scholarly journals Genericity of the multibump dynamics for almost periodic Duffing-like systems

1999 ◽  
Vol 129 (5) ◽  
pp. 885-901 ◽  
Author(s):  
Francesca Alessio ◽  
Paolo Caldiroli ◽  
Piero Montecchiari
Keyword(s):  
Variational Methods ◽  
Dense Subset ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Almost Periodic

In this paper we consider ‘slowly’ oscillating perturbations of almost periodic Duffing-like systems, i.e. systems of the form ü = u − (a(t) + α(wt))W′(u), t ∈ ℝ, u ∈ ℝN, where W ∈ C2N(ℝN, ℝ) is superquadratic and a and α are positive and almost periodic. By variational methods, we prove that if w > 0 is small enough, then the system admits a multibump dynamics. As a consequence we get that the system ü = u − a(t)W′(u), t ∈ ℝ, u ∈ ℝN, admits multibump solutions whenever a belongs to an open dense subset of the set of positive almost periodic continuous functions.

Ergodic optimization of super-continuous functions on shift spaces

2011 ◽  
Vol 32 (6) ◽  
pp. 2071-2082 ◽  
Author(s):  
ANTHONY QUAS ◽  
JASON SIEFKEN
Keyword(s):  
Periodic Orbit ◽  
Dense Subset ◽  
Continuous Functions ◽  
Probability Measures ◽  
Open Dense Subset ◽  
Separable Space ◽  
Ergodic Optimization ◽  
Full Shift ◽  
Separable Spaces ◽  
Positive Results

AbstractErgodic optimization is the process of finding invariant probability measures that maximize the integral of a given function. It has been conjectured that ‘most’ functions are optimized by measures supported on a periodic orbit, and it has been proved in several separable spaces that an open and dense subset of functions is optimized by measures supported on a periodic orbit. All known positive results have been for separable spaces. We give in this paper the first positive result for a non-separable space, the space of super-continuous functions on the full shift, where the set of functions optimized by periodic orbit measures contains an open dense subset.

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

scholarly journals Contra-continuous functions and stronglyS-closed spaces

1996 ◽  
Vol 19 (2) ◽  
pp. 303-310 ◽  
Author(s):  
J. Dontchev
Keyword(s):  
Dense Subset ◽  
Continuous Functions ◽  
Closed Space ◽  
Closed Sets ◽  
Open Set ◽  
Locally Closed Sets ◽  
Continuous Images

In 1989 Ganster and Reilly [6] introduced and studied the notion ofLC-continuous functions via the concept of locally closed sets. In this paper we consider a stronger form ofLC-continuity called contra-continuity. We call a functionf:(X,τ)→(Y,σ)contra-continuous if the preimage of every open set is closed. A space(X,τ)is called stronglyS-closed if it has a finite dense subset or equivalently if every cover of(X,τ)by closed sets has a finite subcover. We prove that contra-continuous images of stronglyS-closed spaces are compact as well as that contra-continuous,β-continuous images ofS-closed spaces are also compact. We show that every stronglyS-closed space satisfies FCC and hence is nearly compact.

Multisymplectic Reduction for Proper Actions

2004 ◽  
Vol 56 (3) ◽  
pp. 638-654 ◽  
Author(s):  
Jędrzej Śniatycki
Keyword(s):  
Symmetry Group ◽  
Dense Subset ◽  
Variational Problems ◽  
Open Dense Subset ◽  
Proper Action ◽  
Partial Derivatives ◽  
Proper Actions ◽  
Reduced Equations ◽  
Definition Of

AbstractWe consider symmetries of the Dedonder equation arising from variational problems with partial derivatives. Assuming a proper action of the symmetry group, we identify a set of reduced equations on an open dense subset of the domain of definition of the fields under consideration. By continuity, the Dedonder equation is satisfied whenever the reduced equations are satisfied.

Non-Extendability of Bounded Continuous Functions

1980 ◽  
Vol 32 (4) ◽  
pp. 867-879
Author(s):  
Ronnie Levy
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Dense Subspace ◽  
Compact Metric Space ◽  
Bounded Continuous Function ◽  
Bounded Functions ◽  
Bounded Continuous Functions ◽  
Completely Metrizable

If X is a dense subspace of Y, much is known about the question of when every bounded continuous real-valued function on X extends to a continuous function on Y. Indeed, this is one of the central topics of [5]. In this paper we are interested in the opposite question: When are there continuous bounded real-valued functions on X which extend to no point of Y – X? (Of course, we cannot hope that every function on X fails to extend since the restrictions to X of continuous functions on Y extend to Y.) In this paper, we show that if Y is a compact metric space and if X is a dense subset of Y, then X admits a bounded continuous function which extends to no point of Y – X if and only if X is completely metrizable. We also show that for certain spaces Y and dense subsets X, the set of bounded functions on X which extend to a point of Y – X form a first category subset of C*(X).

scholarly journals QUANTIZATION OF A MODULI SPACE OF PARABOLIC HIGGS BUNDLES

2004 ◽  
Vol 15 (09) ◽  
pp. 907-917 ◽  
Author(s):  
INDRANIL BISWAS ◽  
AVIJIT MUKHERJEE
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Symplectic Form ◽  
Dense Subset ◽  
Projective Structure ◽  
Open Dense Subset ◽  
Higgs Bundles ◽  
Smooth Variety ◽  
Structure Formula

Let [Formula: see text] be a moduli space of stable parabolic Higgs bundles of rank two over a Riemann surface X. It is a smooth variety defined over [Formula: see text] equipped with a holomorphic symplectic form. Fix a projective structure [Formula: see text] on X. Using [Formula: see text], we construct a quantization of a certain Zariski open dense subset of the symplectic variety [Formula: see text].

The Dynamical Mordell–Lang Conjecture for Skew-Linear Self-Maps. Appendix by Michael Wibmer

2018 ◽  
Vol 2020 (21) ◽  
pp. 7433-7453
Author(s):  
Dragos Ghioca ◽  
Junyi Xie
Keyword(s):  
Linear Transformation ◽  
Difference Equations ◽  
Dense Subset ◽  
Closed Field ◽  
Open Dense Subset ◽  
Irreducible Curve ◽  
Algebraically Closed Field ◽  
Set Of Points ◽  
Dense Set

Abstract Let $k$ be an algebraically closed field of characteristic $0$, let $N\in{\mathbb{N}}$, let $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ be a nonconstant morphism, and let $A:{\mathbb{A}}^N{\longrightarrow } {\mathbb{A}}^N$ be a linear transformation defined over $k({\mathbb{P}}^1)$, that is, for a Zariski-open dense subset $U\subset{\mathbb{P}}^1$, we have that for $x\in U(k)$, the specialization $A(x)$ is an $N$-by-$N$ matrix with entries in $k$. We let $f:{\mathbb{P}}^1\times{\mathbb{A}}^N{\dashrightarrow } {\mathbb{P}}^1\times{\mathbb{A}}^N$ be the rational endomorphism given by $(x,y)\mapsto (\,g(x), A(x)y)$. We prove that if $g$ induces an automorphism of ${\mathbb{A}}^1\subset{\mathbb{P}}^1$, then each irreducible curve $C\subset{\mathbb{A}}^1\times{\mathbb{A}}^N$ that intersects some orbit $\mathcal{O}_f(z)$ in infinitely many points must be periodic under the action of $f$. Furthermore, in the case $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ is an endomorphism of degree greater than $1$, then we prove that each irreducible subvariety $Y\subset{\mathbb{P}}^1\times{\mathbb{A}}^N$ intersecting an orbit $\mathcal{O}_f(z)$ in a Zariski dense set of points must be periodic. Our results provide the desired conclusion in the Dynamical Mordell–Lang Conjecture in a couple new instances. Moreover, our results have interesting consequences toward a conjecture of Rubel and toward a generalized Skolem–Mahler–Lech problem proposed by Wibmer in the context of difference equations. In the appendix it is shown that the results can also be used to construct Picard–Vessiot extensions in the ring of sequences.

V.—Les transformations asymptotiquement presque périodiques discontinues et le lemme ergodique. (Première Note.)

1950 ◽  
Vol 63 (1) ◽  
pp. 61-68
Author(s):  
Maurice Fréchet
Keyword(s):  
Ergodic Theorem ◽  
Continuous Functions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Discontinuous Functions ◽  
Extended Class ◽  
Class Of Functions ◽  
Asymptotically Almost Periodic

SynopsisWith the aim of establishing, under wide conditions, the ergodic theorem of G. D. Birkhoff, the author extends the class of asymptotically almost-periodic functions, considering now not only continuous functions, as he had already done in 1943, but discontinuous functions. Definitions and properties of the extended class of functions are set out, some comparisons being made with almost-periodic functions in the sense of Bohr, Stepanoff, Weyl and Besicovitch. Applications to the ergodic theorem are adumbrated.

On generic hypersurfaces in ℝn

1981 ◽  
Vol 90 (3) ◽  
pp. 389-394 ◽  
Author(s):  
J. W. Bruce
Keyword(s):  
Differential Geometry ◽  
Euclidean Space ◽  
Dense Subset ◽  
Precise Definition ◽  
Open Dense Subset ◽  
Curves And Surfaces ◽  
Definition Of

In this paper we consider certain questions concerning the differential geometry of generic hypersurfaces in ℝn. Our results prove, for example, that the curve of rib points of a generic surface in ℝ3 has transverse self-intersections.In (4) Porteous discussed (amongst other things) the generic geometry of curves and surfaces in ℝ3. Subsequently Looijenga ((3) and see also (5)) gave a more precise definition of the term generic and showed that an open dense subset of smooth embeddings of manifolds in Euclidean space were indeed generic.

scholarly journals Variational Methods for Almost Periodic Solutions of a Class of Neutral Delay Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-13 ◽  
Author(s):  
M. Ayachi ◽  
J. Blot
Keyword(s):  
Variational Methods ◽  
Periodic Solutions ◽  
Delay Equations ◽  
Delay Equation ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Neutral Delay ◽  
Density Result

We provide new variational settings to study the a.p. (almost periodic) solutions of a class of nonlinear neutral delay equations. We extend Shu and Xu (2006) variational setting for periodic solutions of nonlinear neutral delay equation to the almost periodic settings. We obtain results on the structure of the set of the a.p. solutions, results of existence of a.p. solutions, results of existence of a.p. solutions, and also a density result for the forced equations.

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