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Elmas Irmak - One of the best experts on this subject based on the ideXlab platform.
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edge preserving maps of the nonseparating curve graphs curve graphs and rectangle preserving maps of the hatcher thurston graphs
arXiv: Geometric Topology, 2017Co-Authors: Elmas IrmakAbstract:Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components with $g \geq 2$, $n \geq 0$. Let $\mathcal{N}(R)$ be the nonseparating curve graph, $\mathcal{C}(R)$ be the curve graph and $\mathcal{HT}(R)$ be the Hatcher-Thurston graph of $R$. We prove that if $\lambda : \mathcal{N}(R) \rightarrow\mathcal{N}(R)$ is an edge-preserving map, then $\lambda$ is induced by a Homeomorphism of $R$. We prove that if $\theta : \mathcal{C}(R) \rightarrow \mathcal{C}(R)$ is an edge-preserving map, then $\theta$ is induced by a Homeomorphism of $R$. We prove that if $R$ is closed and $\tau: \mathcal{HT}(R) \rightarrow\mathcal{HT}(R)$ is a rectangle preserving map, then $\tau$ is induced by a Homeomorphism of $R$. We also prove that these Homeomorphisms are unique up to isotopy when $(g, n) \neq (2, 0)$.
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Superinjective Simplicial Maps of Complexes of Curves and Injective Homomorphisms of Subgroups of Mapping Class Groups
2012Co-Authors: Elmas IrmakAbstract:Let R be a compact, connected, orientable surface of genus at least 3, possibly with nonempty boundary. Let C(R) be the complex of curves on R and Mod ∗ R be the extended mapping class group of R. We prove that a simplicial map λ: C(R) → C(R) is superinjective if and only if it is induced by a Homeomorphism of R. As a corollary, we prove that if K is a finite index subgroup of Mod ∗ R and f: K → Mod∗R is an injective homomorphism, then f is induced by a Homeomorphism of R and f has a unique extension to an automorphism of Mod ∗ R. This extends the author’s previous results about closed connected orientable surfaces of genus at least 3, to the surface R.
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superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups
Topology, 2004Co-Authors: Elmas IrmakAbstract:Abstract Let S be a closed, connected, orientable surface of genus at least 3, C (S) be the complex of curves on S and Mod S ∗ be the extended mapping class group of S. We prove that a simplicial map, λ : C (S)→ C (S) , preserves nondisjointness (i.e. if α and β are two vertices in C (S) and i(α,β)≠0, then i(λ(α),λ(β))≠0) iff it is induced by a Homeomorphism of S. As a corollary, we prove that if K is a finite index subgroup of Mod S ∗ and f : K→Mod S ∗ is an injective homomorphism, then f is induced by a Homeomorphism of S and f has a unique extension to an automorphism of Mod S ∗ .
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superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups
arXiv: Geometric Topology, 2002Co-Authors: Elmas IrmakAbstract:Let R be a compact, connected, orientable surface of genus g with p boundary components. Let C(R) be the complex of curves on R and Mod_R^* be the extended mapping class group of R. Suppose that either g = 2 and p > 1 or g > 2 and p >= 0. We prove that a simplicial map lambda from C(R) to C(R) is superinjective if and only if it is induced by a Homeomorphism of R. As a corollary, we prove that if K is a finite index subgroup of Mod_R^* and f is an injective homomorphism from K to Mod_R^*, then f is induced by a Homeomorphism of R and f has a unique extension to an automorphism of Mod_R^*. This extends the author's previous results about closed connected orientable surfaces of genus at least 3, to the surface R.
Tal, Fabio Armando - One of the best experts on this subject based on the ideXlab platform.
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Existence of Non-Contractible Periodic Orbits for Homeomorphisms of the Open Annulus
2019Co-Authors: Conejeros Jonathan, Tal, Fabio ArmandoAbstract:In this article we consider Homeomorphisms of the open annulus $\mathbb{A}=\mathbb{R}/\mathbb{Z}\times \mathbb{R}$ which are isotopic to the identity and preserve a Borel probability measure of full support, focusing on the existence of non-contractible periodic orbits. Assume $f$ such Homeomorphism such that the connected components of the set of fixed points of $f$ are all compact. Further assume that there exists $\check{f}$ a lift of $f$ to the universal covering of $\mathbb{A}$ such that the set of fixed points of $\check{f}$ is non-empty and that this set projects into an open topological disk of $\mathbb{A}$. We prove that, in this setting, one of the following two conditions must be satisfied: (1) $f$ has non-contractible periodic points of arbitrarily large prime period, or (2) for every compact set $K$ of $\mathbb{A}$ there exists a constant $M$ (depending on the compact set) such that, if $\check{z}$ and $\check{f}^n(\check{z})$ project on $K$, then their projections on the first coordinate have distance less or equal to $M$. Some consequence for Homeomorphisms of the open annulus whose rotation set is reduced to an integer number are derived.Comment: This version incorporates significant simplifications on the proof of Theorem A as suggested by the referee. To appear in Math.
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Applications of Forcing Theory to Homeomorphisms of the Closed Annulus
2019Co-Authors: Conejeros Jonathan, Tal, Fabio ArmandoAbstract:This paper studies Homeomorphisms of the closed annulus that are isotopic to the identity from the viewpoint of rotation theory, using a newly developed forcing theory for surface Homeomorphisms. Our first result is a solution to the so called strong form of Boyland's Conjecture on the closed annulus: Assume $f$ is a Homeomorphism of $\overline{\mathbb{A}}:=(\mathbb{R}/\mathbb{Z})\times [0,1]$ which is isotopic to the identity and preserves a Borel probability measure $\mu$ with full support. We prove that if the rotation set of $f$ is a non-trivial segment, then the rotation number of the measure $\mu$ cannot be an endpoint of this segment. We also study the case of Homeomorphisms such that $\mathbb{A}=(\mathbb{R}/\mathbb{Z})\times (0,1)$ is a region of instability of $f$. We show that, if the rotation numbers of the restriction of $f$ to the boundary components lies in the interior of the rotation set of $f$, then $f$ has uniformly bounded deviations from its rotation set. Finally, by combining this last result and recent work on realization of rotation vectors for annular continua, we obtain that if $f$ is any area-preserving Homeomorphism of $\overline{\mathbb{A}}$ isotopic to the identity, then for every real number $\rho$ in the rotation set of $f$, there exists an associated Aubry-Mather set, that is, a compact $f$-invariant set such that every point in this set has a rotation number equal to $\rho$. This extends a result by P. Le Calvez previously known only for diffeomorphisms.Comment: 46 pages, 5 figure
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Topological horseshoes for surface Homeomorphisms
2018Co-Authors: Patrice Le Calvez, Tal, Fabio ArmandoAbstract:In this work we develop a new criterion for the existence of topological horseshoes for surface Homeomorphisms in the isotopy class of the identity. Based on our previous work on forcing theory, this new criterion is purely topological and can be expressed in terms of equivariant Brouwer foliations and transverse trajectories. We then apply this new tool in the study of the dynamics of Homeomorphisms of surfaces with zero genus and null topological entropy and we obtain several applications. For Homeomorphisms of the open annulus $\mathbb{A}$ with zero topological entropy, we show that rotation numbers exists for all points with nonempty omega limits, and that if $\mathbb{A}$ is a generalized region of instability then it admits a single rotation vector. We also offer a new proof of a recent result of Passegi, Potrie and Sambarino, showing that zero entropy dissipative Homeomorphisms of the annulus having as an atractor a circloid have a single rotation number. Our work also studies Homeomorphisms of the sphere without horseshoes. For these maps we present a structure theorem in terms of fixed point free invariant sub-annuli, as well as a very restricted description of all possible dynamical behavior in the transitive subsets. This description ensures, for instance, that transitive sets can contain at most $2$ distinct periodic orbits and that, in many cases, the restriction of the Homeomorphism to the transitive set must be an extension of an odometer. In particular, we show that any nontrivial and stable transitive subset of a dissipative diffeomorphism of the plane is always infinitely renormalizable in the sense of Bonatti-Gambaudo-Lion-Tresser
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A triple boundary lemma for surface Homeomorphisms
2018Co-Authors: Koropecki Andres, Patrice Le Calvez, Tal, Fabio ArmandoAbstract:Given an orientation-preserving and area-preserving Homeomorphism $f$ of the sphere, we prove that every point which is in the common boundary of three pairwise disjoint invariant open topological disks must be a fixed point. As an application, if $K$ is an invariant Wada type continuum, then $f^n|_K$ is the identity for some $n>0$. Another application is an elementary proof of the fact that invariant disks for a nonwandering Homeomorphisms homotopic to the identity in an arbitrary surface are homotopically bounded if the fixed point set is inessential. The main results in this article are self-contained.Comment: Minor corrections. To appear in Proc. Amer. Math. So
Saracco Paolo - One of the best experts on this subject based on the ideXlab platform.
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Topological tensor product of bimodules, complete Hopf algebroids and convolution algebras
'World Scientific Pub Co Pte Lt', 2018Co-Authors: El Kaoutit L., Saracco PaoloAbstract:Given a finitely generated and projective LieâRinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an Homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the appendices we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings. © 2018 World Scientific Publishing Companyinfo:eu-repo/semantics/publishe
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Topological tensor product of bimodules, complete Hopf Algebroids and convolution algebras
'World Scientific Pub Co Pte Lt', 2018Co-Authors: Laiachi El Kaoutit, Saracco PaoloAbstract:Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an Homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the Appendix we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings.Comment: Minor changes, 33 pages. To appear in CC
Andrea Marchese - One of the best experts on this subject based on the ideXlab platform.
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residually many bv Homeomorphisms map a null set in a set of full measure
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2019Co-Authors: Andrea MarcheseAbstract:Let Q = (0,1) 2 be the unit square in R 2 . We prove that in a suitable complete metric space of BV Homeomorphisms f : Q → Q with f|@Q = Id, the generical Homeomorphism (in the sense of Baire categories) maps a null set in a set of full measure and vice versa. Moreover we observe that, for 1 ≤ p < 2, in the most reasonable complete metric space for such problem, the family of W 1,p homemomorphisms satisfying the above property is of first category, instead.
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residually many bv Homeomorphisms map a null set in a set of full measure
arXiv: Functional Analysis, 2015Co-Authors: Andrea MarcheseAbstract:Let $Q=(0,1)^2$ be the unit square in $\mathbb{R}^2$. We prove that in a suitable complete metric space of $BV$ Homeomorphisms $f:Q\rightarrow Q$ with $f_{|\partial Q}=Id$, the generical Homeomorphism (in the sense of Baire categories) maps a null set in a set of full measure and vice versa. Moreover we observe that, for $1\leq p<2$, in the most reasonable complete metric space for such problem, the family of $W^{1,p}$ homemomorphisms satisfying the above property is of first category, instead.
Patrice Le Calvez - One of the best experts on this subject based on the ideXlab platform.
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Topological horseshoes for surface Homeomorphisms
2018Co-Authors: Patrice Le Calvez, Tal, Fabio ArmandoAbstract:In this work we develop a new criterion for the existence of topological horseshoes for surface Homeomorphisms in the isotopy class of the identity. Based on our previous work on forcing theory, this new criterion is purely topological and can be expressed in terms of equivariant Brouwer foliations and transverse trajectories. We then apply this new tool in the study of the dynamics of Homeomorphisms of surfaces with zero genus and null topological entropy and we obtain several applications. For Homeomorphisms of the open annulus $\mathbb{A}$ with zero topological entropy, we show that rotation numbers exists for all points with nonempty omega limits, and that if $\mathbb{A}$ is a generalized region of instability then it admits a single rotation vector. We also offer a new proof of a recent result of Passegi, Potrie and Sambarino, showing that zero entropy dissipative Homeomorphisms of the annulus having as an atractor a circloid have a single rotation number. Our work also studies Homeomorphisms of the sphere without horseshoes. For these maps we present a structure theorem in terms of fixed point free invariant sub-annuli, as well as a very restricted description of all possible dynamical behavior in the transitive subsets. This description ensures, for instance, that transitive sets can contain at most $2$ distinct periodic orbits and that, in many cases, the restriction of the Homeomorphism to the transitive set must be an extension of an odometer. In particular, we show that any nontrivial and stable transitive subset of a dissipative diffeomorphism of the plane is always infinitely renormalizable in the sense of Bonatti-Gambaudo-Lion-Tresser
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A triple boundary lemma for surface Homeomorphisms
2018Co-Authors: Koropecki Andres, Patrice Le Calvez, Tal, Fabio ArmandoAbstract:Given an orientation-preserving and area-preserving Homeomorphism $f$ of the sphere, we prove that every point which is in the common boundary of three pairwise disjoint invariant open topological disks must be a fixed point. As an application, if $K$ is an invariant Wada type continuum, then $f^n|_K$ is the identity for some $n>0$. Another application is an elementary proof of the fact that invariant disks for a nonwandering Homeomorphisms homotopic to the identity in an arbitrary surface are homotopically bounded if the fixed point set is inessential. The main results in this article are self-contained.Comment: Minor corrections. To appear in Proc. Amer. Math. So
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forcing theory for transverse trajectories of surface Homeomorphisms
arXiv: Dynamical Systems, 2015Co-Authors: Patrice Le Calvez, Fabio Armando TalAbstract: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.
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about the homological discrete conley index of isolated invariant acyclic continua
Geometry & Topology, 2013Co-Authors: Luis Hernandezcorbato, Patrice Le Calvez, Francisco Romero Ruiz Del PortalAbstract:This article includes an almost self-contained exposition on the discrete Conley index and its duality. We work with a locally defined Homeomorphism f in R d and an acyclic continuum X, such as a cellular set or a fixed point, invariant under f and isolated. We prove that the trace of the first discrete homological Conley index of f and X is greater than or equal to -1 and describe its periodical behavior. If equality holds then the traces of the higher homological indices are 0. In the case of orientation-reversing Homeomorphisms of R 3 , we obtain a characterization of the fixed point index sequence {i(f n ,p)}n�1 for a fixed point p which is isolated as an invariant set. In particular, we obtain that i(f,p) ≤ 1. As a corollary, we prove that there are no minimal orientation-reversing Homeomorphisms in R 3 .
