We study convergence of ε-Nash dynamics in congestion games when delay functions of all resources are similar. Delay functions are said to be similar if their values are within a polynomial factor at every congestion level. We show that for any ε > 0, an ε-Nash dynamics in symmetric congestion games with similar resources converge in steps polynomial in 1/ε and the number of players are resourc...

We investigate the dynamics of semigroups generated by polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials. We show that for such a polynomial semigroup, if A and B are two connected components of the Julia set, then one of A and B surrounds the other. A criterion for the Jul...

We prove a dynamical version of the Mordell-Lang conjecture for subvarieties of the affine space A over a p-adic field, endowed with polynomial actions on each coordinate of A. We use analytic methods similar to the ones employed by Skolem, Chabauty, and Coleman for studying diophantine equations.

Third-order explicit autonomous di€erential equations, commonly called jerky dynamics, constitute a powerful approach to understand the properties of functionally very simple but nonlinear three-dimensional dynamical systems that can exhibit chaotic long-time behavior. In this paper, we investigate the dynamics that can be generated by the two simplest polynomial jerky dynamics that, up to thes...

The question of which terms of a recurrence sequence fail to have primitive prime divisors has been significantly studied for several classes of linear recurrence sequences and for elliptic divisibility sequences. In this paper, we consider the question for sequences generated by the iteration of a polynomial. For two classes of polynomials f(x) ∈ Z[x] and initial values a1 ∈ Z, we show that th...

We extend a recent result on the existence of wandering domains of polynomial functions defined over the p-adic field Cp to any algebraically closed complete non-archimedean field CK with residue characteristic p > 0. We also prove that polynomials with wandering domains form a dense subset of a certain one-dimensional family of degree p + 1 polynomials in CK [z]. Given a rational function φ ∈ ...

Assuming that this is false, one can choose a subsequence, denoted by u j again, such that ‖u j‖ > j . Let z j := u j/‖u j‖. Then ‖z j‖ = 1, z j is orthogonal to N (I + T ), and z j + T z j = f j/‖u j‖ → 0. As before, it follows that z j → z in H , and passing to the limit in the equation for z j one gets z + T z = 0. Since z is orthogonal to N (I + T ), it follows that z = 0. This is a contrad...

Let f : C n ! C be a polynomial function. We deene global polar invariants associated to bres of f and we describe a CW-complex model of a bre. We show how to use aane polar curves in order to study the monodromy around atypical values of f, including the value innnity. We give a zeta-function formula for such a monodromy. Topologie des fonctions polynomiales et dynamique mono-dromique R esum e...