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Register a new userAttracted by an elliptic fixed point
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Bassam Fayad
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We give examples of symplectic diffeomorphisms of R^6 for which the origin is a non-resonant elliptic fixed point which attracts an orbit.
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For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of $n$ variables $K:[0,1]^ntoR$ which are componentwise Lipschitz. The proof is based on coupling and decay of correlation properties of the map. We then give various applications of this inequality to the almost-sure central limit theorem, the kernel density estimation, the empirical measure and the periodogram.
Poincares last geometric theorem (Poincare-Birkhoff Theorem) states that any area-preserving twist map of annulus has at least two fixed points. We replace the area-preserving condition with a weaker intersection property, which states that any essential simple closed curve intersects its image under $f$ at least at one point. The conclusion is that any such map has at least one fixed point. Besides providing a new proof to Poincares geometric theorem, our result also has some applications to reversible systems.
We obtain an extended Reich fixed point theorem for the setting of generalized cone rectangular metric spaces without assuming the normality of the underlying cone. Our work is a generalization of the main result in cite{AAB} and cite{JS}.
We describe the set of all $(3,1)$-rational functions given on the set of complex $p$-adic field $mathbb C_p$ and having a unique fixed point. We study $p$-adic dynamical systems generated by such $(3,1)$-rational functions and show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We obtain Siegel disks of these dynamical systems. Moreover an upper bound for the set of limit points of each trajectory is given. For each $(3,1)$-rational function on $mathbb C_p$ there is a point $hat x=hat x(f)in mathbb C_p$ which is zero in its denominator. We give explicit formulas of radii of spheres (with the center at the fixed point) containing some points that the trajectories (under actions of $f$) of the points after a finite step come to $hat x$. For a class of $(3,1)$-rational functions defined on the set of $p$-adic numbers $mathbb Q_p$ we study ergodicity properties of the corresponding dynamical systems. We show that if $pgeq 3$ then the $p$-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure. For $p=2$, under some conditions we prove non ergodicity and show that there exists a sphere on which the dynamical system is ergodic. Finally, we give a characterization of periodic orbits and some uniformly local properties of the $(3.1)-$rational functions.
This paper introduce a new class of operators and contraction mapping for a cyclical map T on G-metric spaces and the approximately fixed point properties. Also,we prove two general lemmas regarding approximate fixed Point of cyclical contraction mapping on G-metric spaces. Using these results we prove several approximate fixed point theorems for a new class of operators on G-metric spaces (not necessarily complete). These results can be exploited to establish new approximate fixed point theorems for cyclical contraction maps. Further,there is a new class of cyclical operators and contraction mapping on G-metric space (not necessarily complete)which do not need to be continuous.Finally,examples are given to support the usability of our results.
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