The study of minimal Cantor systems and zero entropy dynamical systems provided recently striking results. Topological full groups of minimal subshifts provide finitely generated groups with original properties: they are simple, amenable, may have intermediate growth for some zero entropy subshifts. Frantzikinakis-Host proved the Sarnak conjecture for the logarithmic average and zero entropy dynamical systems with at most countably many invariant measures. Adamczewski-Bugeaud constructed transcendantal numbers from zero entropy subshifts. Hence a deep understanding of zero entropy systems is of particular importance by itself and for other topics like number, group theory but also for applications to quasicristallography, computer science or statistical physics. Despite substantial efforts to understand zero entropy and although many families are well understood few general results have been obtained. We aim to unify parts of existing results and to go deeper into zero entropy.

The main objective of GoodBerry is to provide the necessary knowledge and procedures to facilitate the development of highly productive and top quality berry fruits, even under multiple suboptimal growth conditions, at a competitive cost. The project is based on an integrative multi-actor approach, from cultivation techniques to molecular studies, aiming the development and validation of a range of tools to improve competitiveness of European berry production, and eventually the attraction and confidence of consumers. The selection of the model species can be considered as strategic since strawberry is the most important berry crop in Europe and the production of raspberry and blackcurrant are increasing strongly in recent years. The project will apply the most recent technical advances in: a) The identification of berry germplasm exhibiting advantageous balance of production vs nutritional quality throughout the EU, b) The search of innovative production systems to maintain high yield in a range of European-wide environments, c) The development of standardized and reliable analytical tools to evaluate berry production and fruit quality. As result, it is expected: a) the implementation of modern breeding strategies to accelerate the release of new berry cultivars; b) The adoption by EU-growers of high quality production systems to improve fruit quality. The proposal establishes as obligatory to disseminate and communicate the results to the scientific community, industry, the broad public and interested stakeholders user. The final impact will be to consolidate the emerging needs of high-quality berries, and to boost consumer and market confidence supported by an improved competitiveness of producers. It is a multidisciplinary, collaborative project based on complementary expertise and skills of internationally recognized berry research institutions, and highly involved key berry SMEs that will combine their effort to secure the robustness of the results.