A note on fixed points of abelian actions in dimension one
aa r X i v : . [ m a t h . GN ] A ug A NOTE ON FIXED POINTS OF ABELIAN ACTIONS INDIMENSION ONE
J. P. BORO ´NSKI
Abstract.
The result of Boyce and Huneke gives rise to a 1-dimensionalcontinuum, which is the intersection of a descending family of disks,that admits two commuting homeomorphisms without a common fixedpoint.
Two commuting C -diffeomorphisms of a disk must have a common fixedpoint, and if they preserve a nonseparating plane continuum Z (i.e. a com-pact and connected set such that R \ Z is connected), then there is acommon fixed point in Z [8]. An arc-like continuum is the one that is theinverse limit of arcs. Bing showed that every arc-like continuum embedsin the plane as the intersection of a descending family of disks [2]. Hamil-ton showed that every arc-like continuum has the fixed point property [4].In particular, any Z -action on such a continuum fixes a point. For moregeneral abelian actions on arc-like continua no such result has been known,although it holds for example for dendrites and uniquely arcwise connectedcontinua [9]. For self-maps of the arc the examples of Boyce [3] and Huneke[5] provide two commuting surjective maps without a common fixed point .Noteworthy their result gives the following. Example.
There exists an arc-like continuum X and a pair of homeomor-phisms F, G : X → X such that F ◦ G = G ◦ F and Fix( F ) ∩ Fix( G ) = ∅ .The argument for the existence of the above example is quite short. How-ever, we did not find such a result in the literature, and none of the expertswe consulted had been aware of such a result before. After personal discus-sions during the 52nd Spring Topology and Dynamics Conference, held atAuburn University in March of 2018, we were encouraged to publish thisnote. Mathematics Subject Classification.
Key words and phrases. commuting homeomorphisms, fixed point. I am grateful to Benjamin Vejnar for bringing this result to my attention. An inter-ested reader might want to consult [10] for some related recent results.
Proof.
Let f, g : [0 , → [0 ,
1] be two commuting surjections without acommon fixed point. Let h = f ◦ g and consider the inverse limit space X = lim ← { [0 , , h } = { ( x , x , ... ) ∈ [0 , N : h ( x i +1 ) = x i , for all i ∈ N } . Since f and g commute, all three maps induce maps on X , given by H ( x , x , ... ) = ( h ( x ) , h ( x ) , .... ) ,F ( x , x , ... ) = ( f ( x ) , f ( x ) , .... ) , and G ( x , x , ... ) = ( g ( x ) , g ( x ) , .... ) . To see that F and G are homeomorphisms one can use the following,particularly elegant, argument from [7]. Since H = F ◦ G = G ◦ F , and H isjust the shift homeomorphism, the maps F and G are also homeomorphisms.Since f and g have no common fixed point, the same is true for F and G . (cid:3) We note that by Barge-Martin embedding theorem [1] both F and G extend (up to conjugacy) to homeomorphisms of the plane. Therefore thefollowing question is of interest. Question 1.
Can F and G be extended to two commuting planar homeo-morphisms? An affirmative answer to the above problem would prove the necessityof the C setting in the generalization of the Cartwright-Littlewood fixedpoint theorem in [8]. We also note, that the example in [5] is constructedas a limit of piecewise linear functions, each of which has a well-definedderivative of constant absolute value greater than 3 everywhere, outside ofa finite set of critical points. This gives a good starting point to a potentialmodification, that would result in the continuum X presented here, beinghomeomorphic to the pseudo-arc; see [6]. Question 2.
Can f and g be modified so that the example holds on thepseudo-arc? The interested reader might want to consult [7] for related results.
Acknowledgments.
The author is supported by University of Ostravagrant lRP201824 ”Complex topological structures” and the NPU II projectLQ1602 IT4Innovations excellence in science.
NOTE ON FIXED POINTS OF ABELIAN ACTIONS IN DIMENSION ONE 3
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