In [4], Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map T(x)=1/x mod 1 satisfy a ‘dimension gap’ meaning that for some c > 0, supp dim μp < 1– c, where μp denotes the (pushforward) Bernoulli measure for the countable probability vector p. In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.

Several results in the existing literature establish Euclidean density theorems of the following strong type. These results claim that every set of positive upper Banach density in the Euclidean space of an appropriate dimension contains isometric copies of all sufficiently large elements of a prescribed family of finite point configurations. So far, all results of this type discussed linear isotropic dilates of a fixed point configuration. In this paper, we initiate the study of analogous density theorems for families of point configurations generated by anisotropic dilations, i.e., families with power-type dependence on a single parameter interpreted as their size. More specifically, we prove nonisotropic power-type generalizations of a result by Bourgain on vertices of a simplex, a result by Lyall and Magyar on vertices of a rectangular box, and a result on distance trees, which is a particular case of the treatise of distance graphs by Lyall and Magyar. Another source of motivation for this paper is providing additional evidence for the versatility of the approach stemming from the work of Cook, Magyar, and Pramanik and its modification used recently by Durcik and the present author. Finally, yet another purpose of this paper is to single out anisotropic multilinear singular integral operators associated with the above combinatorial problems, as they are interesting on their own.

Let $S \subset \mathbb {R}^{n}$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\mathbb {R}^{n}$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^{q}(X)} \lesssim \| f \|_{L^{p}(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \lesssim R^{\alpha }$ for all balls $B_R$ in $\mathbb {R}^{n}$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^{q}$ against the measure $\chi _X \,{\textrm {d}}x$. Our approach consists of replacing the characteristic function $\chi _X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted Hölder-type inequality that holds for general non-negative Lebesgue measurable functions on $\mathbb {R}^{n}$ and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du–Zhang theorem in the range $0 < \alpha < n/2$.

Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.

Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let $\Lambda _0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of $g_0$ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let $L_{g,\Lambda ,f}$ (respectively $M_{g,\Lambda ,f}$ ) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively highest) values of f along the geodesics in the hyperbolic continuation $\Lambda $ of $\Lambda _0$ . We prove that for generic choices of g and f, the Hausdorff dimensions of the sets $L_{g,\Lambda , f}\cap (-\infty , t)$ vary continuously with $t\in \mathbb {R}$ and, moreover, $M_{g,\Lambda , f}\cap (-\infty , t)$ has the same Hausdorff dimension as $L_{g,\Lambda , f}\cap (-\infty , t)$ for all $t\in \mathbb {R}$ .

Given a closed, orientable, compact surface S of constant negative curvature and genus $g \geq 2$ , we study the measure-theoretic entropy of the Bowen–Series boundary map with respect to its smooth invariant measure. We obtain an explicit formula for the entropy that only depends on the perimeter of the $(8g-4)$ -sided fundamental polygon of the surface S and its genus. Using this, we analyze how the entropy changes in the Teichmüller space of S and prove the following flexibility result: the measure-theoretic entropy takes all values between 0 and a maximum that is achieved on the surface that admits a regular $(8g-4)$ -sided fundamental polygon. We also compare the measure-theoretic entropy to the topological entropy of these maps and show that the smooth invariant measure is not a measure of maximal entropy.

In this paper we study various aspects of porosities for conformal fractals. We first explore porosity in the general context of infinite graph directed Markov systems (GDMS), and we show that their limit sets are porous in large (in the sense of category and dimension) subsets. We also provide natural geometric and dynamic conditions under which the limit set of a GDMS is upper porous or mean porous. On the other hand, we prove that if the limit set of a GDMS is not porous, then it is not porous almost everywhere. We also revisit porosity for finite graph directed Markov systems, and we provide checkable criteria which guarantee that limit sets have holes of relative size at every scale in a prescribed direction.

We then narrow our focus to systems associated to complex continued fractions with arbitrary alphabet and we provide a novel characterisation of porosity for their limit sets. Moreover, we introduce the notions of upper density and upper box dimension for subsets of Gaussian integers and we explore their connections to porosity. As applications we show that limit sets of complex continued fractions system whose alphabet is co-finite, or even a co-finite subset of the Gaussian primes, are not porous almost everywhere, while they are uniformly upper porous and mean porous almost everywhere.

We finally turn our attention to complex dynamics and we delve into porosity for Julia sets of meromorphic functions. We show that if the Julia set of a tame meromorphic function is not the whole complex plane then it is porous at a dense set of its points and it is almost everywhere mean porous with respect to natural ergodic measures. On the other hand, if the Julia set is not porous then it is not porous almost everywhere. In particular, if the function is elliptic we show that its Julia set is not porous at a dense set of its points.

We study the dynamical Borel–Cantelli lemma for recurrence sets in a measure-preserving dynamical system $(X, \mu , T)$ with a compatible metric d. We prove that under some regularity conditions, the $\mu $ -measure of the following set

$$\begin{align*}R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\mathbb{N} \} \end{align*}$$

$\psi :\mathbb {N}\to \mathbb {R}^+$

$\beta $

We prove a number of results concerning the Hausdorff and packing dimension of sets of points which escape (at least in average) to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we generalize the results proved by Sixsmith in 2016 and answer his question on annular itineraries for exponential maps.

We show that every countable group with infinite finite conjugacy (FC)-center has the Schmidt property, that is, admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As a consequence, every countable, inner amenable group with property (T) has the Schmidt property.

We study projectional properties of Poisson cut-out sets E in non-Euclidean spaces. In the first Heisenbeg group \[\mathbb{H} = \mathbb{C} \times \mathbb{R}\], endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection \[\pi (E)\] (projection along the center of \[\mathbb{H}\]) almost surely equals \[\min \{ 2,{\dim _\operatorname{H} }(E)\} \] and that \[\pi (E)\] has non-empty interior if \[{\dim _{\text{H}}}(E) > 2\]. As a corollary, this allows us to determine the Hausdorff dimension of E with respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension \[{\dim _{\text{H}}}(E)\].

We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere \[{{\text{S}}^3}\] endowed with the visual metric d obtained by identifying \[{{\text{S}}^3}\] with the boundary of the complex hyperbolic plane. In \[{{\text{S}}^3}\], we prove a projection result that holds simultaneously for all radial projections (projections along so called “chains”). This shows that the Poisson cut-outs in \[{{\text{S}}^3}\] satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions.

We consider a strictly substochastic matrix or a stochastic matrix with absorbing states. By using quasi-stationary distributions we show that there is an associated canonical Markov chain that is built from the resurrected chain, the absorbing states, and the hitting times, together with a random walk on the absorbing states, which is necessary for achieving time stationarity. Based upon the 2-stringing representation of the resurrected chain, we supply a stationary representation of the killed and the absorbed chains. The entropies of these representations have a clear meaning when one identifies the probability measure of natural factors. The balance between the entropies of these representations and the entropy of the canonical chain serves to check the correctness of the whole construction.

We show that recurrence conditions do not yield invariant Borel probability measures in the descriptive set-theoretic milieu, in the strong sense that if a Borel action of a locally compact Polish group on a standard Borel space satisfies such a condition but does not have an orbit supporting an invariant Borel probability measure, then there is an invariant Borel set on which the action satisfies the condition but does not have an invariant Borel probability measure.

Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if \[K \subset {\mathbb{R}^n}\] is any set with equal Hausdorff and packing dimensions, then

\begin{equation} \[{\dim _{\text{H}}}{\pi _e}(K) = \min \{ {\dim _{\text{H}}}{\text{ }}K{\text{, 1}}\} \] \end{equation}

for almost every \[e \in {S^{n - 1}}\]. Here \[{\pi _e}\] stands for orthogonal projection to span (\[e\]). The primary purpose of this paper is to present proofs for Lutz and Stull’s projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions of Kaufman’s “potential theoretic” method, the pigeonhole principle, and a lemma of Katz and Tao. A secondary purpose is to generalise Lutz and Stull’s theorems: the versions in this paper apply to orthogonal projections to m-planes in \[{\mathbb{R}^n}\], for all \[0 < m < n\].

Let f be a $C^2$ diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure $\mu $ . We relate those entropies to covering numbers in order to give a new upper bound on the metric entropy of $\mu $ in terms of Lyapunov exponents and topological entropy or volume growth of sub-manifolds. We also discuss extensions to the $C^{1+\alpha },\,\alpha>0$ , case.

We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel chromatic number at least three. Moreover, we characterize the Borel graphs on standard Borel spaces of vertex-degree at most two with this property and show that the analogous result for digraphs fails.

Let $M=$ diag $(\rho _1,\rho _2)\in M_{2}({\mathbb R})$ be an expanding matrix and Let $\{D_n\}_{n=1}^{\infty }$ be a sequence of digit sets with $D_n=\left \{(0, 0)^T,\,\,\,(a_n, 0 )^T, \,\,\, (0, b_n )^T \right \}$ , where $a_n, b_n\in \{-1,1\}$ . The associated Borel probability measure

$$ \begin{align*} \mu_{M,\{D_n\}}:=\delta_{M^{-1}D_1}\ast \delta_{M^{-2}D_2}\ast \delta_{M^{-3}D_3}\ast \cdots \end{align*} $$

$\mu _{M, \{D_n\}}$

$3\mid \rho _i$

$i=1, 2$

$a_n=b_n=1$

$n\in {\mathbb N}$

We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

We investigate, both analytically and numerically, dispersive fractalisation and quantisation of solutions to periodic linear and nonlinear Fermi–Pasta–Ulam–Tsingou systems. When subject to periodic boundary conditions and discontinuous initial conditions, e.g., a step function, both the linearised and nonlinear continuum models for FPUT exhibit fractal solution profiles at irrational times (as determined by the coefficients and the length of the interval) and quantised profiles (piecewise constant or perturbations thereof) at rational times. We observe a similar effect in the linearised FPUT chain at times t where these models have validity, namely t = O(h−2), where h is proportional to the intermass spacing or, equivalently, the reciprocal of the number of masses. For nonlinear periodic FPUT systems, our numerical results suggest a somewhat similar behaviour in the presence of small nonlinearities, which disappears as the nonlinear force increases in magnitude. However, these phenomena are manifested on very long time intervals, posing a severe challenge for numerical integration as the number of masses increases. Even with the high-order splitting methods used here, our numerical investigations are limited to nonlinear FPUT chains with a smaller number of masses than would be needed to resolve this question unambiguously.

For every countable wellordering $\alpha $ greater than $\omega $ , it is shown that clopen determinacy for games of length $\alpha $ with moves in $\mathbb {N}$ is equivalent to determinacy for a class of shorter games, but with more complicated payoff. In particular, it is shown that clopen determinacy for games of length $\omega ^2$ is equivalent to $\sigma $ -projective determinacy for games of length $\omega $ and that clopen determinacy for games of length $\omega ^3$ is equivalent to determinacy for games of length $\omega ^2$ in the smallest $\sigma $ -algebra on $\mathbb {R}$ containing all open sets and closed under the real game quantifier.