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Register a new userExpansivity and Shadowing in Linear Dynamics
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In the early 1970s Eisenberg and Hedlund investigated relationships between expansivity and spectrum of operators on Banach spaces. In this paper we establish relationships between notions of expansivity and hypercyclicity, supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space is $c_0$ or $ell_p$ ($1 leq p < infty$), we give complete characterizations of weighted shifts which satisfy various notions of expansivity. We also establish new relationships between notions of expansivity and spectrum. Moreover, we study various notions of shadowing for operators on Banach spaces. In particular, we solve a basic problem in linear dynamics by proving the existence of nonhyperbolic invertible operators with the shadowing property. This also contrasts with the expected results for nonlinear dynamics on compact manifolds, illuminating the richness of dynamics of infinite dimensional linear operators.
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Weighted shifts are an important concrete class of operators in linear dynamics. In particular, they are an essential tool in distinguishing variety dynamical properties. Recently, a systematic study of dynamical properties of composition operators on $L^p$ spaces has been initiated. This class of operators includes weighted shifts and also allows flexibility in construction of other concrete examples. In this article, we study one such concrete class of operators, namely composition operators induced by measures on odometers. In particular, we study measures on odometers which induce mixing and transitive linear operators on $L^p$ spaces.
Let $f$ be a partially hyperbolic diffeomorphism. $f$ is called has the quasi-shadowing property if for any pseudo orbit ${x_k}_{kin mathbb{Z}}$, there is a sequence ${y_k}_{kin mathbb{Z}}$ tracing it in which $y_{k+1}$ lies in the local center leaf of $f(y_k)$ for any $kin mathbb{Z}$. $f$ is called topologically quasi-stable if for any homeomorphism $g$ $C^0$-close to $f$, there exist a continuous map $pi$ and a motion $tau$ along the center foliation such that $picirc g=taucirc fcircpi$. In this paper we prove that if $f$ is dynamically coherent then it has quasi-shadowing and topological quasi-stability properties.
A central question in verification is characterizing when a system has invariants of a certain form, and then synthesizing them. We say a system has a $k$ linear invariant, $k$-LI in short, if it has a conjunction of $k$ linear (non-strict) inequalities -- equivalently, an intersection of $k$ (closed) half spaces -- as an invariant. We present a sufficient condition -- solely in terms of eigenvalues of the $A$-matrix -- for an $n$-dimensional linear dynamical system to have a $k$-LI. Our proof of sufficiency is constructive, and we get a procedure that computes a $k$-LI if the condition holds. We also present a necessary condition, together with many example linear systems where either the sufficient condition, or the necessary is tight, and which show that the gap between the conditions is not easy to overcome. In practice, the gap implies that using our procedure, we synthesize $k$-LI for a larger value of $k$ than what might be necessary. Our result enables analysis of continuous and hybrid systems with linear dynamics in their modes solely using reasoning in the theory of linear arithmetic (polygons), without needing reasoning over nonlinear arithmetic (ellipsoids).
A numerically effective procedure for determining weakly reversible chemical reaction networks that are linearly conjugate to a known reaction network is proposed in this paper. The method is based on translating the structural and algebraic characteristics of weak reversibility to logical statements and solving the obtained set of linear (in)equalities in the framework of mixed integer linear programming. The unknowns in the problem are the reaction rate coefficients and the parameters of the linear conjugacy transformation. The efficacy of the approach is shown through numerical examples.
We continue the study in [21] of the linearizability near an indif- ferent fixed point of a power series f, defined over a field of prime characteristic p. It is known since the work of Herman and Yoccoz [13] in 1981 that Siegels linearization theorem [27] is true also for non- Archimedean fields. However, they also showed that the condition in Siegels theorem is usually not satisfied over fields of prime character- istic. Indeed, as proven in [21], there exist power series f such that the associated conjugacy function diverges. We prove that if the degrees of the monomials of a power series f are divisible by p, then f is analyt- ically linearizable. We find a lower (sometimes the best) bound of the size of the corresponding linearization disc. In the cases where we find the exact size of the linearization disc, we show, using the Weierstrass degree of the conjugacy, that f has an indifferent periodic point on the boundary. We also give a class of polynomials containing a monomial of degree prime to p, such that the conjugacy diverges.
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