Let A ⊂ R be a set-germ at 0 ∈ R such that 0 ∈ A. We say that r ∈ S is a direction of A at 0 ∈ R if there is a sequence of points {xi} ⊂ A\{0} tending to 0 ∈ R such that xi ‖xi‖ → r as i → ∞. Let D(A) denote the set of all directions of A at 0 ∈ R. Let A, B ⊂ R be subanalytic set-germs at 0 ∈ R such that 0 ∈ A ∩ B. We study the problem of whether the dimension of the common direction set, dim(D...

Surface Homeomorphisms That Do Not Extend to Any Handlebody and the Johnson Filtration

Notes for the summer school Symbolic dynamics and homeomorphisms of the Cantor set,

My research primarily uses close analysis of the dynamics of a group or semigroup action on a space to discern properties of the group or semigroup. In particular, I am primarily interested in groups of automorphisms of a space and in semigroups of endomorphisms of a space. Supporting this, I have a secondary focus on topological and symbolic dynamics, which itself has lead me to an interest in...

Two invertible dynamical systems (X,A, μ, T ) and (Y,B, ν, S) where X, Y are metrizable spaces and T , S are homeomorphisms on X and Y , are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping φ from a subset X0 of X of full measure to a subset Y0 of Y of full measure such that φ|X0 is continuous in the relative topology on X0, φ|Y0 is continuous in t...

Abstract We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms class annulus we describe, Handel–Anosov–Katok (HAK) homeomorphisms, generalize the homeomorphism first described by Handel. Given HAK and Cantor set, pseudo-suspension yields new space combines features both original homeomorphisms. This allows us answer well known open question providing examp...

This paper studies the behavior under iteration of the maps Tab(x, y) = (Fab(x)−y, x) of the plane R2, in which Fab(x) = ax if x ≥ 0 and bx if x < 0. The orbits under iteration correspond to solutions of the nonlinear difference equation xn+2 = 1/2(a− b)|xn+1|+1/2(a+ b)xn+1−xn. This family of piecewise-linear maps has the parameter space (a, b) ∈ R2. These maps are area-preserving homeomorphism...

This paper studies the behavior under iteration of the maps Tab(x, y) = (Fab(x) − y, x) of the plane R2, in which Fab(x) = ax if x ≥ 0 and bx if x < 0. The orbits under iteration correspond to solutions of the difference equation xn+2 = 1/2(a−b)|xn+1|+1/2(a+b)xn+1−xn. This family of piecewise-linear maps of the plane has the parameter space (a, b) ∈ R2. These maps are area-preserving homeomorph...