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Journal articles on the topic 'Ergodic and geometric group theory'

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Published: 31 August 2024

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1

Skripchenko, Alexandra Sergeevna. "Renormalization in one-dimensional dynamics." Russian Mathematical Surveys 78, no. 6 (2023): 983–1021. http://dx.doi.org/10.4213/rm10110e.

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The study of the dynamical and topological properties of interval exchange transformations and their natural generalizations is an important problem, which lies at the intersection of several branches of mathematics, including dynamical systems, low-dimensional topology, algebraic geometry, number theory, and geometric group theory. The purpose of the survey is to make a systematic presentation of the existing results on the ergodic and geometric characteristics of the one-dimensional maps under consideration, as well as on the measured foliations on surfaces and two-dimensional complexes that one can associate with these maps. These results are based on the research of the ergodic properties of the renormalization process - an algorithm that takes an original dynamical system and builds a sequence of equivalent dynamical systems with a smaller support set. For all dynamical systems considered in the paper these renormalization algorithms can be viewed as multidimensional fraction algorithms. Bibiliography: 74 titles.

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2

Hartman, Yair, and Ariel Yadin. "Furstenberg entropy of intersectional invariant random subgroups." Compositio Mathematica 154, no. 10 (September 17, 2018): 2239–65. http://dx.doi.org/10.1112/s0010437x18007261.

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We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen. The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply, for example, for certain extensions of the group of finitely supported permutations and lamplighter groups, hence establishing full realization results for these groups. For the free group, we construct the IRSs via a geometric construction of subgroups, by describing their Schreier graphs. The analysis of the entropy of these spaces is obtained by studying the random walk on the appropriate Schreier graphs.

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3

Guirardel, Vincent, Camille Horbez, and Jean Lécureux. "Cocycle superrigidity from higher rank lattices to $ {{\rm{Out}}}{(F_N)} $." Journal of Modern Dynamics 18 (2022): 291. http://dx.doi.org/10.3934/jmd.2022010.

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<p style='text-indent:20px;'>We prove a rigidity result for cocycles from higher rank lattices to <inline-formula><tex-math id="M2">\begin{document}$ \mathrm{Out}(F_N) $\end{document}</tex-math></inline-formula> and more generally to the outer automorphism group of a torsion-free hyperbolic group. More precisely, let <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula> be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let <inline-formula><tex-math id="M4">\begin{document}$ G \curvearrowright X $\end{document}</tex-math></inline-formula> be an ergodic measure-preserving action on a standard probability space, and let <inline-formula><tex-math id="M5">\begin{document}$ H $\end{document}</tex-math></inline-formula> be a torsion-free hyperbolic group. We prove that every Borel cocycle <inline-formula><tex-math id="M6">\begin{document}$ G\times X\to \mathrm{Out}(H) $\end{document}</tex-math></inline-formula> is cohomologous to a cocycle with values in a finite subgroup of <inline-formula><tex-math id="M7">\begin{document}$ \mathrm{Out}(H) $\end{document}</tex-math></inline-formula>. This provides a dynamical version of theorems of Farb–Kaimanovich–Masur and Bridson–Wade asserting that every homomorphism from <inline-formula><tex-math id="M8">\begin{document}$ G $\end{document}</tex-math></inline-formula> to either the mapping class group of a finite-type surface or the outer automorphism group of a free group, has finite image.</p><p style='text-indent:20px;'>The main new geometric tool is a barycenter map that associates to every triple of points in the boundary of the (relative) free factor graph a finite set of (relative) free splittings.</p>

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4

Kida, Yoshikata. "Ergodic group theory." Sugaku Expositions 35, no. 1 (April 7, 2022): 103–26. http://dx.doi.org/10.1090/suga/470.

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5

Young, Lai-Sang. "Geometric and Ergodic Theory of Hyperbolic Dynamical Systems." Current Developments in Mathematics 1998, no. 1 (1998): 237–78. http://dx.doi.org/10.4310/cdm.1998.v1998.n1.a6.

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6

Orponen, Tuomas, Pablo Shmerkin, and Hong Wang. "Incidence Problems in Harmonic Analysis, Geometric Measure Theory, and Ergodic Theory." Oberwolfach Reports 20, no. 2 (December 21, 2023): 1397–452. http://dx.doi.org/10.4171/owr/2023/25.

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7

Clay, Matt. "Geometric Group Theory." Notices of the American Mathematical Society 69, no. 10 (November 1, 2022): 1. http://dx.doi.org/10.1090/noti2572.

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8

Ziegler, Tamar. "An application of ergodic theory to a problem in geometric ramsey theory." Israel Journal of Mathematics 114, no. 1 (December 1999): 271–88. http://dx.doi.org/10.1007/bf02785583.

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9

BESSA, MÁRIO, and JORGE ROCHA. "Contributions to the geometric and ergodic theory of conservative flows." Ergodic Theory and Dynamical Systems 33, no. 6 (August 22, 2012): 1709–31. http://dx.doi.org/10.1017/etds.2012.110.

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AbstractWe prove the following dichotomy for vector fields in a $C^1$-residual subset of volume-preserving flows: for Lebesgue-almost every point, either all of its Lyapunov exponents are equal to zero or its orbit has a dominated splitting. Moreover, we prove that a volume-preserving and $C^1$-stably ergodic flow can be $C^1$-approximated by another volume-preserving flow which is non-uniformly hyperbolic.

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10

BOWEN, LEWIS, and AMOS NEVO. "Hyperbolic geometry and pointwise ergodic theorems." Ergodic Theory and Dynamical Systems 39, no. 10 (December 12, 2017): 2689–716. http://dx.doi.org/10.1017/etds.2017.128.

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We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real rank one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space.

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11

Kwiatkowski, Jan, and Mariusz Lemańczyk. "The centralizer of ergodic theory group extensions." Banach Center Publications 23, no. 1 (1989): 455–63. http://dx.doi.org/10.4064/-23-1-455-463.

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12

Gutschera, K. Robert. "Ergodic elements for actions of Lie groups." Ergodic Theory and Dynamical Systems 16, no. 4 (August 1996): 703–17. http://dx.doi.org/10.1017/s0143385700009056.

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AbstractGiven a connected Lie group G acting ergodically on a space S with finite invariant measure, one can ask when G will contain single elements (or one-parameter subgroups) that still act ergodically. For a compact simple group or the isometry group of the plane, or any group projecting onto such groups, an ergodic action may have no ergodic elements, but for any other connected Lie group ergodic elements will exist. The proof uses the unitary representation theory of Lie groups and Lie group structure theory.

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13

Bridson, Martin, Linus Kramer, Bertrand Rémy, and Karen Vogtmann. "Geometric Structures in Group Theory." Oberwolfach Reports 10, no. 2 (2013): 1629–75. http://dx.doi.org/10.4171/owr/2013/28.

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Bridson, Martin, Linus Kramer, Bertrand Rémy, and Karen Vogtmann. "Geometric Structures in Group Theory." Oberwolfach Reports 14, no. 2 (April 27, 2018): 1869–915. http://dx.doi.org/10.4171/owr/2017/30.

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15

Bridson, Martin, Cornelia Druţu Badea, Linus Kramer, Bertrand Rémy, and Petra Schwer. "Geometric Structures in Group Theory." Oberwolfach Reports 17, no. 2 (July 1, 2021): 877–918. http://dx.doi.org/10.4171/owr/2020/16.

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16

Hoffman, William C. "Group theory and geometric psychology." Behavioral and Brain Sciences 24, no. 4 (August 2001): 674–76. http://dx.doi.org/10.1017/s0140525x01410089.

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The commentary is in general agreement with Roger Shepard's view of evolutionary internalization of certain procedural memories, but advocates the use of Lie groups to express the invariances of motion and color perception involved. For categorization, the dialectical pair is suggested. [Barlow; Hecht; Kubovy & Epstein; Schwartz; Shepard; Todorovič]

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Bridson, Martin, Cornelia Druţu Badea, Linus Kramer, and Bertrand Rémy. "Geometric Structures in Group Theory." Oberwolfach Reports 19, no. 1 (March 10, 2023): 517–76. http://dx.doi.org/10.4171/owr/2022/11.

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18

Martens, Marco. "Distortion results and invariant Cantor sets of unimodal maps." Ergodic Theory and Dynamical Systems 14, no. 2 (June 1994): 331–49. http://dx.doi.org/10.1017/s0143385700007902.

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AbstractA distortion theory is developed for S-unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of S-unimodal maps is classified according to a distortion property, called the Markov-property.

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19

Galindo, Jorge, and Enrique Jorda. "Ergodic properties of convolution operators." Journal of Operator Theory 86, no. 2 (November 15, 2021): 469–501. http://dx.doi.org/10.7900/jot.2020jun25.2303.

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Let G be a locally compact group and μ be a measure on G. In this paper we find conditions for the convolution operators λp(μ):Lp(G)→Lp(G) to be mean ergodic and uniformly mean ergodic. The ergodic properties of the operators λp(μ) are related to the ergodic properties of the measure μ as well.

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20

Burq, N. "Quantum Ergodicity of Boundary Values of Eigenfunctions: A Control Theory Approach." Canadian Mathematical Bulletin 48, no. 1 (March 1, 2005): 3–15. http://dx.doi.org/10.4153/cmb-2005-001-3.

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AbstractConsider M, a bounded domain in ℝd, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary according to the laws of geometric optics is ergodic. We prove that the boundary value of the eigen-functions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by Gérard and Leichtnam as well as Hassell and Zelditch, obtained under the additional assumption of the convexity of M.

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21

LINERO BAS, ANTONIO, and GABRIEL SOLER LÓPEZ. "Minimal interval exchange transformations with flips." Ergodic Theory and Dynamical Systems 38, no. 8 (April 3, 2017): 3101–44. http://dx.doi.org/10.1017/etds.2017.5.

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We consider interval exchange transformations of$n$intervals with$k$flips, or$(n,k)$-IETs for short, for positive integers$k,n$with$k\leq n$. Our main result establishes the existence of minimal uniquely ergodic$(n,k)$-IETs when$n\geq 4$; moreover, these IETs are self-induced for$2\leq k\leq n-1$. This result extends the work on transitivity in Gutierrezet al[Transitive circle exchange transformations with flips.Discrete Contin. Dyn. Syst. 26(1) (2010), 251–263]. In order to achieve our objective we make a direct construction; in particular, we use the Rauzy induction to build a periodic Rauzy graph whose associated matrix has a positive power. Then we use a result in the Perron–Frobenius theory [Pullman, A geometric approach to the theory of non-negative matrices.Linear Algebra Appl. 4(1971) 297–312] which allows us to ensure the existence of these minimal self-induced and uniquely ergodic$(n,k)$-IETs,$2\leq k\leq n-1$. We then find other permutations in the same Rauzy class generating minimal uniquely ergodic$(n,1)$- and$(n,n)$-IETs.

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22

Diller, Jeffrey, Romain Dujardin, and Vincent Guedj. "Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory." Annales scientifiques de l'École normale supérieure 43, no. 2 (2010): 235–78. http://dx.doi.org/10.24033/asens.2120.

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23

Wang, Hai Yang, Xian Qing Lei, and Jing Wei Cui. "Parabola Error Evaluation Using Geometry Ergodic Searching Algorithm." Applied Mechanics and Materials 333-335 (July 2013): 1465–68. http://dx.doi.org/10.4028/www.scientific.net/amm.333-335.1465.

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A method of parabola error evaluation using Geometry Ergodic Searching Algorithm (GESA) was proposed according to geometric features and fitting characteristics of parabola error. First , the feature points of least-squared parabola are set as reference feature points to layout a group of auxiliary feature grid points. After that, a series of auxiliary parabolas as assumed ideal parabolas are reversed with the auxiliary feature points.The range distance from given points to these assumptions ideal parabolas are calculated successively.The minimum one is parabola profile error.The process of GESA was detailed discribed including the algorithm formula and contrastive results in this paper.Simulation experiment results show that the geometry ergodic searching algorithm is more accurate than the least-square method. The parabola profile error can be evaluated steadily and precisely with this algorithm based on the minimum zone.

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24

MIHAILESCU, EUGEN. "DIMENSIONS OF MEASURES, DEGREES, AND FOLDING ENTROPY IN DYNAMICS." Revue Roumaine Mathematiques Pures Appliquees LXVIII, no. 1-2 (2023): 149–67. http://dx.doi.org/10.59277/rrmpa.2023.149.167.

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In this survey, we present some methods in the dynamics and dimension theory for invariant measures of hyperbolic endomorphisms (smooth non-invertible maps), and for conformal iterated function systems with overlaps. For endomorphisms, we recall the notion of asymptotic degree of an equilibrium measure, which is shown to be related to the folding entropy; this degree is then applied to dimension estimates. For finite iterated function systems, we present the notion of overlap number of a measure, which is related to the folding entropy of a lift transformation, and also give some examples when it can be computed or estimated. We apply overlap numbers to prove the exact dimensionality of invariant measures, and to obtain a geometric formula for their dimension. Then, for countable conformal iterated function systems with overlaps, the projections of ergodic measures are shown to be exact dimensional, and we give a dimension formula. Relations with ergodic number theory, continued fractions, and random dynamical systems are also presented.

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25

BOWEN, LEWIS, and AMOS NEVO. "Pointwise ergodic theorems beyond amenable groups." Ergodic Theory and Dynamical Systems 33, no. 3 (April 16, 2012): 777–820. http://dx.doi.org/10.1017/s0143385712000041.

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AbstractWe prove pointwise and maximal ergodic theorems for probability-measure-preserving (PMP) actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable typeIII$_1$. We show that this class contains all irreducible lattices in connected semi-simple Lie groups without compact factors. We also establish similar results when the stable type isIII$_\lambda $,$0 \lt \lambda \lt 1$, under a suitable hypothesis. Our approach is based on the following two principles. First, we show that it is possible to generalize the ergodic theory of PMP actions of amenable groups to include PMP amenable equivalence relations. Secondly, we show that it is possible to reduce the proof of ergodic theorems for PMP actions of a general group to the proof of ergodic theorems in an associated PMP amenable equivalence relation, provided the group admits an amenable action with the properties stated above.

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26

Kra, Bryna. "Commentary on “Ergodic theory of amenable group actions”: Old and new." Bulletin of the American Mathematical Society 55, no. 3 (April 18, 2018): 343–45. http://dx.doi.org/10.1090/bull/1619.

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27

Ozawa, Narutaka. "A remark on fullness of some group measure space von Neumann algebras." Compositio Mathematica 152, no. 12 (November 2, 2016): 2493–502. http://dx.doi.org/10.1112/s0010437x16007727.

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Recently Houdayer and Isono have proved, among other things, that every biexact group $\unicode[STIX]{x1D6E4}$ has the property that for any non-singular strongly ergodic essentially free action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ on a standard measure space, the group measure space von Neumann algebra $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(X)$ is full. In this paper, we prove the same property for a wider class of groups, notably including $\text{SL}(3,\mathbb{Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $\unicode[STIX]{x1D6E4}\leqslant G$, and any closed non-amenable subgroup $H\leqslant G$, the non-singular action $\unicode[STIX]{x1D6E4}\curvearrowright G/H$ is strongly ergodic and the von Neumann factor $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(G/H)$ is full.

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28

Bufetov, Alexander I., and Yanqi Qiu. "Ergodic measures on spaces of infinite matrices over non-Archimedean locally compact fields." Compositio Mathematica 153, no. 12 (September 11, 2017): 2482–533. http://dx.doi.org/10.1112/s0010437x17007412.

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Let$F$be a non-discrete non-Archimedean locally compact field and${\mathcal{O}}_{F}$the ring of integers in$F$. The main results of this paper are the classification of ergodic probability measures on the space$\text{Mat}(\mathbb{N},F)$of infinite matrices with entries in$F$with respect to the natural action of the group$\text{GL}(\infty ,{\mathcal{O}}_{F})\times \text{GL}(\infty ,{\mathcal{O}}_{F})$and the classification, for non-dyadic$F$, of ergodic probability measures on the space$\text{Sym}(\mathbb{N},F)$of infinite symmetric matrices with respect to the natural action of the group$\text{GL}(\infty ,{\mathcal{O}}_{F})$.

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29

SAPIR, MARK V. "SOME GROUP THEORY PROBLEMS." International Journal of Algebra and Computation 17, no. 05n06 (August 2007): 1189–214. http://dx.doi.org/10.1142/s0218196707003925.

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This is a survey of some problems in geometric group theory that I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some necessary definitions and motivations, problems and some discussions of them. For each problem, I try to mention the author. If the author is not given, the problem, to the best of my knowledge, was formulated by me first.

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30

LE MAÎTRE, FRANÇOIS. "On full groups of non-ergodic probability-measure-preserving equivalence relations." Ergodic Theory and Dynamical Systems 36, no. 7 (June 15, 2015): 2218–45. http://dx.doi.org/10.1017/etds.2015.20.

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This article generalizes our previous results [Le Maître. The number of topological generators for full groups of ergodic equivalence relations. Invent. Math. 198 (2014), 261–268] to the non-ergodic case by giving a formula relating the topological rank of the full group of an aperiodic probability-measure-preserving (pmp) equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups that have a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to provide an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing a result of Giordano and Pestov to the non-ergodic case [Giordano and Pestov. Some extremely amenable groups related to operator algebras and ergodic theory. J. Inst. Math. Jussieu6(2) (2007), 279–315, Theorem 5.7].

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31

Knieper, Gerhard, Leonid Polterovich, and Leonid Potyagailo. "Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry." Oberwolfach Reports 9, no. 3 (2012): 2139–203. http://dx.doi.org/10.4171/owr/2012/35.

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32

Bergelson, Vitaly, and Joseph Rosenblatt. "Joint ergodicity for group actions." Ergodic Theory and Dynamical Systems 8, no. 3 (September 1988): 351–64. http://dx.doi.org/10.1017/s0143385700004508.

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AbstractLet T1,…,Tn be continuous representations of a σ-compact separable locally compact amenable group G as measure-preserving transformations of a non-atomic separable probability space (X, β, m). Let (Kn) be a right Følner sequence of compact sets in G. If T1,…,Tn are pairwise commuting in the sense that Ti(g)Tj(h) = Tj(h)Ti(g) for i ≠ j and g, h ∈ G, then necessary and sufficient conditions can be given, in terms of the ergodicity of certain tensor products, for the following to hold: for all F1,…,Fn∈L∞, the sequence AN(x) whereconverges in L2(X) to . The necessary and sufficient conditions are that each of the following representations are ergodic: Tn, Tn−1⊗Tn−1Tn,…,T2⊗T2T3⊗…⊗T2…Tn, T1⊗T1T2⊗…⊗T1…Tn.In order to prove this theorem, specific properties of the decomposition of L2(X) into its weakly mixing and compact subspaces with respect to a representation Ti are needed. These properties are also used to prove some generalizations of wellknown facts from ergodic theory in the case where G is the integer group Z.

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33

Kerr, David. "Book Review: Group actions in ergodic theory, geometry, and topology: Selected papers." Bulletin of the American Mathematical Society 58, no. 4 (July 6, 2021): 627–34. http://dx.doi.org/10.1090/bull/1741.

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34

KAKU, MICHIO. "STRING FIELD THEORY." International Journal of Modern Physics A 02, no. 01 (February 1987): 1–76. http://dx.doi.org/10.1142/s0217751x87000028.

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String theory has emerged as the leading candidate for a unified field theory of all known forces. However, it is impossible to trust the various phenomenological predictions of superstring theory based on classical solutions alone. It appears that the crucial problem of the theory, breaking ten dimensional space-time down to four dimensions, must be solved nonperturbatively before we can extract reliable predictions. String field theory may be the only formalism in which we can resolve this decisive question. Only a rigorous calculation of the true vacuum of the theory will determine which of the many classical solutions the theory actually predicts. In this review article, we summarize the rapid progress in constructing string field theory actions, such as the development of the covariant BRST theory. We also present the newer geometric formulation of string field theory, from which the BRST theory and the older light cone theory can be derived from first principles. This geometric formulation allows us to derive the complete field theory of strings from two geometric principles, in the same way that general relativity and Yang-Mills theory can be derived from two principles based on global and local symmetry. The geometric formalism therefore reduces string field theory to a problem of finding an invariant under a new local gauge group we call the universal string group (USG). Thus, string field theory is the gauge theory of the universal string group in much the same way that Yang-Mills theory is the gauge theory of SU (N). Thus, the geometric formulation places superstring theory on the same rigorous group theoretical level as general relativity and gauge theory.

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35

Hurder, S. "Problems on rigidity of group actions and cocycles." Ergodic Theory and Dynamical Systems 5, no. 3 (September 1985): 473–84. http://dx.doi.org/10.1017/s0143385700003084.

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AbstractA conference on the interaction of ergodic theory, differential geometry and the theory of Lie Groups was held at the Mathematical Sciences Research Institute from May 24 to June 1, 1984. This is a report of the problem session organized by A. Katok and R. Zimmer and held on May 25, 1984 dealing with the topics in the title. Another problem session was centred on the rigidity of manifolds of non-positive curvature and related topics concerning their geodesic flows. This is reported on by K. Burns and A. Karok separately [2].

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36

Diggans, Christopher Tyler, and Abd AlRahman R. AlMomani. "Geometric Partition Entropy: Coarse-Graining a Continuous State Space." Entropy 24, no. 10 (October 8, 2022): 1432. http://dx.doi.org/10.3390/e24101432.

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Entropy is re-examined as a quantification of ignorance in the predictability of a one dimensional continuous phenomenon. Although traditional estimators for entropy have been widely utilized in this context, we show that both the thermodynamic and Shannon’s theory of entropy are fundamentally discrete, and that the limiting process used to define differential entropy suffers from similar problems to those encountered in thermodynamics. In contrast, we consider a sampled data set to be observations of microstates (unmeasurable in thermodynamics and nonexistent in Shannon’s discrete theory), meaning, in this context, it is the macrostates of the underlying phenomenon that are unknown. To obtain a particular coarse-grained model we define macrostates using quantiles of the sample and define an ignorance density distribution based on the distances between quantiles. The geometric partition entropy is then just the Shannon entropy of this finite distribution. Our measure is more consistent and informative than histogram-binning, especially when applied to complex distributions and those with extreme outliers or under limited sampling. Its computational efficiency and avoidance of negative values can also make it preferable to geometric estimators such as k-nearest neighbors. We suggest applications that are unique to this estimator and illustrate its general utility through an application to time series in the approximation of an ergodic symbolic dynamics from limited observations.

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37

Pourhaghani, Asieh, and Hamid Torabi. "A (Discrete) Homotopy Theory for Geometric Spaces." Journal of Mathematics 2023 (November 14, 2023): 1–19. http://dx.doi.org/10.1155/2023/4158190.

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We define the concepts of homotopy and fundamental group for geometric spaces as a generalization of metric spaces, digital spaces, and graphs; then, we compare them with corresponding concepts in these spaces. Also, we state some properties of the fundamental group of geometric spaces and some theorems to calculate them.

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38

DE BEER, RICHARD J. "TAUBERIAN THEOREMS AND SPECTRAL THEORY IN TOPOLOGICAL VECTOR SPACES." Glasgow Mathematical Journal 55, no. 3 (February 25, 2013): 511–32. http://dx.doi.org/10.1017/s0017089512000699.

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AbstractWe investigate the spectral theory of integrable actions of a locally compact abelian group on a locally convex vector space. We start with an analysis of various spectral subspaces induced by the action of the group. This is applied to analyse the spectral theory of operators on the space, generated by measures on the group. We apply these results to derive general Tauberian theorems that apply to arbitrary locally compact abelian groups acting on a large class of locally convex vector spaces, which includes Fréchet spaces. We show how these theorems simplify the derivation of Mean Ergodic theorems.

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39

Ralston, David, and Serge Troubetzkoy. "Ergodic infinite group extensions of geodesic flows on translation surfaces." Journal of Modern Dynamics 6, no. 4 (2012): 477–97. http://dx.doi.org/10.3934/jmd.2012.6.477.

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Chen, W. W. L. "Davenport's theorem in geometric discrepancy theory." International Journal of Number Theory 11, no. 05 (August 2015): 1437–49. http://dx.doi.org/10.1142/s1793042115400047.

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Davenport's theorem was established nearly a lifetime ago, but there have been some very interesting recent developments. The various proofs over the years bring in different ideas from number theory, probability theory, analysis and group theory. In this short survey, we shall not present complete proofs, but will describe instead some of these underlying ideas.

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Deroin, Bertrand, Victor Kleptsyn, and Andrés Navas. "On the ergodic theory of free group actions by real-analytic circle diffeomorphisms." Inventiones mathematicae 212, no. 3 (December 18, 2017): 731–79. http://dx.doi.org/10.1007/s00222-017-0779-4.

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Holik, Federico, Cesar Massri, and A. Plastino. "Geometric probability theory and Jaynes’s methodology." International Journal of Geometric Methods in Modern Physics 13, no. 03 (March 2016): 1650025. http://dx.doi.org/10.1142/s0219887816500250.

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We provide a generalization of the approach to geometric probability advanced by the great mathematician Gian Carlo Rota, in order to apply it to generalized probabilistic physical theories. In particular, we use this generalization to provide an improvement of the Jaynes’ MaxEnt method. The improvement consists in providing a framework for the introduction of symmetry constraints. This allows us to include group theory within MaxEnt. Some examples are provided.

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Farber, Michael, Ross Geoghegan, and Dirk Schütz. "Closed 1-forms in topology and geometric group theory." Russian Mathematical Surveys 65, no. 1 (January 1, 2010): 143–72. http://dx.doi.org/10.1070/rm2010v065n01abeh004663.

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WANG, QIUDONG, and LAI-SANG YOUNG. "Dynamical profile of a class of rank-one attractors." Ergodic Theory and Dynamical Systems 33, no. 4 (May 8, 2012): 1221–64. http://dx.doi.org/10.1017/s014338571200020x.

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AbstractThis paper contains results on the geometric and ergodic properties of a class of strange attractors introduced by Wang and Young [Towards a theory of rank one attractors. Ann. of Math. (2) 167 (2008), 349–480]. These attractors can live in phase spaces of any dimension, and have been shown to arise naturally in differential equations that model several commonly occurring phenomena. Dynamically, such systems are chaotic; they have controlled non-uniform hyperbolicity with exactly one unstable direction, hence the name rank-one. In this paper we prove theorems on their Lyapunov exponents, Sinai–Ruelle–Bowen (SRB) measures, basins of attraction, and statistics of time series, including central limit theorems, exponential correlation decay and large deviations. We also present results on their global geometric and combinatorial structures, symbolic coding and periodic points. In short, we build a dynamical profile for this class of dynamical systems, proving that these systems exhibit many of the characteristics normally associated with ‘strange attractors’.

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Liu, Wei Wei, and Huan Yun Dai. "Research on Stochastic Stability of Wheelsets with Primary Suspension." Applied Mechanics and Materials 249-250 (December 2012): 672–77. http://dx.doi.org/10.4028/www.scientific.net/amm.249-250.672.

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A new model for elastic constraint wheelset system of rail vehicle is proposed. Assuming the stochastic excitation as Gauss white noise, a stochastic model is built for elastic constraint wheelset system. Here two kinds of stochastic excitations are considered: one is the internal multiplicative excitation inherited in the internal system such as the spring and wheelset/rail contact geometric relationship, the other is the external excitation induced by track random irregularities. The model defined here is considered as a weak damping, weak excitation quasi non-integrable Hamiltonian system. The maximal Lyapunov exponent is calculated by quasi non-integrable Hamiltonian theory and oseledec multiplicative ergodic theory, and the stochastic local stability conditions are obtained. Meanwhile, the stochastic global stability conditions are derived by considering the modality of the singular boundary condition.

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Heida, Martin. "Stochastic homogenization on perforated domains III–General estimates for stationary ergodic random connected Lipschitz domains." Networks and Heterogeneous Media 18, no. 4 (2023): 1410–33. http://dx.doi.org/10.3934/nhm.2023062.

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<abstract><p>This is Part III of a series on the existence of uniformly bounded extension operators on randomly perforated domains in the context of homogenization theory. Recalling that randomly perforated domains are typically not John and hence extension is possible only from $ W^{1, p} $ to $ W^{1, r} $, $ r < p $, we will show that the existence of such extension operators can be guaranteed if the weighted expectations of four geometric characterizing parameters are bounded: The local Lipschitz constant $ M $, the local inverse Lipschitz radius $ \delta^{-1} $ resp. $ \rho^{-1} $, the mesoscopic Voronoi diameter $ {\mathfrak{d}} $ and the local connectivity radius $ {\mathscr{R}} $.</p></abstract>

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Yoldaş, Havva. "On quantitative hypocoercivity estimates based on Harris-type theorems." Journal of Mathematical Physics 64, no. 3 (March 1, 2023): 031101. http://dx.doi.org/10.1063/5.0089698.

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This Review concerns recent results on the quantitative study of convergence toward the stationary state for spatially inhomogeneous kinetic equations. We focus on analytical results obtained by means of certain probabilistic techniques from the ergodic theory of Markov processes. These techniques are sometimes referred to as Harris-type theorems. They provide constructive proofs for convergence results in the L1 (or total variation) setting for a large class of initial data. The convergence rates can be made explicit (for both geometric and sub-geometric rates) by tracking the constants appearing in the hypotheses. Harris-type theorems are particularly well-adapted for equations exhibiting non-explicit and non-equilibrium steady states since they do not require prior information on the existence of stationary states. This allows for significant improvements of some already-existing results by relaxing assumptions and providing explicit convergence rates. We aim to present Harris-type theorems, providing a guideline on how to apply these techniques to kinetic equations at hand. We discuss recent quantitative results obtained for kinetic equations in gas theory and mathematical biology, giving some perspectives on potential extensions to nonlinear equations.

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MARGALIT, DAN, and JON McCAMMOND. "GEOMETRIC PRESENTATIONS FOR THE PURE BRAID GROUP." Journal of Knot Theory and Its Ramifications 18, no. 01 (January 2009): 1–20. http://dx.doi.org/10.1142/s0218216509006859.

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We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is why we describe them as geometric presentations. Motivated by a presentation for the full braid group that we call the "rotation presentation", we introduce presentations for the pure braid group that we call the "twist presentation" and the "swing presentation". From the point of view of mapping class groups, the swing presentation can be interpreted as stating that the pure braid group is generated by a finite number of Dehn twists and that the only relations needed are the disjointness relation and the lantern relation.

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AHN, CHANGHYUN, and KYUNGSUNG WOO. "${\mathcal N}=8$ GAUGED SUPERGRAVITY THEORY AND ${\mathcal N}=6$ SUPERCONFORMAL CHERN–SIMONS MATTER THEORY." International Journal of Modern Physics A 25, no. 17 (July 10, 2010): 3407–44. http://dx.doi.org/10.1142/s0217751x10049232.

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By studying the previously known holographic [Formula: see text] supersymmetric renormalization group flow (Gowdigere–Warner) in four dimensions, we find the mass deformed Chern–Simons matter theory which has [Formula: see text] supersymmetry by adding the four mass terms among eight adjoint fields. The geometric superpotential from the 11 dimensions is found and provides the M2-brane probe analysis. As second example, we consider known holographic [Formula: see text] supersymmetric renormalization group flow (Pope–Warner) in four dimensions. The eight mass terms are added and similar geometric superpotential is obtained.

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Mj, Mahan. "Cannon–Thurston maps in Kleinian groups and geometric group theory." Surveys in Differential Geometry 25, no. 1 (2020): 281–318. http://dx.doi.org/10.4310/sdg.2020.v25.n1.a8.

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