Finite Type Research Articles

High pointwise emergence via saturated sets

In this article, we prove that the set of points whose pointwise emergence have same degree in with repect to polynomial has strong dynamical complexity in sense of entropy, residuality and density, and distributional chaos for a β-shift, a C 1 diffeomorphism having a nontrival homoclinic class, or a diffeomorphism preserving a hyperbolic ergodic measure. In this process, we construct nonempty compact connected set with fixed box-counting dimension in the space of invariant measures. Then combining with the relation between pointwise emergence of saturated set and box-counting dimension, we get the goal from two frameworks about dynamical complexity of saturated set. One framework was introduced by C. Pfister and W. Sullivan and can be applicable to transitive Anosov diffeomorphisms, mixing expanding maps, β-shifts, transitive subshifts of finite type, mixing sofic subshifts, and mixing S-gap shifts. The other framework is applicable to general homoclinic classes and diffeomorphisms preserving hyperbolic ergodic measures. We also get some combined observations with irregular set.

Idempotents, free products and quandle coverings

In this paper, we investigate idempotents in quandle rings and relate them with quandle coverings. We prove that integral quandle rings of quandles of finite type that are nontrivial coverings over nice base quandles admit infinitely many nontrivial idempotents, and give their complete description. We show that the set of all these idempotents forms a quandle in itself. As an application, we deduce that the quandle ring of the knot quandle of a nontrivial long knot admit nontrivial idempotents. We consider free products of quandles and prove that integral quandle rings of free quandles have only trivial idempotents, giving an infinite family of quandles with this property. We also give a description of idempotents in quandle rings of unions and certain twisted unions of quandles.

Generalized Fibonacci shifts in the Lorenz attractor

In this article we deal with symmetric Lorenz attractors having a homoclinic loop that exhibits a well ordered orbit. We show the symmetry implies a very regular behaviour on the dynamic in the topological and metric sense. Let ([−1,1],f) be the one-dimensional reduction Lorenz map satisfying a well ordered orbit and ([−1,0],f̃) be the quotient map, given by the equivalence relation x∼−x, the dynamic of f˜ is described explicitly as a subshift of finite type which generalizes the Fibonacci shifts and this fact is used to compute topological entropy of f.Moreover we show that in general ([−1,0],f̃) is related to a factor of the k-bonacci shift. In particular we found that the 1-dimensional Lorenz map replicates an interesting duplicating behaviour of the k-bonacci shift found in Sirvent (1996, 2011).

The entropy of multiplicative subshifts on trees

The aim of this paper is to study the topological entropy of a q-multiplicative subshift on tree (q-MST), that is, every boundary of the tree follows the rule of some one-dimensional multiplicative integer systems defined by Kenyon et al. [20]. Supposing the number of vertices in the tree T grows exponentially, the explicit formula for the topological entropy of the q-MST is presented in Theorem 1.1. Next, we consider the coupled q-MST, namely, every boundary of T satisfies the q-multiple constraint and is characterized by subshift of finite type. The comparison of topological entropy for the coupled and non-coupled q-MSTs is provided in Theorem 1.4. Furthermore, the explicit formula of some coupled q-MSTs are also obtained herein. Finally, the entropy formula for the general q-MSTs, known as XΩ(S), is described in Theorem 1.6.

Positive cluster complexes and τ-tilting simplicial complexes of cluster-tilted algebras of finite type

In this study, we consider the positive cluster complex, a full subcomplex of a cluster complex the vertices of which are all non-initial cluster variables. In particular, we provide a formula for the difference in face vectors of positive cluster complexes caused by a mutation for finite type. Moreover, we explicitly describe specific positive cluster complexes of finite type and calculate their face vectors. We also provide a method to compute the face vector of an arbitrary positive cluster complex of finite type using these results. Furthermore, we apply our results to the -tilting theory of cluster-tilted algebras of finite representation type using the correspondence between clusters and support -tilting modules.

Hardwiring truth in functional interpretations

We present four different approaches to prove the soundness theorem for variants with t-truth of functional interpretations. To showcase our different methods we focus on the intuitionistic nonstandard bounded functional interpretation of the nonstandard extensional Heyting arithmetic in all finite types because a version with t-truth for this interpretation has not been given before. Also, because it is a more involved interpretation than others since it includes both nonstandard principles and majorisability. This leads us to believe that if the approaches work for this more complicated functional interpretation, then they should also work for simpler functional interpretations (and realisabilities).

Simple transitive 2‐representations of Soergel bimodules for finite Coxeter types

AbstractIn this paper, we show that Soergel bimodules for finite Coxeter types have only finitely many equivalence classes of simple transitive 2‐representations and we complete their classification in all types but and .

Construction and applications of proximal maps for typical cocycles

AbstractFor typical cocycles over subshifts of finite type, we show that for any given orbit segment, we can construct a periodic orbit such that it shadows the given orbit segment and that the product of the cocycle along its orbit is a proximal linear map. Using this result, we show that suitable assumptions on the periodic orbits have consequences over the entire subshift.

Topological dynamics of Markov multi-maps of the interval

We study Markov multi-maps of the interval from the point of view of topological dynamics. Specifically, we define the forward trajectory system of a Markov multi-map, and we investigate whether it has properties such as topological transitivity, topological mixing, density of periodic points, and specification. To each Markov multi-map we associate a shift of finite type (SFT), and our main results relate the properties of the SFT with those of the forward trajectory system. Under a general coding condition, we establish necessary and sufficient conditions for topological transitivity and topological mixing of the forward trajectory system. These results complement existing work showing a relationship between the topological entropy of a Markov multi-map and its associated SFT. We also characterize when the inverse limit systems associated to the Markov multi-maps have the properties mentioned above.

Zeta Function for Commuting Matrix Shift of Finite Type

In this paper we show two-dimensional shifts of finite type whose transition matrices commutes and such that the entries of their product also consist of 0’s and 1’s. In particular, the form of associated zeta function of the two-dimensional shift was obtained. Unlike in the one-dimensional shifts of finite type; we gather that zeta functions for them have not closed form. This result derived by obtaining a formula for the number of periodic points of the subshift of finite type. This formula is then incorporated in the corresponding zeta function.

Weak Faces of Highest Weight Modules and Root Systems

Chari and Greenstein (Adv. Math. 220(4), 1193–1221, 2009) introduced certain combinatorial subsets of the roots of a finite-dimensional simple Lie algebra $\mathfrak {g}$ , which were important in studying Kirillov–Reshetikhin modules over $U_{q}(\widehat {\mathfrak {g}})$ and their specializations. Later, Khare (J. Algebra. 455, 32–76, 2016) studied these subsets for large classes of highest weight $\mathfrak {g}$ -modules (in finite type), under the name of weak- $\mathbb {A}$ -faces (for a subgroup $\mathbb {A}$ of $(\mathbb {R},+)$ ), and more generally, ({2};{1,2})-closed subsets. These notions extend and unify the faces of Weyl polytopes as well as the aforementioned combinatorial subsets. In this paper, we consider these “discrete” notions for an arbitrary Kac–Moody Lie algebra $\mathfrak {g}$ , in three prominent settings: (a) the weights of an arbitrary highest weight $\mathfrak {g}$ -module V; (b) the convex hull of the weights of V; (c) the weights of the adjoint representation. For (a) (respectively, (b)) for all highest weight $\mathfrak {g}$ -modules V, we show that the weak- $\mathbb {A}$ -faces and ({2};{1,2})-closed subsets agree, and equal the sets of weights on the exposed faces (respectively, equal the exposed faces) of the convex hull of weights conv(wtV ). This completes the partial progress of Khare in finite type, and is novel in infinite type. Our proofs are type-free and self-contained. For (c) involving the root system, we similarly achieve complete classifications. For the weights of $\text {ad} \mathfrak {g}$ for any indecomposable Kac–Moody $\mathfrak {g}$ , we show that the weak- $\mathbb {A}$ -faces and ({2};{1,2})-closed subsets agree, and equal the Weyl group translates of the sets of weights in certain “standard parabolic faces” (which also holds for highest weight modules). This was proved by Chari and her coauthors for Δ ⊔{0} in finite type, but is novel for other types.

The group of self-homotopy equivalences of a rational space cannot be a free abelian group

In this paper, we prove that a free abelian group cannot occur as the group of self-homotopy equivalences of a rational CW-complex of finite type. Thus, we generalize a result due to Sullivan–Wilkerson showing that if $X$ is a rational CW-complex of finite type such that $\mathrm{dim}\,H^{*}(X, \mathbb{Z}) < \infty$ or $\mathrm{dim}\,\pi_{*}(X) < \infty$, then the group of self-homotopy equivalences of $X$ is isomorphic to a linear algebraic group defined over $\mathbb{Q}$.

Non-intrusive implementation of Multiscale Finite Element Methods: An illustrative example

Multiscale Finite Element Methods (MsFEM) are finite element type approaches dedicated to multiscale problems. They first compute local, oscillatory, problem-dependent basis functions which generate a specific discretization space, and next perform a Galerkin approximation of the problem on that space. We investigate here how these approaches can be implemented in a non-intrusive way, in order to facilitate their dissemination within industrial codes or non academic environments.

Locally finitely presented and coherent hearts

Starting with a Grothendieck category \mathcal{G} and a torsion pair \mathbf{t}=(\mathcal{T},\mathcal{F}) in \mathcal{G} , we study the local finite presentability and local coherence of the heart \mathcal{H}_{\mathbf{t}} of the associated Happel–Reiten–Smalø t -structure in the derived category \mathbf{D}(\mathcal{G}) . We start by showing that, in this general setting, the torsion pair \mathbf{t} is of finite type, if and only if it is quasi-cotilting, if and only if it is cosilting. We then proceed to study those \mathbf{t} for which \mathcal{H}_{\mathbf{t}} is locally finitely presented, obtaining a complete answer under some additional assumptions on the ground category \mathcal{G} , which are general enough to include all locally coherent Grothendieck categories, all categories of modules and several categories of quasi-coherent sheaves over schemes. The third problem that we tackle is that of local coherence. In this direction, we characterize those torsion pairs \mathbf{t}=(\mathcal{T},\mathcal{F}) in a locally finitely presented \mathcal{G} for which \mathcal{H}_{\mathbf{t}} is locally coherent in two cases: when the tilted t -structure in \mathcal{H}_{\mathbf{t}} is assumed to restrict to finitely presented objects, and when \mathcal{F} is cogenerating. In the last part of the paper, we concentrate on the case when \mathcal{G} is a category of modules over a small preadditive category, giving several examples and obtaining very neat (new) characterizations in this more classical setting, underlying connections with the notion of an elementary cogenerator.

On von Neumann regularity of cellular automata

We show that a cellular automaton on a one-dimensional two-sided mixing subshift of finite type is a von Neumann regular element in the semigroup of cellular automata if and only if it is split epic onto its image in the category of sofic shifts and block maps. It follows from previous joint work of the author and Törmä that von Neumann regularity is a decidable condition, and we decide it for all elementary CA, obtaining the optimal radii for weak generalized inverses. Two sufficient conditions for non-regularity are having a proper sofic image or having a point in the image with no preimage of the same period. We show that the non-regular ECA 9 and 28 cannot be proven non-regular using these methods. We also show that a random cellular automaton is non-regular with high probability.

Affine Kac-Moody groups and Lie algebras in the language of SGA3

In infinite-dimensional Lie theory, the affine Kac-Moody Lie algebras and groups play a distinguished role due to their many applications to various areas of mathematics and physics. Underlying these infinite-dimensional objects there are closely related group schemes and Lie algebras of finite type over Laurent polynomial rings. The language of SGA3 is perfectly suited to describe such objects. The purpose of this short article is to provide a natural description of the affine Kac-Moody groups and Lie algebras using this language.

Essential finite generation of extensions of valuation rings

AbstractGiven a generically finite local extension of valuation rings , the question of whether W is the localization of a finitely generated V‐algebra is significant for approaches to the problem of local uniformization of valuations using ramification theory. Hagen Knaf proposed a characterization of when W is essentially of finite type over V in terms of classical invariants of the extension of associated valuations. Knaf's conjecture has been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankar valuations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky for valuation rings of function fields of characteristic 0 using embedded resolution of singularities. In this paper, we prove Knaf's conjecture in full generality.

Spectral properties of the finite system of Klein-Gordon S-wave equations with general boundary condition

The spectral characteristics of the operator L is studied where L is defined within the Hilbert space L2(R+, CV) given by a finite system of Klein-Gordon type differential equations and boundary condition at general form. The research of the Klein-Gordon type operator continues to be an important topic for researchers due to the range of applicability of them in numerous branches of mathematics and quantum physics. Contrary to the previous works, we take the potential as complex valued and generalize the problem to the matrix Klein-Gordon operator case. The spectrum is derived by determining the Jost function and resolvent operator of the prescribed operator. Further, we provide the conditions that must be met for the certain quantitative properties of the spectrum.

A functorial approach to regular homomorphisms

Classically, regular homomorphisms have been defined as a replacement for Abel--Jacobi maps for smooth varieties over an algebraically closed field. In this work, we interpret regular homomorphisms as morphisms from the functor of families of algebraically trivial cycles to abelian varieties and thereby define regular homomorphisms in the relative setting, e.g., families of schemes parameterized by a smooth variety over a given field. In that general setting, we establish the existence of an initial regular homomorphism, going by the name of algebraic representative, for codimension-2 cycles on a smooth proper scheme over the base. This extends a result of Murre for codimension-2 cycles on a smooth projective scheme over an algebraically closed field. In addition, we prove base change results for algebraic representatives as well as descent properties for algebraic representatives along separable field extensions. In the case where the base is a smooth variety over a subfield of the complex numbers we identify the algebraic representative for relative codimension-2 cycles with a subtorus of the intermediate Jacobian fibration which was constructed in previous work. At the heart of our descent arguments is a base change result along separable field extensions for Albanese torsors of separated, geometrically integral schemes of finite type over a field.

Spectrum of weighted Birkhoff average

Let $\{s_n\}_{n\in \mathbb N}$ be a decreasing nonsummable sequence of positive reals. We investigate the weighted Birkhoff average $\frac {1}{S_n}\sum _{k=0}^{n-1}s_k\phi (T^kx)$ on an aperiodic irreducible subshift $\Sigma _{\bf A}$ of finite type where