We investigate to what extend the density of $Z_n$-maps in the characterization of $Q$-manifolds, and the density of maps $fin C(mathbb Ntimes Q,X)$ having discrete images in the $l_2$-manifolds characterization can be weakened to the density of homological $Z_n$-maps and homological $Z$-maps, respectively. As a result, we obtain homological characterizations of $Q$-manifolds and $l_2$-manifolds.
{"title":"Homological characterizations of\u0000$Q$-manifolds and $l_2$-manifolds","authors":"A. Karassev, V. Valov","doi":"10.4064/fm68-3-2022","DOIUrl":"https://doi.org/10.4064/fm68-3-2022","url":null,"abstract":"We investigate to what extend the density of $Z_n$-maps in the characterization of $Q$-manifolds, and the density of maps $fin C(mathbb Ntimes Q,X)$ having discrete images in the $l_2$-manifolds characterization can be weakened to the density of homological $Z_n$-maps and homological $Z$-maps, respectively. As a result, we obtain homological characterizations of $Q$-manifolds and $l_2$-manifolds.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48495102","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}
Regularity properties of the pressure are related to phase transitions. In this article we study thermodynamic formalism for systems defined in non-compact phase spaces, our main focus being countable Markov shifts. We produce metric compactifications of the space which allow us to prove that the pressure is differentiable on a residual set and outside an Aronszajn null set in the space of uniformly continuous functions. We establish a criterion, the so called sectorially arranged property, which implies that the pressure in the original system and in the compactification coincide. Examples showing that the compactifications can have rich boundaries, for example a Cantor set, are provided.
{"title":"Differentiability of the pressure in non-compact spaces","authors":"G. Iommi, M. Todd","doi":"10.4064/fm182-3-2022","DOIUrl":"https://doi.org/10.4064/fm182-3-2022","url":null,"abstract":"Regularity properties of the pressure are related to phase transitions. In this article we study thermodynamic formalism for systems defined in non-compact phase spaces, our main focus being countable Markov shifts. We produce metric compactifications of the space which allow us to prove that the pressure is differentiable on a residual set and outside an Aronszajn null set in the space of uniformly continuous functions. We establish a criterion, the so called sectorially arranged property, which implies that the pressure in the original system and in the compactification coincide. Examples showing that the compactifications can have rich boundaries, for example a Cantor set, are provided.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43175832","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 article, we give a dynamical and elementary proof of a result of Giordano, Putnam and Skau which establishes a necessary and sufficient condition for two minimal homeomorphisms of a Cantor space to be strong orbit equivalent. Our argument is based on a detailed study of some countable locally finite groups attached to minimal homeomorphisms. This approach also enables us to prove that the Borel complexity of the isomorphism relation on simple locally finite groups is a universal relation arising from a Borel S∞-action.
{"title":"Strong orbit equivalence in Cantor dynamics and simple locally finite groups","authors":"Simon Robert","doi":"10.4064/fm227-7-2022","DOIUrl":"https://doi.org/10.4064/fm227-7-2022","url":null,"abstract":"In this article, we give a dynamical and elementary proof of a result of Giordano, Putnam and Skau which establishes a necessary and sufficient condition for two minimal homeomorphisms of a Cantor space to be strong orbit equivalent. Our argument is based on a detailed study of some countable locally finite groups attached to minimal homeomorphisms. This approach also enables us to prove that the Borel complexity of the isomorphism relation on simple locally finite groups is a universal relation arising from a Borel S∞-action.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43509356","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 consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman bracket function defined on link diagrams on the surface. In the case that the 3-manifold is a thickened surface, this Kauffman bracket function leads to a Jones-type polynomial that is an isotopy invariant of links. We show that coefficients of this polynomial provide 2-sided linear bounds on the volume of hyperbolic alternating links in the thickened surface. As a corollary of the proof of this result, we deduce that the twist number of a reduced, twist reduced, checkerboard alternating link projection with disk regions, is an invariant of the link.
{"title":"Guts, volume and skein modules of 3-manifolds","authors":"Brandon Bavier, Efstratia Kalfagianni","doi":"10.4064/FM996-1-2021","DOIUrl":"https://doi.org/10.4064/FM996-1-2021","url":null,"abstract":"We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman bracket function defined on link diagrams on the surface. \u0000In the case that the 3-manifold is a thickened surface, this Kauffman bracket function leads to a Jones-type polynomial that is an isotopy invariant of links. We show that coefficients of this polynomial provide 2-sided linear bounds on the volume of hyperbolic alternating links in the thickened surface. As a corollary of the proof of this result, we deduce that the twist number of a reduced, twist reduced, checkerboard alternating link projection with disk regions, is an invariant of the link.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41849657","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 collect several foundational results regarding the interaction between locally compact spaces, probability spaces and probability algebras, and commutative $C^*$-algebras and von Neumann algebras equipped with traces, in the "uncountable" setting in which no separability, metrizability, or standard Borel hypotheses are placed on these spaces and algebras. In particular, we review the Gelfand dualities and Riesz representation theorems available in this setting. We also introduce a canonical model that represents (opposite) probability algebras as compact Hausdorff probability spaces in a completely functorial fashion, and apply this model to obtain a canonical disintegration theorem and to readily construct various product measures. These tools will be used in future papers by the authors and others in various applications to "uncountable" ergodic theory.
{"title":"Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration","authors":"Asgar Jamneshan, T. Tao","doi":"10.4064/fm226-7-2022","DOIUrl":"https://doi.org/10.4064/fm226-7-2022","url":null,"abstract":"We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and probability algebras, and commutative $C^*$-algebras and von Neumann algebras equipped with traces, in the \"uncountable\" setting in which no separability, metrizability, or standard Borel hypotheses are placed on these spaces and algebras. In particular, we review the Gelfand dualities and Riesz representation theorems available in this setting. We also introduce a canonical model that represents (opposite) probability algebras as compact Hausdorff probability spaces in a completely functorial fashion, and apply this model to obtain a canonical disintegration theorem and to readily construct various product measures. These tools will be used in future papers by the authors and others in various applications to \"uncountable\" ergodic theory.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45333204","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 notion of a textbf{$boldsymbol{mathcal{C}}$-filtered} object, where $mathcal{C}$ is some (typically small) collection of objects in a Grothendieck category, has become ubiquitous since the solution of the Flat Cover Conjecture around the year 2000. We introduce the textbf{$boldsymbol{mathcal{C}}$-Filtration Game of length $boldsymbol{omega_1}$} on a module, paying particular attention to the case where $mathcal{C}$ is the collection of all countably presented, projective modules. We prove that Martin's Maximum implies the determinacy of many $mathcal{C}$-Filtration Games of length $omega_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fraisse games of length $omega_1$; this allows a significant strengthening of a theorem of Mekler-Shelah-Vaananen cite{MR1191613}. Also, Martin's Maximum implies that if $R$ is a countable hereditary ring, the class of textbf{$boldsymbol{sigma}$-closed potentially projective modules}---i.e., those modules that are projective in some $sigma$-closed forcing extension of the universe---is closed under $
{"title":"Filtration games and potentially projective modules","authors":"Sean D. Cox","doi":"10.4064/fm237-10-2022","DOIUrl":"https://doi.org/10.4064/fm237-10-2022","url":null,"abstract":"The notion of a textbf{$boldsymbol{mathcal{C}}$-filtered} object, where $mathcal{C}$ is some (typically small) collection of objects in a Grothendieck category, has become ubiquitous since the solution of the Flat Cover Conjecture around the year 2000. We introduce the textbf{$boldsymbol{mathcal{C}}$-Filtration Game of length $boldsymbol{omega_1}$} on a module, paying particular attention to the case where $mathcal{C}$ is the collection of all countably presented, projective modules. We prove that Martin's Maximum implies the determinacy of many $mathcal{C}$-Filtration Games of length $omega_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fraisse games of length $omega_1$; this allows a significant strengthening of a theorem of Mekler-Shelah-Vaananen cite{MR1191613}. Also, Martin's Maximum implies that if $R$ is a countable hereditary ring, the class of textbf{$boldsymbol{sigma}$-closed potentially projective modules}---i.e., those modules that are projective in some $sigma$-closed forcing extension of the universe---is closed under $<aleph_2$-directed limits. We also give an example of a (ZFC-definable) class of abelian groups that, under the ordinary subgroup relation, constitutes an Abstract Elementary Class (AEC) with Lowenheim-Skolem number $aleph_1$ in some models in set theory, but fails to be an AEC in other models of set theory.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48872626","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 study the notion of Salem set from the point of view of descriptive set theory. We first work in the hyperspace $mathbf{K}([0,1])$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a $boldsymbol{Pi}^0_3$-complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in $mathbf{K}([0,1]^d)$. We then generalize the results by relaxing the compactness of the ambient space, and show that the closed Salem sets are still $boldsymbol{Pi}^0_3$-complete when we endow $mathbf{F}(mathbb{R}^d)$ with the Fell topology. A similar result holds also for the Vietoris topology. We apply our results to characterize the Weihrauch degree of the functions computing the Hausdorff and Fourier dimensions.
{"title":"On the descriptive complexity of Salem sets","authors":"A. Marcone, Manlio Valenti","doi":"10.4064/fm997-7-2021","DOIUrl":"https://doi.org/10.4064/fm997-7-2021","url":null,"abstract":"In this paper we study the notion of Salem set from the point of view of descriptive set theory. We first work in the hyperspace $mathbf{K}([0,1])$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a $boldsymbol{Pi}^0_3$-complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in $mathbf{K}([0,1]^d)$. We then generalize the results by relaxing the compactness of the ambient space, and show that the closed Salem sets are still $boldsymbol{Pi}^0_3$-complete when we endow $mathbf{F}(mathbb{R}^d)$ with the Fell topology. A similar result holds also for the Vietoris topology. We apply our results to characterize the Weihrauch degree of the functions computing the Hausdorff and Fourier dimensions.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"57 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41259532","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 describe the fermionic and bosonic Fock representation of the Lie super-algebra of endomorphisms of the exterior algebra of the ${mathbb Q}$-vector space of infinite countable dimension, vanishing at all but finitely many basis elements. We achieve the goal by exploiting the extension of the Schubert derivations to the Fermionic Fock space.
{"title":"Bosonic and fermionic representations of endomorphisms of exterior algebras","authors":"Ommolbanin Behzad, Letterio Gatto","doi":"10.4064/fm9-12-2020","DOIUrl":"https://doi.org/10.4064/fm9-12-2020","url":null,"abstract":"We describe the fermionic and bosonic Fock representation of the Lie super-algebra of endomorphisms of the exterior algebra of the ${mathbb Q}$-vector space of infinite countable dimension, vanishing at all but finitely many basis elements. We achieve the goal by exploiting the extension of the Schubert derivations to the Fermionic Fock space.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43426243","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 prove that for any holomorphic map, and any bounded orbit which does not accumulate to a singular set or to an attracting cycle, its lower Lyapunov exponent is non-negative. The same result holds for unbounded orbits, for maps with a bounded singular set. Furthermore, the orbit may accumulate to infinity or to a singular set, as long as it is slow enough.
{"title":"On the pointwise Lyapunov exponent of holomorphic maps","authors":"I. Weinstein","doi":"10.4064/fm847-1-2020","DOIUrl":"https://doi.org/10.4064/fm847-1-2020","url":null,"abstract":"We prove that for any holomorphic map, and any bounded orbit which does not accumulate to a singular set or to an attracting cycle, its lower Lyapunov exponent is non-negative. The same result holds for unbounded orbits, for maps with a bounded singular set. Furthermore, the orbit may accumulate to infinity or to a singular set, as long as it is slow enough.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47281745","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 investigate an extension of ZFC set theory (in an extended language) that stipulates the existence of a proper class of indiscernibles over the universe. One of the main results of the paper shows that the purely set-theoretical consequences of this extension of ZFC coincide with the theorems of the system of set theory obtained by augmenting ZFC with the (Levy) scheme whose instances assert, for each natural number $n$ in the metatheory, that there is an $n$-Mahlo cardinal $kappa$ with the property that the initial segment of the universe determined by $kappa$ is a $Sigma_n$-elementary submodel of the universe.
{"title":"Set theory with a proper class of indiscernibles","authors":"A. Enayat","doi":"10.4064/fm999-2-2022","DOIUrl":"https://doi.org/10.4064/fm999-2-2022","url":null,"abstract":"We investigate an extension of ZFC set theory (in an extended language) that stipulates the existence of a proper class of indiscernibles over the universe. One of the main results of the paper shows that the purely set-theoretical consequences of this extension of ZFC coincide with the theorems of the system of set theory obtained by augmenting ZFC with the (Levy) scheme whose instances assert, for each natural number $n$ in the metatheory, that there is an $n$-Mahlo cardinal $kappa$ with the property that the initial segment of the universe determined by $kappa$ is a $Sigma_n$-elementary submodel of the universe.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44920630","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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