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In this note, we give a geometric proof of the Lefschetz fixed point formula for not-necessarily isometric map. The proof uses a modified Getzler rescaling and the associated rescaled bundle construction (Higson and Yi [Doc. Math. 24 (2019), pp. 1677–1720]).

In this work, we introduce a variant form of uniform convexity in partially ordered Banach spaces. This uniform convexity property is more adequate than norm uniform convexity when studying the fixed point problem for monotone nonexpansive mappings.

We introduce the moduli space of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over a compact Riemann surface $\Sigma$ and consider a natural involution, studying its fixed point locus when $\Sigma$ is $\mathbb{C} \mathbb{P}^1$ and establishing an identification with a moduli space of null polygons in Minkowski $3$-space., Comment: Appendix added. To appear in Trans. Amer. Math. Soc

In this work, we give a graphical version of the Ekeland variational principle which enables us to discover a new version of the Caristi fixed point theorem in weighted digraphs not necessarily generated by a partial order. Then we show that both graphical versions of the Ekeland variational principle and Caristi’s fixed point theorem are equivalent. In addition, we applied our main result on a differential structure Banach space.

We determine the Gross–Hopkins duals of certain higher real K K –theory spectra. More specifically, let p p be an odd prime, and consider the Morava E E –theory spectrum of height n = p − 1 n=p-1 . It is known, in expert circles, that for certain finite subgroups G G of the Morava stabilizer group, the homotopy fixed point spectra E n h G E_n^{hG} are Gross–Hopkins self-dual up to a shift. In this paper, we determine the shift for those finite subgroups G G which contain p p –torsion. This generalizes previous results for n = 2 n=2 and p = 3 p=3 .

We show that, if $f\colon \mathbb{B}^q\to \mathbb{B}^q$ is a holomorphic self-map of the unit ball in $\mathbb{C}^q$ and $\zeta\in \partial \mathbb{B}^q$ is a boundary repelling fixed point with dilation $\lambda>1$, then there exists a backward orbit converging to $\zeta$ with step $\log \lambda$. Morever, any two backward orbits converging to the same boundary repelling fixed point stay at finite distance. As a consequence there exists a unique canonical pre-model $(\mathbb{B}^k,\ell, \tau)$ associated with $\zeta$ where $1\leq k\leq q$, $\tau$ is a hyperbolic automorphism of $\mathbb{B}^k$, and whose image $\ell(\mathbb{B}^k)$ is precisely the set of starting points of backward orbits with bounded step converging to $\zeta$. This answers questions in [8] and [3,4]., Comment: 9 pages

An element g g of a group G G is said to be right Engel if for every x ∈ G x\in G there is a number n = n ( g , x ) n=n(g,x) such that [ g , n x ] = 1 [g,{}_{n}x]=1 . We prove that if a profinite group G G admits a coprime automorphism φ \varphi of prime order such that every fixed point of φ \varphi is a right Engel element, then G G is locally nilpotent.

We classify p-toral subgroups of U(n) that can have non-contractible fixed points under the action of U(n) on the com- plex Ln of partitions of C n into mutually orthogonal subspaces.

We study the loxodromic elements for the action of $Out(F_n)$ on the free splitting complex of the rank $n$ free group $F_n$. We prove that each outer automorphism is either loxodromic, or has bounded orbits without any periodic point, or has a periodic point; and we prove that all three possibilities can occur. We also prove that two loxodromic elements are either co-axial or independent, meaning that their attracting/repelling fixed point pairs on the Gromov boundary of the free splitting complex are either equal or disjoint as sets. Each of the alternatives in these results is also characterized in terms of the attracting/repelling lamination pairs of an outer automorphism. As an application, each attracting lamination determines its corresponding repelling lamination independent of the outer automorphism. As part of this study we describe the structure of the subgroup of $Out(F_n)$ that stabilizes the fixed point pair of a given loxodromic outer automorphism, and we give examples which show that this subgroup need not be virtually cyclic. As an application, the action of $Out(F_n)$ on the free splitting complex is not acylindrical, and its loxodromic elements do not all satisfy the WPD property of Bestvina and Fujiwara., Comment: 53 pages. This version contains many improvements including a simpler proof of Theorem 1.1 (2) characterizing nonloxodromic elements of the action of Out(F_n) on the free splitting complex. Also, certain results have been updated for purposes of application to computations of second bounded cohomology (in later works by the same authors)

In this paper we engage in a general study of the asymptoticexpansion of the Witten–Reshetikhin–Turaev invariants of mapping tori ofsurface mapping class group elements. We use the geometric constructionof the Witten–Reshetikhin–Turaev topological quantum field theory via thegeometric quantization of moduli spaces of flat connections on surfaces. Weidentify assumptions on the mapping class group elements that allow us toprovide a full asymptotic expansion. In particular, we show that our resultsapply to all pseudo-Anosov mapping classes on a punctured torus and show byexample that our assumptions on the mapping class group elements are strictlyweaker than hitherto successfully considered assumptions in this context. Theproof of our main theorem relies on our new results regarding asymptoticexpansions of oscillatory integrals, which allows us to go significantly beyondthe standard transversely cut out assumption on the fixed point set. Thismakes use of the Picard–Lefschetz theory for Laplace integrals.

Given a quiver automorphism with nice properties, we give a presentation of the fixed subalgebra of the associated cyclotomic quiver Hecke algebra. Generalising an isomorphism of Brundan and Kleshchev between the cyclotomic Hecke algebra of type G(r,1,n) and the cyclotomic quiver Hecke algebra of type A, we apply the previous result to find a presentation of the cyclotomic Hecke algebra of type G(r,p,n) which looks very similar to the one of a cyclotomic quiver Hecke algebra. In the meanwhile, we give an explicit isomorphism which realises a well-known Morita equivalence between Ariki-Koike algebras., Comment: 35 pages, 2 figures. v2: include suggestions and minor corrections from the referee. Final version, to appear in Transactions of the AMS. v3: minor changes

We construct a smooth, area preserving, mixing flow with finitely many nondegenerate fixed points and no saddle connections on a closed surface of genus 5 5 . This resolves a problem that has been open for four decades.

Assume that X X is a Banach space of measurable functions for which Komlós’ Theorem holds. We associate to any closed convex bounded subset C C of X X a coefficient t ( C ) t(C) which attains its minimum value when C C is closed for the topology of convergence in measure and we prove some fixed point results for affine Lipschitzian mappings, depending on the value of t ( C ) ∈ [ 1 , 2 ] t(C)\in [1,2] and the value of the Lipschitz constants of the iterates. As a first consequence, for every L > 2 L>2 , we deduce the existence of fixed points for affine uniformly L L -Lipschitzian mappings defined on the closed unit ball of L 1 [ 0 , 1 ] L_1[0,1] . Our main theorem also provides a wide collection of convex closed bounded sets in L 1 ( [ 0 , 1 ] ) L^1([0,1]) and in some other spaces of functions which satisfy the fixed point property for affine nonexpansive mappings. Furthermore, this property is still preserved by equivalent renormings when the Banach-Mazur distance is small enough. In particular, we prove that the failure of the fixed point property for affine nonexpansive mappings in L 1 ( μ ) L_1(\mu ) can only occur in the extremal case t ( C ) = 2 t(C)=2 . Examples are given proving that our fixed point theorem is optimal in terms of the Lipschitz constants and the coefficient t ( C ) t(C) .

Suppose that G G is a finite, unitary reflection group acting on a complex vector space V V and X X is the fixed point subspace of an element of G G . Define N N to be the setwise stabilizer of X X in G G , Z Z to be the pointwise stabilizer, and C = N / Z C=N/Z . Then restriction defines a homomorphism from the algebra of G G -invariant polynomial functions on V V to the algebra of C C -invariant functions on X X . Extending earlier work by Douglass and Röhrle for Coxeter groups, we characterize when the restriction mapping is surjective for arbitrary unitary reflection groups G G in terms of the exponents of G G and C C and their reflection arrangements. A consequence of our main result is that the variety of G G -orbits in the G G -saturation of X X is smooth if and only if it is normal.

We study non-stationary stochastic processes arising from sequential dynamical systems built on maps with a neutral fixed points and prove the existence of Extreme Value Laws for such processes. We use an approach developed in \cite{FFV16}, where we generalised the theory of extreme values for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. The present work is an extension of our previous results for concatenations of uniformly expanding maps obtained in \cite{FFV16}., Comment: To appear in Proceedings of the American Mathematical Society. arXiv admin note: substantial text overlap with arXiv:1510.04357

We propose a generalization of the Poincaré-Birkhoff Theorem on area-preserving twist maps to area-preserving twist maps F F that are random with respect to an ergodic probability measure. In this direction, we will prove several theorems concerning existence, density, and type of the fixed points. To this end first we introduce a randomized version of generalized generating functions, and verify the correspondence between its critical points and the fixed points of F F , a fact which we successively exploit in order to prove the theorems. The study we carry out needs to combine probabilistic techniques with methods from nonlinear PDE, and from differential geometry, notably Moser’s method and Conley-Zehnder theory. Our stochastic model in the periodic case coincides with the classical setting of the Poincaré-Birkhoff Theorem.

In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the K\"ahler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting, and elucidates the key role played by the following fact: the moment image of $e^t \cdot x$ increases as $t \in \mathbb{R}$ increases., Comment: 25 pages

Consider a finite l-group acting on the affine space of dimension n over a field k, whose characteristic differs from l. We prove the existence of a fixed point, rational over k, in the following cases: --- The field k is p-special for some prime p different from its characteristic. --- The field k is perfect and fertile, and n = 3.

In a recent work Malkiewich and Merling proposed a definition of the equivariant K K -theory of spaces for spaces equipped with an action of a finite group. We show that the fixed points of this spectrum admit a tom Dieck-type splitting. We also show that this splitting is compatible with the splitting of the equivariant suspension spectrum. The first of these results has been obtained independently by John Rognes.

Let p p be a prime and A A a finite group of exponent p p acting by automorphisms on a finite p ′ p’ -group G G . Assume that A A has order at least p 3 p^3 and C G ( a ) C_G(a) is nilpotent of class at most c c for any a ∈ A # a\in A^{\#} . It is shown that G G is nilpotent with class bounded solely in terms of c c and p p .

We study the local geometry of irreducible parabolic geometries admitting strongly essential flows; these are flows by local automorphisms with higher-order fixed points. We prove several new rigidity results, and recover some old ones for projective and conformal structures, which show that in many cases the existence of a strongly essential flow implies local flatness of the geometry on an open set having the fixed point in its closure. For almost c-projective and almost quaternionic structures we can moreover show flatness of the geometry on a neighborhood of the fixed point.

We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information. In particular, this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary. The local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces., Stability result added in the latest version

Let $M$ be a Kobayashi hyperbolic homogenous manifold. Let $\mathcal F$ be a holomorphic foliation on $M$ invariant under a transitive group $G$ of biholomorphisms. We prove that the leaves of $\mathcal F$ are the fibers of a holomorphic $G$-equivariant submersion $\pi \colon M \to N$ onto a $G$-homogeneous complex manifold $N$. We also show that if $\mathcal Q$ is an automorphism family of a hyperbolic convex (possibly unbounded) domain $D$ in $\mathbb C^n$, then the fixed point set of $\mathcal Q$ is either empty or a connected complex submanifold of $D$., Comment: final version

For each prime number p p , the dynamical behavior of the square mapping on the ring Z p \mathbb {Z}_p of p p -adic integers is studied. For p = 2 p=2 , there are only attracting fixed points with their attracting basins. For p ≥ 3 p\geq 3 , there are a fixed point 0 0 with its attracting basin, finitely many periodic points around which there are countably many minimal components and some balls of radius 1 / p 1/p being attracting basins. All these minimal components are precisely exhibited for different primes p p .

In his paper "A Construction for Coisotropic Subalgebras of Lie Bialgebras", Marco Zambon gave a way to use a long root of a complex semisimple Lie biaglebra $\mathfrak{g}$ to construct a coisotropic subalgebra of $\mathfrak{g}$. In this paper, we generalize Zambon's construction. Our construction is based on the theory of Lagrangian subalgebras of the double $\mathfrak{g}\oplus\mathfrak{g}$ of $\mathfrak{g}$, and our coisotropic subalgebras correspond to torus fixed points in the variety $\mathcal{L}(\mathfrak{g}\oplus\mathfrak{g})$ of Lagrangian subalgebras of $\mathfrak{g}\oplus\mathfrak{g}$.

It is well-known that a holomorphic function near a repelling fixed point may be conjugated to a linear function. The function which conjugates is called a Poincar\'e linearizer and may be extended to a transcendental entire function in the plane. In this paper, we study the dynamics of a higher dimensional generalization of Poincar\'e linearizers. These arise by conjugating a uniformly quasiregular mapping in $\R^m$ near a repelling fixed point to the mapping $x\mapsto 2x$. In particular, we show that the fast escaping set of such a linearizer has a spider's web structure., Comment: 14 pages, 1 figure

We study the boundary behaviour of a meromorphic map $f: \mathbb C \to \widehat{\mathbb C}$ on its invariant simply connected Fatou component $U$. To this aim, we develop the theory of accesses to boundary points of $U$ and their relation to the dynamics of $f$. In particular, we establish a correspondence between invariant accesses from $U$ to infinity or weakly repelling points of $f$ and boundary fixed points of the associated inner function on the unit disc. We apply our results to describe the accesses to infinity from invariant Fatou components of the Newton maps., Comment: 31 pages, 8 figures

For an action of a finite group on a C*-algebra, we present some conditions under which properties of the C*-algebra pass to the crossed product or the fixed point algebra. We mostly consider the ideal property, the projection property, topological dimension zero, and pure infiniteness. In many of our results, additional conditions are necessary on the group, the algebra, or the action. Sometimes the action must be strongly pointwise outer, and in a few results it must have the Rokhlin property. When the group is finite abelian, we prove that crossed products and fixed point algebras preserve topological dimension zero with no condition on the action. We give an example to show that the ideal property and the projection property do not pass to fixed point algebras (even for the two element group). The construction also gives an example of a C*-algebra which does not have the ideal property but such that the algebra of 2 by 2 matrices over it does have the ideal property; in fact, this matrix algebra has the projection property.

Let L | K L|K be a Galois extension of fields with finite Galois group G G . Greither and Pareigis showed that there is a bijection between Hopf Galois structures on L | K L|K and regular subgroups of P e r m ( G ) Perm(G) normalized by G G , and Byott translated the problem into that of finding equivalence classes of embeddings of G G in the holomorph of groups N N of the same cardinality as G G . In 2007 we showed, using Byott’s translation, that fixed point free endomorphisms of G G yield Hopf Galois structures on L | K L|K . Here we show how abelian fixed point free endomorphisms yield Hopf Galois structures directly, using the Greither-Pareigis approach and, in some cases, also via the Byott translation. The Hopf Galois structures that arise are “twistings” of the Hopf Galois structure by H λ H_{\lambda } , the K K -Hopf algebra that arises from the left regular representation of G G in P e r m ( G ) Perm(G) . The paper concludes with various old and new examples of abelian fixed point free endomorphisms.

We consider multi-valued dynamical systems with continuous time of the form ẋ ∈ F (x), where F (x) is a set-valued function. Such models have been studied recently in mathematical economics. We provide a definition for chaos, ω-chaos and topological entropy for these differential inclusions that is in terms of the natural R-action on the space of all solutions of the model. By considering this more complicated topological space and its R-action we show that chaos is the ‘typical’ behavior in these models by showing that near any hyperbolic fixed point there is a region where the system is chaotic, ω-chaotic, and has infinite topological entropy.

Let G be a simple algebraic group over an algebraically closed field k of bad characteristic. We classify the spherical unipotent conjugacy classes of G. We also show that if the characteristic of k is 2, then the fixed point subgroup of every involutorial automorphism (involution) of G is a spherical subgroup of G.

We prove that a random group has fixed points when it isometrically acts on a CAT(0) cube complex. We do not assume that the action is simplicial.

Let k > 3(> 2) be an integer and F be a family of functions meromorphic in a domain D C C, all of whose poles have multiplicity at least 2 (at least 3). If in D each f ∈ F has neither repelling fixed points nor repelling periodic points of period k, then F is a normal family in D. Examples are given to show that the conditions on poles are necessary and sharp.

We prove several results on products of conjugacy classes in finite simple groups. The first result is that for any finite nonabelian simple groups, there exists a triple of conjugate elements with product 1 1 which generate the group. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite nonabelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a nonabelian finite simple group can be written as a product of two r r th powers for any prime power r r (in particular, a product of two squares answering a conjecture of Larsen, Shalev and Tiep).

Let E be a k-local profinite G-Galois extension of an E_infty-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite setting. We show the function spectrum F_A((E^hH)_k, (E^hK)_k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]])^hK)_k where H and K are closed subgroups of G. Applications to Morava E-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points., Comment: 60 Pages

Two results concerning orientation-reversing homeomorphisms of the plane are proved. Let h : R 2 → R 2 h:\mathbb {R}^2\rightarrow \mathbb {R}^2 be an orientation-reversing planar homeomorphism with a continuum X X invariant (i.e. h ( X ) = X h(X)=X ). First, suppose there are at least n n bounded components of R 2 ∖ X \mathbb {R}^2\setminus X that are invariant under h h . Then there are at least n + 1 n+1 components of the fixed point set of h h in X X . This provides an affirmative answer to a question posed by K. Kuperberg. Second, suppose there is a k k -periodic orbit in X X with k > 2 k>2 . Then there is a 2-periodic orbit in X X , or there is a 2-periodic component of R 2 ∖ X \mathbb {R}^2\setminus X . The second result is based on a recent result of M. Bonino concerning linked periodic orbits of orientation-reversing homeomorphisms of the 2-sphere S 2 \mathbb {S}^2 . These results generalize to orientation-reversing homeomorphisms of S 2 \mathbb {S}^2 .

The aim of this paper is to study and compare the dynamics of two classes of periodic point free homeomorphisms of the open annulus, homotopic to the identity. First, we consider skew products over irrational rotations (often called quasiperiodically forced monotone maps) and derive a decomposition of the phase space that strengthens a classification given by J. Stark. There exists a sequence of invariant essential embedded open annuli on which the dynamics are either topologically transitive or wandering (from one of the boundary components to the other). The remaining regions between these annuli are densely filled by so-called invariant minimal strips, which serve as natural analogues for fixed points of one-dimensional maps in this context. Secondly, we study homeomorphisms of the open annulus which have neither periodic points nor wandering open sets. Somewhat surprisingly, there are remarkable analogies to the case of skew product transformations considered before. Invariant minimal strips can be replaced by a class of objects which we call invariant circloids, and using this concept we arrive again at a decomposition of the phase space. There exists a sequence of invariant essential embedded open annuli with transitive dynamics, and the remaining regions are densely filled by invariant circloids. In particular, the dynamics on the whole phase space are transitive if and only if there exists no invariant circloid and if and only if there exists an orbit which is unbounded both above and below.

Let M M be an indecomposable k k -junctioned tree-like continuum. Let f f be a map of M M that sends each composant of M M into itself. Using an argument of O. H. Hamilton, we prove that f f has a fixed point.

We define the notion of a positive path of a homeomorphism of a topological space. It seems to be a natural object to understand Birkhoff's arguments in his proof of the celebrated Poincare-Birkhoff theorem. We write the proof of this theorem, by using positive paths, and the proof of its generalization due to P. Carter. We will also explain the links with the free disk chains introduced in the subject by J. Franks. We will finish the paper by studying the local versions where the upper curve is not invariant and will explain why this curve or its image must be a graph to get such a generalization.

For smooth actions of G G on spheres with exactly two fixed points, the Laitinen Conjecture proposed an answer to the Smith question about the G G -modules determined on the tangent spaces at the two fixed points. Morimoto obtained the first counterexample to the Laitinen Conjecture for G = Aut ( A 6 ) G = \textrm {Aut}(A_6) . By answering the Smith question for some finite solvable Oliver groups G G , we obtain new counterexamples to the Laitinen Conjecture, presented for the first time in the case where G G is solvable.

If X is a finite dimensional real normed space, C is a closed convex subset of X and f: C → C is nonexpansive with respect to the norm on X, then we show that either f has a fixed point in C or there is a linear functional ϕ ∈ X * such that lim k→∞ ϕ(f k (x)) = ∞ for all x ∈ C.