Dimension of the Torelli group for Out(F_n)

Let T_n be the kernel of the natural map from Out(F_n) to GL(n,Z). We use combinatorial Morse theory to prove that T_n has an Eilenberg-MacLane space which is (2n-4)-dimensional and that H_{2n-4}(T_n,Z) is not finitely generated (n at least 3). In particular, this recovers the result of Krstic-McCool that T_3 is not finitely presented. We also give a new proof of the fact, due to Magnus, that T_n is finitely generated.Comment: 27 pages, 9 figure

Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents

This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$. One can associate to such an automorphism two currents $\mu^\pm$ and the equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some geometric and dynamical properties of these objects. First, we characterize $\mu$ as the unique measure of maximal entropy. Then we show that the measure $\mu$ has a local product structure and that the currents $\mu^\pm$ have a laminar structure. This allows us to deduce information about periodic points and heteroclinic intersections. For example, we prove that the support of $\mu$ coincides with the closure of the set of saddle points. The methods used combine the pluripotential theory with the theory of non-uniformly hyperbolic dynamical systems

Autologous osteochondral plug transplantation for osteochondrosis of the second metatarsal head: a case report

<p>Abstract</p> <p>Introduction</p> <p>Osteochondrosis of the second or third metatarsal head is a rare condition called Freiberg's disease. To relieve foot pain, conservative treatment with a foot orthosis to reduce weight-bearing and immobilize the foot are recommended. In cases in which such treatments have proved to be ineffective, several surgical treatments have been performed. The appropriate surgical treatment for Freiberg's disease remains controversial.</p> <p>Case presentation</p> <p>We describe the case of a 20-year-old Japanese woman with a three-year history of right forefoot pain and no history of trauma. Two years after treatment by autologous osteochondral plug transplantation, she has neither complaints nor symptoms.</p> <p>Conclusion</p> <p>Autologous osteochondral plug transplantation represents a potentially successful surgical arthroplastic option in preserving the metatarsophalangeal joint in patients with Freiberg's disease.</p