Related papers: J-Stability in non-archimedean dynamics

We prove the Existential Closedness conjecture for the differential equation of the $j$-function and its derivatives. It states that in a differentially closed field certain equations involving the differential equation of the $j$-function…

Following the ideas of A.~Douady, we give an alternative proof of the authors' result: for any boundary point $c_0$ of the Mandelbrot set $M$, we can find small quasiconformal copies of $M$ in $M$ that are encaged in nested quasiconformal…

Let f be a transcendental entire map that is subhyperbolic, i.e., the intersection of the Fatou set F(f) and the postsingular set P(f) is compact and the intersection of the Julia set J(f) and P(f) is finite. Assume that no asymptotic value…

It is an open problem whether repelling periodic points are dense in the classical Julia set of a non-archimedean rational function of degree more than one. We give a partial positive answer to this question based on a study of a…

We study the J-flow from the point of view of an algebro-geometric stability condition. In terms of this we give a lower bound for the natural associated energy functional, and we show that the blowup behavior found by Fang-Lai is reflected…

We study the dynamics of a generic automorphism $f$ of a Stein manifold with the density property. Such manifolds include all linear algebraic groups. Even in the special case of $\mathbb C^n$, $n\geq 2$, most of our results are new. We…

Let G be a connected reductive group defined over a non-archimedean local field of characteristic 0. We assume G is quasi-split, adjoint and absolutly simple. Let g be the Lie algebra of G. We consider the space of the invariant…

We extend the definition of $n$-dimensional difference equations to complex order $\alpha\in \mathbb{C} $. We investigate the stability of linear systems defined by an $n$-dimensional matrix $A$ and derive conditions for the stability of…

We study the ground states of the one-dimensional non-self-adjoint Jacobi operators in the almost periodic media by using the method of dynamical systems. We show the existence of the ground state. Particularly, in the quasi-periodic media,…

We show that $J-$ stability is open and dense in natural families of meromorphic maps of one complex variable with a finite number of singular values, and even more generally, to finite type maps. This extends the results of…

We show analytically that the dynamical equations for an underdamped ladder of coupled small Josephson junctions can be approximately reduced to the discrete sine-Gordon equation. As numerical confirmation, we solve the coupled Josephson…

We explore the connected/disconnected dichotomy for the Julia set of polynomial automorphisms of C^2. We develop several aspects of the question, which was first studied by Bedford-Smillie. We introduce a new sufficient condition for the…

Consider a polynomial $f$ of degree $d \geq 2$ that has a Siegel disk $\Delta_f$ with a rotation number of bounded type. We prove that there does not exist a hedgehog containing $\Delta_f$. Moreover, if the Julia set $J_f$ of $f$ is…

We study the stability of fluctuations around a homogeneous non-Abelian electric field background that is of a form that is protected from Schwinger pair production. Our analysis identifies the unstable modes and we find a limiting set of…

Let $K$ be an algebraically closed field of characteristic 0. When the Jacobian $({\partial f}/{\partial x})({\partial g}/{\partial y}) - ({\partial g}/{\partial x})({\partial f}/{\partial y})$ is a constant for $f,g\in K[x,y]$, Magnus'…

A Fej\'{e}r-type theorem is proved within the framework of $C^*$-algebras associated with certain irreversible algebraic dynamical systems. This makes it possible to strengthen a result on the structure of the relative commutant of a family…

Let $\Theta=(\theta_{j,k})_{3\times 3}$ be a non-degenerate real skew-symmetric $3\times 3$ matrix, where $\theta_{j,k}\in [0,1).$ For any $\varepsilon>0$, we prove that there exists $\delta>0$ satisfying the following: if $v_1,v_2,v_3$ are…

In recent years, researchers have discovered various large algebraic structures that have surprising finiteness properties, such as FI-modules and Delta-modules. In this paper, we add another example to the growing list: we show that…

Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to…

This paper studies the asymptotic convergence properties of the primal-dual dynamics designed for solving constrained concave optimization problems using classical notions from stability analysis. We motivate the need for this study by…