Related papers: Schmidt's Game on Certain Fractals

We prove that in any Hausdorff space, the Rothberger game is equivalent to the $k$-Rothberger game, i.e. the game in which player II chooses $k$ open sets in each move. This result follows from a more general theorem in which we show these…

We introduce and study Maker/Breaker-type positional games on random graphs. Our main concern is to determine the threshold probability $p_{F}$ for the existence of Maker's strategy to claim a member of $F$ in the unbiased game played on…

We study non-invertible piecewise hyperbolic maps in the plane. The Hausdorff dimension of the attractor is calculated in terms of the Lyapunov exponents, provided that the map satisfies a transversality condition. Explicit examples of maps…

Subtraction games are played with one or more heaps of tokens, with players taking turns removing from a single heap a number of tokens belonging to a specified subtraction set; the last player to move wins. We describe how to compute the…

The classical Besicovitch-Federer projection theorem implies that the d-dimensional Hausdorff measure of a set in Euclidean space with non-negligible d-unrectifiable part will strictly decrease under orthogonal projection onto almost every…

We prove that for any $s,t\ge0$ with $s+t=1$ and any $\theta\in\mathbb{R}$ with $\inf_{q\in\mathbb{N}}q^{\frac{1}{s}}\|q\theta\|>0$, the set of $y\in\mathbb{R}$ for which $(\theta,y)$ is $(s,t)$-badly approximable is 1/2-winning for…

We show that if K is a self-similar set in the plane with positive length, then the distance set of K has Hausdorff dimension one.

Generalising a construction of Falconer, we consider classes of $G_\delta$-subsets of $\mathbb{R}^d$ with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We…

Let $I=[0,1)$, $b\in \{2,3,\ldots\}$ and $f:I\to I$ be an injective piecewise $\frac{1}{b}$-affine map, that is, assume that there exists a partition of $I$ into intervals $I_1,\ldots,I_n$ such that $\vert f(x)-f(y)\vert\le\frac1b \vert…

In this paper we study the geometry of the attractors of holomorphic maps with an irrationally indifferent fixed point. We prove that for an open set of such holomorphic systems, the local attractor at the fixed point has Hausdorff…

We characterize when a compact, invariant, asymptotically stable attractor on a locally compact Hausdorff space is a strong deformation retract of its domain of attraction.

For any pair of real numbers $(i,j)$ with $0<i,j<1$ and $i+j=1$, we prove that the set of $p$-adic mixed $(i,j)$-badly approximable numbers $\bad_p(i, j)$ is 1/2-winning in the sense of Schmidt's game. This improves a recent result of…

We study the topological properties of attractors of Iterated Function Systems (I.F.S.) on the real line, consisting of affine maps of homogeneous contraction ratio. These maps define what we call a second generation I.F.S.: they are…

In order to compute the Schmidt decomposition of $A\in M_k\otimes M_m$, we must consider an associated self-adjoint map. Here, we show that if $A$ is positive under partial transposition (PPT) or symmetric with positive coefficients (SPC)…

Let $E \subseteq R^n$ be a closed set of Hausdorff dimension $\alpha$. For $m \geq n$, let $\{B_1,\ldots,B_k\}$ be $n \times (m-n)$ matrices. We prove that if the system of matrices $B_j$ is non-degenerate in a suitable sense, $\alpha$ is…

In this paper, we use algorithmic tools, effective dimension and Kolmogorov complexity, to study the fractal dimension of distance sets. We show that, for any analytic set $E\subseteq\R^2$ of Hausdorff dimension strictly greater than one,…

In this paper geometric properties of infinitely renormalizable real H\'enon-like maps $F$ in $\R^2$ are studied. It is shown that the appropriately defined renormalizations $R^n F$ converge exponentially to the one-dimensional…

Let $\Lambda$ be a countable index set and $S=\{\phi_i: i\in \Lambda\}$ be a conformal iterated function system on $[0,1]^d$ satisfying the open set condition. Denote by $J$ the attractor of $S$. With each sequence $(w_1,w_2,...)\in…

Let $(\Sigma, \sigma)$ be the one-sided shift space with $m$ symbols and $R_n(x)$ be the first return time of $x\in\Sigma$ to the $n$-th cylinder containing $x$. Denote $$E^\varphi_{\alpha,\beta}=\left\{x\in\Sigma:…

Bivariant (equivariant) K-theory is the standard setting for non-commutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from…