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Register a new user{alpha}-limit sets and Lyapunov function for maps with one topological attractor
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We consider the topological behaviors of continuous maps with one topological attractor on compact metric space $X$. This kind of map is a generalization of maps such as topologically expansive Lorenz map, unimodal map without homtervals and so on. We provide a leveled $A$-$R$ pair decomposition for such maps, and characterize $alpha$-limit set of each point. Based on weak Morse decomposition of $X$, we construct a bounded Lyapunov function $V(x)$, which give a clear description of orbit behavior of each point in $X$ except a meager set.
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A rigorous proof of a theorem on the coexistence of smooth Lyapunov function and smooth planar dynamical system with one arbitrary limit cycle is given, combining with a novel decomposition of the dynamical system from the perspective of mechanics. We base on this dynamic structure incorporating several efforts of this dynamic structure on fixed points, limit cycles and chaos, as well as on relevant known results, such as Schoenflies theorem, Riemann mapping theorem, boundary correspondence theorem and differential geometry theory, to prove this coexistence. We divide our procedure into three steps. We first introduce a new definition of Lyapunov function for these three types of attractors. Next, we prove a lemma that arbitrary simple closed curve in plane is diffeomorphic to the unit circle. Then, the strict construction of smooth Lyapunov function of the system with circle as limit cycle is given by the definition of a potential function. And then, a theorem is hence obtained: The smooth Lyapunov function always exists for the smooth planar dynamical system with one arbitrary limit cycle. Finally, by discussing the two criteria for system dissipation(divergence and dissipation power), we find they are not equal, and explain the meaning of dissipation in an infinitely repeated motion of limit cycle.
We prove that in an open and dense set, Symplectic linear cocycles over time one maps of Anosov flows, have positive Lyapunov exponents for SRB measures.
We study the attractor of Iterated Function Systems composed of infinitely many affine, homogeneous maps. In the special case of second generation IFS, defined herein, we conjecture that the attractor consists of a finite number of non-overlapping intervals. Numerical techniques are described to test this conjecture, and a partial rigorous result in this direction is proven.
We prove that there exists an open and dense subset $mathcal{U}$ in the space of $C^{2}$ expanding self-maps of the circle $mathbb{T}$ such that the Lyapunov minimizing measures of any $Tin{mathcal U}$ are uniquely supported on a periodic orbit.This answers a conjecture of Jenkinson-Morris in the $C^2$ topology.
Let $C(mathbf I)$ be the set of all continuous self-maps from ${mathbf I}=[0,1]$ with the topology of uniformly convergence. A map $fin C({mathbf I})$ is called a transitive map if for every pair of non-empty open sets $U,V$ in $mathbf{I}$, there exists a positive integer $n$ such that $Ucap f^{-n}(V) ot=emptyset.$ We note $T(mathbf{I})$ and $overline{T(mathbf{I})}$ to be the sets of all transitive maps and its closure in the space $C(mathbf I)$. In this paper, we show that $T(mathbf{I})$ and $overline{T(mathbf{I})}$ are homeomorphic to the separable Hilbert space $ell_2$.
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