{"title":"Complexities and representations of $mathcal F$-Borel spaces","authors":"Vojtěch Kovařík","doi":"10.4064/DM794-2-2019","DOIUrl":"https://doi.org/10.4064/DM794-2-2019","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70158077","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 present a unifying approach to the study of entropies in Mathematics, such as measure entropy, topological entropy, algebraic entropy, set-theoretic entropy. We take into account discrete dynamical systems, that is, pairs $(X,T)$, where $X$ is the underlying space and $T:Xto X$ a transformation. We see entropies as functions $h:mathfrak Xto mathbb R_+$, associating to each flow $(X,T)$ of a category $mathfrak X$ either a non negative real or $infty$. We introduce the notion of semigroup entropy $h_mathfrak S:mathfrak Stomathbb R_+$, which is a numerical invariant attached to endomorphisms of the category $mathfrak S$ of normed semigroups. Then, for a functor $F:mathfrak Xtomathfrak S$ from any specific category $mathfrak X$ to $mathfrak S$, we define the functorial entropy $h_F:mathfrak Xtomathbb R_+$ as the composition $h_{mathfrak S}circ F$. Clearly, $h_F$ inherits many of the properties of $h_mathfrak S$, depending also on the properties of $F$. Such general scheme permits to obtain relevant known entropies as functorial entropies $h_F$, for appropriate categories $mathfrak X$ and functors $F$, and to establish the properties shared by them. In this way we point out their common nature. Finally, we discuss and deeply analyze through the looking glass of our unifying approach the relations between pairs of entropies. To this end we formalize the notion of Bridge Theorem between two entropies $h_i:mathfrak X_ito mathbb R_+$, $i=1,2$, with respect to a functor $varepsilon:mathfrak X_1tomathfrak X_2$. Then, for pairs of functorial entropies we use the above scheme to introduce the notion and the related scheme of Strong Bridge Theorem, which allows us to put under the same umbrella various relations between pairs of entropies.
{"title":"Entropy on normed semigroups (towards a unifying approach to entropy)","authors":"D. Dikranjan, A. Bruno","doi":"10.4064/dm791-2-2019","DOIUrl":"https://doi.org/10.4064/dm791-2-2019","url":null,"abstract":"We present a unifying approach to the study of entropies in Mathematics, such as measure entropy, topological entropy, algebraic entropy, set-theoretic entropy. We take into account discrete dynamical systems, that is, pairs $(X,T)$, where $X$ is the underlying space and $T:Xto X$ a transformation. We see entropies as functions $h:mathfrak Xto mathbb R_+$, associating to each flow $(X,T)$ of a category $mathfrak X$ either a non negative real or $infty$. We introduce the notion of semigroup entropy $h_mathfrak S:mathfrak Stomathbb R_+$, which is a numerical invariant attached to endomorphisms of the category $mathfrak S$ of normed semigroups. Then, for a functor $F:mathfrak Xtomathfrak S$ from any specific category $mathfrak X$ to $mathfrak S$, we define the functorial entropy $h_F:mathfrak Xtomathbb R_+$ as the composition $h_{mathfrak S}circ F$. Clearly, $h_F$ inherits many of the properties of $h_mathfrak S$, depending also on the properties of $F$. Such general scheme permits to obtain relevant known entropies as functorial entropies $h_F$, for appropriate categories $mathfrak X$ and functors $F$, and to establish the properties shared by them. In this way we point out their common nature. Finally, we discuss and deeply analyze through the looking glass of our unifying approach the relations between pairs of entropies. To this end we formalize the notion of Bridge Theorem between two entropies $h_i:mathfrak X_ito mathbb R_+$, $i=1,2$, with respect to a functor $varepsilon:mathfrak X_1tomathfrak X_2$. Then, for pairs of functorial entropies we use the above scheme to introduce the notion and the related scheme of Strong Bridge Theorem, which allows us to put under the same umbrella various relations between pairs of entropies.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48523837","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 introduce and study multivariate generalizations of the classical BV spaces of Jordan, F. Riesz and Wiener. The family of the introduced spaces contains or is intimately related to a considerable class of function spaces of modern analysis including BMO, BV, Morrey spaces and those of Sobolev of arbitrary smoothness, Besov and Triebel-Lizorkin spaces. We prove under mild restrictions that the BV spaces of this family are dual and present constructive characterizations of their preduals via atomic decompositions. Moreover, we show that under additional restrictions such a predual space is isometrically isomorphic to the dual space of the separable subspace of the related BV space generated by $C^infty$ functions. As a corollary we obtain the "two stars theorem" asserting that the second dual of this separable subspace is isometrically isomorphic to the BV space. An essential role in the proofs play approximation properties of the BV spaces under consideration, in particular, weak$^*$ denseness of their subspaces of $C^infty$ functions. Our results imply the similar ones (old and new) for the classical function spaces listed above obtained by the unified approach.
{"title":"On the Banach structure of multivariate BV spaces","authors":"A. Brudnyi, Y. Brudnyi","doi":"10.4064/dm801-7-2019","DOIUrl":"https://doi.org/10.4064/dm801-7-2019","url":null,"abstract":"We introduce and study multivariate generalizations of the classical BV spaces of Jordan, F. Riesz and Wiener. The family of the introduced spaces contains or is intimately related to a considerable class of function spaces of modern analysis including BMO, BV, Morrey spaces and those of Sobolev of arbitrary smoothness, Besov and Triebel-Lizorkin spaces. We prove under mild restrictions that the BV spaces of this family are dual and present constructive characterizations of their preduals via atomic decompositions. Moreover, we show that under additional restrictions such a predual space is isometrically isomorphic to the dual space of the separable subspace of the related BV space generated by $C^infty$ functions. As a corollary we obtain the \"two stars theorem\" asserting that the second dual of this separable subspace is isometrically isomorphic to the BV space. An essential role in the proofs play approximation properties of the BV spaces under consideration, in particular, weak$^*$ denseness of their subspaces of $C^infty$ functions. Our results imply the similar ones (old and new) for the classical function spaces listed above obtained by the unified approach.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44710544","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 the stochastic dissipative quasi-geostrophic equation with space-time white noise on the two-dimensional torus. This equation is highly singular and basically ill-posed in its original form. The main objective of the present paper is to formulate and solve this equation locally in time in the framework of paracontrolled calculus when the differential order of the main term, the fractional Laplacian, is larger than $7/4$. No renormalization has to be done for this model.
{"title":"Paracontrolled quasi-geostrophic equation with space-time white noise","authors":"Y. Inahama, Y. Sawano","doi":"10.4064/dm806-7-2020","DOIUrl":"https://doi.org/10.4064/dm806-7-2020","url":null,"abstract":"We study the stochastic dissipative quasi-geostrophic equation with space-time white noise on the two-dimensional torus. This equation is highly singular and basically ill-posed in its original form. The main objective of the present paper is to formulate and solve this equation locally in time in the framework of paracontrolled calculus when the differential order of the main term, the fractional Laplacian, is larger than $7/4$. No renormalization has to be done for this model.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45504995","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 show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental components are separately, complex, orthogonal, and symplectic holonomy reductions of the canonical projective tractor/Cartan connection. This leads to a characterisation of Sasaki-Einstein structures as projective structures with certain unitary holonomy reductions. As an immediate application, this is used to describe the projective compactification of indefinite (suitably) complete non-compact Sasaki-Einstein structures and to prove that the boundary at infinity is a Fefferman conformal manifold that thus fibres over a nondegenerate CR manifold (of hypersurface type). We prove that this CR manifold coincides with the boundary at infinity for the c-projective compactification of the K"ahler-Einstein manifold that arises, in the usual way, as a leaf space for the defining Killing field of the given Sasaki-Einstein manifold. A procedure for constructing examples is given. The discussion of symplectic holonomy reductions of projective structures leads us moreover to a new and simplifying approach to contact projective geometry. This is of independent interest and is treated in some detail.
{"title":"Projective geometry of Sasaki–Einstein structures and their compactification","authors":"A. Gover, Katharina Neusser, T. Willse","doi":"10.4064/dm786-7-2019","DOIUrl":"https://doi.org/10.4064/dm786-7-2019","url":null,"abstract":"We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental components are separately, complex, orthogonal, and symplectic holonomy reductions of the canonical projective tractor/Cartan connection. This leads to a characterisation of Sasaki-Einstein structures as projective structures with certain unitary holonomy reductions. As an immediate application, this is used to describe the projective compactification of indefinite (suitably) complete non-compact Sasaki-Einstein structures and to prove that the boundary at infinity is a Fefferman conformal manifold that thus fibres over a nondegenerate CR manifold (of hypersurface type). We prove that this CR manifold coincides with the boundary at infinity for the c-projective compactification of the K\"ahler-Einstein manifold that arises, in the usual way, as a leaf space for the defining Killing field of the given Sasaki-Einstein manifold. A procedure for constructing examples is given. The discussion of symplectic holonomy reductions of projective structures leads us moreover to a new and simplifying approach to contact projective geometry. This is of independent interest and is treated in some detail.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45015547","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}
T. Banakh, Szymon Glkab, Eliza Jablo'nska, J. Swaczyna
Generalizing Christensen's notion of a Haar-null set and Darji's notion of a Haar-meager set, we introduce and study the notion of a Haar-$mathcal I$ set in a Polish group. Here $mathcal I$ is an ideal of subsets of some compact metrizable space $K$. A Borel subset $Bsubset X$ of a Polish group $X$ is called Haar-$mathcal I$ if there exists a continuous map $f:Kto X$ such that $f^{-1}(B+x)inmathcal I$ for all $xin X$. Moreover, $B$ is generically Haar-$mathcal I$ if the set of witness functions ${fin C(K,X):forall xin X;;f^{-1}(B+x)inmathcal I}$ is comeager in the function space $C(K,X)$. We study (generically) Haar-$mathcal I$ sets in Polish groups for many concrete and abstract ideals $mathcal I$, and construct the corresponding distinguishing examples. Also we establish various Steinhaus properties of the families of (generically) Haar-$mathcal I$ sets in Polish groups for various ideals $mathcal I$.
推广了Christensen的Haar空集概念和Darji的Haar贫集概念，引入并研究了波兰群中Haar-$mathcalI$集的概念。这里$mathcal I$是某个紧致可度量空间$K$的子集的理想。波兰群$X$的Borel子集$B子集X$称为Haar-$mathcal I$，如果存在到X$的连续映射$f:K，使得对于X$中的所有$X，$f^｛-1｝（B+X）inmathcal I$。此外，如果C（K，X）中的见证函数$｛f： for all X in X；；f^｛-1｝（B+X） in mathcal I｝$的集合在函数空间$C（K，X）$中是comeager，则$B$一般是Haar-$mathcal I$。我们（一般地）研究了波兰群中许多具体和抽象理想$mathcal I$的Haar-$mathical I$集，并构造了相应的区别例子。此外，我们还为各种理想$mathcalI$建立了波兰群中（一般）Haar-$mathcal I$集合族的各种Steinhaus性质。
{"title":"Haar-$mathcal I$ sets: looking at small sets in Polish groups through compact glasses","authors":"T. Banakh, Szymon Glkab, Eliza Jablo'nska, J. Swaczyna","doi":"10.4064/dm812-2-2021","DOIUrl":"https://doi.org/10.4064/dm812-2-2021","url":null,"abstract":"Generalizing Christensen's notion of a Haar-null set and Darji's notion of a Haar-meager set, we introduce and study the notion of a Haar-$mathcal I$ set in a Polish group. Here $mathcal I$ is an ideal of subsets of some compact metrizable space $K$. A Borel subset $Bsubset X$ of a Polish group $X$ is called Haar-$mathcal I$ if there exists a continuous map $f:Kto X$ such that $f^{-1}(B+x)inmathcal I$ for all $xin X$. Moreover, $B$ is generically Haar-$mathcal I$ if the set of witness functions ${fin C(K,X):forall xin X;;f^{-1}(B+x)inmathcal I}$ is comeager in the function space $C(K,X)$. We study (generically) Haar-$mathcal I$ sets in Polish groups for many concrete and abstract ideals $mathcal I$, and construct the corresponding distinguishing examples. Also we establish various Steinhaus properties of the families of (generically) Haar-$mathcal I$ sets in Polish groups for various ideals $mathcal I$.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47412491","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}
Many special functions are solutions of both, a differential and a functional equation. We use this duality to solve a large class of abstract Sturm-Liouville equations, initiating a theory of Sturm-Liouville operator functions; cosine, Bessel, and Legendre operator functions are contained as special cases. This is part of a general concept of operator functions being multiplicative with respect to convolution of a hypergroup - containing all representations of (hyper)groups, and further abstract Cauchy problems.
{"title":"Sturm-Liouville Operator Functions","authors":"Felix Früchtl","doi":"10.4064/DM763-3-2018","DOIUrl":"https://doi.org/10.4064/DM763-3-2018","url":null,"abstract":"Many special functions are solutions of both, a differential and a functional equation. We use this duality to solve a large class of abstract Sturm-Liouville equations, initiating a theory of Sturm-Liouville operator functions; cosine, Bessel, and Legendre operator functions are contained as special cases. This is part of a general concept of operator functions being multiplicative with respect to convolution of a hypergroup - containing all representations of (hyper)groups, and further abstract Cauchy problems.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157722","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}
{"title":"Bilinear decompositions for products of Hardy and Lipschitz spaces on spaces of homogeneous type","authors":"Liguang Liu, Dachun Yang, Wen Yuan","doi":"10.4064/DM774-2-2018","DOIUrl":"https://doi.org/10.4064/DM774-2-2018","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157806","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 a class of embeddings of tent inverse limit spaces. We introduce techniques relying on the Milnor-Thurston kneading theory and use them to study sets of accessible points and prime ends of given embeddings. We completely characterize accessible points and prime ends of standard embeddings arising from the Barge-Martin construction of global attractors. In other (non-extendable) embeddings we find phenomena which do not occur in the standard embeddings.
{"title":"Accessible points of planar embeddings of tent inverse limit spaces","authors":"A. Anušić, Jernej Činč","doi":"10.4064/DM776-1-2019","DOIUrl":"https://doi.org/10.4064/DM776-1-2019","url":null,"abstract":"In this paper we study a class of embeddings of tent inverse limit spaces. We introduce techniques relying on the Milnor-Thurston kneading theory and use them to study sets of accessible points and prime ends of given embeddings. We completely characterize accessible points and prime ends of standard embeddings arising from the Barge-Martin construction of global attractors. In other (non-extendable) embeddings we find phenomena which do not occur in the standard embeddings.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45565200","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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