Slow chaos in surface flows

@article{Ulcigrai2020SlowCI,
  title={Slow chaos in surface flows},
  author={Corinna Ulcigrai},
  journal={Bollettino dell'Unione Matematica Italiana},
  year={2020},
  volume={14},
  pages={231-255}
}
  • C. Ulcigrai
  • Published 13 October 2020
  • Mathematics
  • Bollettino dell'Unione Matematica Italiana
Flows on surfaces describe many systems of physical origin and are one of the most fundamental examples of dynamical systems, studied since Poincará. In the last decade, there have been a lot of advances in our understanding of the chaotic properties of smooth area-preserving flows (a class which include locally Hamiltonian flows), thanks to the connection to Teichmueller dynamics and, very recently, to the influence of the work of Marina Ratner in homogeneous dynamics. We motivate and survey… 
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References

SHOWING 1-10 OF 123 REFERENCES

Multiple mixing for a class of conservative surface flows

Arnold and Kochergin mixing conservative flows on surfaces stand as the main and almost only natural class of mixing transformations for which higher order mixing has not been established, nor

On disjointness properties of some parabolic flows

The Ratner property, a quantitative form of divergence of nearby trajectories, is a central feature in the study of parabolic homogeneous flows. Discovered by Marina Ratner and used in her 1980th

Invariant distributions and time averages for horocycle flows

There are infinitely many obstructions to existence of smooth solutions of the cohomological equation Uu = f , where U is the vector field generating the horocycle flow on the unit tangent bundle SM

Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces

We consider typical area preserving flows on higher genus surfaces and prove that the flow restricted to mixing minimal components is mixing of all orders, thus answering affimatively to Rohlin's

Time-changes of horocycle flows

We consider smooth time-changes of the classical horocycle flows on the unit tangent bundle of a compact hyperbolic surface and prove sharp bounds on the rate of equidistribution and the rate of

Absolutely Continuous Spectrum for Parabolic Flows/Maps

We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely

SOLUTIONS OF THE COHOMOLOGICAL EQUATION FOR AREA-PRESERVING FLOWS ON COMPACT SURFACES OF HIGHER GENUS

Let S,(M, E) be the set of smooth vector fields on a compact orientable surface M of genus g > 2, which preserve a smooth area form w and have a finite set E C M of saddle-type singularities. Our

Absence of mixing in area-preserving flows on surfaces

We prove that minimal area-preserving ows locally given by a smooth Hamiltonian on a closed surface of genus g 2 are typically (in the measure-theoretical sense) not mixing. The result is obtained by

On isomorphism problem for von Neumann flows with one discontinuity

A von Neumann flow is a special flow over an irrational rotation of the circle and under a piecewise C1 roof function with a non-zero sum of jumps. We prove that the absolute value of the jump is a

Flows on 2-dimensional Manifolds: An Overview

1 Definitions and Examples: Preliminaries Basic constructions Basic examples. 2 Poncare-Bendixon's theory: Existence of closed transversal Absence of non-trivial recurrent trajectories on some
...