We show that the Birkhoff normal form near a positive definite KAM torus is given by the function of Mather. This observation is due to Siburg [Si2], [Si1] in dimension 2. It clarifies the link between the Birkhoff invariants and the action spectrum near the torus. Our extension to high dimension is made possible by a simplification of the proof given in [Si2].
We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...
We determine the asymptotic behaviour of the iterates of the Perron-Frobenius operator for specific interval maps with an indifferent fixed point which gives rise to an infinite invariant measure.
We prove that the automorphism group of the random lattice is not amenable, and we identify the universal minimal flow for the automorphism group of the random distributive lattice.
By a dynamical system we mean the action of the semigroup on a metrizable topological space induced by a continuous selfmap . Let denote the set of all compatible metrics on the space . Our main objective is to show that a selfmap of a compact space is a Banach contraction relative to some if and only if there exists some which, regarded as a -cocycle of the system , is a coboundary.
In this paper piecewise monotonic maps are considered. Let be a finite union of open intervals, and consider the set of all points whose orbits omit . The influence of small perturbations of the endpoints of the intervals in on the dynamical system is investigated. The decomposition of the nonwandering set into maximal topologically transitive subsets behaves very unstably. Nonetheless, it is shown that a maximal topologically transitive subset cannot be completely destroyed by arbitrary...
We give a brief introduction to the Bernoulli shift map as a basic chaotic dynamical system. We give several examples where the iterates of a~mapping can be understood using the Bernoulli shift. Namely, the iteration of real interval maps and iteration of quadratic functions in the complex plain.