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The Ehrenfest modeland entropy zero
deterministic random walks
Corinna Ulcigrai
University of Bristol
Probability, Analysis and Dynamics
Bristol, 23 April 2014
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The Ehrenfest modelThe Ehrenfest windtree model:Z2-periodic array of rectangular scatterersBilliard trajectory: straight line + elastic reflections(angle of incidence = angle of reflection)
Paul and Tatjana Ehrenfest, 1912 (periodic version by Hardy-Weber)
I Is a typical trajectory recurrent?(i.e. do points come back?)
I Are there dense trajectories?
I Is the billiard ergodic?(i.e. no invariant sets)
I What is the diffusion speed?
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The Ehrenfest modelThe Ehrenfest windtree model:Z2-periodic array of rectangular scatterersBilliard trajectory: straight line + elastic reflections(angle of incidence = angle of reflection)
Paul and Tatjana Ehrenfest, 1912 (periodic version by Hardy-Weber)
I Is a typical trajectory recurrent?(i.e. do points come back?)
I Are there dense trajectories?
I Is the billiard ergodic?(i.e. no invariant sets)
I What is the diffusion speed?
-
The Ehrenfest modelThe Ehrenfest windtree model:Z2-periodic array of rectangular scatterersBilliard trajectory: straight line + elastic reflections(angle of incidence = angle of reflection)
Paul and Tatjana Ehrenfest, 1912 (periodic version by Hardy-Weber)
I Is a typical trajectory recurrent?(i.e. do points come back?)
I Are there dense trajectories?
I Is the billiard ergodic?(i.e. no invariant sets)
I What is the diffusion speed?
-
The Ehrenfest modelThe Ehrenfest windtree model:Z2-periodic array of rectangular scatterersBilliard trajectory: straight line + elastic reflections(angle of incidence = angle of reflection)
Paul and Tatjana Ehrenfest, 1912 (periodic version by Hardy-Weber)
I Is a typical trajectory recurrent?(i.e. do points come back?)
I Are there dense trajectories?
I Is the billiard ergodic?(i.e. no invariant sets)
I What is the diffusion speed?
-
The Ehrenfest modelThe Ehrenfest windtree model:Z2-periodic array of rectangular scatterersBilliard trajectory: straight line + elastic reflections(angle of incidence = angle of reflection)
Paul and Tatjana Ehrenfest, 1912 (periodic version by Hardy-Weber)
I Is a typical trajectory recurrent?(i.e. do points come back?)
I Are there dense trajectories?
I Is the billiard ergodic?(i.e. no invariant sets)
I What is the diffusion speed?
-
The Ehrenfest modelThe Ehrenfest windtree model:Z2-periodic array of rectangular scatterersBilliard trajectory: straight line + elastic reflections(angle of incidence = angle of reflection)
Paul and Tatjana Ehrenfest, 1912 (periodic version by Hardy-Weber)
I Is a typical trajectory recurrent?(i.e. do points come back?)
I Are there dense trajectories?
I Is the billiard ergodic?(i.e. no invariant sets)
I What is the diffusion speed?
-
The Ehrenfest modelThe Ehrenfest windtree model:Z2-periodic array of rectangular scatterersBilliard trajectory: straight line + elastic reflections(angle of incidence = angle of reflection)
Paul and Tatjana Ehrenfest, 1912 (periodic version by Hardy-Weber)
I Is a typical trajectory recurrent?(i.e. do points come back?)
I Are there dense trajectories?
I Is the billiard ergodic?(i.e. no invariant sets)
I What is the diffusion speed?
-
The Ehrenfest modelThe Ehrenfest windtree model:Z2-periodic array of rectangular scatterersBilliard trajectory: straight line + elastic reflections(angle of incidence = angle of reflection)
Paul and Tatjana Ehrenfest, 1912 (periodic version by Hardy-Weber)
I Is a typical trajectory recurrent?(i.e. do points come back?)
I Are there dense trajectories?
I Is the billiard ergodic?(i.e. no invariant sets)
I What is the diffusion speed?
-
Lorentz gas versus Ehrenfest model
I the periodic Lorentz Gas
H. A. Lorentz, 1905
Very well studied, many results fromthe 80s onwards(Buminovich, Sinai, Bleher, Szasz,Varju, Chernov, Dolgopyat, Melbourne,
Nicol, Dettmann, Marklof,
Strombergson, Toth...)
Hyperbolic: positive entropy
I the Ehrenfest windtree model
Paul and Tatjana Ehrenfest, 1912
(periodic version by Hardy-Weber)numerical simulations, almost no
rigourous results to now...
several recent breakthroughs (last 2-3
years) via Teichmueller dynamics
Flat: entropy zero!
-
Lorentz gas versus Ehrenfest model
I the periodic Lorentz Gas
H. A. Lorentz, 1905
Very well studied, many results fromthe 80s onwards(Buminovich, Sinai, Bleher, Szasz,Varju, Chernov, Dolgopyat, Melbourne,
Nicol, Dettmann, Marklof,
Strombergson, Toth...)
Hyperbolic: positive entropy
I the Ehrenfest windtree model
Paul and Tatjana Ehrenfest, 1912
(periodic version by Hardy-Weber)
numerical simulations, almost no
rigourous results to now...
several recent breakthroughs (last 2-3
years) via Teichmueller dynamics
Flat: entropy zero!
-
Lorentz gas versus Ehrenfest model
I the periodic Lorentz Gas
H. A. Lorentz, 1905
Very well studied, many results fromthe 80s onwards(Buminovich, Sinai, Bleher, Szasz,Varju, Chernov, Dolgopyat, Melbourne,
Nicol, Dettmann, Marklof,
Strombergson, Toth...)
Hyperbolic: positive entropy
I the Ehrenfest windtree model
Paul and Tatjana Ehrenfest, 1912
(periodic version by Hardy-Weber)
numerical simulations, almost no
rigourous results to now...
several recent breakthroughs (last 2-3
years) via Teichmueller dynamics
Flat: entropy zero!
-
Lorentz gas versus Ehrenfest model
I the periodic Lorentz Gas
H. A. Lorentz, 1905
Very well studied, many results fromthe 80s onwards(Buminovich, Sinai, Bleher, Szasz,Varju, Chernov, Dolgopyat, Melbourne,
Nicol, Dettmann, Marklof,
Strombergson, Toth...)
Hyperbolic: positive entropy
I the Ehrenfest windtree model
Paul and Tatjana Ehrenfest, 1912
(periodic version by Hardy-Weber)
numerical simulations, almost no
rigourous results to now...
several recent breakthroughs (last 2-3
years) via Teichmueller dynamics
Flat: entropy zero!
-
Lorentz gas versus Ehrenfest model
I the periodic Lorentz Gas
H. A. Lorentz, 1905
Very well studied, many results fromthe 80s onwards(Buminovich, Sinai, Bleher, Szasz,Varju, Chernov, Dolgopyat, Melbourne,
Nicol, Dettmann, Marklof,
Strombergson, Toth...)
Hyperbolic: positive entropy
I the Ehrenfest windtree model
Paul and Tatjana Ehrenfest, 1912
(periodic version by Hardy-Weber)
numerical simulations, almost no
rigourous results to now...
several recent breakthroughs (last 2-3
years) via Teichmueller dynamics
Flat: entropy zero!
-
Some recent results on the Ehrenfest model
Notation: Let 0 0) divergent trajectories; (Delecroix)
Theorem (Delecroix-Hubert-Lelievre)For all a, b the billiard is superdiffusive: the distance d(bθt (p), p) from theinitial point p after time t satisfies
lim sup d(bθt (p), p) ∼ t2/3.
More precisely:lim sup log d(bθt (p),p)
log t = 2/3.
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Some recent results on the Ehrenfest model
Notation: Let 0 0) divergent trajectories; (Delecroix)
Theorem (Delecroix-Hubert-Lelievre)For all a, b the billiard is superdiffusive: the distance d(bθt (p), p) from theinitial point p after time t satisfies
lim sup d(bθt (p), p) ∼ t2/3.
More precisely:lim sup log d(bθt (p),p)
log t = 2/3.
-
Some recent results on the Ehrenfest model
Notation: Let 0 0) divergent trajectories; (Delecroix)
Theorem (Delecroix-Hubert-Lelievre)For all a, b the billiard is superdiffusive: the distance d(bθt (p), p) from theinitial point p after time t satisfies
lim sup d(bθt (p), p) ∼ t2/3.
More precisely:lim sup log d(bθt (p),p)
log t = 2/3.
-
Some recent results on the Ehrenfest model
Notation: Let 0 0) divergent trajectories; (Delecroix)
Theorem (Delecroix-Hubert-Lelievre)For all a, b the billiard is superdiffusive: the distance d(bθt (p), p) from theinitial point p after time t satisfies
lim sup d(bθt (p), p) ∼ t2/3.
More precisely:lim sup log d(bθt (p),p)
log t = 2/3.
-
Some recent results on the Ehrenfest model
Notation: Let 0 0) divergent trajectories; (Delecroix)
Theorem (Delecroix-Hubert-Lelievre)For all a, b the billiard is superdiffusive: the distance d(bθt (p), p) from theinitial point p after time t satisfies
lim sup d(bθt (p), p) ∼ t2/3.
More precisely:lim sup log d(bθt (p),p)
log t = 2/3.
-
Some recent results on the Ehrenfest model
Notation: Let 0 0) divergent trajectories; (Delecroix)
Theorem (Delecroix-Hubert-Lelievre)For all a, b the billiard is superdiffusive: the distance d(bθt (p), p) from theinitial point p after time t satisfies
lim sup d(bθt (p), p) ∼ t2/3.
More precisely:lim sup log d(bθt (p),p)
log t = 2/3.
-
Some recent results on the Ehrenfest model
Notation: Let 0 0) divergent trajectories; (Delecroix)
Theorem (Delecroix-Hubert-Lelievre)For all a, b the billiard is superdiffusive: the distance d(bθt (p), p) from theinitial point p after time t satisfies
lim sup d(bθt (p), p) ∼ t2/3.
More precisely:lim sup log d(bθt (p),p)
log t = 2/3.
-
Some recent results on the Ehrenfest model
Notation: Let 0 0) divergent trajectories; (Delecroix)
Theorem (Delecroix-Hubert-Lelievre)For all a, b the billiard is superdiffusive: the distance d(bθt (p), p) from theinitial point p after time t satisfies
lim sup d(bθt (p), p) ∼ t2/3.
More precisely:lim sup log d(bθt (p),p)
log t = 2/3.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Ergodicity for bounded polygonal billiards
I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)
I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.
E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.
I Fact: the flow bθt preserves µ = Leb ×∑k
i=1 δθi .
I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A
c) = 0.
Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.
CorollaryBilliard trajectories in a random direction are dense.
-
Non-ergodicity for the Ehrenfest model
Theorem (Fraczek-Ulcigrai)For any lengths 0 < a, b < 1 of the sides,for a.e. direction θ, bθt in the Ehrenfest model is:
I not transitive (i.e. no trajectory is dense)
I NOT ergodic (uncountably many erg. comp.).
[Ref: Fraczek-U, Inventiones for a.e. (a,b) + Eskin-Chaika for all (a,b)]
More in general: criterium for non ergodicitywhich holds for a class of periodic billiards
E.g.: billiard in a tube with periodic barriers
I Tools: Non-ergodicity and superdiffusion both exploit:
I deterministic random walks(driven by interval exchange transformations);
I Lyapunov exponents of product of matrices(given by the Kontsevich-Zorich cocycle).
-
Non-ergodicity for the Ehrenfest model
Theorem (Fraczek-Ulcigrai)For any lengths 0 < a, b < 1 of the sides,for a.e. direction θ, bθt in the Ehrenfest model is:
I not transitive (i.e. no trajectory is dense)
I NOT ergodic (uncountably many erg. comp.).
[Ref: Fraczek-U, Inventiones for a.e. (a,b) + Eskin-Chaika for all (a,b)]
More in general: criterium for non ergodicitywhich holds for a class of periodic billiards
E.g.: billiard in a tube with periodic barriers
I Tools: Non-ergodicity and superdiffusion both exploit:
I deterministic random walks(driven by interval exchange transformations);
I Lyapunov exponents of product of matrices(given by the Kontsevich-Zorich cocycle).
-
Non-ergodicity for the Ehrenfest model
Theorem (Fraczek-Ulcigrai)For any lengths 0 < a, b < 1 of the sides,for a.e. direction θ, bθt in the Ehrenfest model is:
I not transitive (i.e. no trajectory is dense)
I NOT ergodic (uncountably many erg. comp.).
[Ref: Fraczek-U, Inventiones for a.e. (a,b) + Eskin-Chaika for all (a,b)]
More in general: criterium for non ergodicitywhich holds for a class of periodic billiards
E.g.: billiard in a tube with periodic barriers
I Tools: Non-ergodicity and superdiffusion both exploit:
I deterministic random walks(driven by interval exchange transformations);
I Lyapunov exponents of product of matrices(given by the Kontsevich-Zorich cocycle).
-
Non-ergodicity for the Ehrenfest model
Theorem (Fraczek-Ulcigrai)For any lengths 0 < a, b < 1 of the sides,for a.e. direction θ, bθt in the Ehrenfest model is:
I not transitive (i.e. no trajectory is dense)
I NOT ergodic (uncountably many erg. comp.).
[Ref: Fraczek-U, Inventiones for a.e. (a,b) + Eskin-Chaika for all (a,b)]
More in general: criterium for non ergodicitywhich holds for a class of periodic billiards
E.g.: billiard in a tube with periodic barriers
I Tools: Non-ergodicity and superdiffusion both exploit:
I deterministic random walks(driven by interval exchange transformations);
I Lyapunov exponents of product of matrices(given by the Kontsevich-Zorich cocycle).
-
Non-ergodicity for the Ehrenfest model
Theorem (Fraczek-Ulcigrai)For any lengths 0 < a, b < 1 of the sides,for a.e. direction θ, bθt in the Ehrenfest model is:
I not transitive (i.e. no trajectory is dense)
I NOT ergodic (uncountably many erg. comp.).
[Ref: Fraczek-U, Inventiones for a.e. (a,b) + Eskin-Chaika for all (a,b)]
More in general: criterium for non ergodicitywhich holds for a class of periodic billiards
E.g.: billiard in a tube with periodic barriers
I Tools: Non-ergodicity and superdiffusion both exploit:
I deterministic random walks(driven by interval exchange transformations);
I Lyapunov exponents of product of matrices(given by the Kontsevich-Zorich cocycle).
-
Non-ergodicity for the Ehrenfest model
Theorem (Fraczek-Ulcigrai)For any lengths 0 < a, b < 1 of the sides,for a.e. direction θ, bθt in the Ehrenfest model is:
I not transitive (i.e. no trajectory is dense)
I NOT ergodic (uncountably many erg. comp.).
[Ref: Fraczek-U, Inventiones for a.e. (a,b) + Eskin-Chaika for all (a,b)]
More in general: criterium for non ergodicitywhich holds for a class of periodic billiards
E.g.: billiard in a tube with periodic barriers
I Tools: Non-ergodicity and superdiffusion both exploit:
I deterministic random walks(driven by interval exchange transformations);
I Lyapunov exponents of product of matrices(given by the Kontsevich-Zorich cocycle).
-
Non-ergodicity for the Ehrenfest model
Theorem (Fraczek-Ulcigrai)For any lengths 0 < a, b < 1 of the sides,for a.e. direction θ, bθt in the Ehrenfest model is:
I not transitive (i.e. no trajectory is dense)
I NOT ergodic (uncountably many erg. comp.).
[Ref: Fraczek-U, Inventiones for a.e. (a,b) + Eskin-Chaika for all (a,b)]
More in general: criterium for non ergodicitywhich holds for a class of periodic billiards
E.g.: billiard in a tube with periodic barriers
I Tools: Non-ergodicity and superdiffusion both exploit:
I deterministic random walks(driven by interval exchange transformations);
I Lyapunov exponents of product of matrices(given by the Kontsevich-Zorich cocycle).
-
Non-ergodicity for the Ehrenfest model
Theorem (Fraczek-Ulcigrai)For any lengths 0 < a, b < 1 of the sides,for a.e. direction θ, bθt in the Ehrenfest model is:
I not transitive (i.e. no trajectory is dense)
I NOT ergodic (uncountably many erg. comp.).
[Ref: Fraczek-U, Inventiones for a.e. (a,b) + Eskin-Chaika for all (a,b)]
More in general: criterium for non ergodicitywhich holds for a class of periodic billiards
E.g.: billiard in a tube with periodic barriers
I Tools: Non-ergodicity and superdiffusion both exploit:
I deterministic random walks(driven by interval exchange transformations);
I Lyapunov exponents of product of matrices(given by the Kontsevich-Zorich cocycle).
-
Non-ergodicity for the Ehrenfest model
Theorem (Fraczek-Ulcigrai)For any lengths 0 < a, b < 1 of the sides,for a.e. direction θ, bθt in the Ehrenfest model is:
I not transitive (i.e. no trajectory is dense)
I NOT ergodic (uncountably many erg. comp.).
[Ref: Fraczek-U, Inventiones for a.e. (a,b) + Eskin-Chaika for all (a,b)]
More in general: criterium for non ergodicitywhich holds for a class of periodic billiards
E.g.: billiard in a tube with periodic barriers
I Tools: Non-ergodicity and superdiffusion both exploit:
I deterministic random walks(driven by interval exchange transformations);
I Lyapunov exponents of product of matrices(given by the Kontsevich-Zorich cocycle).
-
Reduction to a straight-line flow
Fix θ. Consider 4 copies of the Ehrenfest table, one per direction.
Billiard trajectories become straight-line trajectories on 4 tables withpairs of sides glued together (=infinite flat surface).
-
Reduction to a straight-line flow
Fix θ. Consider 4 copies of the Ehrenfest table, one per direction.
Billiard trajectories become straight-line trajectories on 4 tables withpairs of sides glued together (=infinite flat surface).
-
Reduction to a straight-line flow
Fix θ. Consider 4 copies of the Ehrenfest table, one per direction.
Billiard trajectories become straight-line trajectories on 4 tables withpairs of sides glued together (=infinite flat surface).
-
Reduction to a straight-line flow
Fix θ. Consider 4 copies of the Ehrenfest table, one per direction.
Billiard trajectories become straight-line trajectories on 4 tables withpairs of sides glued together (=infinite flat surface).
-
Reduction to a straight-line flow
Fix θ. Consider 4 copies of the Ehrenfest table, one per direction.
Billiard trajectories become straight-line trajectories on 4 tables withpairs of sides glued together (=infinite flat surface).
-
Reduction to a straight-line flow
Fix θ. Consider 4 copies of the Ehrenfest table, one per direction.
Billiard trajectories become straight-line trajectories on 4 tables withpairs of sides glued together (=infinite flat surface).
-
Reduction to a straight-line flow
Fix θ. Consider 4 copies of the Ehrenfest table, one per direction.
Billiard trajectories become straight-line trajectories on 4 tables withpairs of sides glued together (=infinite flat surface).
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).
Consider the red section ∼= [0, 1]×Z.
Given x ∈ [0, 1], the next hittingT (x) is given by a rotation
T (x) = x − α mod 1,
α = cot θ
Associated deterministic walk on Z:
f (x) =
{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =
n−1∑k=0
f (T ix)
Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)
-
Walks driven by interval exchange transformations
Similarly: the vertical (or horizontal) Z-motionin the Ehrenfest billiard is also given bya deterministic random walk, with:
Snf (x) =n−1∑k=0
f (T ix)
I T is an interval exchange transformation (orIET), i.e a piecewise isometry of the interval:
I f : [0, 1]→ Z is a piecewise constant functionon each exchanged subinterval with E(f ) = 0
Say: T random IET = any irreducible permuation, a.e. choice of lenghts
-
Walks driven by interval exchange transformations
Similarly: the vertical (or horizontal) Z-motionin the Ehrenfest billiard is also given bya deterministic random walk, with:
Snf (x) =n−1∑k=0
f (T ix)
I T is an interval exchange transformation (orIET), i.e a piecewise isometry of the interval:
I f : [0, 1]→ Z is a piecewise constant functionon each exchanged subinterval with E(f ) = 0
Say: T random IET = any irreducible permuation, a.e. choice of lenghts
-
Walks driven by interval exchange transformations
Similarly: the vertical (or horizontal) Z-motionin the Ehrenfest billiard is also given bya deterministic random walk, with:
Snf (x) =n−1∑k=0
f (T ix)
I T is an interval exchange transformation (orIET), i.e a piecewise isometry of the interval:
A B C D
ABD C
T
I f : [0, 1]→ Z is a piecewise constant functionon each exchanged subinterval with E(f ) = 0
Say: T random IET = any irreducible permuation, a.e. choice of lenghts
-
Walks driven by interval exchange transformations
Similarly: the vertical (or horizontal) Z-motionin the Ehrenfest billiard is also given bya deterministic random walk, with:
Snf (x) =n−1∑k=0
f (T ix)
I T is an interval exchange transformation (orIET), i.e a piecewise isometry of the interval:
A B C D
ABD C
T
I f : [0, 1]→ Z is a piecewise constant functionon each exchanged subinterval with E(f ) = 0
Say: T random IET = any irreducible permuation, a.e. choice of lenghts
-
Walks driven by interval exchange transformations
Similarly: the vertical (or horizontal) Z-motionin the Ehrenfest billiard is also given bya deterministic random walk, with:
Snf (x) =n−1∑k=0
f (T ix)
I T is an interval exchange transformation (orIET), i.e a piecewise isometry of the interval:
A B C D
ABD C
T
I f : [0, 1]→ Z is a piecewise constant functionon each exchanged subinterval with E(f ) = 0
Say: T random IET = any irreducible permuation, a.e. choice of lenghts
-
Limsup behaviour of walks driven by IETs
Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0
-
Limsup behaviour of walks driven by IETs
Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0
-
Limsup behaviour of walks driven by IETs
Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0
-
Limsup behaviour of walks driven by IETs
Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0
-
Limsup behaviour of walks driven by IETs
Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0
-
Limsup behaviour of walks driven by IETs
Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0
-
Limsup behaviour of walks driven by IETs
Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0
-
Limsup behaviour of walks driven by IETs
Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0
-
Limsup behaviour of walks driven by IETs
Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0
-
Limsup behaviour of walks driven by IETs
Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0
-
Limsup behaviour of walks driven by IETs
Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0
-
Limsup behaviour of walks driven by IETs
Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.
I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)
(same # exchanged intervals)
I I(n)1 , . . . , I
(n)d exchanged intervals;
I r(n)1 , . . . , r
(n)d return times;
I f = (f1, . . . , fd), fi value on I(0)i ;
I set f (n) = (f(n)1 , . . . , f
(n)d ) where
f(n)j =
r(n)i∑
k=0
f (T k(xi )), where xi ∈ I (n)i .
I Growth of f (n): f (n+1) = An f(n), where An = A(T
(n)) ∈ SL(d ,Z)
⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)
⇒ use Oseledetes Thm/Lyapunov exponents.
-
Deterministic walks on R with singularities of type 1x
Consider Xn = Snf , where
I Snf (x) =∑n−1
k=0 f (Tn(x)),
I T is an interval exchangetransformation;
I f is R-valued with 1x -type singularities,[i.e. f (x − xi ) ∼ cix−xinear (a subset of) discontinuities xi of T ]
A B C D
ABD C
T
Motivation:
ergodic theory of smooth area-preservingflows on surfaces:
limiting behaviour of Snf determineschaotic properties (in particular mixing)of locally Hamiltonian flows
-
Deterministic walks on R with singularities of type 1x
Consider Xn = Snf , where
I Snf (x) =∑n−1
k=0 f (Tn(x)),
I T is an interval exchangetransformation;
I f is R-valued with 1x -type singularities,[i.e. f (x − xi ) ∼ cix−xinear (a subset of) discontinuities xi of T ]
A B C D
ABD C
T
Motivation:
ergodic theory of smooth area-preservingflows on surfaces:
limiting behaviour of Snf determineschaotic properties (in particular mixing)of locally Hamiltonian flows
-
Deterministic walks on R with singularities of type 1x
Consider Xn = Snf , where
I Snf (x) =∑n−1
k=0 f (Tn(x)),
I T is an interval exchangetransformation;
I f is R-valued with 1x -type singularities,[i.e. f (x − xi ) ∼ cix−xinear (a subset of) discontinuities xi of T ]
Motivation:
ergodic theory of smooth area-preservingflows on surfaces:
limiting behaviour of Snf determineschaotic properties (in particular mixing)of locally Hamiltonian flows
-
Deterministic walks on R with singularities of type 1x
Consider Xn = Snf , where
I Snf (x) =∑n−1
k=0 f (Tn(x)),
I T is an interval exchangetransformation;
I f is R-valued with 1x -type singularities,[i.e. f (x − xi ) ∼ cix−xinear (a subset of) discontinuities xi of T ]
Motivation:
ergodic theory of smooth area-preservingflows on surfaces:
limiting behaviour of Snf determineschaotic properties (in particular mixing)of locally Hamiltonian flows
-
Deterministic walks on R with singularities of type 1x
Consider Xn = Snf , where
I Snf (x) =∑n−1
k=0 f (Tn(x)),
I T is an interval exchangetransformation;
I f is R-valued with 1x -type singularities,[i.e. f (x − xi ) ∼ cix−xinear (a subset of) discontinuities xi of T ]
Motivation:
ergodic theory of smooth area-preservingflows on surfaces:
limiting behaviour of Snf determineschaotic properties (in particular mixing)of locally Hamiltonian flows
-
Deterministic walks on R with singularities of type 1x
Consider Xn = Snf , where
I Snf (x) =∑n−1
k=0 f (Tn(x)),
I T is an interval exchangetransformation;
I f is R-valued with 1x -type singularities,[i.e. f (x − xi ) ∼ cix−xinear (a subset of) discontinuities xi of T ]
Motivation:
ergodic theory of smooth area-preservingflows on surfaces:
limiting behaviour of Snf determineschaotic properties (in particular mixing)of locally Hamiltonian flows
-
Limit theorems for 1x -type of singularitiesBehaviour of Xn = Snf is different if
1x singuarities of f are:
asymmetrice.g. f = 1x
E(Xn) = ∞
Theorem (U’, EDTS ’07)T random IET,f with symmetric 1x -sing,
Xnn log n
P→ constant.
Key to prove: Mixing components inloc. Hamiltonian flows with traps
symmetric, e.g.
f = 1x −1
1−x
E(Xn) =∞,but
limδ→0
∫ 1−δδ
f
-
Limit theorems for 1x -type of singularitiesBehaviour of Xn = Snf is different if
1x singuarities of f are:
asymmetrice.g. f = 1x
E(Xn) = ∞
Theorem (U’, EDTS ’07)T random IET,f with symmetric 1x -sing,
Xnn log n
P→ constant.
Key to prove: Mixing components inloc. Hamiltonian flows with traps
symmetric, e.g.
f = 1x −1
1−x
E(Xn) =∞,but
limδ→0
∫ 1−δδ
f
-
Limit theorems for 1x -type of singularitiesBehaviour of Xn = Snf is different if
1x singuarities of f are:
asymmetrice.g. f = 1x
E(Xn) = ∞
Theorem (U’, EDTS ’07)T random IET,f with symmetric 1x -sing,
Xnn log n
P→ constant.
Key to prove: Mixing components inloc. Hamiltonian flows with traps
symmetric, e.g.
f = 1x −1
1−x
E(Xn) =∞,but
limδ→0
∫ 1−δδ
f
-
Limit theorems for 1x -type of singularitiesBehaviour of Xn = Snf is different if
1x singuarities of f are:
asymmetrice.g. f = 1x
E(Xn) = ∞
Theorem (U’, EDTS ’07)T random IET,f with symmetric 1x -sing,
Xnn log n
P→ constant.
Key to prove: Mixing components inloc. Hamiltonian flows with traps
symmetric, e.g.
f = 1x −1
1−x
E(Xn) =∞,but
limδ→0
∫ 1−δδ
f
-
Limit theorems for 1x -type of singularitiesBehaviour of Xn = Snf is different if
1x singuarities of f are:
asymmetrice.g. f = 1x
E(Xn) = ∞
Theorem (U’, EDTS ’07)T random IET,f with symmetric 1x -sing,
Xnn log n
P→ constant.
Key to prove: Mixing components inloc. Hamiltonian flows with traps
symmetric, e.g.
f = 1x −1
1−x
E(Xn) =∞,but
limδ→0
∫ 1−δδ
f
-
Limit theorems for 1x -type of singularitiesBehaviour of Xn = Snf is different if
1x singuarities of f are:
asymmetrice.g. f = 1x
E(Xn) = ∞
Theorem (U’, EDTS ’07)T random IET,f with symmetric 1x -sing,
Xnn log n
P→ constant.
Key to prove: Mixing components inloc. Hamiltonian flows with traps
symmetric, e.g.
f = 1x −1
1−x
E(Xn) =∞,but
limδ→0
∫ 1−δδ
f
-
Limit theorems for 1x -type of singularitiesBehaviour of Xn = Snf is different if
1x singuarities of f are:
asymmetrice.g. f = 1x
E(Xn) = ∞
Theorem (U’, EDTS ’07)T random IET,f with symmetric 1x -sing,
Xnn log n
P→ constant.
Key to prove: Mixing components inloc. Hamiltonian flows with traps
symmetric, e.g.
f = 1x −1
1−x
E(Xn) =∞,but
limδ→0
∫ 1−δδ
f
-
Limit theorems for 1x -type of singularitiesBehaviour of Xn = Snf is different if
1x singuarities of f are:
asymmetrice.g. f = 1x
E(Xn) = ∞
Theorem (U’, EDTS ’07)T random IET,f with symmetric 1x -sing,
Xnn log n
P→ constant.
Key to prove: Mixing components inloc. Hamiltonian flows with traps
symmetric, e.g.
f = 1x −1
1−x
E(Xn) =∞,but
limδ→0
∫ 1−δδ
f
-
Limit theorems for 1x -type of singularities
Behaviour of Xn = Snf is different if1x singuarities of f are:
asymmetrice.g. f = 1x
E(Xn) = ∞
Theorem (U’, EDTS ’07)T random IET,f with symmetric 1x -sing,
Xnn log n
P→ constant.
Key to prove: Mixing components inloc. Hamiltonian flows with traps
symmetric, e.g.
f = 1x −1
1−x
E(Xn) =∞,but
limδ→0
∫ 1−δδ
f
-
Limiting behaviour for random walks driven by IETsI Back to: Snf (x) =
∑n−1k=0 f (T
n(x)), with:
I T interval exchange transformation,
I f piecewise constant, E(f ) = 0.
I Consider the rescaled r.v.
Xn =Snf√
Var(Snf ).
A B C D
ABD C
T
Bufetov (Annals of Math., ’13) has recently shown that:
I Xn do NOT converge in distribution;
I Consider exponential scales (given 0 < s < 1, consider Yn = X[sn]):
I Accumulation points of Yn contain both Dirac deltas (for dense setof s) and non-degenerate distributions.
I Convergence to a moving distribution (driven by the Teichmueller
geodesic flow, e.g. Ynd→ X (gln sS), S associated surface).
-
Limiting behaviour for random walks driven by IETsI Back to: Snf (x) =
∑n−1k=0 f (T
n(x)), with:
I T interval exchange transformation,
I f piecewise constant, E(f ) = 0.
I Consider the rescaled r.v.
Xn =Snf√
Var(Snf ).
A B C D
ABD C
T
Bufetov (Annals of Math., ’13) has recently shown that:
I Xn do NOT converge in distribution;
I Consider exponential scales (given 0 < s < 1, consider Yn = X[sn]):
I Accumulation points of Yn contain both Dirac deltas (for dense setof s) and non-degenerate distributions.
I Convergence to a moving distribution (driven by the Teichmueller
geodesic flow, e.g. Ynd→ X (gln sS), S associated surface).
-
Limiting behaviour for random walks driven by IETsI Back to: Snf (x) =
∑n−1k=0 f (T
n(x)), with:
I T interval exchange transformation,
I f piecewise constant, E(f ) = 0.
I Consider the rescaled r.v.
Xn =Snf√
Var(Snf ).
A B C D
ABD C
T
Bufetov (Annals of Math., ’13) has recently shown that:
I Xn do NOT converge in distribution;
I Consider exponential scales (given 0 < s < 1, consider Yn = X[sn]):
I Accumulation points of Yn contain both Dirac deltas (for dense setof s) and non-degenerate distributions.
I Convergence to a moving distribution (driven by the Teichmueller
geodesic flow, e.g. Ynd→ X (gln sS), S associated surface).
-
Limiting behaviour for random walks driven by IETsI Back to: Snf (x) =
∑n−1k=0 f (T
n(x)), with:
I T interval exchange transformation,
I f piecewise constant, E(f ) = 0.
I Consider the rescaled r.v.
Xn =Snf√
Var(Snf ).
A B C D
ABD C
T
Bufetov (Annals of Math., ’13) has recently shown that:
I Xn do NOT converge in distribution;
I Consider exponential scales (given 0 < s < 1, consider Yn = X[sn]):
I Accumulation points of Yn contain both Dirac deltas (for dense setof s) and non-degenerate distributions.
I Convergence to a moving distribution (driven by the Teichmueller
geodesic flow, e.g. Ynd→ X (gln sS), S associated surface).
-
Limiting behaviour for random walks driven by IETsI Back to: Snf (x) =
∑n−1k=0 f (T
n(x)), with:
I T interval exchange transformation,
I f piecewise constant, E(f ) = 0.
I Consider the rescaled r.v.
Xn =Snf√
Var(Snf ).
A B C D
ABD C
T
Bufetov (Annals of Math., ’13) has recently shown that:
I Xn do NOT converge in distribution;
I Consider exponential scales (given 0 < s < 1, consider Yn = X[sn]):
I Accumulation points of Yn contain both Dirac deltas (for dense setof s) and non-degenerate distributions.
I Convergence to a moving distribution (driven by the Teichmueller
geodesic flow, e.g. Ynd→ X (gln sS), S associated surface).
-
Limiting behaviour for random walks driven by IETsI Back to: Snf (x) =
∑n−1k=0 f (T
n(x)), with:
I T interval exchange transformation,
I f piecewise constant, E(f ) = 0.
I Consider the rescaled r.v.
Xn =Snf√
Var(Snf ).
A B C D
ABD C
T
Bufetov (Annals of Math., ’13) has recently shown that:
I Xn do NOT converge in distribution;
I Consider exponential scales (given 0 < s < 1, consider Yn = X[sn]):
I Accumulation points of Yn contain both Dirac deltas (for dense setof s) and non-degenerate distributions.
I Convergence to a moving distribution (driven by the Teichmueller
geodesic flow, e.g. Ynd→ X (gln sS), S associated surface).
-
Limiting behaviour for random walks driven by IETsI Back to: Snf (x) =
∑n−1k=0 f (T
n(x)), with:
I T interval exchange transformation,
I f piecewise constant, E(f ) = 0.
I Consider the rescaled r.v.
Xn =Snf√
Var(Snf ).
A B C D
ABD C
T
Bufetov (Annals of Math., ’13) has recently shown that:
I Xn do NOT converge in distribution;
I Consider exponential scales (given 0 < s < 1, consider Yn = X[sn]):
I Accumulation points of Yn contain both Dirac deltas (for dense setof s) and non-degenerate distributions.
I Convergence to a moving distribution (driven by the Teichmueller
geodesic flow, e.g. Ynd→ X (gln sS), S associated surface).
-
Limiting behaviour for random walks driven by IETsI Back to: Snf (x) =
∑n−1k=0 f (T
n(x)), with:
I T interval exchange transformation,
I f piecewise constant, E(f ) = 0.
I Consider the rescaled r.v.
Xn =Snf√
Var(Snf ).
A B C D
ABD C
T
Bufetov (Annals of Math., ’13) has recently shown that:
I Xn do NOT converge in distribution;
I Consider exponential scales (given 0 < s < 1, consider Yn = X[sn]):
I Accumulation points of Yn contain both Dirac deltas (for dense setof s) and non-degenerate distributions.
I Convergence to a moving distribution (driven by the Teichmueller
geodesic flow, e.g. Ynd→ X (gln sS), S associated surface).
-
Limiting behaviour for random walks driven by IETsI Back to: Snf (x) =
∑n−1k=0 f (T
n(x)), with:
I T interval exchange transformation,
I f piecewise constant, E(f ) = 0.
I Consider the rescaled r.v.
Xn =Snf√
Var(Snf ).
A B C D
ABD C
T
Bufetov (Annals of Math., ’13) has recently shown that:
I Xn do NOT converge in distribution;
I Consider exponential scales (given 0 < s < 1, consider Yn = X[sn]):
I Accumulation points of Yn contain both Dirac deltas (for dense setof s) and non-degenerate distributions.
I Convergence to a moving distribution (driven by the Teichmueller
geodesic flow, e.g. Ynd→ X (gln sS), S associated surface).
-
Limiting behaviour for random walks driven by IETsI Back to: Snf (x) =
∑n−1k=0 f (T
n(x)), with:
I T interval exchange transformation,
I f piecewise constant, E(f ) = 0.
I Consider the rescaled r.v.
Xn =Snf√
Var(Snf ).
A B C D
ABD C
T
Bufetov (Annals of Math., ’13) has recently shown that:
I Xn do NOT converge in distribution;
I Consider exponential scales (given 0 < s < 1, consider Yn = X[sn]):
I Accumulation points of Yn contain both Dirac deltas (for dense setof s) and non-degenerate distributions.
I Convergence to a moving distribution (driven by the Teichmueller
geodesic flow, e.g. Ynd→ X (gln sS), S associated surface).
-
Some results on existence of limiting distributions
I Tα(x) = x + α mod 1 rotation;
I f : I → R real valued r.v.;I Xn(x , α) = Snf (x ,Tα) =
∑n−1k=0 f (T
kαx)
as r.v. jointly in x and α.
Theorem (Kesten)f piecewise constant with 2 values, Ef = 0Xn(x ,α)
log nd→ X , with X Cauchy r.v.
E.g. f (x) = χI − |I |
E.g. f (x) = 1x =1
1−x
Theorem (Sinai-U’)If f with symmetric 1x singularities
Xn(x ,α)n
d→ X (limiting distribution) .
Open: Similar limit theorems for T IET? for random θ in Ehrenfest?
-
Some results on existence of limiting distributions
I Tα(x) = x + α mod 1 rotation;
I f : I → R real valued r.v.;I Xn(x , α) = Snf (x ,Tα) =
∑n−1k=0 f (T
kαx)
as r.v. jointly in x and α.
Theorem (Kesten)f piecewise constant with 2 values, Ef = 0Xn(x ,α)
log nd→ X , with X Cauchy r.v.
E.g. f (x) = χI − |I |
E.g. f (x) = 1x =1
1−x
Theorem (Sinai-U’)If f with symmetric 1x singularities
Xn(x ,α)n
d→ X (limiting distribution) .
Open: Similar limit theorems for T IET? for random θ in Ehrenfest?
-
Some results on existence of limiting distributions
I Tα(x) = x + α mod 1 rotation;
I f : I → R real valued r.v.;I Xn(x , α) = Snf (x ,Tα) =
∑n−1k=0 f (T
kαx)
as r.v. jointly in x and α.
Theorem (Kesten)f piecewise constant with 2 values, Ef = 0Xn(x ,α)
log nd→ X , with X Cauchy r.v. E.g. f (x) = χI − |I |
E.g. f (x) = 1x =1
1−x
Theorem (Sinai-U’)If f with symmetric 1x singularities
Xn(x ,α)n
d→ X (limiting distribution) .
Open: Similar limit theorems for T IET? for random θ in Ehrenfest?
-
Some results on existence of limiting distributions
I Tα(x) = x + α mod 1 rotation;
I f : I → R real valued r.v.;I Xn(x , α) = Snf (x ,Tα) =
∑n−1k=0 f (T
kαx)
as r.v. jointly in x and α.
Theorem (Kesten)f piecewise constant with 2 values, Ef = 0Xn(x ,α)
log nd→ X , with X Cauchy r.v. E.g. f (x) = χI − |I |
E.g. f (x) = 1x =1
1−x
Theorem (Sinai-U’)If f with symmetric 1x singularities
Xn(x ,α)n
d→ X (limiting distribution) .
Open: Similar limit theorems for T IET? for random θ in Ehrenfest?
-
Some results on existence of limiting distributions
I Tα(x) = x + α mod 1 rotation;
I f : I → R real valued r.v.;I Xn(x , α) = Snf (x ,Tα) =
∑n−1k=0 f (T
kαx)
as r.v. jointly in x and α.
Theorem (Kesten)f piecewise constant with 2 values, Ef = 0Xn(x ,α)
log nd→ X , with X Cauchy r.v. E.g. f (x) = χI − |I |
E.g. f (x) = 1x =1
1−x
Theorem (Sinai-U’)If f with symmetric 1x singularities
Xn(x ,α)n
d→ X (limiting distribution) .
Open: Similar limit theorems for T IET? for random θ in Ehrenfest?
-
Some results on existence of limiting distributions
I Tα(x) = x + α mod 1 rotation;
I f : I → R real valued r.v.;I Xn(x , α) = Snf (x ,Tα) =
∑n−1k=0 f (T
kαx)
as r.v. jointly in x and α.
Theorem (Kesten)f piecewise constant with 2 values, Ef = 0Xn(x ,α)
log nd→ X , with X Cauchy r.v. E.g. f (x) = χI − |I |
E.g. f (x) = 1x =1
1−x
Theorem (Sinai-U’)If f with symmetric 1x singularities
Xn(x ,α)n
d→ X (limiting distribution) .
Open: Similar limit theorems for T IET? for random θ in Ehrenfest?
-
Some results on existence of limiting distributions
I Tα(x) = x + α mod 1 rotation;
I f : I → R real valued r.v.;I Xn(x , α) = Snf (x ,Tα) =
∑n−1k=0 f (T
kαx)
as r.v. jointly in x and α.
Theorem (Kesten)f piecewise constant with 2 values, Ef = 0Xn(x ,α)
log nd→ X , with X Cauchy r.v. E.g. f (x) = χI − |I |
E.g. f (x) = 1x =1
1−x
Theorem (Sinai-U’)If f with symmetric 1x singularities
Xn(x ,α)n
d→ X (limiting distribution) .
Open: Similar limit theorems for T IET? for random θ in Ehrenfest?
-
Some results on existence of limiting distributions
I Tα(x) = x + α mod 1 rotation;
I f : I → R real valued r.v.;I Xn(x , α) = Snf (x ,Tα) =
∑n−1k=0 f (T
kαx)
as r.v. jointly in x and α.
Theorem (Kesten)f piecewise constant with 2 values, Ef = 0Xn(x ,α)
log nd→ X , with X Cauchy r.v. E.g. f (x) = χI − |I |
E.g. f (x) = 1x =1
1−x
Theorem (Sinai-U’)If f with symmetric 1x singularities
Xn(x ,α)n
d→ X (limiting distribution) .
Open: Similar limit theorems for T IET? for random θ in Ehrenfest?