Graph towers, laminations and their invariant measures

In this paper we present a combinatorial machinery, consisting of a graph tower $\overleftarrow \Gamma$ and vector towers $\overleftarrow v$ on $\overleftarrow \Gamma$, which allows us to efficiently describe all invariant measures $\mu = \mu^{\overleftarrow v}$ on any given shift space over a finite alphabet. The new technology admits a number of direct applications, in particular concerning invariant measures on non-primitive substitution subshifts, minimal subshifts with many ergodic measures, or an efficient calculation of the measure of a given cylinder. It also applies to currents on a free group $F_N$, and in particular the set of projectively fixed currents under the action of a (possibly reducible) endomorphism $\varphi: F_N \to F_N$ is determined, when $\varphi$ is represented by a train track map.Comment: 52 pages, 3 figures. This is rather a new paper than a new version of the old one. The setting is much more general, and also closer to a symbolic dynamics spirit. Also, some of the work from the original paper has been removed and will be taken up in a forthcoming paper. Accepted in Journal of London Mathematical society. To appear 201

Generic substitutions

Up to equivalence, a substitution in propositional logic is an endomorphism of its free algebra. On the dual space, this results in a continuous function, and whenever the space carries a natural measure one may ask about the stochastic properties of the action. In classical logic there is a strong dichotomy: while over finitely many propositional variables everything is trivial, the study of the continuous transformations of the Cantor space is the subject of an extensive literature, and is far from being a completed task. In many-valued logic this dichotomy disappears: already in the finite-variable case many interesting phenomena occur, and the present paper aims at displaying some of these.Comment: 22 pages, 2 figures. Revised version according to the referee's suggestions. To appear in the J. of Symbolic Logi

Toric Representation and Positive Cone of Picard Group and Deformation Space in Mirror Symmetry of Calabi-Yau Hypersurfaces in Toric Varieties

We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry is established for a pair of Calabi-Yau (CY) ${\sf n}$-spaces in toric varieties defined by reflexive polytopes for an arbitrary dimension ${\sf n}$. We further identify the Kahler cone of the toric variety and degeneration cone of CY hypersurfaces, by which the Kahler cone and degeneration cone for a mirror CY pair are interchangeable under mirror symmetry. In particular, different degeneration cones of a CY 3-fold are corresponding to flops of its mirror 3-fold.Comment: Latex 29 pag

Note on the Core-Walras Equivalence Problem when the Commodity Space is a Banach Lattice

The core-Walras equivalence problem for an atomless economy is considered in the commodity space setting of Banach lattices. In particular, necessary and sufficient conditions on the commodity space in order for core-Walras equivalence to hold are established. In general, these conditions can be regarded as implying that an economy with a continuum of agents has indeed "many more agents than commodities". However, it turns out that there are special commoditiy spaces in which core-Walras equivalence holds for every atomless economy satisfying certain standard assumptions, but in which an atomless economy does not have the meaning of there being "many more agents than commodities."