Classifying C1+ structures on dynamical fractals: 1. The moduli space of solenoid functions for Markov maps on train tracks

1995 ◽  
Vol 15 (4) ◽  
pp. 697-734 ◽  
Author(s):  
A. A. Pinto ◽  
D. A. Rand
Keyword(s):  
Moduli Space ◽  
Scaling Function ◽  
Matching Condition ◽  
Continuous Functions ◽  
Conjugacy Classes ◽  
Train Track ◽  
Hölder Continuous ◽  
Expanding Circle ◽  
Train Tracks ◽  
Smooth Conjugacy

AbstractSullivan's scaling function provides a complete description of the smooth conjugacy classes of cookie-cutters. However, for smooth conjugacy classes of Markov maps on a train track, such as expanding circle maps and train track mappings induced by pseudo-Anosov systems, the generalisation of the scaling function suffers from a deficiency. It is difficult to characterise the structure of the set of those scaling functions which correspond to smooth mappings. We introduce a new invariant for Markov maps called the solenoid function. We prove that for any prescribed topological structure, there is a one-to-one correspondence between smooth conjugacy classes of smooth Markov maps and pseudo-Hölder solenoid functions. This gives a characterisation of the moduli space for smooth conjugacy classes of smooth Markov maps. For smooth expanding maps of the circle with degree d this moduli space is the space of Hölder continuous functions on the space {0,…, d − 1}ℕ satisfying the matching condition.

Classifying C1+ structures on dynamical fractals: 2. Embedded trees

1995 ◽  
Vol 15 (5) ◽  
pp. 969-992 ◽  
Author(s):  
A. A. Pinto ◽  
D. A. Rand
Keyword(s):  
Moduli Spaces ◽  
Binary Tree ◽  
Conjugacy Classes ◽  
Expanding Maps ◽  
Bounded Geometry ◽  
Arbitrary Topology ◽  
Contact Points ◽  
Train Tracks ◽  
Smooth Conjugacy

AbstractWe classify the C1+α structures on embedded trees. This extends the results of Sullivan on embeddings of the binary tree to trees with arbitrary topology and to embeddings without bounded geometry and with contact points. We used these results in an earlier paper to describe the moduli spaces of smooth conjugacy classes of expanding maps and Markov maps on train tracks. In later papers we will use those results to do the same for pseudo-Anosov diffeomorphisms of surfaces. These results are also used in the classification of renormalisation limits of C1+α diffeomorphisms of the circle.

scholarly journals Continuous solutions to two iterative functional equations

2021 ◽  
Author(s):  
Karol Baron
Keyword(s):  
Probability Measure ◽  
Functional Equations ◽  
Continuous Functions ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Hölder Continuous Functions

AbstractBased on iteration of random-valued functions we study the problem of solvability in the class of continuous and Hölder continuous functions $$\varphi $$ φ of the equations $$\begin{aligned} \varphi (x)=F(x)-\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ),\\ \varphi (x)=F(x)+\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ), \end{aligned}$$ φ ( x ) = F ( x ) - ∫ Ω φ ( f ( x , ω ) ) P ( d ω ) , φ ( x ) = F ( x ) + ∫ Ω φ ( f ( x , ω ) ) P ( d ω ) , where P is a probability measure on a $$\sigma $$ σ -algebra of subsets of $$\Omega $$ Ω .

scholarly journals Dirichlet problem for Poisson equations in Jordan domains

2018 ◽  
Vol 32 ◽  
pp. 30-41
Author(s):  
Vladimir Gutlyanskii ◽  
Vladimir Ryazanov ◽  
Eduard Yakubov
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data ◽  
Continuous Functions ◽  
Logarithmic Capacity ◽  
Hölder Continuous ◽  
Arbitrary Boundary ◽  
Poisson Equations ◽  
Local Properties ◽  
Continuous Boundary ◽  
Angular Limits

First, we study the Dirichlet problem for the Poisson equations \(\triangle u(z) = g(z)\) with \(g\in L^p\), \(p>1\), and continuous boundary data \(\varphi :\partial D\to\mathbb{R}\) in arbitrary Jordan domains \(D\) in \(\mathbb{C}\) and prove the existence of continuous solutions \(u\) of the problem in the class \(W^{2,p}_{\rm loc}\). Moreover, \(u\in W^{1,q}_{\rm loc}\) for some \(q>2\) and \(u\) is locally Hölder continuous. Furthermore, \(u\in C^{1,\alpha}_{\rm loc}\) with \(\alpha = (p-2)/p\) if \(p>2\). Then, on this basis and applying the Leray-Schauder approach, we obtain the similar results for the Dirichlet problem with continuous data in arbitrary Jordan domains to the quasilinear Poisson equations of the form \(\triangle u(z) = h(z)\cdot f(u(z))\) with the same assumptions on \(h\) as for \(g\) above and continuous functions \(f:\mathbb{R}\to\mathbb{R}\), either bounded or with nondecreasing \(|f\,|\) of \( |t\,|\) such that \(f(t)/t \to 0\) as \(t\to\infty\). We also give here applications to mathematical physics that are relevant to problems of diffusion with absorbtion, plasma and combustion. In addition, we consider the Dirichlet problem for the Poisson equations in the unit disk \(\mathbb{D}\subset\mathbb{C}\) with arbitrary boundary data \(\varphi :\partial\mathbb{D}\to\mathbb{R}\) that are measurable with respect to logarithmic capacity. Here we establish the existence of continuous nonclassical solutions \(u\) of the problem in terms of the angular limits in \(\mathbb D\) a.e. on \(\partial\mathbb D\) with respect to logarithmic capacity with the same local properties as above. Finally, we extend these results to almost smooth Jordan domains with qusihyperbolic boundary condition by Gehring-Martio.

scholarly journals Yang–Mills Measure on the Two-Dimensional Torus as a Random Distribution

2019 ◽  
Vol 372 (3) ◽  
pp. 1027-1058
Author(s):  
Ilya Chevyrev
Keyword(s):  
Random Distribution ◽  
Random Variable ◽  
Continuous Functions ◽  
Small Scale ◽  
Dimensional Torus ◽  
Hölder Continuous ◽  
Line Segments ◽  
Yang Mills ◽  
Hölder Continuous Functions ◽  
On Line

Abstract We introduce a space of distributional 1-forms $$\Omega ^1_\alpha $$Ωα1 on the torus $$\mathbf {T}^2$$T2 for which holonomies along axis paths are well-defined and induce Hölder continuous functions on line segments. We show that there exists an $$\Omega ^1_\alpha $$Ωα1-valued random variable A for which Wilson loop observables of axis paths coincide in law with the corresponding observables under the Yang–Mills measure in the sense of Lévy (Mem Am Math Soc 166(790), 2003). It holds furthermore that $$\Omega ^1_\alpha $$Ωα1 embeds into the Hölder–Besov space $$\mathcal {C}^{\alpha -1}$$Cα-1 for all $$\alpha \in (0,1)$$α∈(0,1), so that A has the correct small scale regularity expected from perturbation theory. Our method is based on a Landau-type gauge applied to lattice approximations.

scholarly journals NONCLASSICAL SPECTRAL ASYMPTOTICS AND DIXMIER TRACES: FROM CIRCLES TO CONTACT MANIFOLDS

2017 ◽  
Vol 5 ◽  
Author(s):  
HEIKO GIMPERLEIN ◽  
MAGNUS GOFFENG
Keyword(s):  
Multiplication Operator ◽  
Continuous Functions ◽  
Spectral Asymptotics ◽  
Hölder Continuous ◽  
Contact Manifolds ◽  
Integral Formulas ◽  
Weyl Law ◽  
Wide Range ◽  
Hölder Continuous Functions ◽  
Geometric Origin

We consider the spectral behavior and noncommutative geometry of commutators$[P,f]$, where$P$is an operator of order 0 with geometric origin and$f$a multiplication operator by a function. When$f$is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes’ residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Hölder continuous functions$f$, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.

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