Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Unique equilibrium states for geodesic flows in
nonpositive curvature
Keith Burns
Northwestern University
School and Conference on Dynamical Systems
ICTP Trieste
Joint work with V. Climenhaga, T. Fisher, and
D. Thompson
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Outline
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Topological entropy
�
F = ft : X → X smooth flow on a compact manifold
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Topological entropy
�
�
F = ft : X → X smooth flow on a compact manifold
Bowen ball : BT (x; �) = {y : d(ft y , ft x) < � for 0 ≤ t ≤ T }
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Topological entropy
�
�
�
F = ft : X → X smooth flow on a compact manifold
Bowen ball : BT (x; �) = {y : d(ft y , ft x) < � for 0 ≤ t ≤ T }
�
x1 , . . . , xn are (T , �)-spanning if ni=1 BT (xi ; �) = X
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Topological entropy
�
�
�
�
F = ft : X → X smooth flow on a compact manifold
Bowen ball : BT (x; �) = {y : d(ft y , ft x) < � for 0 ≤ t ≤ T }
�
x1 , . . . , xn are (T , �)-spanning if ni=1 BT (xi ; �) = X
x1 , . . . , xn are (T , 2�)-separated if BT (xi ; �) ∩ BT (xj ; �) �= ∅
for i �= j
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Topological entropy
�
�
�
�
�
F = ft : X → X smooth flow on a compact manifold
Bowen ball : BT (x; �) = {y : d(ft y , ft x) < � for 0 ≤ t ≤ T }
�
x1 , . . . , xn are (T , �)-spanning if ni=1 BT (xi ; �) = X
x1 , . . . , xn are (T , 2�)-separated if BT (xi ; �) ∩ BT (xj ; �) �= ∅
for i �= j
Λsep (T , �) = sup {#(E ) | E ⊂ X is (T , �)-separated}
Λspan (T , �) = inf {#(E ) | E ⊂ X is (T , �)-spanning}
1
log Λsep (T , �)
T
1
= lim lim sup log Λspan (T , �)
�→0 T →∞ T
htop (F) = lim lim sup
�→0 T →∞
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Measure entropy and the variational principle
�
Let µ be a probability measure that is ergodic for the flow
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Measure entropy and the variational principle
�
�
Let µ be a probability measure that is ergodic for the flow
Λsep
µ (T , �) =
sup {#(E ) | E ⊂ X is (T , �)-separated and covers measure ≤ 1/2}
Λspan
(T , �) =
µ
inf {#(E ) | E ⊂ X is (T , �)-spanning and covers measure ≥ 1/2}
1
log Λsep
µ (T , �)
�→0 T →∞ T
1
(T , �)
= lim lim sup log Λspan
µ
�→0 T →∞ T
hµ = lim lim sup
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Measure entropy and the variational principle
�
�
Let µ be a probability measure that is ergodic for the flow
Λsep
µ (T , �) =
sup {#(E ) | E ⊂ X is (T , �)-separated and covers measure ≤ 1/2}
Λspan
(T , �) =
µ
inf {#(E ) | E ⊂ X is (T , �)-spanning and covers measure ≥ 1/2}
1
log Λsep
µ (T , �)
�→0 T →∞ T
1
(T , �)
= lim lim sup log Λspan
µ
�→0 T →∞ T
hµ = lim lim sup
�
htop (F) = supµ∈Erg hµ
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Topological pressure
�
ϕ : X → R a continuous potential function
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Topological pressure
�
�
ϕ : X → R a continuous potential function
Λsep (ϕ; T�, �) =
�
�T
�
0 ϕ(ft x) dt | E ⊂ X is (T , �)-separated
sup
e
x∈E
Λspan (ϕ;�T , �) =
�
�T
�
0 ϕ(ft x) dt | E ⊂ X is (T , �)-spanning
inf
e
x∈E
1
log Λsep (ϕ; T , �)
�→0 T →∞ T
1
= lim lim sup log Λspan (ϕ; T , �)
�→0 T →∞ T
P(ϕ) = lim lim sup
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
The variational principle for pressure
�
Let µ be a probability measure that is ergodic for the flow
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
The variational principle for pressure
�
Let µ be a probability measure that is ergodic for the flow
�
Pµ (ϕ) = hµ +
�
ϕ dµ
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
The variational principle for pressure
�
Let µ be a probability measure that is ergodic for the flow
�
Pµ (ϕ) = hµ +
�
P(ϕ) = supµ∈Erg Pµ (ϕ)
�
µ ∈ Erg is an equilibrium state for ϕ if Pµ (ϕ) = P(ϕ).
�
ϕ dµ
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
The variational principle for pressure
�
Let µ be a probability measure that is ergodic for the flow
�
Pµ (ϕ) = hµ +
�
P(ϕ) = supµ∈Erg Pµ (ϕ)
�
µ ∈ Erg is an equilibrium state for ϕ if Pµ (ϕ) = P(ϕ).
�
If the flow is C ∞ there is an equilibrium state for any
continuous potential (Newhouse: upper semi continuity of
µ �→ hµ )
�
ϕ dµ
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Geodesic flow in negative curvature
M compact Riemannian manifold with negative sectional
curvatures.
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Geodesic flow in negative curvature
M compact Riemannian manifold with negative sectional
curvatures.
ft : T 1 M → T 1 M geodesic flow
Anosov and volume preserving (Liouville measure)
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Geodesic flow in negative curvature
M compact Riemannian manifold with negative sectional
curvatures.
ft : T 1 M → T 1 M geodesic flow
Anosov and volume preserving (Liouville measure)
Bowen: any Hölder continuous potential ϕ has a unique
equilibrium state
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Geodesic flow in negative curvature
M compact Riemannian manifold with negative sectional
curvatures.
ft : T 1 M → T 1 M geodesic flow
Anosov and volume preserving (Liouville measure)
Bowen: any Hölder continuous potential ϕ has a unique
equilibrium state
Geometric potential ϕu = limt→0 1t log �Dft |E u �
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Pressure of −qϕu
Pµ (−qϕu ) = hµ − q
Keith Burns
Equilibrium States
�
ϕu dµ
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Outline
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Nonpositive curvature
M compact manifold with nonpositive curvature
ft : T 1 M → T 1 M geodesic flow
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Nonpositive curvature
M compact manifold with nonpositive curvature
ft : T 1 M → T 1 M geodesic flow
Two types of geodesic:
singular: horospheres make second order contact
regular: horospheres do not make second order contact
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Nonpositive curvature
M compact manifold with nonpositive curvature
ft : T 1 M → T 1 M geodesic flow
Two types of geodesic:
singular: horospheres make second order contact
regular: horospheres do not make second order contact
For a surface: regular = geodesic passes through negative
curvature
singular = curvature is zero everywhere along the geodesic
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Previous results
T 1 M = Reg ∪ Sing
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Previous results
T 1 M = Reg ∪ Sing
Reg is open and dense. Geodesic flow is ergodic on Reg.
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Previous results
T 1 M = Reg ∪ Sing
Reg is open and dense. Geodesic flow is ergodic on Reg.
Open problem: What is Liouville measure of Reg?
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Previous results
T 1 M = Reg ∪ Sing
Reg is open and dense. Geodesic flow is ergodic on Reg.
Open problem: What is Liouville measure of Reg?
Knieper: There is a unique measure of maximal entropy. It is
supported on Reg.
Obtained by Patterson-Sullivan construction or as limit of uniform
measures on closed regular orbits.
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Previous results
T 1 M = Reg ∪ Sing
Reg is open and dense. Geodesic flow is ergodic on Reg.
Open problem: What is Liouville measure of Reg?
Knieper: There is a unique measure of maximal entropy. It is
supported on Reg.
Obtained by Patterson-Sullivan construction or as limit of uniform
measures on closed regular orbits.
Entropy on Sing is zero for sufaces, < htop (F) in general
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
P(−qϕu ) for a surface with Sing �= ∅
The graph of q �→ P(−qϕu ) has a corner at (1, 0) created by µLiou
and measures supported on Sing.
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
P(−qϕu ) for a surface with Sing �= ∅
The graph of q �→ P(−qϕu ) has a corner at (1, 0) created by µLiou
and measures supported on Sing.
KB, Climenhaga, Fisher, Thompson: The graph has a tangent line
at all other points.
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
P(−qϕu ) for a surface with Sing �= ∅
The graph of q �→ P(−qϕu ) has a corner at (1, 0) created by µLiou
and measures supported on Sing.
KB, Climenhaga, Fisher, Thompson: The graph has a tangent line
at all other points.
In fact: −qϕu has a unique equilibrium state for each q < 1.
Related result: If ϕ : T 1 M → (0, htop (F)) is Hölder continuous,
then ϕ has a unique equilibrium state.
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Outline
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Bowen’s result
There is a unique equilibrium state for ϕ if
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Bowen’s result
There is a unique equilibrium state for ϕ if
�
F is expansive
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Bowen’s result
There is a unique equilibrium state for ϕ if
�
F is expansive
�
F has specification
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Bowen’s result
There is a unique equilibrium state for ϕ if
�
F is expansive
�
F has specification
�
ϕ has the Bowen property: there exist K > 0 and � > 0 such
that for any T > 0, if d(ft x, ft y ) < � for 0 ≤ t ≤ T , then
�� T
�
� T
�
�
�
ϕ(ft x)dt −
ϕ(ft y )dt �� < K .
�
0
0
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Decomposing orbit segments
F = ft flow on X
O = X × [0, ∞) = {finite orbit segments}
∗ = concatenation of orbit segments
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Decomposing orbit segments
F = ft flow on X
O = X × [0, ∞) = {finite orbit segments}
∗ = concatenation of orbit segments
Three subsets of O: P, G, S
O =P ∗G∗S
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Outline theorem
Suppose we have:
�
expansivity and specification for orbit segments in G
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Outline theorem
Suppose we have:
�
expansivity and specification for orbit segments in G
�
ϕ has the Bowen property on segments in G
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Outline theorem
Suppose we have:
�
expansivity and specification for orbit segments in G
�
ϕ has the Bowen property on segments in G
�
P(ϕ; P ∪ S) < P(ϕ)
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Outline theorem
Suppose we have:
�
expansivity and specification for orbit segments in G
�
ϕ has the Bowen property on segments in G
�
P(ϕ; P ∪ S) < P(ϕ)
Then ϕ has a unique equilibrium state
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Outline theorem
Suppose we have:
�
expansivity and specification for orbit segments in G
�
ϕ has the Bowen property on segments in G
�
P(ϕ; P ∪ S) < P(ϕ)
Then ϕ has a unique equilibrium state
Climenhaga-Thompson: symbolic systems such as β-shifts
Climenhaga-Fisher-Thompson: partially hyperbolic examples
Climenhaga-Thompson: Flow version of the theorem
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Outline
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
κ(v ) = minimium curvature of the two horocycles othogonal to v
(v , t) is δ, η-bad if Leb{s ∈ [0, t] : κ(fs (v )) ≥ η} ≤ δt.
P = S = {(v , t) ∈ O : (v , t) is δ, η-bad}
To decompose (v , t) ∈ O:
�
Find longest initial segment that is P
�
Find longest tail segment of the remainder that is P
�
What is left is in G
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Keith Burns
Equilibrium States
Introduction
Surfaces with nonpositive curvature
The Climinhaga-Thompson program
Decomposition for a surface of nonpositive curvature
Thanks!
Keith Burns
Equilibrium States