The arborification-coarborification transform : analytic, combinatorial, and algebraic aspects
The arithmetic of curves defined by iteration
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...
The Art and Science of Symmetric Design
The asymptotic behavior of invariant collective motion.
The asymptotics of the Perron-Frobenius operator of a class of interval maps preserving infinite measures
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.
The attractors of the common differential operator are determined by hyperbolic and lacunary functions.
The automorphism group of the random lattice is not amenable
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.
The average length of a trajectory in a certain billiard in a flat two-torus.
The Banach contraction mapping principle and cohomology
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.
The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations
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...
The Bernoulli shift as a basic chaotic dynamical system
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.
The Bifurcation Set and Logarithmic Vector Fields.
The bifurcation set for the resonance problem.
The Billiard in a Regular Polygon.
The Birkhoff-Lewis Fixed Point Theorem and a Conjecture of V.I. Arnold.
The bitransitive continuous maps of the interval are conjugate to maps extremely chaotic a. e.
The bounce problem, on n-dimensional Riemannian manifolds
In questo lavoro vengono generalizzati i risultati relativi al problema del rimbalzo unidimensionale studiato in [5]. Precisamente si considera un punto mobile su una varietà Riemanniana -dimensionale, soggetto all’azione di un potenziale variabile nel tempo e vincolato a restare in una parte di avente un bordo di classe contro cui il punto «rimbalza»....
The branch locus for one-dimensional Pisot tiling spaces
If φ is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Φ on the tiling space factors via geometric realization onto a d-dimensional solenoid. Under this realization, the collection of Φ-periodic asymptotic tilings corresponds to a finite set that projects onto the branch locus in a d-torus. We prove that if two such tiling spaces are homeomorphic, then the resulting branch loci are the same up to the action of certain affine maps on the torus.
The C 1 generic diffeomorphism has trivial centralizer
Answering a question of Smale, we prove that the space of C 1 diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.
The interior of zero entropy diffeomorphisms.