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Fixed points of discrete nilpotent group actions on S 2

Suely Druck, Fuquan Fang, Sebastião Firmo (2002)

Annales de l’institut Fourier

We prove that for each integer k ≥ 2 there is an open neighborhood 𝒱 k of the identity map of the 2-sphere S 2 , in C 1 topology such that: if G is a nilpotent subgroup of Diff 1 ( S 2 ) with length k of nilpotency, generated by elements in 𝒱 k , then the natural G -action on S 2 has nonempty fixed point set. Moreover, the G -action has at least two fixed points if the action has a finite nontrivial orbit.

Flowability of plane homeomorphisms

Frédéric Le Roux, Anthony G. O’Farrell, Maria Roginskaya, Ian Short (2012)

Annales de l’institut Fourier

We describe necessary and sufficient conditions for a fixed point free planar homeomorphism that preserves the standard Reeb foliation to embed in a planar flow that leaves the foliation invariant.

Flows of flowable Reeb homeomorphisms

Shigenori Matsumoto (2012)

Annales de l’institut Fourier

We consider a fixed point free homeomorphism h of the closed band B = ℝ × [ 0 , 1 ] which leaves each leaf of a Reeb foliation on B invariant. Assuming h is the time one of various topological flows, we compare the restriction of the flows on the boundary.

For almost every tent map, the turning point is typical

Henk Bruin (1998)

Fundamenta Mathematicae

Let T a be the tent map with slope a. Let c be its turning point, and μ a the absolutely continuous invariant probability measure. For an arbitrary, bounded, almost everywhere continuous function g, it is shown that for almost every a, ʃ g d μ a = l i m n → ∞ 1 n ∑ i = 0 n - 1 g ( T a i ( c ) ) . As a corollary, we deduce that the critical point of a quadratic map is generically not typical for its absolutely continuous invariant probability measure, if it exists.

Forcing relation on interval patterns

Jozef Bobok (2005)

Fundamenta Mathematicae

We consider-without restriction to the piecewise monotone case-a forcing relation on interval (transitive, roof, bottom) patterns. We prove some basic properties of this type of forcing and explain when it is a partial ordering. Finally, we show how our approach relates to the results known from the literature.

Forcing relation on minimal interval patterns

Jozef Bobok (2001)

Fundamenta Mathematicae

Let ℳ be the set of pairs (T,g) such that T ⊂ ℝ is compact, g: T → T is continuous, g is minimal on T and has a piecewise monotone extension to convT. Two pairs (T,g),(S,f) from ℳ are equivalent if the map h: orb(minT,g) → orb(minS,f) defined for each m ∈ ℕ₀ by h ( g m ( m i n T ) ) = f m ( m i n S ) is increasing on orb(minT,g). An equivalence class of this relation-a minimal (oriented) pattern A-is exhibited by a continuous interval map f:I → I if there is a set T ⊂ I such that (T,f|T) = (T,f) ∈ A. We define the forcing relation on...

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