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Michał Rams

Publications and source records attributed to Michał Rams.

30 records · Page 2Linked to original sources
arXiv2017

Hausdorff dimension of the set approximated by irrational rotations

Let $θ$ be an irrational number and $φ: {\mathbb N} \to {\mathbb R}^{+}$ be a monotone decreasing function tending to zero. Let $$E_φ(θ) =\Big\{y \in \mathbb R: \|nθ- y\|<φ(n), \ {\text{for infinitely many}}\ n\in {\mathbb N} \Big\}, $$ i.e. the set of points which are approximated by the irrational rotation with respect to the error function $φ(n)$. In this article, we give a complete description of the Hausdorff dimension of $E_φ(θ)$ for any monotone function $φ$ and any irrational $θ$.

math.NT↗
arXiv2016

The entropy of Lyapunov-optimizing measures of some matrix cocycles

We consider one-step cocycles of $2 \times 2$ matrices, and we are interested in their Lyapunov-optimizing measures, i.e., invariant probability measures that maximize or minimize a Lyapunov exponent. If the cocycle is dominated, that is, the two Lyapunov exponents are uniformly separated along all orbits, then Lyapunov-optimizing measures always exist, and are characterized by their support. Under an additional hypothesis of nonoverlapping between the cones that characterize domination, we prove that the Lyapunov-optimizing measures have zero entropy. This conclusion certainly fails without the domination assumption, even for typical one-step $\mathrm{SL}(2,\mathbb{R})$-cocycles; indeed we show that in the latter case there are measures of positive entropy with zero Lyapunov exponent.

math.DS↗
arXiv2016

Dimension maximizing measures for self-affine systems

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.

math.DS↗
arXiv2015

On the dimension of self-affine sets and measures with overlaps

In this paper we consider diagonally affine, planar IFS $Φ=\left\{S_i(x,y)=(α_ix+t_{i,1},β_iy+t_{i,2})\right\}_{i=1}^m$. Combining the techniques of Hochman and Feng, Hu we compute the Hausdorff dimension of the self-affine attractor and measures and we give an upper bound for the dimension of the exceptional set of parameters.

math.DS↗
arXiv2015

On Shrinking Targets for Piecewise Expanding Interval Maps

For a map $T \colon [0,1] \to [0,1]$ with an invariant measure $μ$, we study, for a $μ$-typical $x$, the set of points $y$ such that the inequality $|T^n x - y| < r_n$ is satisfied for infinitely many $n$. We give a formula for the Hausdorff dimension of this set, under the assumption that $T$ is piecewise expanding and $μ_ϕ$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.

math.DS↗
arXiv2014

On entropy of dynamical systems with almost specification

We construct a family of shift spaces with almost specification and multiple measures of maximal entropy. This answers a question from Climenhaga and Thompson [Israel J. Math. 192 (2012), no. 2, 785--817]. Elaborating on our examples we also prove that some sufficient conditions for every subshift factor of a shift space to be intrinsically ergodic given by Climenhaga and Thompson are in some sense best possible, moreover, the weak specification property neither implies intrinsic ergodicity, nor follows from almost specification. We also construct a dynamical system with the weak specification property, which does not have the almost specification property. We prove that the minimal points are dense in the support of any invariant measure of a system with the almost specification property. Furthermore, if a system with almost specification has an invariant measure with non-trivial support, then it also has uniform positive entropy over the support of any invariant measure and can not be minimal.

math.DS↗
arXiv2014

Packing spectra for Bernoulli measures supported on Bedford-McMullen carpets

In this paper we consider the packing spectra for local dimension of Bernoulli measures supported on Bedford-McMullen carpets. We show that typically the packing dimension of the regular set is smaller than the packing dimension of the attractor. We also consider a specific class of measures for which we are able to calculate the packing spectrum exactly and we show that the packing spectrum is discontinuous as a function on the space of Bernoulli measures.

math.DS↗
arXiv2012

Dimension of self-affine sets with holes

In this paper we compute the dimension of a class of dynamically defined non-conformal sets. Let $X\subseteq\mathbb{T}^2$ denote a Bedford-McMullen set and $T:X\to X$ the natural expanding toral endomorphism which leaves $X$ invariant. For an open set $U\subset X$ we let X_U={x\in X : T^k(x)\not\in U \text{for all}k}. We investigate the box and Hausdorff dimensions of $X_U$ for both a fixed Markov hole and also when $U$ is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through $U$, while the Hausdorff dimension depends on the escape rate of the measure of maximal dimension.

math.DS↗
arXiv2011

Rich phase transitions in step skew-products

We present examples of partially hyperbolic and topologically transitive local diffeomorphisms defined as skew products over a horseshoe which exhibit rich phase transitions for the topological pressure. This phase transition follows from a gap in the spectrum of the central Lyapunov exponents. It is associated to the coexistence of two equilibrium states with positive entropy. The diffeomorphisms mix hyperbolic behavior of different types. However, in some sense the expanding behavior is not dominating which is indicated by the existence of a measure of maximal entropy with nonpositive central exponent.

math.DS↗