We prove that every infinite nowhere dense compact subset of the interval $I$ is an $\omega$-limit set of homoclinic type for a continuous function from $I$ to $I$.

Let ${\left(f_{\lambda }\right)}_{\lambda \in \Lambda }$ be a holomorphic family of rational mappings of degree $d$ on ${\mathbb{P}}^1\left(ℂ\right)$, with $k$ marked critical points $c_1,...,c_k$. To this data is associated a closed positive current $T_1\wedge \cdots \wedge T_k$ of bidegree $(k,k)$ on $\Lambda$, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which $c_1,...,c_k$ eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of $\mathrm{Supp}(T_1\wedge \cdots \wedge T_k)$.

Equivalence is established between a special class of Painlevé VI equations parametrized by a conformal dimension $\mu$, time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The isomonodromic tau function and solutions of the Euler top equations are explicitly constructed in terms of Wronskian solutions of the 2-vector 1-constrained symplectic Kadomtsev-Petviashvili (CKP) hierarchy by means of Grassmannian formulation. These Wronskian solutions give...

The Teichmüller geodesic flow is the flow obtained by quasiconformal deformation of Riemann surface structures. The goal of this lecture is to show the strong connection between the geometry of the Hodge bundle (a vector bundle over the moduli space of Riemann surfaces) and the dynamics of the Teichmüller geodesic flow. In particular, we shall provide geometric criterions (based on the variational formulas derived by G. Forni) to detect some special orbits (“totally degenerate”) of the Teichmüller...

In this paper we describe the close relationship between invariant evolutions of projective curves and the Hamiltonian evolutions of Adler, Gel'fand and Dikii. We also show how KdV evolutions are related as well to invariant evolutions of projective surfaces.

Let $L:TM\to ℝ$ be a Tonelli Lagrangian function (with $M$ compact and connected and $dimM\ge 2$). The tiered Aubry set (resp. Mañé set) ${𝒜}^T\left(L\right)$ (resp. ${𝒩}^T\left(L\right)$) is the union of the Aubry sets (resp. Mañé sets) $𝒜(L+\lambda )$ (resp. $𝒩(L+\lambda )$) for $\lambda$ closed 1-form. Then1.the set ${𝒩}^T\left(L\right)$ is closed, connected and if $dim{H}^1\left(M\right)\ge 2$, its intersection with any energy level is connected and chain transitive;2.for $L$ generic in the Mañé sense, the sets $\overline{{𝒜}^T\left(L\right)}$ and $\overline{{𝒩}^T\left(L\right)}$ have no interior;3.if the interior of $\overline{{𝒜}^T\left(L\right)}$ is non empty, it contains a dense subset of periodic points.We then give an example...

We answer affirmatively Coven's question [PC]: Suppose f: I → I is a continuous function of the interval such that every point has at least two preimages. Is it true that the topological entropy of f is greater than or equal to log 2?

Let f: [a,b] → [a,b] be a continuous function of the compact real interval such that (i) $card{f}^{-1}\left(y\right)\ge 2$ for every y ∈ [a,b]; (ii) for some m ∈ ∞,2,3,... there is a countable set L ⊂ [a,b] such that $card{f}^{-1}\left(y\right)\ge m$ for every y ∈ [a,b]∖L. We show that the topological entropy of f is greater than or equal to log m. This generalizes our previous result for m = 2.