Related papers: Schmidt's Game and Nonuniformly Expanding Interval…

We show that the sets of weighted badly approximable vectors in $\Bbb R^n$ are winning sets of certain games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in 1966. The latter winning property is stable with…

We consider expanding maps such that the unit interval can be represented as a full symbolic shift space with bounded distortion. There are already theorems about the Hausdorff dimension for sets defined by the set of accumulation points…

We obtain large deviation results for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space…

Given $b > 1$ and $y \in \mathbb{R}/\mathbb{Z}$, we consider the set of $x\in \mathbb{R}$ such that $y$ is not a limit point of the sequence $\{b^n x \bmod 1: n\in\mathbb{N}\}$. Such sets are known to have full Hausdorff dimension, and in…

We provide a criterion for determining the winner in two-player win-lose alternating-move games on trees, in terms of the Hausdorff dimension of the target set. We focus our study on special cases, including the Gale-Stewart game on the…

We prove that a large class of expanding maps of the unit interval with a $C^2$-regular indifferent point in 0 and full increasing branches are global-local mixing. This class includes the standard Pomeau-Manneville maps $T(x) = x +…

Schmidt's game is generally used to deduce qualitative information about the Hausdorff dimensions of fractal sets and their intersections. However, one can also ask about quantitative versions of the properties of winning sets. In this…

We prove that for every two natural numbers M and N, if Tau is a Borel, finite, absolutely friendly measure on a compact set K of R^MN, then the intersection of K and BA(M,N) is a winning set in Schmidt's game sense played on K, where…

Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of…

Let $G$ be a Lie group, $\Gamma\subset G$ a discrete subgroup, $X=G/\Gamma$, and $f$ an affine map from $X$ to itself. We give conditions on a submanifold $Z$ of $X$ guaranteeing that the set of points $x\in X$ with $f$-trajectories…

We give necessary and sufficient conditions for a nonexpansive map on a finite dimensional normed space to have a nonempty, bounded set of fixed points. Among other results we show that if $f : V \rightarrow V$ is a nonexpansive map on a…

We consider random perturbations of non-uniformly expanding maps, possibly having a non-degenerate critical set. We prove that, if the Lebesgue measure of the set of points failing the non-uniform expansion or the slow recurrence to the…

We present a new variant of the potential game and show that certain compact subsets of $\R^n$, including a large class of self-affine sets, are winning in our game. We prove that sets with sufficiently strong winning conditions are…

For any i,j>0 with i+j =1, let Bad(i,j) denote the set of points (x,y) \in R^2 such that max \{ ||qx||^{1/i}, \, ||qy||^{1/j} \} > c/q for some positive constant c = c(x,y) and all q in N. We show that \Bad(i,j) \cap C is winning in the…

We consider dynamical systems given by interval maps with a finite number of turning points (including critical points, discontinuities) possibly of different critical orders from two sides. If such a map $f$ is continuous and piecewise…

We explore and refine techniques for estimating the Hausdorff dimension of exceptional sets and their diffeomorphic images. Specifically, we use a variant of Schmidt's game to deduce the strong C^1 incompressibility of the set of badly…

We consider semidifferentiable (possibly nonsmooth) maps, acting on a subset of a Banach space, that are nonexpansive either in the norm of the space or in the Hilbert's or Thompson's metric inherited from a convex cone. We show that the…

We consider piecewise expanding maps of the interval with finitely many branches of monotonicity and show that they are generically combinatorially stable, i.e., the number of ergodic attractors and their corresponding mixing periods do not…

We study the expanding properties of random perturbations of regular interval maps satisfying the summability condition of exponent one. Under very general conditions on the interval maps and perturbation types, we prove strong stochastic…

We study shrinking target problems and the set $\mathcal{E}_{\text{ah}}$ of eventually always hitting points. These are the points whose first $n$ iterates will never have empty intersection with the $n$-th target for sufficiently large…