Permutations of Integers and pseudo-Anosov maps (original) (raw)

Creating pseudo-Anosov Maps from Permutations and Matrices

2019

We will prove that an ordered block permutation (OBP) (a permutation of n positive integers) when admissible, corresponds to an oriented-fixed (OF) pseudo-Anosov homeomorphism of a closed Riemann surface (with respect to an Abelian differential and fixing all critical trajectories); and conversely, every OF pseudo-Anosov homeomorphism gives rise to an admissible OBP. In particular, a bounded power of any homeomorphism of an oriented surface (after possibly having taken a branched double cover) corresponds to an admissible OBP and is determined by the OBP up to (independent) scaling in the horizontal and vertical directions, which once fixed, the homeomorphism is determined.

A construction of pseudo-Anosov homeomorphisms

Transactions of the American Mathematical Society, 1988

We describe a generalization of Thurston’s original construction of pseudo-Anosov maps on a surface F F of negative Euler characteristic. In fact, we construct whole semigroups of pseudo-Anosov maps by taking appropriate compositions of Dehn twists along certain families of curves; our arguments furthermore apply to give examples of pseudo-Anosov maps on nonorientable surfaces. For each self-map f : F → F f:F \to F arising from our recipe, we construct an invariant "bigon track" (a slight generalization of train track) whose incidence matrix is Perron-Frobenius. Standard arguments produce a projective measured foliation invariant by f f . To finally prove that f f is pseudo-Anosov, we directly produce a transverse invariant projective measured foliation using tangential measures on bigon tracks. As a consequence of our argument, we derive a simple criterion for a surface automorphism to be pseudo-Anosov.

Anosov Diffeomorphisms and {\gamma}$$ γ -Tilings

Communications in Mathematical Physics, 2016

We consider a toral Anosov automorphism G γ : T γ → T γ given by G γ (x, y) = (ax + y, x) in the < v, w > base, where a ∈ N\{1}, γ = 1/(a + 1/(a + 1/. . .)), v = (γ , 1) and w = (−1, γ) in the canonical base of R 2 and T γ = R 2 /(vZ × wZ). We introduce the notion of γ-tilings to prove the existence of a one-to-one correspondence between (i) marked smooth conjugacy classes of Anosov diffeomorphisms, with invariant measures absolutely continuous with respect to the Lebesgue measure, that are in the isotopy class of G γ ; (ii) affine classes of γ-tilings; and (iii) γ-solenoid functions. Solenoid functions provide a parametrization of the infinite dimensional space of the mathematical objects described in these equivalences.

Some invariants for o-permutation maps

1999

In this paper we obtain some formulas which allow us to compute topological and metric entropy and topological pressure for a new class of maps. It is also shown that similar formulas do not hold for metric and topological sequence entropy and a new commutativity problem is posed.

Combinatorics and topology of straightening maps I: compactness and bijectivity

arXiv (Cornell University), 2008

We study the parameter space structure of degree d ≥ 3 one complex variable polynomials as dynamical systems acting on C. We introduce and study straightening maps. These maps are a natural higher degree generalization of the ones introduced by Douady and Hubbard to prove the existence of small copies of the Mandelbrot set inside itself. We establish that straightening maps are always injective and that their image contains all the corresponding hyperbolic systems. Also, we characterize straightening maps with compact domain. Moreover, we give two classes of bijective straightening maps. The first produces an infinite collection of embedded copies of the (d − 1)-fold product of the Mandelbrot set in the connectedness locus of degree d ≥ 3. The second produces an infinite collection of full families of quadratic connected filled Julia sets in the cubic connectedness locus, such that each filled Julia set is quasiconformally embedded.

On the fixed-point set of automorphisms of non-orientable surfaces without boundary

The Epstein Birthday Schrift

Macbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the fixed-point set of each non-identity element of a cyclic group of automorphisms acting on a closed non-orientable surface with one exception; namely, when this element has order 2. In this case the fixed-point set may have simple closed curves (called ovals) as well as fixed points. In this note we extend Macbeath's results to include the number of ovals and also determine whether they are twisted or not.