Consider the quadratic family $T_a(x) = a x (1 - x)$ for $x \in [0, 1]$ and mixing Collet–Eckmann (CE) parameters $a \in (2,4)$. For bounded $\varphi $, set $\tilde \varphi _{a} := \varphi - \int \varphi \, d\mu _a$, with $\mu _a$ the unique acim of $T_a$, and put $(\sigma _a (\varphi ))^2 := \int \tilde \varphi _{a}^2 \, d\mu _a + 2 \sum _{i>0} \int \tilde \varphi _{a} (\tilde \varphi _{a} \circ T^i_{a}) \, d\mu _a$. For any mixing Misiurewicz parameter $a_{*}$, we find a positive measure set $\Omega _{*}$ of mixing CE parameters, containing $a_{*}$ as a Lebesgue density point, such that for any Hölder $\varphi $ with $\sigma _{a_{*}}(\varphi )\ne 0$, there exists $\epsilon _\varphi>0$ such that, for normalized Lebesgue measure on $\Omega _{*}\cap [a_{*}-\epsilon _\varphi , a_{*}+\epsilon _\varphi ]$, the functions $\xi _i(a)=\tilde \varphi _a(T_a^{i+1}(1/2))/\sigma _a (\varphi )$ satisfy an almost sure invariance principle (ASIP) for any error exponent $\gamma>2/5$. (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the lines of Schnellmann’s proof for piecewise expanding maps. We need to introduce a variant of Benedicks–Carleson parameter exclusion and to exploit fractional response and uniform exponential decay of correlations from Baladi et al [Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. Math. 201 (2015), 773–844].

Researchers have long discussed whether Scandinavian rock art reflects narratives. Their interpretations have frequently been based on inspections of rock art panels combined with knowledge from ethnographic and historical sources. Here, the authors adopt a more focused narratological approach that takes the concept of (visual) narrativity into consideration and draws on studies by literary analysts, cognitive psychologists, and semioticians. Images of spear use in the provinces of Bohuslän and Östergötland in Sweden, given their diversity and indexical qualities, are well-suited to such a study. They reveal different kinds of indexical relationships, i.e. how the spears direct attention to possible targets, arguably corresponding to action scripts well-known to Bronze Age communities. Many spear images may be regarded as mini-narratives and mnemonic devices intended to represent schematized action sequences. The authors suggest that concepts such as iconicity, indexical relationships, scripts, and mini-narratives could be fruitfully employed in research on Scandinavian rock art and beyond.

This paper discusses rock art in southern Scandinavia as a multisensory format, where both sight and touch would have contributed to the comprehension of the images. From a structural semiotic point of view, we suggest that rock art can be construed as an organised set of features, such as visual and tactile elements, organised into heterogeneous unities with dynamic relations between elements that can change over time with respect to how they are experienced. We argue that in order to understand the rock art medium, it is crucial to take into consideration the multisensory interaction between the perceiver and the qualities of the rock art surface. The reason for including tactile elements in our interpretation of the conception of rock art is the way it was created: by hands interacting with tools and rock surfaces, as well as the spontaneous human tendency to explore the physical world through touch. One can identify key features in the images that would arguably facilitate tactile recognition, as well as be better explained from a multisensorial perspective. This includes the position of the images on horizontal outcrops, the moderate size of the images, the application of an orthographic perspective, the use of ‘tactile markers’ (ie crucial features having a strategic function for understanding images by touch), and the occurrence of incomplete images. A multisensorial perspective on rock art furthermore has semiotic implications. Incomplete images, for example, can be understood as indexical stand-ins for the whole imagined picture, ie as iconic indices. A multisensorial approach to Scandinavian rock art thus allows for new explanations for certain design choices, as well as a new understanding of how the images could relay meaning to a perceiver.

In Europe, Scandinavia holds the largest concentration of rock art (i.e. petroglyphs), created c. 5000–first century bc, many of them showing figurative and seemingly narrative representations. In this paper, we will discuss possible narratological approaches applied to these images. We might reasonably distinguish between three levels of pictorial narrativity: representations of (i) single events, understood as the transition from one state of affairs to another, usually involving (groups of) agents interacting; (ii) stories, e.g. particular sequences of related events that are situated in the past and retold for e.g. ideological or religious purposes; and (iii) by implication, master-narratives deeply embedded in a culture, which provide and consolidate cosmological explanations and social structures. Some concrete examples of petroglyphs will be presented and analysed from narratological and iconographical perspectives. We will as a point of departure focus on (i), i.e. single events, though we shall also further consider the possibility of narrative interpretations according to (ii) and (iii).

We consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdorff dimension of the set of parameters for which the orbit of a fixed point has a certain shrinking target property. This generalises several similar results for β-transformations to more general non-linear families. The proofs are based on a result by Schnellmann on typicality in parametrised families.

For a map $T:[0,1]\rightarrow [0,1]$ with an invariant measure $\unicode[STIX]{x1D707}$, we study, for a $\unicode[STIX]{x1D707}$-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 $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D719}}$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.

We consider classes of subsets of [0, 1], originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least s. We provide a Frostman-type lemma to determine if a limsup set is in such a class. Suppose that E = lim sup En ⊂ [0, 1], and that μn are probability measures with support in En. If there exists a constant C such that

for all n, then, under suitable conditions on the limit measure of the sequence (μn), we prove that the set E is in the class .

As an application we prove that, for α > 1 and almost all λ ∈ (½, 1), the set

where and ak ∈ {0, 1}}, belongs to the class . This improves one of our previously published results.

For and α, we consider sets of numbers x such that for infinitely many n, x is 2−αn-close to some ∑ ni=1ωiλi, where ωi∈{0,1}. These sets are in Falconer’s intersection classes for Hausdorff dimension s for some s such that −(1/α)(log λ /log   2 )≤s≤1/α. We show that for almost all , the upper bound of s is optimal, but for a countable infinity of values of λ the lower bound is the best possible result.

We find problems with Vaesen's treatment of the primatological research, in particular his analysis of foresight, function representation, and social intelligence. We argue that his criticism of research on foresight in great apes is misguided. His claim that primates do not attach functions to particular objects is also problematic. Finally, his analysis of theory of mind neglects many distinctions.

Let (Bi) be a sequence of measurable sets in a probability space (X,ℬ,μ) such that ∑ ∞n=1μ(Bi)=∞. The classical Borel–Cantelli lemma states that if the sets Bi are independent, then μ({x∈X:x∈Bi infinitely often})=1. Suppose (T,X,μ) is a dynamical system and (Bi) is a sequence of sets in X. We consider whether Tix∈Bi infinitely often for μ almost every x∈X and, if so, is there an asymptotic estimate on the rate of entry? If Tix∈Bi infinitely often for μ almost every x, we call the sequence (Bi) a Borel–Cantelli sequence. If the sets Bi :=B(p,ri) are nested balls of radius ri about a point p, then the question of whether Tix∈Bi infinitely often for μ almost every x is often called the shrinking target problem. We show, under certain assumptions on the measure μ, that for balls Bi if μ(Bi)≥i−γ, 0<γ<1, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observables implies that the sequence is Borel–Cantelli. If μ(Bi)≥C1 /i, then exponential decay of correlations implies that the sequence is Borel–Cantelli. We give conditions in terms of return time statistics which quantify Borel–Cantelli results for sequences of balls such that μ(Bi)≥C/i. Corollaries of our results are that for planar dispersing billiards and Lozi maps, sequences of nested balls B(p,1/i) are Borel–Cantelli. We also give applications of these results to a variety of non-uniformly hyperbolic dynamical systems.

A class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.

We find that the nature and origin of the proposed “dialogical cognitive representations” in the target article is not sufficiently clear. Our proposal is that (triadic) bodily mimesis and in particular mimetic schemas – prelinguistic representational, intersubjective structures, emerging through imitation but subsequently interiorized – can provide the necessary link between private sensory-motor experience and public language. In particular, we argue that shared intentionality requires triadic mimesis.