0,1]$, for $k$-dimensional affine subspaces ($1 \leq k \leq n-1$) in place of hyperplanes. We also prove a lower estimate for the Hausdorff dimension of generalized Furstenberg-type sets: sets intersecting every element of a given family of $k$-dimensional affine subspaces in a set of Hausdorff dimension at least $\alpha$, where $0 < \alpha \leq k$.

We also study the following closely related questions: What is the smallest possible Hausdorff dimension of a set which contains the $k$-skeleton of an $n$-dimensional axis-parallel scaled cube / rotated and scaled cube / rotated unit cube around every point of $\mathbb{R}^n$?

Joint work with Alan Chang, Marianna Csörnyei, Tamás Keleti and András Máthé.

Joint work with Jonathan Fraser. The Assouad dimension is a notion of dimension considers local coverings of sets and can be strictly larger than Hausdorff and box-dimenision. This has been shown to be the typical case for several families of self-affine sets with grid structure (such as Bedford-McMullen carpets). We consider simple families of self-affine sets based on diagonal matrices but without grid structure and show how computing the Assouad dimension relates to the local dimension and L^q spectra of certain self-similar measures with overlap. This allows us to show that for 'most' sets in our class the Assouad dimension is the same as the Hausdorff dimension but allows us to also give new examples where the Assouad dimension is strictly larger than the Hausdorff and box counting dimension.

We show that the class of quasisymmetric maps between horizontal self-affine carpets is rigid. Such maps can only exist when the dimensions of the carpets coincide, and in this case, the quasisymmetric maps are quasi-Lipschitz. We also show that horizontal self-affine carpets are minimal for the conformal Assouad dimension.

In 1988, Falconer proved that, for given matrices, the Hausdorff dimension of the self-affine set is the affinity dimension for Lebesgue almost every choice of translation vectors. Similar statement was proven by Jordan, Pollicott, and Simon in 2007 for the dimension of self-affine measures. I discuss an orthogonal approach, introducing a class of self-affine systems in which, given translation vectors, the same dimension results hold for Lebesgue almost all matrices. Our proofs rely on Ledrappier-Young theory that was recently verified for affine iterated function systems by Barany and Kaenmaki, and a new transversality condition, and in particular they do not depend on properties of the Furstenberg measure. This allows our results to hold for self-affine sets and measures in any Euclidean space.

The work is joint with Balazs Barany and Antti Kaenmaki.

In my talk I will present joint work with M. Einsiedler regarding the following natural question posed by Bourgain:

Given a number $\alpha$ in $\mathbb{R}$, how well can alpha be approximated by a rational $p/(q_1q_2)$ with $q_1,q_2 < Q$? Or more generally $q_1 < Q, q_2 < \phi(Q)$? We show that outside a set of $Q$’s of logarithmic density $0$, the trivial bound on how well one can approximate can be improved.

The proof uses a new measure classification theorem.

The affinity dimension, first defined by Falconer in 1988, is the analogue of the similarity dimension for affine iterated function systems. I will discuss some simple properties of the similarity dimension - namely, continuity, computability and monotonicity - and discuss the nontrivial matter of extending these properties to the affinity dimension. I will also describe some of my recent work on the equilibrium states of the singular value potential with Antti Käenmäki and Jairo Bochi.

ABSTRACT

Let $A_1,\ldots, A_k$ be $SL(2,\mathbb{R})$ matrices. Given a sequence $\omega \in \{1,\ldots, k\}^\mathbb{N}$, define the Lyapunov exponent \[ \chi(\omega) = \lim_{n\to\infty} \frac 1n \log ||A_{\omega_n}\cdot\ldots\cdot A_{\omega_1}|| \] (if such a limit exists). Let \[ L(\alpha) = \{\omega; \chi(\omega)=\alpha\}. \] The function $\alpha \to f(\alpha) = h_{\rm top}(L(\alpha))$ is called Lyapunov spectrum of the matrix cocycle $(A_1,\ldots, A_k)$.

We give a description of the Lyapunov spectrum for generic $SL(2,\mathbb{R})$ matrix cocycles. Of more interesting properties: it is a concave function, a Legendre transform of some properly defined pressure function, it has a maximum $\log k$ which is achieved only at one point, for an elliptic cocycle generically we have $f(0)\in (0,\log k)$. It is a joint work with Lorenzo Diaz and Katrin Gelfert.

I will present a construction of a planar homogeneous self-similar measure, with strong separation, dense rotations and dimension greater than 1, such that there exist lines for which dimension conservation does not hold and the projection of the measure is singular.

The distance set conjecture of Falconer asserts that the set of distances of a planar set of Hausdorff dimension $\ge 1$ has Hausdorff dimension $1$. I will present a strong verification of this conjecture for sets of equal Hausdorff and packing dimension $>1$ (in particular, for Ahlfors-regular sets of dimension $>1$). The proof is based on multiscale decompositions for energies and entropy, the behavior of energies under projection, and a quantitative version of a circular projection theorem due to Mattila and Orponen.

Recently considerable attention has been paid to the study of the arithmetic sums of two planar sets $A+\Gamma:=\left\{a+g:a\in A, g\in \Gamma\right\}$. We focus on the case when $\Gamma$ is a piecewise $\mathcal{C}^2$ curve, in particular when $\Gamma$ is the unit circle. In this case there is a natural guess what the size (Hausdorff dimension, Lebesgue measure) of $A+\Gamma$ should be. We verify this under some simple natural assumptions. We also address the more difficult question: under which condition does the set $A+\Gamma$ has non-empty interior?

Joint work with Krystal Taylor (OSU).

In this talk, I will discuss a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension, as well as my recent result showing that the answer is negative. The counterexample is a self-affine sponge in $\mathbb R^3$ coming from an affine iterated function system whose coordinate subspace projections satisfy the strong separation condition. Its dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge, which implies that sponges with a dimension gap represent a nonempty open subset of the parameter space. This work is joint with Tushar Das (Wisconsin -- La Crosse).

We extend results of Y. Benoist and J.-F. Quint concerning random walks on homogeneous spaces of simple Lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily Zariski dense, but satisfies some expansion properties for the adjoint action. Using these dynamical results, we study Diophantine properties of typical points on some self-similar fractals. Joint work with David Simmons

We study the orthogonal projections of a large class of self-affine carpets, including the Feng-Wang carpets. We show that for a self-affine measure \mu on such carpet, under separation condition and certain natural irreducuble conditions, the projection of \mu in a non-principal direction has dimension  \min(1,dim(\mu)).  Joint work with Antti Kaenmaki