Multiplicity of fixed points and growth of ε-neighborhoods of orbits

Mardešić, Pavao; Resman, Maja; Županović, Vesna

doi:10.1016/j.jde.2012.06.020

Cited by 26 publications

(50 citation statements)

References 16 publications

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“…In the famous Weyl-Berry conjecture, the box dimension and the Minkowski content of the boundary for Laplace equation are related to the eigenvalue counting function, see [9]. In discrete systems, box dimension and Minkowski content of orbits which accumulate at a fixed point reveal multiplicity of the generating function, moment of bifurcation or the complexity of bifurcation, see [7], [13], [20].…”

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confidence: 99%

ε-neighbourhoods of orbits of parabolic diffeomorphisms and cohomological equations

Resman¹

2014

Nonlinearity

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In this article, we study analyticity properties of (directed) areas of ε-neighborhoods of orbits of parabolic germs. The article is motivated by the question of analytic classification using ε-neighborhoods of orbits in the simplest formal class.We show that the coefficient in front of ε 2 term in the asymptotic expansion in ε, which we call the principal part of the area, is a sectorially analytic function in the initial point of the orbit. It satisfies a cohomological equation similar to the standard trivialization equation for parabolic diffeomorphisms.We give necessary and sufficient conditions on a diffeomorphism f for the existence of globally analytic solution of this equation. Furthermore, we introduce new classification type for diffeomorphisms implied by this new equation and investigate the relative position of its classes with respect to the analytic classes.

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confidence: 99%

ε-neighbourhoods of orbits of parabolic diffeomorphisms and cohomological equations

Resman¹

2014

Nonlinearity

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“…For details, see the proof of Proposition 5. 11. Note that the formal normal form of a strongly hyperbolic transseries cannot be expressed as the formal time-one map of a vector field in L. The exponential of a parabolic vector field does not converge in L in any of the three topologies that we mentioned on page 11.…”

Section: Consider Two Transfinite Sequencesmentioning

confidence: 96%

“…Proof of Proposition 5. 11. The assumption ord(ξ) guarantees that X = ξ d dx is a small operator in the sense of Definition 5.…”

Section: 2mentioning

confidence: 99%

“…The question that we pose is if we can recognize a germ by fractal properties of its realizations (orbits). Fractal properties of orbits of Poincaré maps around limit periodic sets have been studied in [11] and [20]. In the differentiable cases of elliptic points and limit cycles, it was proven in [20] that fractal analysis of orbits of Poincaré maps gives the multiplicity and the cyclicity.…”

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Normal forms and embeddings for power-log transseries

Mardešić

Resman

Rolin

et al. 2016

Advances in Mathematics

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International audienceDulac series are asymptotic expansions of first return maps in a neighborhood of a hyperbolic polycycle. In this article, we consider two algebras of power-log transseries (generalized series) which extend the algebra of Dulac series. We give a formal normal form and prove a formal embedding theorem for transseries in these algebras. (C) 2016 Elsevier Inc. All rights reserved

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“…The Poincaré map near a saddle loop, although it is not analytic, shows its cyclycity (see [Roussarie(1998)], [Zhao & Wang(2009)]). In [Mardešić et al(2011)] this was investigated from the point of view of fractal geometry. The classical box dimension was not fine enough to distinguish between all the cases which could appear, so a generalisation called the critical Minkowski order has been introduced.…”

Section: Introductionmentioning

confidence: 99%