Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls raised by some fixed power α > 0. We characterize the limiting behavior of the proportion of balls in the bins.The proof uses a dynamical approach to relate the proportion of balls to a vector field. Our main result is that the limit set of the proportion of balls is contained in the equilibria set of the vector field. We also prove that if α < 1 then there is a single point v = v(G, α) with non-zero entries such that the proportion converges to v almost surely.A special case is when G is regular and α ≤ 1. We show e.g. that if G is non-bipartite then the proportion of balls in the bins converges to the uniform measure almost surely.

We construct symbolic dynamics on sets of full measure (with respect to an ergodic measure of positive entropy) for C 1+ε flows on closed smooth three dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a closed C ∞ surface has at least const ×(e hT /T ) simple closed orbits of period less than T , whenever the topological entropy h is positive -and without further assumptions on the curvature. Kat82]. It extends to flows of lesser regularity, under the additional assumption that they possess a measure of maximal entropy (Theorem 8.1). The lower bound Ce ht /T is sharp in many special cases [Hub59, Mar69, PP83, Kni97], but not in the general setup of this paper. For more on this, see §8. The theorem strengthens Katok's bound lim infWe obtain Theorem 1.1 by constructing a symbolic model that is a finite-to-one extension of ϕ. The orbits of this model are easier to understand than those of the original flow. This technique, called "symbolic dynamics", can be traced back to the work of Hadamard, Morse, Artin, and Hedlund.We proceed to describe the symbolic models used in this work. Let G be a directed graph with a countable set of vertices V . We write v → w if there is an edge from v to w, and we assume throughout that for every v there are u, w s.t.Birkhoff cocycle: Suppose r : Σ → R is a function. The Birkhoff sums of r are r n := r + r • σ + · · · + r • σ n−1 (n ≥ 1). There is a unique way to extend the definition to n ≤ 0 in such a way that the cocycle identity r m+n = r n + r m • σ n holds for all m, n ∈ Z: r 0 := 0 and r n := −r |n| • σ −|n| (n < 0). Topological Markov flow: Suppose r : Σ → R + is Hölder continuous and bounded away from zero and infinity. The topological Markov flow with roof function r and base map σ : Σ → Σ is the flow σUniform Poincaré section: The Poincaré section Λ is called uniform if its roof function is bounded away from zero and infinity. If Λ is uniform, then µ Λ is finite and it can be normalized. With this normalization, for every Borel subset E ⊂ Λ and 0All the Poincaré sections considered in this paper will be uniform, and each of them will be the disjoint union of finitely many embedded smooth two dimensional discs. Let ∂Λ denote the union of the boundaries of these discs. The set ∂Λ will introduce discontinuities to the Poincaré map of Λ. Singular set: The singular set of a Poincaré section Λ is S(Λ) :=    p ∈ Λ : p does not have a relative neighborhood V ⊂ Λ \ ∂Λ s.t. V is diffeomorphic to an open disc, and f Λ : V → f Λ (V ) and f −1 Λ : V → f −1 Λ (V ) are diffeomorphisms    . Regular set: Λ := Λ \ S(Λ). SYMBOLIC DYNAMICS FOR FLOWS 5 Basic constructions. Let ϕ be a flow satisfying our standing assumptions.Canonical transverse disc:The following lemmas are standard, see the appendix for proofs.Lemma 2.1. There is a constant r s > 0 which only depends on M and ϕ s.t. for every p ∈ M and 0 < r < r s , S := S r (p) is a C ∞ embedded closed disc, | (X q , T q S)| ≥ 1 2 radians for all q ∈ S, and dist M (·, ·) ≤ dist S (·, ·) ≤ 2 d...

Abstract. Let {T t } be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let µ be an ergodic measure of maximal entropy. We show that either {T t } is Bernoulli, or {T t } is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.

We propose a counting dimension for subsets of Z and prove that, under certain conditions on E, F ⊂ Z, for Lebesgue almost every λ ∈ R the counting dimension of E + ⌊λF ⌋ is at least the minimum between 1 and the sum of the counting dimensions of E and F . Furthermore, if the sum of the counting dimensions of E and F is larger than 1, then E + ⌊λF ⌋ has positive upper Banach density for Lebesgue almost every λ ∈ R. The result has direct consequences when E, F are arithmetic sets, e.g. the integer values of a polynomial with integer coefficients.

An infection spreads in a binary tree Tn of height n as follows: initially, each leaf is either infected by one of k states or it is not infected at all. The infection state of each leaf is independently distributed according to a probability vector p = (p 1 , . . . , p k+1 ). The remaining nodes become infected or not via annihilation and coalescence: nodes whose two children have the same state (infected or not) are infected (or not) by this state; nodes whose two children have different states are not infected; nodes whose only one of the children is infected are infected by this state. In this note we characterize, for every p, the limiting distribution at the root node of Tn as n goes to infinity.We also consider a variant of the model when k = 2 and a mutation can happen, with a fixed probability q, at each infection step. We characterize, in terms of p and q, the limiting distribution at the root node of Tn as n goes to infinity.The distribution at the root node is driven by a dynamical system, and the proofs rely on the analysis of this dynamics.