Inverse limit

Abstract: We consider the dynamical systems arising from substitution tilings. Under some hypotheses, we show that the dynamics of the substitution or inflation map on the space of tilings is topologically conjugate to a shift on a stationary inverse limit, i.e. one of R. F. Williams' generalized solenoids. The underlying space in the inverse limit construction is easily computed in most examples and frequently has the structure of a CW-complex. This allows us to compute the cohomology and K-theory of the space of tilings. This is done completely for several one- and two-dimensional tilings, including the Penrose tilings. This approach also allows computation of the zeta function for the substitution. We discuss -algebras related to these dynamical systems and show how the above methods may be used to compute the K-theory of these.

Abstract: We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the tropicalizations of its quasiprojective embeddings.

Abstract: In May 1967 I had suggested in my Chicago lectures certain applications of category theory to smooth geometry and dynamics, reviving a direct approach to function spaces and therefore to functionals. Making that suggestion more explicit led later to elementary topos theory as well as to the line of research now known as synthetic differential geometry. The fuller development of those subjects turned out to involve a truth value object that classifies subobjects, but in the present paper (presented in the 1968 Battelle conference in Seattle) I refer only to weak properties of such an object; it is the other axiom, cartesian closure, that plays the central role. Daniel Kan had recognized that the function space construction for simplicial sets and other categories is a right adjoint, thus unique. Because this uniqueness property of adjoints implies their main calculational rules, I took the further axiomatic step of defining functor categories as a right adjoint to the finite product construction in my 1963 thesis. In 1965, Eilenberg and Kelly introduced the term closed to mean that there is a hom functor valued in the category itself. Such a hom functor is characterized in a relative way as right adjoint to a given tensor product functor; we concentrate here on the absolute case where the tensor is cartesian. Although the cartesian-closed view of function spaces and functionals was intuitively obvious in all but name to Volterra and Hurewicz (and implicitly to Bernoulli), it has counterexamples within the rigid framework advocated by Dieudonné and others. According to that framework the only acceptable fundamental structure for expressing the cohesiveness of space is a contravariant algebra of open sets or possibly of functions. Even though such algebras are of course extremely important invariants, their nature is better seen as a consequence of the covariant geometry of figures. Specific cases of this determining role of figures were obvious in the work of Kan and in the popularizations of Hurewicz’s k-spaces by Kelley, Brown, Spanier, and Steenrod, but in the present paper I made this role a matter of principle: the Yoneda embedding was shown to preserve

Abstract: Introduction. In 1954 C. E. Capel proved [1] the following theorems: Let S be the inverse limit of a sequence of arcs (simple closed curves) where the bounding maps are onto and monotone. Then S is an arc (simple closed curve). It may be noted that if f is a monotone map of an arc (simple closed curve) onto itself, then f is the uniform limit of a sequence of onto homeomorphisms.2 We call such a map a nearhomeomorphism. In this paper we prove the following two theorems: (1) If S is the inverse limit of a sequence of copies of a given compact metric space X and the bonding maps are near-homeomorphisms, then S is homeomorphic to X. (2) Let f: XY, g: Y-*X, where f, g are maps and X, Y are compact metric spaces. Suppose fg and gf are nearhomeomorphisms. Then X is homeomorphic to Y. The second theorem follows directly from the first. In order to establish the first theorem we develop an approximation theorem which has interest in its own right. DEFINITIONS AND NOTATION. Let Xi be a sequence of compact metric spaces, and for i ? 2 letf* map Xi into Xi-,. Then the subspace3 S= {zEz -II Xilfij(zj) =zi} of I1 Xi is the limit space of the inverse system (Xi, fi); in notation S = Lim (Xi, ft). Let f map X into Y where X, Y are compact metric spaces. Then for e>O:L(e, f)=Sup13 cn.