Ergodic Theory of Generic Continuous Maps

Abdenur, Flavio; Andersson, Martin

doi:10.1007/s00220-012-1622-9

Cited by 17 publications

(44 citation statements)

References 24 publications

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“…And by the same construction, since the dynamics of f N converge to that of f , and in particular the sets U N j and {w N j,i } converge to the sets U j and W j,i , the measures µ f N ,U tend to the measures µ f U . † In other words, a generic homeomorphism is weird; see also [AA13]. (Figure 11), but we can at least observe that convergence of this would be a good indication that the homeomorphism considered is dissipative.…”

Section: P-a Guihéneufmentioning

confidence: 90%

“…that there exists a Borel set of full measure Dynamical properties of spatial discretizations of a generic homeomorphism 1511 whose image under f is zero measure [AA13]. Again, it reflects the regularity of the behaviour of the discretizations of a dissipative homeomorphism: generically, we can describe the behaviour of all (sufficiently fine) discretizations.…”

Section: P-a Guihéneufmentioning

confidence: 97%

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Dynamical properties of spatial discretizations of a generic homeomorphism

Guihéneuf¹

2014

Ergod. Th. Dynam. Sys.

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This paper concerns the link between the dynamical behaviour of a dynamical system and the dynamical behaviour of its numerical simulations. Here, we model numerical truncation as a spatial discretization of the system. Some previous works on well-chosen examples (such as Gambaudo and Tresser [Some difficulties generated by small sinks in the numerical study of dynamical systems: two examples. Phys. Lett. A 94(9) (1983), 412-414]) show that the dynamical behaviours of dynamical systems and of their discretizations can be quite different. We are interested in generic homeomorphisms of compact manifolds. So our aim is to tackle the following question: can the dynamical properties of a generic homeomorphism be detected on the spatial discretizations of this homeomorphism? We will prove that the dynamics of a single discretization of a generic conservative homeomorphism does not depend on the homeomorphism itself, but rather on the grid used for the discretization. Therefore, dynamical properties of a given generic conservative homeomorphism cannot be detected using a single discretization. Nevertheless, we will also prove that some dynamical features of a generic conservative homeomorphism (such as the set of the periods of all periodic points) can be read on a sequence of finer and finer discretizations.

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Section: P-a Guihéneufmentioning

confidence: 90%

Section: P-a Guihéneufmentioning

confidence: 97%

Dynamical properties of spatial discretizations of a generic homeomorphism

Guihéneuf¹

2014

Ergod. Th. Dynam. Sys.

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“…[15] or [42]) and the set of all quasi-regular points is denoted by Q(f ). For x ∈ X, let pω(x) denote the set of all limit points of the sequence Abdenur and Andersson showed in [3] that on manifolds with dim M ≥ 2, for C 0 generic maps f : M → M (the same for homeomorphisms), Lebesgue a.e. x ∈ M is quasi-regular, but f admits no SRB measure.…”

Section: Introductionmentioning

confidence: 99%

On the irregular points for systems with the shadowing property

Dong¹,

Oprocha

Tian³

2017

Ergod. Th. Dynam. Sys.

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We prove that when f is a continuous self-map acting on a compact metric space (X, d) which satisfies the shadowing property, then the set of irregular points (i.e. points with divergent Birkhoff averages) has full entropy.Using this fact we prove that in the class of C 0 -generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergent Birkhoff averages. In particular, the time average of atomic measures along orbit of such points converges to some SRB-like measure in the weak * topology. Moreover, such points carry zero entropy. In contrast, irregular points are nonobservable but carry infinite entropy.

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“…By hypothesis Leb(A ε (µ)) > 0 (see Equation (1)), then ε = 1 2 min(ε, Leb(A ε (µ))) > 0. As f is generic it satisfies the conclusions of the shredding lemma of F. Abdenur and M. Andersson (see [AA13]) applied to f and ε , in particular there exists B ⊂ A ε (µ) and O ⊂ T 2 such that:…”

Section: Generic Propertiesmentioning

confidence: 75%

How roundoff errors help to compute the rotation set of torus homeomorphisms

Guihéneuf

2015

Topology and its Applications

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The goals of this paper are to obtain theoretical models of what happens when a computer calculates the rotation set of a homeomorphism, and to find a good algorithm to perform simulations of this rotation set. To do that we introduce the notion of observable rotation set, which takes into account the fact that we can only detect phenomenon appearing on positive Lebesgue measure sets; we also define the asymptotic discretized rotation set which in addition takes into account the fact that the computer calculates with a finite number of digits It appears that both theoretical results and simulations suggest that the asymptotic discretized rotation set is a much better approximation of the rotation set than the observable rotation set, in other words we need to do coarse roundoff errors to obtain numerically the rotation set.

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Ergodic Theory of Generic Continuous Maps

Cited by 17 publications

References 24 publications

Dynamical properties of spatial discretizations of a generic homeomorphism

Dynamical properties of spatial discretizations of a generic homeomorphism

On the irregular points for systems with the shadowing property

How roundoff errors help to compute the rotation set of torus homeomorphisms

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