A profinite group equipped with an expansive endomorphism is equivalent to a one-sided group shift. We show that these groups have a very restricted structure. More precisely, we show that any such group can be decomposed into a finite sequence of full one-sided group shifts and two finite groups.
{"title":"Expansive dynamics on profinite groups","authors":"M. Wibmer","doi":"10.4064/FM15-1-2021","DOIUrl":"https://doi.org/10.4064/FM15-1-2021","url":null,"abstract":"A profinite group equipped with an expansive endomorphism is equivalent to a one-sided group shift. We show that these groups have a very restricted structure. More precisely, we show that any such group can be decomposed into a finite sequence of full one-sided group shifts and two finite groups.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42653265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bestvina introduced a $mathcal{Z}$-structure for a group $G$ to generalize the boundary of a CAT(0) or hyperbolic group. A refinement of this notion, introduced by Farrell and Lafont, includes a $G$-equivariance requirement, and is known as an $mathcal{E}mathcal{Z}$-structure. In this paper, we show that fundamental groups of graphs of nonpositively curved Riemannian $n$-manifolds admit $mathcal{Z}$-structures and graphs of negatively curved or flat $n$-manifolds admit $mathcal{E}mathcal{Z}$-structures. This generalizes a recent result of the first two authors with Tirel, which put $mathcal{E}mathcal{Z}$-structures on Baumslag-Solitar groups and $mathcal{Z}$-structures on generalized Baumslag-Solitar groups.
{"title":"Compressible spaces and $mathcal{E}mathcal{Z}$-structures","authors":"C. Guilbault, Molly A. Moran, Kevin Schreve","doi":"10.4064/fm972-7-2021","DOIUrl":"https://doi.org/10.4064/fm972-7-2021","url":null,"abstract":"Bestvina introduced a $mathcal{Z}$-structure for a group $G$ to generalize the boundary of a CAT(0) or hyperbolic group. A refinement of this notion, introduced by Farrell and Lafont, includes a $G$-equivariance requirement, and is known as an $mathcal{E}mathcal{Z}$-structure. In this paper, we show that fundamental groups of graphs of nonpositively curved Riemannian $n$-manifolds admit $mathcal{Z}$-structures and graphs of negatively curved or flat $n$-manifolds admit $mathcal{E}mathcal{Z}$-structures. This generalizes a recent result of the first two authors with Tirel, which put $mathcal{E}mathcal{Z}$-structures on Baumslag-Solitar groups and $mathcal{Z}$-structures on generalized Baumslag-Solitar groups.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42355979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Roitman's combinatorial principle $Delta$ is equivalent to monotone normality of the nabla product, $nabla (omega +1)^omega$. If ${ X_n : nin omega}$ is a family of metrizable spaces and $nabla_n X_n$ is monotonically normal, then $nabla_n X_n$ is hereditarily paracompact. Hence, if $Delta$ holds then the box product $square (omega +1)^omega$ is paracompact. Large fragments of $Delta$ hold in $mathsf{ZFC}$, yielding large subspaces of $nabla (omega+1)^omega$ that are `really' monotonically normal. Countable nabla products of metrizable spaces which are respectively: arbitrary, of size $le mathfrak{c}$, or separable, are monotonically normal under respectively: $mathfrak{b}=mathfrak{d}$, $mathfrak{d}=mathfrak{c}$ or the Model Hypothesis. It is consistent and independent that $nabla A(omega_1)^omega$ and $nabla (omega_1+1)^omega$ are hereditarily normal (or hereditarily paracompact, or monotonically normal). In $mathsf{ZFC}$ neither $nabla A(omega_2)^omega$ nor $nabla (omega_2+1)^omega$ is hereditarily normal.
{"title":"Monotone normality and nabla products","authors":"H. Barriga-Acosta, P. Gartside","doi":"10.4064/FM926-10-2020","DOIUrl":"https://doi.org/10.4064/FM926-10-2020","url":null,"abstract":"Roitman's combinatorial principle $Delta$ is equivalent to monotone normality of the nabla product, $nabla (omega +1)^omega$. If ${ X_n : nin omega}$ is a family of metrizable spaces and $nabla_n X_n$ is monotonically normal, then $nabla_n X_n$ is hereditarily paracompact. Hence, if $Delta$ holds then the box product $square (omega +1)^omega$ is paracompact. Large fragments of $Delta$ hold in $mathsf{ZFC}$, yielding large subspaces of $nabla (omega+1)^omega$ that are `really' monotonically normal. Countable nabla products of metrizable spaces which are respectively: arbitrary, of size $le mathfrak{c}$, or separable, are monotonically normal under respectively: $mathfrak{b}=mathfrak{d}$, $mathfrak{d}=mathfrak{c}$ or the Model Hypothesis. \u0000It is consistent and independent that $nabla A(omega_1)^omega$ and $nabla (omega_1+1)^omega$ are hereditarily normal (or hereditarily paracompact, or monotonically normal). In $mathsf{ZFC}$ neither $nabla A(omega_2)^omega$ nor $nabla (omega_2+1)^omega$ is hereditarily normal.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45145811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a natural generalization of covering projections defined in terms of unique lifting properties. A map $p:Eto X$ has the "continuous path-covering property" if all paths in $X$ lift uniquely and continuously (rel. basepoint) with respect to the compact-open topology. We show that maps with this property are closely related to fibrations with totally path-disconnected fibers and to the natural quotient topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering projections to a classification of maps with the continuous path-covering property in terms of topological $pi_1$: for any path-connected Hausdorff space $X$, maps $Eto X$ with the continuous path-covering property are classified up to weak equivalence by subgroups $Hleq pi_1(X,x_0)$ with totally path-disconnected coset space $pi_1(X,x_0)/H$. Here, "weak equivalence" refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.
{"title":"On maps with continuous path lifting","authors":"Jeremy Brazas, A. Mitra","doi":"10.4064/fm977-3-2023","DOIUrl":"https://doi.org/10.4064/fm977-3-2023","url":null,"abstract":"We study a natural generalization of covering projections defined in terms of unique lifting properties. A map $p:Eto X$ has the \"continuous path-covering property\" if all paths in $X$ lift uniquely and continuously (rel. basepoint) with respect to the compact-open topology. We show that maps with this property are closely related to fibrations with totally path-disconnected fibers and to the natural quotient topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering projections to a classification of maps with the continuous path-covering property in terms of topological $pi_1$: for any path-connected Hausdorff space $X$, maps $Eto X$ with the continuous path-covering property are classified up to weak equivalence by subgroups $Hleq pi_1(X,x_0)$ with totally path-disconnected coset space $pi_1(X,x_0)/H$. Here, \"weak equivalence\" refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42650263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
If $X$ is a finite tree and $f colon X longrightarrow X$ is a continuous function, as the Main Theorem of this paper (Theorem 8), we find eight conditions, each of which is equivalent to the fact that $f$ is equicontinuous. Our results either generalize ones shown by Vidal-Escobar and Garcia-Ferreira, or complement those of Bruckner and Ceder, Mai and Camargo, Rincon and Uzcategui. Some of the methods, however, have not been used previously in this context (for example, in one of our proofs we apply the Ramsey-theoretic result known as Hindman's theorem). To name just a few of the results obtained: the equicontinuity of $f$ is equivalent to the fact that there is no arc $A subseteq X$ satisfying $A subsetneq f^n[A]$ for some $nin mathbb{N}$. It is also equivalent to the fact that for some nonprincial ultrafilter $u$, the function $f^u colon X longrightarrow X$ is continuous (in other words, failure of equicontinuity of $f$ is equivalent to the failure of continuity of every element of the Ellis remainder $E^*(X,f)$).
{"title":"Equicontinuous mappings on finite trees","authors":"G. Acosta, David J. Fernández-Bretón","doi":"10.4064/FM923-9-2020","DOIUrl":"https://doi.org/10.4064/FM923-9-2020","url":null,"abstract":"If $X$ is a finite tree and $f colon X longrightarrow X$ is a continuous function, as the Main Theorem of this paper (Theorem 8), we find eight conditions, each of which is equivalent to the fact that $f$ is equicontinuous. Our results either generalize ones shown by Vidal-Escobar and Garcia-Ferreira, or complement those of Bruckner and Ceder, Mai and Camargo, Rincon and Uzcategui. Some of the methods, however, have not been used previously in this context (for example, in one of our proofs we apply the Ramsey-theoretic result known as Hindman's theorem). To name just a few of the results obtained: the equicontinuity of $f$ is equivalent to the fact that there is no arc $A subseteq X$ satisfying $A subsetneq f^n[A]$ for some $nin mathbb{N}$. It is also equivalent to the fact that for some nonprincial ultrafilter $u$, the function $f^u colon X longrightarrow X$ is continuous (in other words, failure of equicontinuity of $f$ is equivalent to the failure of continuity of every element of the Ellis remainder $E^*(X,f)$).","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44980598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a pair of weakenings of the classical partition relation $nu rightarrow (mu)^2_lambda$ recently introduced by Bergfalk-Hrusak-Shelah and Bergfalk, respectively. Given an edge-coloring of the complete graph on $nu$-many vertices, these weakenings assert the existence of monochromatic subgraphs exhibiting high degrees of connectedness rather than the existence of complete monochromatic subgraphs asserted by the classical relations. As a result, versions of these weakenings can consistently hold at accessible cardinals where their classical analogues would necessarily fail. We prove some complementary positive and negative results indicating the effect of large cardinals, forcing axioms, and square principles on these partition relations. We also prove a consistency result indicating that a non-trivial instance of the stronger of these two partition relations can hold at the continuum.
{"title":"A note on highly connected and well-connected Ramsey theory","authors":"C. Lambie-Hanson","doi":"10.4064/fm141-9-2022","DOIUrl":"https://doi.org/10.4064/fm141-9-2022","url":null,"abstract":"We study a pair of weakenings of the classical partition relation $nu rightarrow (mu)^2_lambda$ recently introduced by Bergfalk-Hrusak-Shelah and Bergfalk, respectively. Given an edge-coloring of the complete graph on $nu$-many vertices, these weakenings assert the existence of monochromatic subgraphs exhibiting high degrees of connectedness rather than the existence of complete monochromatic subgraphs asserted by the classical relations. As a result, versions of these weakenings can consistently hold at accessible cardinals where their classical analogues would necessarily fail. We prove some complementary positive and negative results indicating the effect of large cardinals, forcing axioms, and square principles on these partition relations. We also prove a consistency result indicating that a non-trivial instance of the stronger of these two partition relations can hold at the continuum.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42960135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a sequence $(lambda_n)$ of positive real numbers we consider the exponential functions $f_{lambda_n} (z) = lambda_n e^z$ and the compositions $F_n = f_{lambda_n} circ f_{lambda_{n-1}} circ ... circ f_{lambda_1}$. For such a non-autonomous family we can define the Fatou and Julia sets analogously to the usual case of autonomous iteration. The aim of this document is to study how the Julia set depends on the sequence $(lambda_n)$. Among other results, we prove the Julia set for a random sequence ${lambda_n }$, chosen uniformly from a neighbourhood of $frac{1}{e}$, is the whole plane with probability $1$. We also prove the Julia set for $frac{1}{e} + frac{1}{n^p}$ is the whole plane for $p < frac{1}{2}$, and give an example of a sequence ${lambda_n } $ for which the iterates of $0$ converge to infinity starting from any index, but the Fatou set is non-empty.
{"title":"Julia sets of random exponential maps","authors":"Krzysztof Lech","doi":"10.4064/FM959-10-2020","DOIUrl":"https://doi.org/10.4064/FM959-10-2020","url":null,"abstract":"For a sequence $(lambda_n)$ of positive real numbers we consider the exponential functions $f_{lambda_n} (z) = lambda_n e^z$ and the compositions $F_n = f_{lambda_n} circ f_{lambda_{n-1}} circ ... circ f_{lambda_1}$. For such a non-autonomous family we can define the Fatou and Julia sets analogously to the usual case of autonomous iteration. The aim of this document is to study how the Julia set depends on the sequence $(lambda_n)$. Among other results, we prove the Julia set for a random sequence ${lambda_n }$, chosen uniformly from a neighbourhood of $frac{1}{e}$, is the whole plane with probability $1$. We also prove the Julia set for $frac{1}{e} + frac{1}{n^p}$ is the whole plane for $p < frac{1}{2}$, and give an example of a sequence ${lambda_n } $ for which the iterates of $0$ converge to infinity starting from any index, but the Fatou set is non-empty.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42465129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The first two sections of the paper provide a convenient scheme and additional diagrammatics for working with Frobenius extensions responsible for key flavors of equivariant SL(2) link homology theories. The goal is to clarify some basic structures in the theory and propose a setup to work over sufficiently non-degenerate base rings. The third section works out two related SL(2) evaluations for seamed surfaces.
{"title":"Link homology and Frobenius extensions II","authors":"M. Khovanov, Louis-Hadrien Robert","doi":"10.4064/fm912-6-2021","DOIUrl":"https://doi.org/10.4064/fm912-6-2021","url":null,"abstract":"The first two sections of the paper provide a convenient scheme and additional diagrammatics for working with Frobenius extensions responsible for key flavors of equivariant SL(2) link homology theories. The goal is to clarify some basic structures in the theory and propose a setup to work over sufficiently non-degenerate base rings. The third section works out two related SL(2) evaluations for seamed surfaces.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47187009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Coarse geometry studies metric spaces on the large scale. Our goal here is to study dynamics from a coarse point of view. To this end we introduce a coarse version of topological entropy, suitable for unbounded metric spaces, consistent with the coarse perspective on such spaces. As is the case with the usual topological entropy, the coarse entropy measures the divergence of orbits. Following Bowen's ideas, we use $(n,varepsilon)$-separated or $(n,varepsilon)$-spanning sets. However, we have to let $varepsilon$ go to infinity rather than to zero.
{"title":"Coarse entropy","authors":"W. Geller, M. Misiurewicz","doi":"10.4064/fm932-12-2020","DOIUrl":"https://doi.org/10.4064/fm932-12-2020","url":null,"abstract":"Coarse geometry studies metric spaces on the large scale. Our goal here is to study dynamics from a coarse point of view. To this end we introduce a coarse version of topological entropy, suitable for unbounded metric spaces, consistent with the coarse perspective on such spaces. As is the case with the usual topological entropy, the coarse entropy measures the divergence of orbits. Following Bowen's ideas, we use $(n,varepsilon)$-separated or $(n,varepsilon)$-spanning sets. However, we have to let $varepsilon$ go to infinity rather than to zero.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49388254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we're considering the question of whether or not a partial order on a finite set is realisable as a Smale order of a structurally stable diffeomorphism or flow acting on a closed manifold. We classify the orders that are realisable by 1) an $Omega$-stable diffeomorphism acting on a closed surface 2) an Anosov flow on a closed 3-manifold 3) a stable diffeomorphism with trivial attractors and repellers acting on a closed surface.
{"title":"On the realisability of Smale orders","authors":"Ioannis Iakovoglou","doi":"10.4064/fm178-4-2022","DOIUrl":"https://doi.org/10.4064/fm178-4-2022","url":null,"abstract":"In this paper we're considering the question of whether or not a partial order on a finite set is realisable as a Smale order of a structurally stable diffeomorphism or flow acting on a closed manifold. We classify the orders that are realisable by 1) an $Omega$-stable diffeomorphism acting on a closed surface 2) an Anosov flow on a closed 3-manifold 3) a stable diffeomorphism with trivial attractors and repellers acting on a closed surface.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48345056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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