A little bit more about ergodicity Jana Rodriguez Hertz ICTP 2018 - - PowerPoint PPT Presentation

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A little bit more about ergodicity Jana Rodriguez Hertz ICTP 2018 - - PowerPoint PPT Presentation

Jun 18, 2023 306 likes •1.46k views

counting fish mixing properties A little bit more about ergodicity Jana Rodriguez Hertz ICTP 2018 counting fish mixing properties mark and recapture method mark and recapture method mark and recapture counting fish mixing properties

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counting fish mixing properties

A little bit more about ergodicity

Jana Rodriguez Hertz

ICTP

2018

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counting fish mixing properties mark and recapture method

mark and recapture method

mark and recapture

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counting fish mixing properties mark and recapture method

mark and recapture method

mark and recapture ↓

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counting fish mixing properties mark and recapture method

mark and recapture method

mark and recapture ↓ ecology

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counting fish mixing properties mark and recapture method

mark and recapture method

mark and recapture ↓ ecology epidemiology

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counting fish mixing properties mark and recapture method

mark and recapture method

mark and recapture ↓ ecology epidemiology ↓

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counting fish mixing properties mark and recapture method

mark and recapture method

mark and recapture ↓ ecology epidemiology ↓ estimate populations

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counting fish mixing properties mark and recapture method

counting fish

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counting fish mixing properties mark and recapture method

counting fish

procedure

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counting fish mixing properties mark and recapture method

counting fish

procedure

capture 1000 fishes

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counting fish mixing properties mark and recapture method

counting fish

procedure

capture 1000 fishes tag them

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counting fish mixing properties mark and recapture method

counting fish

procedure

capture 1000 fishes tag them release them

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counting fish mixing properties mark and recapture method

counting fish

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counting fish mixing properties mark and recapture method

counting fish

procedure

let them get mixed

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counting fish mixing properties mark and recapture method

counting fish

procedure

let them get mixed

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counting fish mixing properties mark and recapture method

counting fish

procedure

let them get mixed

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counting fish mixing properties mark and recapture method

counting fish

procedure

let them get mixed recapture

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counting fish mixing properties mark and recapture method

estimate

recaptured sample

recaptured fish

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counting fish mixing properties mark and recapture method

estimate

recaptured sample

recaptured fish

tagged

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counting fish mixing properties mark and recapture method

estimate

recaptured sample

1000

tagged

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counting fish mixing properties mark and recapture method

estimate

recaptured sample

1000 10

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counting fish mixing properties mark and recapture method

estimate

recaptured sample

100% 1%

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counting fish mixing properties mark and recapture method

estimate

recaptured sample

100% 1%

estimated population

100% 1%

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counting fish mixing properties mark and recapture method

estimate

recaptured sample

1000 10

estimated population

100% 1%

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counting fish mixing properties mark and recapture method

estimate

recaptured sample

1000 10

estimated population

100% 1% initial sample

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counting fish mixing properties mark and recapture method

estimate

recaptured sample

1000 10

estimated population

100% initial sample

1000

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counting fish mixing properties mark and recapture method

estimate

recaptured sample

1000 10

estimated population

100,000 initial sample

1000

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counting fish mixing properties implicit hypotheses

assumptions

implicit hypotheses

fish amount or measure is not altered

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counting fish mixing properties implicit hypotheses

assumptions

implicit hypotheses

fish amount or measure is not altered tagged fish move around the lake and come back

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counting fish mixing properties implicit hypotheses

assumptions

implicit hypotheses

fish amount or measure is not altered tagged fish move around the lake and come back tagged and non-tagged fish get well mixed

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counting fish mixing properties possible failure 1

possible failure 1

possibility 1

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counting fish mixing properties possible failure 1

estimate

recaptured sample

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counting fish mixing properties possible failure 1

estimate

recaptured sample

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counting fish mixing properties possible failure 1

estimate

recaptured sample

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counting fish mixing properties possible failure 1

estimate

recaptured sample

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counting fish mixing properties possible failure 1

estimate

recaptured sample

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counting fish mixing properties possible failure 1

estimate

recaptured sample

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counting fish mixing properties possible failure 1

estimate

recaptured sample estimated population?

? 1000

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counting fish mixing properties possible failure 1

preservation of the measure

measure preserving process

f : L → L process where

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counting fish mixing properties possible failure 1

preservation of the measure

measure preserving process

f : L → L process where for any sample A ⊂ L

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counting fish mixing properties possible failure 1

preservation of the measure

measure preserving process

f : L → L process where for any sample A ⊂ L we have m(f(A)) = m(A)

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counting fish mixing properties possible failure 2

possible failure 2

possibility 2

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counting fish mixing properties possible failure 2

possible failure 2 - trace

possibility 2 - trace

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counting fish mixing properties possible failure 2

possibility 2

recaptured sample

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counting fish mixing properties possible failure 2

possibility 2

recaptured sample

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counting fish mixing properties possible failure 2

possibility 2

recaptured sample

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counting fish mixing properties possible failure 2

possibility 2

recaptured sample

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counting fish mixing properties possible failure 2

possibility 2

recaptured sample

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counting fish mixing properties possible failure 2

possibility 2

recaptured sample

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counting fish mixing properties possible failure 2

possibility 2

recaptured sample estimated population

1000 ?

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counting fish mixing properties possible failure 3

possibility 3

possibility 3

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counting fish mixing properties possible failure 3

possibility 3

possibility 3 - trace

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counting fish mixing properties possible failure 3

estimate

recaptured sample

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counting fish mixing properties possible failure 3

estimate

recaptured sample

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counting fish mixing properties possible failure 3

estimate

recaptured sample

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counting fish mixing properties possible failure 3

estimate

recaptured sample

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counting fish mixing properties possible failure 3

estimate

recaptured sample

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counting fish mixing properties possible failure 3

estimate

recaptured sample estimated population

1000

?

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counting fish mixing properties mixing lake population

possibility 4

possibility 4

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counting fish mixing properties mixing lake population

estimate - mixing situation

recaptured sample

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counting fish mixing properties mixing lake population

estimate - mixing situation

recaptured sample

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counting fish mixing properties mixing lake population

estimate - mixing situation

recaptured sample

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counting fish mixing properties mixing lake population

estimate - mixing situation

recaptured sample

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counting fish mixing properties mixing lake population

estimate - mixing situation

recaptured sample

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counting fish mixing properties mixing lake population

estimate - mixing situation

recaptured sample estimated population

1%

100%

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counting fish mixing properties mixing lake population

estimate - mixing situation

recaptured sample estimated population

1000

100%

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counting fish mixing properties mixing lake population

estimate - mixing situation

recaptured sample estimated population

1000

100,000

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counting fish mixing properties mixing lake population

mixing property

mixing property

f : L → L is mixing

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counting fish mixing properties mixing lake population

mixing property

mixing property

f : L → L is mixing if every pair of samples A, B ⊂ L satisfy

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counting fish mixing properties mixing lake population

mixing property

mixing property

f : L → L is mixing if every pair of samples A, B ⊂ L satisfy m(A ∩ f n(B)) → m(A)m(B)

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counting fish mixing properties mixing lake population

mixing property

in our context

if the lake population is mixing

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counting fish mixing properties mixing lake population

mixing property

in our context

if the lake population is mixing A ⊂ L initial sample

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counting fish mixing properties mixing lake population

mixing property

in our context

if the lake population is mixing A ⊂ L initial sample m(A ∩ f n(A)) m(A)

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counting fish mixing properties mixing lake population

mixing property

in our context

if the lake population is mixing A ⊂ L initial sample m(A ∩ f n(A)) m(A) → m(A)

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counting fish mixing properties mixing lake population

mixing property

mixing property

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counting fish mixing properties mixing lake population

mixing property

mixing property ⇓ mark and recapture method works

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counting fish mixing properties mixing lake population

mixing property

NO mixing property ? mark and recapture method works

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counting fish mixing properties ergodicity

possibility 3

possibility 3

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counting fish mixing properties ergodicity

possibility 3

possibility 3 - trace

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counting fish mixing properties ergodicity

estimate

recaptured sample

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counting fish mixing properties ergodicity

estimate

recaptured sample

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counting fish mixing properties ergodicity

estimate

recaptured sample

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counting fish mixing properties ergodicity

estimate

recaptured sample

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counting fish mixing properties ergodicity

estimate

recaptured sample

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counting fish mixing properties ergodicity

estimate

recaptured sample estimated population

1000

?

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counting fish mixing properties ergodicity

possibility 3

in this example

∃ lim m(A ∩ f n(A)) m(A)

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counting fish mixing properties ergodicity

possibility 3

in this example

∃ lim m(A ∩ f n(A)) m(A)

what about this:

1 N

N−1

  • n=0

m(A ∩ f n(A)) m(A) → ?

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counting fish mixing properties ergodicity

ergodicity

ergodicity (exercise)

f : L → L is ergoodic

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counting fish mixing properties ergodicity

ergodicity

ergodicity (exercise)

f : L → L is ergoodic if every pair of samples A, B ⊂ L satisfy

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counting fish mixing properties ergodicity

ergodicity

ergodicity (exercise)

f : L → L is ergoodic if every pair of samples A, B ⊂ L satisfy 1 N

N−1

  • n=0

m(A ∩ f n(B)) → m(A)m(B)

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counting fish mixing properties ergodicity

ergodic property

in our context

if the lake population is ergodic

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counting fish mixing properties ergodicity

ergodic property

in our context

if the lake population is ergodic A ⊂ L initial sample

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counting fish mixing properties ergodicity

ergodic property

in our context

if the lake population is ergodic A ⊂ L initial sample 1 N

N−1

  • n=0

m(A ∩ f n(A)) m(A)

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counting fish mixing properties ergodicity

ergodic property

in our context

if the lake population is ergodic A ⊂ L initial sample 1 N

N−1

  • n=0

m(A ∩ f n(A)) m(A) → m(A)

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counting fish mixing properties ergodicity

1 N

N−1

n=0 m(A∩f n(A)) m(A)

→ m(A)

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counting fish mixing properties ergodicity

1 N

N−1

n=0 m(A∩f n(A)) m(A)

→ m(A)

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counting fish mixing properties ergodicity

1 N

N−1

n=0 m(A∩f n(A)) m(A)

→ m(A)

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counting fish mixing properties ergodicity

1 N

N−1

n=0 m(A∩f n(A)) m(A)

→ m(A)

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counting fish mixing properties ergodicity

1 N

N−1

n=0 m(A∩f n(A)) m(A)

→ m(A)

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counting fish mixing properties ergodicity

1 N

N−1

n=0 m(A∩f n(A)) m(A)

→ m(A)

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counting fish mixing properties ergodicity

1 N

N−1

n=0 m(A∩f n(A)) m(A)

→ m(A)

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counting fish mixing properties ergodicity

1 N

N−1

n=0 m(A∩f n(A)) m(A)

→ m(A)

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counting fish mixing properties ergodicity

1 N

N−1

n=0 m(A∩f n(A)) m(A)

→ m(A)

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counting fish mixing properties ergodicity

1 N

N−1

n=0 m(A∩f n(A)) m(A)

→ m(A)

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counting fish mixing properties ergodicity

exercise

exercise

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counting fish mixing properties ergodicity

exercise

exercise

show that f ergodic ⇔

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counting fish mixing properties ergodicity

exercise

exercise

show that f ergodic ⇔ m  

n≥0

f n(A)   = 1 ∀A s.t. m(A) > 0

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counting fish mixing properties stable ergodicity

fragility of ergodicity

irrational traslation on T2

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counting fish mixing properties stable ergodicity

fragility of ergodicity

rational traslation on T2

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counting fish mixing properties stable ergodicity

stable ergodicity

stable ergodicity

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counting fish mixing properties stable ergodicity

stable ergodicity

stable ergodicity

f is stably ergodic

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counting fish mixing properties stable ergodicity

stable ergodicity

stable ergodicity

f is stably ergodic if g ∼ f ⇒ g ergodic

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counting fish mixing properties stable ergodicity

  • pen problem
  • pen problem

stably ergodic ⇒ mixing ?

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counting fish mixing properties stable ergodicity

thank you

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counting fish mixing properties stable ergodicity

thank you

thank you!