Chronic disturbance is widely recognized as one of main triggers of diversity loss in seasonally dry tropical forests (SDTFs). However, the pathways through which diffuse disturbance is acting on the forest are little understood. This information is especially demanded in the case of vanishing Neotropical seasonally dry forests such as the Tumbesian ones. We proposed a conceptual model to analyze the factors behind the loss of woody species richness along a forest disturbance gradient, explicitly considering the existence of direct and indirect causal pathways of biodiversity loss. We hypothesized that the chronic disturbance can act on the woody species richness directly, either by selective extraction of resources or by browsing of palatable species for livestock, or indirectly, by modifying characteristics of the forest structure and productivity. To test our model, we sampled forest remnants in a very extensive area submitted to long standing chronic pressure. Our forests cells (200 × 200 m) were characterized both in terms of woody species composition, structure, and human pressure. Our structural equation models (SEMs) showed that chronic disturbance is driving a loss of species richness. This was done mainly by indirect effects through the reduction of large trees density. We assume that changes in tree density modify the environmental conditions, thus increasing the stress and finally filtering some specific species. The analysis of both, direct and indirect, allows us to gain a better understanding of the processes behind plant species loss in this SDTF.
A nilspace system is a generalization of a nilsystem, consisting of a compact nilspace X equipped with a group of nilspace translations acting on X. Nilspace systems appear in different guises in several recent works, and this motivates the study of these systems per se as well as their relation to more classical types of systems. In this paper we study morphisms of nilspace systems, i.e., nilspace morphisms with the additional property of being consistent with the actions of the given translations. A nilspace morphism does not necessarily have this property, but one of our main results shows that it factors through some other morphism which does have the property. As an application we obtain a strengthening of the inverse limit theorem for compact nilspaces, valid for nilspace systems. This is used in work of the first and third named authors to generalize the celebrated structure theorem of Host and Kra on characteristic factors.
In this paper, the nilspace approach to higher-order Fourier analysis is developed in the setting of vector spaces over a prime field $\mathbb {F}_p$ , with applications mainly in ergodic theory. A key requisite for this development is to identify a class of nilspaces adequate for this setting. We introduce such a class, whose members we call p-homogeneous nilspaces. One of our main results characterizes these objects in terms of a simple algebraic property. We then prove various further results on these nilspaces, leading to a structure theorem describing every finite p-homogeneous nilspace as the image, under a nilspace fibration, of a member of a simple family of filtered finite abelian p-groups. The applications include a description of the Host–Kra factors of ergodic $\mathbb {F}_p^\omega $ -systems as p-homogeneous nilspace systems. This enables the analysis of these factors to be reduced to the study of such nilspace systems, with central questions on the factors thus becoming purely algebraic problems on finite nilspaces. We illustrate this approach by proving that for $k\leq p+1$ the kth Host–Kra factor is an Abramov system of order at most k, extending a result of Bergelson–Tao–Ziegler that holds for $k< p$ . We illustrate the utility of p-homogeneous nilspaces also by showing that the structure theorem yields a new proof of the Tao–Ziegler inverse theorem for Gowers norms on $\mathbb {F}_p^n$ .
In this paper, the nilspace approach to higher-order Fourier analysis is developed in the setting of vector spaces over a prime field F p , with applications mainly in ergodic theory. A key requisite for this development is to identify a class of nilspaces adequate for this setting. We introduce such a class, whose members we call p-homogeneous nilspaces. One of our main results characterizes these objects in terms of a simple algebraic property. We then prove various further results on these nilspaces, leading to a structure theorem describing every finite p-homogeneous nilspace as the image, under a nilspace fibration, of a member of a simple family of filtered finite abelian p-groups.The applications include a description of the Host-Kra factors of ergodic F ω p -systems as p-homogeneous nilspace systems. This enables the analysis of these factors to be reduced to the study of such nilspace systems, with central questions on the factors thus becoming purely algebraic problems on finite nilspaces. We illustrate this approach by proving that for k ≤ p + 1 the k-th Host-Kra factor is an Abramov system of order ≤ k, extending a result of Bergelson-Tao-Ziegler that holds for k < p. We illustrate the utility of p-homogeneous nilspaces also by showing that the above-mentioned structure theorem yields a new proof of the Tao-Ziegler inverse theorem for Gowers norms on F n p .
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