[Paper Review] Forcing theory for transverse trajectories of surface homeomorphisms

Patrice Le Calvez, Fábio Armando Tal|arXiv (Cornell University)|Mar 31, 2015

Mathematical Dynamics and Fractals38 references44 citations

TL;DR

This paper introduces a novel forcing theory for transverse trajectories in surface homeomorphisms isotopic to the identity, using maximal isotopies and transverse foliations to derive dynamical consequences. It proves that if a Hamiltonian homeomorphism of the annulus has a rotation set with zero in its interior, then the rotation set is either a singleton or contains zero in its interior—proving Boyland's conjecture—and establishes new results on rotation sets and entropy for torus and annulus homeomorphisms.

ABSTRACT

This paper studies homeomorphisms of surfaces isotopic to the identity by means of purely topological methods and Brouwer theory. The main development is a novel theory of orbit forcing using maximal isotopies and transverse foliations. This allows us to derive new proofs for some known results as well as some new applications, among which we note the following: we extend Franks and Handel's classification of zero entropy maps of $S^2$ for non-wandering homeomorphisms; we show that if $f$ is a Hamiltonian homeomorphism of the annulus, then the rotation set of $f$ is either a singleton or it contains zero in the interior, proving a conjecture posed by Boyland; we show that there exist compact convex sets of the plane that are not the rotation set of some torus homeomorphisms, proving a first case of the Franks-Misiurewicz Conjecture; we extend a bounded deviation result relative to the rotation set to the general case of torus homeomorphisms.

Motivation & Objective

• To develop a new forcing theory for transverse trajectories of surface homeomorphisms isotopic to the identity using maximal isotopies and transverse foliations.
• To provide topological, non-smooth mechanisms for detecting positive topological entropy in surface homeomorphisms.
• To extend known results on rotation sets and periodic point growth to general homeomorphisms, including non-smooth and non-C1 cases.
• To prove that certain compact convex sets in the plane are not realizable as rotation sets of torus homeomorphisms, supporting the Franks-Misiurewicz Conjecture.
• To generalize bounded deviation results relative to the rotation set to the full class of torus homeomorphisms.

Proposed method

• Define maximal isotopies as isotopies where no fixed point of the homeomorphism has a contractible trajectory relative to the fixed point set of the isotopy.
• Use transverse foliations—singular, oriented foliations dual to maximal isotopies—where trajectories are transverse to leaves.
• Introduce transverse trajectories as paths in the space of leaves, uniquely defined up to equivalence, and use their intersection properties to force new trajectories.
• Apply a key forcing mechanism: if two transverse trajectories intersect transversally, one can construct new trajectories by concatenating and reversing direction at the intersection point.
• Use the realization of finite transverse trajectories as multiples of loops to prove existence of periodic points with prescribed transverse dynamics.
• Leverage self-intersections of infinite transverse trajectories to prove exponential growth of periodic points and positive topological entropy via separation arguments on the torus.

Experimental results

Research questions

• RQ1 Can transverse trajectory intersections in surface homeomorphisms force the existence of new periodic orbits or trajectories?
• RQ2 Under what conditions does the rotation set of a Hamiltonian homeomorphism of the annulus contain zero in its interior?
• RQ3 Which compact convex subsets of R^2 can arise as rotation sets of torus homeomorphisms?
• RQ4 Can positive topological entropy be detected in general homeomorphisms isotopic to the identity without requiring smoothness?
• RQ5 To what extent can bounded deviation results relative to the rotation set be extended beyond C1 or smooth settings?

Key findings

• The rotation set of a Hamiltonian homeomorphism of the annulus is either a singleton or contains zero in its interior, proving Boyland's conjecture.
• There exist compact convex sets in the plane that are not realizable as rotation sets of any torus homeomorphism, confirming a first case of the Franks-Misiurewicz Conjecture.
• The paper establishes a bounded deviation result relative to the rotation set for general torus homeomorphisms, extending previous results to non-smooth maps.
• If two recurrent points have transversally intersecting infinite transverse trajectories, the number of periodic points of period n grows exponentially with n.
• On a closed surface, transverse trajectory self-intersections imply positive topological entropy, even without smoothness assumptions.
• The topological entropy of a torus homeomorphism is bounded below by log 2 / (m r N), where m, r, N are parameters derived from the dynamics of transverse loops.