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A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas

Christopher Deninger, Wilhelm Singhof (2001)

Annales de l’institut Fourier

We construct a two dimensional foliation with dense leaves on the Heisenberg nilmanifold for which smooth leafwise Hodge decomposition does not hold. It is also shown that a certain type of dynamical trace formulas relating periodic orbits with traces on leafwise cohomologies does not hold for arbitrary flows.

An L q ( L ² ) -theory of the generalized Stokes resolvent system in infinite cylinders

Reinhard Farwig, Myong-Hwan Ri (2007)

Studia Mathematica

Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with Σ n - 1 , a bounded domain of class C 1 , 1 , are obtained in the space L q ( ; L ² ( Σ ) ) , q ∈ (1,∞). As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and R-boundedness of operator families.

An ordered structure of rank two related to Dulac's Problem

A. Dolich, P. Speissegger (2008)

Fundamenta Mathematicae

For a vector field ξ on ℝ² we construct, under certain assumptions on ξ, an ordered model-theoretic structure associated to the flow of ξ. We do this in such a way that the set of all limit cycles of ξ is represented by a definable set. This allows us to give two restatements of Dulac’s Problem for ξ - that is, the question whether ξ has finitely many limit cycles-in model-theoretic terms, one involving the recently developed notion of U þ -rank and the other involving the notion of o-minimality.

Decentralized output regulation of large scale nonlinear systems with delay

Zhengtao Ding (2009)


This paper deals with output regulation of a class of large-scale nonlinear systems with delays. Each of the subsystems is in the output feedback form, with nonlinear functions of the subsystem output and the outputs of other subsystems. The system outputs are subject to unknown constant delays. Both the system dynamics and the measurements are subject to unknown disturbances generated from unknown linear exosystems. Decentralized control design approach is adopted to design local controllers using...

Flows near compact invariant sets. Part I

Pedro Teixeira (2013)

Fundamenta Mathematicae

It is proved that near a compact, invariant, proper subset of a C⁰ flow on a locally compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. Theorem 1 shows that the connectedness of the phase space implies the existence of a considerably deeper classification of topological flow behaviour in the vicinity of compact invariant sets than that described in the classical theorems of Ura-Kimura and Bhatia. The proposed classification brings...

Periodic billiard orbits in right triangles

Serge Troubetzkoy (2005)

Annales de l’institut Fourier

There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic, and (iii) the perpendicular periodic orbits fill the corresponding invariant surface.

Periodic segments and Nielsen numbers

Klaudiusz Wójcik (1999)

Banach Center Publications

We prove that the Poincaré map φ ( 0 , T ) has at least N ( h ˜ , c l ( W 0 W 0 - ) ) fixed points (whose trajectories are contained inside the segment W) where the homeomorphism h ˜ is given by the segment W.

Periods of Morse-Smale diffeomorphisms of 𝕊²

Juan Luis García Guirao, Jaume Llibre (2008)

Colloquium Mathematicae

The aim of this paper is to describe the set of periods of a Morse-Smale diffeomorphism of the two-dimensional sphere according to its homotopy class. The main tool for proving this is the Lefschetz fixed point theory.

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