Related papers: Livsic theorem for matrix cocycles
In this paper we propose some continuation theorems for the periodic problem \begin{equation*} \begin{cases} \, x_{i}' = g_{i}(t,x_{i+1}), &i=1,\ldots,n-1, \\ \, x_{n}' = h(t,x_{1},\ldots,x_{n}), \\ \, x_{i}(0)=x_{i}(T), &i=1,\ldots,n,…
A Lie algebra is Frobenius if it admits a linear functional F such that the Kirillov form F([x,y]) is non-degenerate. If g is the m-th maximal parabolic subalgebra P(n,m) of sl(n) this occurs precisely when (n,m) = 1. We define a "cyclic"…
We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of $\mathrm{GL}(2,\mathbb{C})$ cocycles and establish a Johnson-type theorem for extended CMV matrices, relating the spectrum to the…
We study linear co-cycles in GL(d,R) (or C) depending on a parameter (in a Lipschitz or Holder fashion) and establish Holder regularity of the Lyapunov exponents for the shift dynamics on the base. We also obtain rates of convergence of the…
In this article, we will prove a full topological version of Popa's measurable cocycle superrigidity theorem for full shifts. More precisely, we prove that every H\"older continuous cocycle for the full shifts of every finitely generated…
We consider the linear cocycle $(T,A)$ induced by a measure preserving dynamical system $T:X \to X$ and a map $A:X \to \mathit{SL}(2,\mathbb{R})$. We address the dependence of the upper Lyapunov exponent of $(T,A)$ on the dynamics $T$ when…
We consider finite families of SL(2,R) matrices whose products display uniform exponential growth. These form open subsets of (SL(2,R))^N, and we study their components, boundary, and complement. We also consider the more general situation…
We will show that if a dynamical system has enough constants of motion then a Moser-Weinstein type theorem can be applied for proving the existence of periodic orbits in the case when the linearized system is degenerate.
We use the homological perturbation lemma to give an explicit proof of the cyclic Eilenberg-Zilber theorem for cylindrical modules.
In this paper we generalize [3] and prove that the class of accessible and saddle-conservative cocycles (a wide class which includes cocycles evolving in GL(d,R), SL(d,R) and Sp(d,R) Lp-densely have a simple spectrum. We also generalize [3,…
We consider one-parameter families of smooth circle cocycles over an ergodic transformation in the base, and show that their rotation numbers must be log-H\"older regular with respect to the parameter. As an immediate application, we get a…
We prove that the Lyapunov exponent of quasi-periodic cocyles with singularities behaves continuously over the analytic category. We thereby generalize earlier results, where singularities were either excluded completely or constrained by…
Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the i-th and (i+1)-th Lyapunov exponents for all invariant measures implies the existence of a dominated splitting…
We define the periodic Full Kostant-Toda on every simple Lie algebra, and show its Liouville integrability. More precisely we show that this lattice is given by a Hamiltonian vector field, associated to a Poisson bracket which results from…
The pairings between the cyclic cohomologies and the K-theories of separable $C^\ast$-algebras supply topological invariants that often relate to physical response coefficients of materials. Using three numerical simulations, we exemplify…
The purpose of these notes is to discuss the advances in the theory of Lyapunov exponents of linear $\text{SL}_2(\mathbb{R})$ cocycles over hyperbolic maps. The main focus is around results regarding the positivity of the Lyapunov exponent…
We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $\phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $\phi$…
We prove lifting theorems for complex representations $V$ of finite groups $G$. Let $\sigma=(\sigma_1,\dots,\sigma_n)$ be a minimal system of homogeneous basic invariants and let $d$ be their maximal degree. We prove that any continuous map…
The $H~\mathrm{mod}~K$ theorem gives all possible periodic solutions in a $\Gamma-$equivariant dynamical system, based on the group-theoretical aspects. In addition, it classifies the spatio temporal symmetries that are possible. By the…
Cyclic proof theory studies proofs where cycles are allowed. This is useful for developing proof theory for logics with fixpoint operators: cycles can be used to represent the unfolding of a fixpoint. However, this cyclic character is not…