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This paper offers a critical assessment of the value and utility of the evolving City Region Food Systems (CRFS) approach to improve our insights into flows of resources-food, waste, people, and knowledge-from rural to peri-urban to urban and back again, and the policies and process needed to enable sustainability. This paper reflects on (1) CRFS merits compared to other approaches; (2) the operational potential of applying the CRFS approach to existing projects through case analysis; (3) how to make the CRFS approach more robust and ways to further operationalize the approach; and (4) the potential for the CRFS approach to address complex challenges including integrated governance, territorial development, metabolic flows, and climate change. The paper begins with the rationale for CRFS as both a conceptual framework and an integrative operational approach, as it helps to build increasingly coherent transformational food systems. CRFS is differentiated from existing approaches to understand the context and gaps in theory and practice. We then explore the strength of CRFS through the conceptual building blocks of 'food systems' and 'city-regions' as appropriate, or not, to address pressing complex challenges. As both a multi-stakeholder, sustainability-building approach and process, CRFS provides a collective voice for food actors across scales and could provide coherence across jurisdictions, policies, and scales, including the Milan Urban Food Policy Pact, the Sustainable Development Goals, the Habitat III New Urban Agenda, and the Conference of the Parties (COP) 21. CRFS responds directly to calls in the literature to provide a conceptual and practical framing for policy through wide engagement across sectors that enables the co-construction of a relevant policy frame that can be enacted through sufficiently integrated policies and programs that achieve increasingly sustainable food systems.

We prove several new results about AF-equivalence relations, and relate these to Cantor minimal systems (i.e. to minimal Z-actions). The results we obtain turn out to be crucial for the study of the topological orbit structure of more general countable group actions (as homeomorphisms) on Cantor sets, which will be the topic of a forthcoming paper. In all this, Bratteli diagrams and their dynamical interpretation, are indispensable tools.

We show that every minimal action of any finitely generated abelian group on the Cantor set is (topologically) orbit equivalent to an AF relation. As a consequence, this extends the classification up to orbit equivalence of minimal dynamical systems on the Cantor set to include AF relations and Z d -actions. *

A topological group G is called extremely amenable if every continuous action of G on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a C * -algebra A is nuclear if and only if the unitary group U (A) with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of L p (0, 1), 1 p < ∞, extending a classical result of Gromov and Milman (p = 2). We show that a measure class preserving equivalence relation R on a standard Borel space is amenable if and only if the full group [R], equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a non-trivial space in the sense of Gromov occurring in the automorphism groups of injective factors of type III.