Exact Sequences and Closed Model Categories

Pinillos, Mónica García; Paricio, Luis Javier Hernández; Rodríguez, María Teresa Rivas

doi:10.1007/s10485-008-9176-x

Cited by 7 publications

(7 citation statements)

References 33 publications

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“…ending at dimension zero in the above diagram. These higher exterior homotopy invariants are powerful tools for the study and classification of exterior spaces, see [3], [5], [13]. In the next sections, we will only consider the zero dimensional part of this sequence for the study of end points (and their basins) of an exterior discrete semi-flow.…”

Section: Natural Transformations For Limit and End Spaces Of Exteriormentioning

confidence: 99%

Regions of attraction, limits and end points of an exterior discrete semi-flow

Calcines

Paricio

Grandes

et al. 2019

Homology, Homotopy and Applications

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An exterior space is a topological space provided with a quasifilter of open subsets (closed by finite intersections). In this work, we analyze some relations between the notion of an exterior space and the notion of a discrete semi-flow.On the one hand, for an exterior space, one can consider limits, bar-limits and different sets of end points (Steenrod,Čech, Brown-Grossman). On the other hand, for a discrete semi-flow, one can analyze fixed points, periodic points, omega-limits, et cetera.In this paper, we introduce a notion of exterior discrete semi-flow, which is a mix of exterior space and discrete semi-flow. We see that a discrete semi-flow can be provided with the structure of an exterior discrete semi-flow by taking as structure of exterior space the family of right-absorbing open subsets, which can be used to study the relation between limits and periodic points and connections between bar-limits and omega-limits. The different notions of end points are used to decompose the region of attraction of an exterior discrete semi-flow as a disjoint union of basins of end points. We also analyze the exterior discrete semi-flow structure induced by the family of open neighborhoods of a given sub-semi-flow.2010 Mathematics Subject Classification. 54H20, 37B99, 18B99, 18A40. Key words and phrases. Discrete semi-flow, periodic point, omega limit, positively Lagrange stable, region of attraction, exterior space, limit space, end point, end space, exterior discrete semi-flow, basin of an end point.Partially supported by the University of La Rioja PROFAI13/15 and an FPI grant from the Government of La Rioja.

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Section: Natural Transformations For Limit and End Spaces Of Exteriormentioning

confidence: 99%

Regions of attraction, limits and end points of an exterior discrete semi-flow

Calcines

Paricio

Grandes

et al. 2019

Homology, Homotopy and Applications

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“…The new category of exterior spaces and maps is complete and cocomplete and contains as a full subcategory the category of spaces and proper maps, see [9,11]. For more properties and applications of exterior homotopy categories we refer the reader to [10,7,4,12,13] and for a survey of proper homotopy to [21]. Definition 2.1.…”

Section: Preliminaries On Exterior Spaces and Dynamical Systemsmentioning

confidence: 99%

Limit and end functors of dynamical systems via exterior spaces

Calcines¹,

Paricio²,

Rodríguez³

2013

Bull. Belg. Math. Soc. Simon Stevin

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In this paper we analyze some applications of the category of exterior spaces to the study of dynamical systems (flows). We study the notion of an absorbing open subset of a dynamical system; i.e., an open subset that contains the "future part" of all the trajectories. The family of all absorbing open subsets is a quasi-filter which gives the structure of an exterior space to the flow. The limit space and end space of an exterior space are used to construct the limit spaces and end spaces of a dynamical system. On the one hand, for a dynamical system two limits spaces L r (X) andL r (X) are constructed and their relations with the subflows of periodic, Poisson stable points and Ω r -limits of X are analyzed. On the other hand, different end spaces are also associated to a dynamical system having the property that any positive semi-trajectory has an end point in these end spaces. This type of construction permits us to consider the subflow containing all trajectories finishing at an end point a. When a runs over the set of all end points, we have an induced decomposition of a dynamical system as a disjoint union of stable (at infinity) subflows.Proof. If x ∈ Ω r (X), by the lemma above, there exists V x ∈ (t X ) x such that X \ V x is r-exterior. By Proposition 6.8, it follows that x ∈L r (X). Theorem 6.13. Let X be a flow. If X is a locally compact regular space, then L r (X) = Ω r (X).Proof. By the corollary aboveL r (X) ⊂ Ω r (X) and by Lemma 6.7, Ω r (X) ⊂L(X). Corollary 6.14. Let X be a flow. If X is a locally compact T 3 space, then L r (X) = P (X),L r (X) = Ω r (X) and L r (X) = P (X) ⊂ P r (X) ⊂ Ω r (X) ⊂ Ω r (X) =L r (X).Proof. This is a consequence of Theorems 6.3 and 6.13.20J.M. GARCÍA CALCINES, L. HERNÁNDEZ PARICIO AND M. TERESA RIVAS RODRÍGUEZ

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“…The category of exterior spaces has been provided with a well developed homotopy theory ( [12,[16][17][18]21]). The study of the exterior and proper homotopy invariants has proved to be useful in the study of non-compact manifolds ( [8,30]), the study of the shape of some compact spaces ( [22]), the L-S proper category ( [14,15]), et cetera.…”

Section: Introductionmentioning

confidence: 99%

“…The new category of exterior spaces and maps is complete and cocomplete and contains as a full subcategory the category of spaces and proper maps, see [16,17]. We refer to [9,11,12,21,22] for further properties and applications of exterior homotopy, and to [26] for a survey of proper homotopy.…”

mentioning

confidence: 99%

Omega limits, prolongational limits and almost periodic points of a continuous flow via exterior spaces

Calcines

Paricio

Rodríguez

2017

Glas. Mat. Ser. III

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Abstract. In this paper we analyse some applications of the category of exterior spaces to the study of dynamical systems (flows). The limit space and end space of an exterior space are used to construct different types of limit spaces and end spaces of a dynamical system. In this work we analyse the relationships between the notions and constructions given by the exterior structures of a continuous flow and the more usual notions of omega-limits, first prolongational limits and several types of almost periodic points (Poisson-stable points, non-wandering points) of a flow.

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Exact Sequences and Closed Model Categories

Cited by 7 publications

References 33 publications

Regions of attraction, limits and end points of an exterior discrete semi-flow

Regions of attraction, limits and end points of an exterior discrete semi-flow

Limit and end functors of dynamical systems via exterior spaces

Omega limits, prolongational limits and almost periodic points of a continuous flow via exterior spaces

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