An example of a pathological random perturbation of the Cat Map

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An example of a pathological random
perturbationof the Cat Map

• The unique invariant measure is supported on a
  line segment that acts as a "global statistical
  attractor."

Tatiana Yarmola yarmola_at_cims.nyu.edu University
of Maryland, College Park, MD Sept 30, 2008.
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Random Perturbations
-some probability distribution

• Random perturbations are Markov Chains with
  transition probabilities .
• Intuitively describes how the image of
  misses .

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Some Definitions

• Let be the perturbed dynamics generated by
• and a family of
  distributions
• The push forward of a measure on under
  
• is
• is called invariant if

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Facts

• Fact 1
• M compact depend continuously on
  x
• MC admits stationary/invariant measures.
• Fact 2
• ANY invariant measure
• m Riemmaniam measure on M
• If interesting
  problems arise.

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Rank one Random Perturbations

• compact n-dim differentiable manifold
  (torus).
• diffeomorphism
• unit vector field,
• fixed.
• the uniform measure on the -long
  interval along the vector field V centered at x

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Randomly perturbed uniform contraction in
1-dimension

• Let
  
• uniform on
• The invariant density is supported on

Invariant density under random dynamics
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Acquiring Density
If all the points acquire density All
invariant measures If not, singular invariant
measures exist.
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• Coexistence of singular invariant measures with
  acim generally does no harm.
• There can exist singular invariant measures that
  attract sets of positive Lebesgue measure (cf.
  physical measures)
• Call such measures statistical attractors.

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The Pathological Example

• Motivation Suppose
• the dynamics of are rich
• strong hyperbolicity
• mixing properties
• -a.e. acquires density.
• Q Is it still possible for a statistical
  attractor or a global statistical attractor to
  occur?
• Yes rank 1 perturbation of the Cat Map s.t.
• the unique invariant measure is supported on
  line segment, a local stable manifold of the
  fixed point.
• for any measure ,

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The Cat Map

• Let be Cat Map defined by the iterations of

on the 2-dimensional torus .

• Then and

are the contraction and expansion rates along
the stable and unstable manifolds.

• The invariant measure for the Cat Map is .

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Vector Field

• Consider V such that all vectors V(x) form some
  constant angle a with the unstable direction.

unstable
fixed point
stable

• The dynamics
• stretches in y-direction
• contracts in x-direction

Every point acquires density in 2 steps
All invariant measures
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Add a small kink of a vector field near zero
no acim
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perturbations along Gaussian-shaped curves
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Layers in W

• Let
• Define the layers and for

• For in W such that is in
  , fully lies in .
• If
  

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Dynamics on the Layers
need regular returns
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The Markov Chain

• Transition probabilities
• for ,
• P ,
  where is as defined in assumption (A2).
• for all other , .
• This Markov Chain is transient.

k
0
1
2


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Assumptions and Proposition

• (A1) is large enough
• (A2) and V are extended outside W in
  some fashion to satisfy the following
• For any finite measure on the torus at
  least q- fraction of gets to W each
  step the measure is pushed forward by .
• Proposition If , V and satisfy (A1) and
  (A2), then the only invariant measure under the
  perturbed dynamics is singular and is supported
  on the interval of the x-axis.

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Dynamics on layers

• The length of Gaussian curve from
  to

is bounded above by

• Pick

• Then at least of any -long piece of
  the
• Gaussian-shaped curve in will be
  mapped
• by to

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Conditional Measures Transitions

• are uniform on -long curves along the
  vector field centered at .
• Conditional densities on the vector field lines
  cannot exceed .
• The length of the Gaussian on
• does not exceed .
• So at most of the conditional
  measure
• can be supported on a piece of Gaussian in
• At least of the conditional measure on
  in will be mapped by to

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Measure Transitions on Layers

• If we start with any finite measure supported
  on , at least 2/3 of it will be pushed
  forward to

under the perturbed dynamics.

• Assumption (A2) states that for any finite
  measure ? on the torus at least q-fraction of
  ?WC gets to W each step the measure is pushed
  forward by the perturbed dynamics.
• The dynamics on the layers is similar to our
  countable state Markov Chain.

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Random Dynamics vs MC

• For any measure on define
  on the Markov Chain states by ,

0
1
2
n


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• Markov Chain is transient
• The measures on the states converge to 0 as
  
• Stationary (prob.) measures do not exist.
• Measure on the first layers is always
  the measure on the first states of MC.
• There does not exist an invariant measure for
  the perturbed dynamics on .

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Modifications

• (A2) does not sound realistic.
• However for the Cat Map we have regular return
  rates

B
unstable direction
B
A
A
stable direction
N depends on side lengths of A and B only
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Lower bound on the return rates

• If measure gets to , of it stays
  after each subsequent iteration of .
• Proposition There exist and
  s.t. For any finite measure on the torus, at
  least -fraction of gets to in
  steps under .

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Markov Chain Modifications

• (A2) change we have to wait N steps.
• The Markov Chain required is roughly the Nth
  power of the Markov Chain S.
• Transition probabilities are only different on
  first N states with
• This Markov Chain is Transient.

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• holds in a similar fashion
• The unique invariant measure for the perturbed
  dynamics is the singular measure
  supported on .
• For any measure , some
  distribution on
• and

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Anosov Diffeomorphisms

• For such examples to occur
• The vector field flow must coincide with the
  stable foliation on some interval
• (around fixed point or along the orbit of the
  periodic point).
• Given a coincidence, pathological examples occur
  for an open set of conditions.

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• Q1 Would ruling out such coincidences ensure
  that all invariant ?
• NO even for the linear case
• there may be proper invariant subsets that
  contain curves (Franks 77).

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Genericity Results

• Theorem
• Linear Anosov or
• Anosov Diffeomorphism with 1D unst. mfd.
• for a residual set of smooth vector fields,
  all the invariant .
• in particular there are no coincidences.