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First we survey and explain the strategy of some recent results that construct holomorphic sl (2 , ℂ) -differential systems over some Riemann surfaces Σ g of genus g ≥ 2 , satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, Fuchsian sl (2 , ℂ) -systems of compact Riemann surfaces [with an appendix by Takuro Mochizuki], preprint, arXiv:org/abs/2104.04818] or some cocompact Kleinian subgroup Γ ⊂ SL (2 , ℂ) as in [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of SL (2 , ℂ) , preprint, arXiv:org/abs/2112.03131]. As a consequence, there exist holomorphic maps from Σ g to the quotient space SL (2 , ℂ) / Γ , where Γ ⊂ SL (2 , ℂ) is a cocompact lattice, that do not factor through any elliptic curve [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of SL (2 , ℂ) , preprint, arXiv:org/abs/2112.03131]. This answers positively a question of Ghys in [E. Ghys, Déformations des structures complexes sur les espaces homogènes de SL (2 , ℂ) , J. Reine Angew. Math. 468 (1995) 113–138]; the question was also raised by Huckleberry and Winkelmann in [A. H. Huckleberry and J. Winkelmann, Subvarieties of parallelizable manifolds, Math. Ann. 295 (1993) 469–483]. Then we prove that when M is a Riemann surface, a Torelli-type theorem holds for the affine group scheme over ℂ obtained from the category of holomorphic connections on étale trivial holomorphic bundles. After that, we explain how to compute in a simple way the holonomy of a holomorphic connection on a free vector bundle. Finally, for a compact Kähler manifold M , we investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over M. If ϖ (respectively, Θ) stands for the affine group scheme over ℂ obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism v : (Θ) → (ϖ) of Hopf algebras. We present a description of the transpose of v in terms of the iterated integrals. [ABSTRACT FROM AUTHOR]

It is known that multiple zeta values can be written in terms of certain iterated log-sine integrals. Conversely, we evaluate iterated log-sine integrals in terms of multiple zeta values and multiple polylogarithms in this paper. We also suggest some conjectures on multiple zeta values, multiple Clausen values, multiple Glaisher values and iterated log-sine integrals. [ABSTRACT FROM AUTHOR]

Classical multiple zeta values can be viewed as iterated integrals of the differentials d t t , d t 1 − t from 0 to 1. In this paper, we reprove Brown's theorem: For a i , b i , c i j ∈ ℤ , the iterated integral of the form ∫ ⋯ ∫ 0 < t 1 < ⋯ < t N < 1 ∏ i t i a i (1 − t i) b i ∏ i < j (t j − t i) c i j d t 1 ⋯ d t N is a ℚ -linear combination of multiple zeta values of weight ≤ N if convergent. What is more, we show that if p i (t) , 1 ≤ i ≤ N , are in a ℚ [ t , 1 / t , 1 / (1 − t) ] -algebra generated by multiple polylogarithms and their dual, and if q i j (t) , 1 ≤ i < j ≤ N , are in a ℚ [ t , 1 / t ] -algebra generated by logarithm, then the iterated integral ∫ ⋯ ∫ 0 < t 1 < ⋯ < t N < 1 ∏ i p i (t i) ∏ i < j q i j (t j − t i) d t 1 ⋯ d t N is a ℚ -linear combination of multiple zeta values. As an application of our main results, we show that the coefficients of the Taylor expansions of the Selberg integrals ∫ ⋯ ∫ 0 < t 1 < ⋯ < t N < 1 f ∏ i t i α i (1 − t i) β i ∏ i < j (t j − t i) γ i j d t 1 ⋯ d t N (with respect to α i , β i , γ i j ) at the integral points in some product of right half complex plane are ℚ -linear combinations of multiple zeta values for any f ∈ ℚ [ t i , t i − 1 , (t i − t j) − 1 | 1 ≤ i ≤ N , 1 ≤ i < j ≤ N ]. This statement generalizes Terasoma's original result. [ABSTRACT FROM AUTHOR]

We revisit the bifurcation theory of the Lotka–Volterra quadratic system X 0 : ẋ = − y − x 2 + y 2 , ẏ = x − 2 x y with respect to arbitrary quadratic deformations. The system has a double center, which is moreover isochronous. We show that the deformed system can have at most two limit cycles on the finite plane, with possible distribution (i , j) , where i + j ≤ 2. Our approach is based on the study of pairs of bifurcation functions associated to the centers, expressed in terms of iterated path integrals of length two. [ABSTRACT FROM AUTHOR]

In this paper, we show that the iterated integrals on products of one variable multiple polylogarithms from 0 to 1 are actually in the algebra of multiple zeta values if they are convergent. In the divergent case, we define the regularized iterated integrals from 0 to 1. By the same method, we show that the regularized iterated integrals are also in the algebra of multiple zeta values. As an application, we give new series representations for multiple zeta values and calculate some interesting examples of iterated integrals. [ABSTRACT FROM AUTHOR]

Polylogarithms are those multiple polylogarithms that factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. Building on work of Dan-Cohen, Wewers, and Brown, we push the computational boundary of our explicit motivic version of Kim's method in the case of the thrice punctured line over an open subscheme of Spec  ℤ. To do so, we develop a greatly refined version of the algorithm of Dan-Cohen tailored specifically to this case, and we focus attention on the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-one part of the mixed Tate Galois group studied extensively by Goncharov. We also discover an interesting consequence of the symmetry-breaking nature of the polylog quotient that forces us to symmetrize our polylogarithmic version of Kim's conjecture. In this first part of a two-part series, we focus on a specific example, which allows us to verify an interesting new case of Kim's conjecture. [ABSTRACT FROM AUTHOR]

The period for a compact Riemann surface, defined by the integral of differential 1-forms, is a classical complex analytic invariant, strongly related to the complex structure of the surface. In this paper, we treat another complex analytic invariant called the pointed harmonic volume. As a natural extension of the period defined using Chen's iterated integrals, it captures more detailed information of the complex structure. It is also one of a few explicitly computable examples of complex analytic invariants. We obtain its new value for a certain pointed hyperelliptic curve. An application of the pointed harmonic volume is presented. We explain the relationship between the harmonic volume and first extended Johnson homomorphism on the mapping class group of a pointed oriented closed surface. Moreover, we explicitly compute a certain restriction of this homomorphism. [ABSTRACT FROM AUTHOR]

We give a new proof of the duality of multiple zeta values, which makes no use of the iterated integrals. The same method is also applicable to Ohno's relation for (q -)multiple zeta values. [ABSTRACT FROM AUTHOR]

In this paper, we prove a family of identities among multiple zeta values, which contains as a special case a conjectural identity of Hoffman. We use the iterated integrals on ℙ 1 ∖ { 0 , 1 , ∞ , z } for our proof. [ABSTRACT FROM AUTHOR]

This paper is devoted to study some chaotic properties of iterated function systems (IFSs). Specially, a new notion named thick chaotic IFSs is introduced. The relationship between thick chaos and another properties of some notions in dynamical systems are studied. [ABSTRACT FROM AUTHOR]

Given , , and compact metric spaces, we consider two iterated function systems and , where and are contractions. Let be the set of probabilities with -marginal being holonomic with respect to and -marginal being holonomic with respect to . Given and , let be the set of probabilities in having -marginal and -marginal . Let be the relative entropy of with respect to and be the relative entropy of with respect to . Given a cost function , let . We will prove the duality equation: In particular, if and are single points and we drop the entropy, the equation above can be rewritten as the Kantorovich duality for the compact spaces and a continuous cost function . [ABSTRACT FROM AUTHOR]

Liu and Lin ( Statist. Probab. Lett. 2006) introduced a kind of complete moment convergence which includes complete convergence as a special case. In this paper, we study the convergence rates of the precise asymptotics for complete moment convergence introduced by Liu and Lin (2006) and get the corresponding convergence rates. [ABSTRACT FROM AUTHOR]

We investigate combinatorial issues relating to the use of random orbit approximations to the attractor of an iterated function system with the aim of clarifying the role of the stochastic process during the generation of the orbit. A Baire category counterpart of almost sure convergence is presented. [ABSTRACT FROM AUTHOR]

The evaluation of iterated primitives of powers of logarithms is expressed in closed form. The expressions contain polynomials with coefficients given in terms of the harmonic numbers and their generalizations. The logconcavity of these polynomials is established. [ABSTRACT FROM AUTHOR]