Related papers: Is a typical bi-Perron number a pseudo-Anosov dila…

For a surface $S$ with $n$ marked points and fixed genus $g\geq2$, we prove that the logarithm of the minimal dilatation of a pseudo-Anosov homeomorphism of $S$ is on the order of $(\log n)/n$. This is in contrast with the cases of genus…

For any nonorientable closed surface, we determine the minimal dilatation among pseudo-Anosov mapping classes arising from Penner's construction. We deduce that the sequence of minimal Penner dilatations has exactly two accumulation points,…

We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our technique also simplifies Cho and Ham's…

This paper describes a family of pseudo-Anosov braids with small dilatation. The smallest dilatations occurring for braids with 3, 4 and 5 strands appear in this family. A pseudo-Anosov braid with 2g+1 strands determines a hyperelliptic…

We construct sequences of pseudo-Anosov mapping classes whose dilatations behave asymptotically like the inverse of the Euler characteristic of the surface they are defined on. These sequences are used to show that if the genus, g, and…

We prove a new lower bound for the dilatation of an arbitrary pseudo-Anosov map on a surface of genus g with n punctures. Our bound improves the former super-exponential dependence on the genus by a polynomial dependence.

This paper concerns a family of pseudo-Anosov braids with dilatations arbitrarily close to one. The associated graph maps and train tracks have stable "star-like" shapes, and the characteristic polynomials of their transition matrices form…

In this paper, we give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface. We give an explicit construction of the surface and the map…

We prove that the dilatation of any pseudo-Anosov homeomorphism on a translation surface that belong to a hyperelliptic component is bounded from below uniformly by sqrt{2}. This is in contrast to Penner's asymptotic. Penner proved that the…

The theme of this paper is that algebraic complexity implies dynamical complexity for pseudo-Anosov homeomorphisms of a closed surface S_g of genus g. Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov…

The motivation for this paper is to justify a remark of Thurston that the algebraic degree of stretch factors of pseudo-Anosov maps on a surface $S$ can be as high as the dimension of the Teichm\"uller space of $S$. In addition to proving…

We present an explicit sequence of pseudo-Anosov maps $\phi_k: S_{2k}\to S_{2k}$ of surfaces of genus $2k$ whose growth rates converge to one.

We consider the hyperelliptic handlebody group on a closed surface of genus $g$. This is the subgroup of the mapping class group on a closed surface of genus $g$ consisting of isotopy classes of homeomorphisms on the surface that commute…

Let $S$ be a closed Riemann surface of genus $p>1$ with one point removed. In this paper, we identify those point-pushing pseudo-Anosov maps on $S$ that preserve at least one bi-infinite geodesic in the curve complex.

This note gives a brief survey of the minimum dilatation problem for pseudo-Anosov mapping classes, and the first explicit train track description of an infinite family of pseudo-Anosov mapping classes with orientable stable foliations and…

We establish bounds on the minimal asymptotic pseudo-Anosov translation lengths on the complex of curves of orientable surfaces. In particular, for a closed surface with genus $g \geqslant 2$, we show that there are positive constants $a_1…

In [J.Birman, V.Gebhardt, J.Gonzalez-Meneses, Conjugacy in Garside groups I: cyclings, powers and rigidity] authors asked: (open question 2) is the size of USS of a rigid pseudo-Anosov braid is bounded above by some polynomial in the number…

In this project, we develop a new connection between the dynamics of quadratic polynomials on the complex plane and the dynamics of homeomorphisms of surfaces. In particular, given a quadratic polynomial, we investigate whether one can…

We study the minimal dilatation of pseudo-Anosov pure surface braids and provide upper and lower bounds as a function of genus and the number of punctures. For a fixed number of punctures, these bounds tend to infinity as the genus does. We…

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their…