[Paper Review] Transitivity of Surface Dynamics Lifted to Abelian Covers

Philip Boyland|ArXiv.org|Apr 14, 2008

Mathematical Dynamics and Fractals16 references3 citations

TL;DR

This paper establishes the equivalence of three conditions for rel pseudo-Anosov maps on compact surfaces: (a) $H_1$-transitivity, (b) spectral radius one of the induced action on first homology, and (c) existence of a dense leaf in the lifted unstable foliation to the universal Abelian cover. The proof relies on characterizing transitivity in twisted ${\mathbb{Z}}^d$-extensions of transitive subshifts of finite type, using rotation sets and the finite lifting property.

ABSTRACT

A homeomorphism f of a manifold M is called H_1-transitive if there is a transitive lift of an iterate of f to the universal Abelian cover M. Roughly speaking, this means that f has orbits which repeatedly and densely explore all elements of H_1(M). For a rel pseudo-Anosov map ϕof a compact surface M we show that the following are equivalent: (a) ϕis H_1-transitive, (b) the action of ϕon H_1(M) has spectral radius one, and (c) the lifts of the invariant foliations of ϕto M have dense leaves. The proof relies on a characterization of transitivity for twisted \Z^d-extensions of a transitive subshift of finite type.

Motivation & Objective

• To characterize $H_1$-transitivity for rel pseudo-Anosov maps on compact surfaces using algebraic, dynamical, and topological invariants.
• To establish a precise link between the spectral radius of the map's action on $H_1(M)$ and the transitivity of its lift to the universal Abelian cover.
• To determine when the leaves of the lifted unstable foliation are dense in the universal Abelian cover.
• To extend known results on twisted skew products to the case of nontrivial homological twisting arising from rel pseudo-Anosov dynamics.

Proposed method

• Characterize transitivity of twisted ${\mathbb{Z}}^d$-extensions of transitive subshifts of finite type via rotation sets and the finite lifting property.
• Use the Fried quotient construction to analyze the dynamics of lifts to the universal Abelian cover.
• Apply the concept of rotation vectors and rotation sets to formalize the 'all directions' condition necessary for transitivity.
• Leverage the quasi-isometry between the symbolic model and the lifted manifold to relate dynamics on the cover to homological behavior.
• Utilize the finite lifting property as a key tool to ensure that transitivity in the base implies transitivity in the cover under suitable conditions.
• Employ the Baire category theorem to show that the set of points whose leaves are dense in the universal cover is a dense $G_\delta$-set when at least one leaf is dense.

Experimental results

Research questions

• RQ1 Under what conditions is a rel pseudo-Anosov map $H_1$-transitive?
• RQ2 How does the spectral radius of the map's action on $H_1(M)$ relate to the transitivity of its lift to the universal Abelian cover?
• RQ3 When do the leaves of the lifted unstable foliation become dense in the universal Abelian cover?
• RQ4 What role does the rotation set of the lifted dynamics play in determining transitivity?
• RQ5 Can the finite lifting property be used to characterize transitivity in nontrivial twisted extensions of subshifts of finite type?

Key findings

• A rel pseudo-Anosov map $\phi$ is $H_1$-transitive if and only if the spectral radius of $\phi_*$ on $H_1(M)$ is exactly one.
• If the spectral radius of $\phi_*$ is one, then there exists a leaf of the lifted unstable foliation $\tilde{\mathcal{F}}^u$ that is dense in the universal Abelian cover $\tilde{M}$.
• When $\phi$ is $H_1$-transitive, the set of points in $\tilde{M}$ whose unstable leaf is dense is a dense $G_\delta$-set.
• Nontrivial leaves of the lifted foliations are always unbounded in $\tilde{M}$, even when they are not dense.
• The lifted foliations always contain nontrivial leaves that are not dense, even when the spectral radius is one.
• If $\rho(\phi_*) \neq 1$, then no leaf of the lifted foliations is dense in $\tilde{M}$.