Sofic Shifts via Conley Index Theory: Computing Lower Bounds on Recurrent Dynamics for Maps

Day, Sarah; Frongillo, Rafael M.

doi:10.1137/18m1192007

Cited by 9 publications

(2 citation statements)

References 29 publications

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“…To understand the recurrent behavior, we study the dynamics on individual Morse sets. (The Conley index can rigorously relate these multivalued map dynamics to the dynamics of the underlying system; see, for example, [5,9,11,12,15,17,18,28,37,46]. )…”

Section: Local Persistencementioning

confidence: 99%

Persistence of Morse decompositions over grid resolution for maps and time series

Wiseman¹

2020

Preprint

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We can approximate a continuous map f : X → X of a compact metric space by discretizing the space into a grid. Through either the map itself or a time series, f induces a multivalued grid map F . The dynamical properties of F depend on the resolution of the grid, and we study the persistence of these properties as we change the resolution. In particular, we look at the persistence of Morse decompositions, at both the global (Morse graph) and local (individual Morse set) levels, using several notions of persistence -graph structure, persistent homology, and mixing properties.

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Section: Local Persistencementioning

confidence: 99%

Persistence of Morse decompositions over grid resolution for maps and time series

Wiseman¹

2020

Preprint

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“…They also have an array of applications both within and outside of dynamical systems, including billiards, ergodic theory, continuous dynamics, and information theory, automata theory, and matrix theory [19]. In particular, one motivation for the present work is the set of computational problems that arise in application to continuous maps via Conley index theory [8,15,16]. Table 1: Natural decision problems for sofic shifts.…”

Section: Introductionmentioning

confidence: 99%

Computational complexity of problems for deterministic presentations of sofic shifts

Cai¹,

Frongillo²

2021

Preprint

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Sofic shifts are symbolic dynamical systems defined by the set of bi-infinite sequences on an edge-labeled directed graph, called a presentation. We study the computational complexity of an array of natural decision problems about presentations of sofic shifts, such as whether a given graph presents a shift of finite type, or an irreducible shift; whether one graph presents a subshift of another; and whether a given presentation is minimal, or has a synchronizing word. Leveraging connections to automata theory, we first observe that these problems are all decidable in polynomial time when the given presentation is irreducible (strongly connected), via algorithms both known and novel to this work. For the general (reducible) case, however, we show they are all PSPACE-complete. All but one of these problems (subshift) remain polynomial-time solvable when restricting to synchronizing deterministic presentations. We also study the size of synchronizing words and synchronizing deterministic presentations.

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Morse Predecomposition of an Invariant Set

Lipiński,

Mischaikow,

Mrozek

2024

Qual. Theory Dyn. Syst.

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Motivated by the study of recurrent orbits and dynamics within a Morse set of a Morse decomposition we introduce the concept of Morse predecomposition of an isolated invariant set within the setting of both combinatorial and classical dynamical systems. While Morse decomposition summarizes solely the gradient part of a dynamical system, the developed generalization extends to the recurrent component as well. In particular, a chain recurrent set, which is indecomposable in terms of Morse decomposition, can be represented more finely in the Morse predecomposition framework. This generalization is achieved by forgoing the poset structure inherent to Morse decomposition and relaxing the notion of connection between Morse sets (elements of Morse decomposition) in favor of what we term ’links’. We prove that a Morse decomposition is a special case of Morse predecomposition indexed by a poset. Additionally, we show how a Morse predecomposition may be condensed back to retrieve a Morse decomposition.

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Sofic Shifts via Conley Index Theory: Computing Lower Bounds on Recurrent Dynamics for Maps

Cited by 9 publications

References 29 publications

Persistence of Morse decompositions over grid resolution for maps and time series

Persistence of Morse decompositions over grid resolution for maps and time series

Computational complexity of problems for deterministic presentations of sofic shifts

Morse Predecomposition of an Invariant Set

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