scholarly journals Regular variation and rates of mixing for infinite measure preserving almost Anosov diffeomorphisms

2018 ◽  
Vol 40 (3) ◽  
pp. 663-698 ◽  
Author(s):  
HENK BRUIN ◽  
DALIA TERHESIU
Keyword(s):  
Regular Variation ◽  
Return Time ◽  
Main Task ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Anosov Diffeomorphisms ◽  
Infinite Measure ◽  
Mixing Rates ◽  
First Return Time ◽  
Almost Anosov

The purpose of this paper is to establish mixing rates for infinite measure preserving almost Anosov diffeomorphisms on the two-dimensional torus. The main task is to establish regular variation of the tails of the first return time to the complement of a neighbourhood of the neutral fixed point.

Arithmetical theory of Anosov diffeomorphisms

1987 ◽  
Vol 413 (1844) ◽  
pp. 97-107 ◽  
Keyword(s):  
Nineteenth Century ◽  
Periodic Orbits ◽  
Algebraic Number ◽  
Number Fields ◽  
Dynamical Behaviour ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Algebraic Number Fields ◽  
Anosov Diffeomorphisms ◽  
The Way

Nineteenth century arithmetic is used to study periodic orbits of Anosov diffeomorphisms of the two-dimensional torus. We find that the period of the orbits, as well as their dynamical behaviour, are intimately related to the way ideals factorize in algebraic number fields.

scholarly journals Upper and lower bounds for the correlation function via inducing with general return times

2016 ◽  
Vol 38 (1) ◽  
pp. 34-62 ◽  
Author(s):  
HENK BRUIN ◽  
DALIA TERHESIU
Keyword(s):  
Correlation Function ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Return Time ◽  
Upper And Lower Bounds ◽  
Interval Maps ◽  
Return Times ◽  
Infinite Measure ◽  
Higher Order Asymptotics ◽  
Mixing Rates

For non-uniformly expanding maps inducing with a general return time to Gibbs Markov maps, we provide sufficient conditions for obtaining higher-order asymptotics for the correlation function in the infinite measure setting. Along the way, we show that these conditions are sufficient to recover previous results on sharp mixing rates in the finite measure setting for non-Markov maps, but for a larger class of observables. The results are illustrated by (finite and infinite measure-preserving) non-Markov interval maps with an indifferent fixed point.

scholarly journals Circular quiver gauge theories, isomonodromic deformations and $$W_N$$ fermions on the torus

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Giulio Bonelli ◽  
Fabrizio Del Monte ◽  
Pavlo Gavrylenko ◽  
Alessandro Tanzini
Keyword(s):  
Integrable System ◽  
Gauge Theories ◽  
The Self ◽  
Free Fermion ◽  
Two Dimensional ◽  
Isomonodromic Deformation ◽  
Specific Time ◽  
Dimensional Torus ◽  
Isomonodromic Deformations ◽  
Deformation Problem

AbstractWe study the relation between class $$\mathcal {S}$$ S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual $$\Omega $$ Ω -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $$\tau $$ τ -function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $$W_N$$ W N free fermion correlators on the torus.

On the integrability of intermediate distributions for Anosov diffeomorphisms

1995 ◽  
Vol 15 (2) ◽  
pp. 317-331 ◽  
Author(s):  
M. Jiang ◽  
Ya B. Pesin ◽  
R. de la Llave
Keyword(s):  
Finite Number ◽  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Three Dimensional ◽  
Dimensional Torus ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Stable Foliation

AbstractWe study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.

scholarly journals Stable Arcs Connecting Polar Cascades on a Torus

2021 ◽  
Vol 17 (1) ◽  
pp. 23-37
Author(s):  
O. V. Pochinka ◽  
◽  
E. V. Nozdrinova ◽  
Keyword(s):  
Two Dimensional ◽  
Dimensional Torus ◽  
Positive Orientation ◽  
Stable Isotopic ◽  
Orientation Type

In the article, the components of the stable isotopic connection of polar gradient-like diffeomorphisms on a two-dimensional torus are found under the assumption that all non-wandering points are fixed and have a positive orientation type.

scholarly journals Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

2020 ◽  
pp. 2150006
Author(s):  
Denis Bonheure ◽  
Jean Dolbeault ◽  
Maria J. Esteban ◽  
Ari Laptev ◽  
Michael Loss
Keyword(s):  
Magnetic Field ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Euclidean Spaces ◽  
Hardy Inequalities ◽  
Interpolation Inequalities ◽  
Nonlinear Interpolation ◽  
Magnetic Potentials ◽  
Aharonov Bohm

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions [Formula: see text] and [Formula: see text].

Mixing for invertible dynamical systems with infinite measure

2015 ◽  
Vol 15 (02) ◽  
pp. 1550012 ◽  
Author(s):  
Ian Melbourne
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Large Class ◽  
Renewal Theory ◽  
Additional Structure ◽  
Infinite Measure ◽  
Higher Order Asymptotics ◽  
Mixing Rates ◽  
Noninvertible Maps ◽  
Invent Math

In a recent paper, Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math.189 (2012) 61–110] obtained results on mixing and mixing rates for a large class of noninvertible maps preserving an infinite ergodic invariant measure. Here, we are concerned with extending these results to the invertible setting. Mixing is established for a large class of infinite measure invertible maps. Assuming additional structure, in particular exponential contraction along stable manifolds, it is possible to obtain good results on mixing rates and higher order asymptotics.

scholarly journals Thermodynamics of the Katok map

2017 ◽  
Vol 39 (3) ◽  
pp. 764-794 ◽  
Author(s):  
Y. PESIN ◽  
S. SENTI ◽  
K. ZHANG
Keyword(s):  
Limit Theorem ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Existence Of Equilibrium ◽  
Equilibrium Measures ◽  
The Central Limit Theorem ◽  
Exponential Decay Of Correlations ◽  
Continuous Potential ◽  
Uniqueness Of Equilibrium ◽  
Unstable Direction

We effect the thermodynamical formalism for the non-uniformly hyperbolic $C^{\infty }$ map of the two-dimensional torus known as the Katok map [Katok. Bernoulli diffeomorphisms on surfaces. Ann. of Math. (2)110(3) 1979, 529–547]. It is a slow-down of a linear Anosov map near the origin and it is a local (but not small) perturbation. We prove the existence of equilibrium measures for any continuous potential function and obtain uniqueness of equilibrium measures associated to the geometric $t$-potential $\unicode[STIX]{x1D711}_{t}=-t\log \mid df|_{E^{u}(x)}|$ for any $t\in (t_{0},\infty )$, $t\neq 1$, where $E^{u}(x)$ denotes the unstable direction. We show that $t_{0}$ tends to $-\infty$ as the domain of the perturbation shrinks to zero. Finally, we establish exponential decay of correlations as well as the central limit theorem for the equilibrium measures associated to $\unicode[STIX]{x1D711}_{t}$ for all values of $t\in (t_{0},1)$.

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