In this work, we study concentration properties for vector valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in Rk. To this task, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions to earlier results by Pinelis in the Gaussian case, ...

Definition 1. Let N ′ be a Carnot group and N be a subgroup of N ′ with Hausdorff dimension k. A subset E of another Carnot Group (M,dM ) is said to be N -rectifiable if there exists U , a positive measure subset of N , and a Lipschitz map f : U → M such that H k M (E \ f(U)) = 0. We say E is countably N -rectifiable if there exist a countable number of pairs of subsets Ui and Lipschitz maps fi...

Methods of estimating (Riemannian and Finsler) filling volumes by using nonexpanding maps to Banach spaces of L∞-type are developed and generalized. For every Finsler volume functional (such as the Busemann volume or the Holmes– Thompson volume), a natural extension is constructed from the class of Finsler metrics to all Lipschitz metrics, and the notion of area is defined for Lipschitz surface...

An iterated function system on X R d is deened by successively iterating an i.i.d. sequence of random Lipschitz functions from X to X. This paper shows how Fn = f 1 fn may converge even in the absence of the strong contraction conditions | for instance, Lipschitz constant smaller than 1 on average | which earlier work has required. Instead, it is posited that there be a region of contraction wh...

Definition 1. Let N ′ be a Carnot group and N be a subgroup of N ′ with Hausdorff dimension k. A subset E of another Carnot Group (M,dM ) is said to be N -rectifiable if there exists U , a positive measure subset of N , and a Lipschitz map f : U → M such that H k M (E \ f(U)) = 0. We say E is countably N -rectifiable if there exist a countable number of pairs of subsets Ui and Lipschitz maps fi...

For a large class of expanding maps of the interval, we prove that partial sums of Lipschitz observables satisfy an almost sure central limit theorem (ASCLT). In fact, we provide a speed of convergence in the Kantorovich metric. Maxima of partial sums are also shown to obey an ASCLT. The key-tool is an exponential inequality recently obtained. Then we derive almost-sure convergence rates for th...

In this paper we discuss the computational power of Lipschitz dynamical systems which are robust to infinitesimal perturbations. Whereas the study in [1] was done only for not-so-natural systems from a classical mathematical point of view (discontinuous differential equation systems, discontinuous piecewise affine maps, or perturbed Turing machines), we prove that the results presented there ca...

We show that there is an operator space notion of Lipschitz embeddability between spaces which strictly weaker than its linear counterpart but still strong enough to impose restrictions on structures. This shows a nontrivial theory nonlinear geometry for and it answers question in Braga et al. (Proc Am Math Soc 149(3):1139–1149, 2021). For that, we introduce the version Lipschitz-free Banach pr...

We show that a separable Banach space has the RadonNikodym Property (RNP) if and only if it is isomorphic to the limit of an inverse system, V1 ← V2 ← . . . ← Vk ← . . . , where the Vi’s are finite dimensional Banach spaces, and the bonding maps Vk−1 ← Vk are quotient maps. We also show that the inverse system can be chosen to be a good finite dimensional approximation (GFDA), a notion introduc...

We establish Erdös-Rényi limit laws for Lipschitz observations on a class of non-uniformly expanding dynamical systems, including logistic-like maps. These limit laws give the maximal average of a time series over a time window of logarithmic length. We also give results on maximal averages of a time series arising from Hölder observations on intermittent-type maps over a time window of polynom...