Unique equilibrium states for geodesic flows in nonpositive curvature Todd Fisher Department of Mathematics Brigham Young University Fractal Geometry, Hyperbolic Dynamics and Thermodynamical Formalism Joint work with K. Burns, V. Climenhaga, and D. Thompson Todd FIsher (BYU) Equilibrium States 2016
Outline Introduction 1 Surfaces with nonpositive curvature 2 Climenhaga-Thompson program 3 Decomposition for a surface of nonpositive curvature 4 Todd FIsher (BYU) Equilibrium States 2016
Topological entropy F = f t : X → X a smooth flow on a compact manifold Bowen ball : B T ( x ; ǫ ) = { y : d ( f t y , f t x ) < ǫ for 0 ≤ t ≤ T } x 1 , . . . , x n are ( T , ǫ )-spanning if � n i =1 B T ( x i ; ǫ ) = X Todd FIsher (BYU) Equilibrium States 2016
Topological entropy F = f t : X → X a smooth flow on a compact manifold Bowen ball : B T ( x ; ǫ ) = { y : d ( f t y , f t x ) < ǫ for 0 ≤ t ≤ T } x 1 , . . . , x n are ( T , ǫ )-spanning if � n i =1 B T ( x i ; ǫ ) = X Λ span ( T , ǫ ) = inf { #( E ) | E ⊂ X is ( T , ǫ )-spanning } 1 T log Λ span ( T , ǫ ) h top ( F ) = lim ǫ → 0 lim sup T →∞ Remark: An equivalent definition is h top ( F ) = h top ( f 1 ) where the second term is the entropy of the time-1 map. Todd FIsher (BYU) Equilibrium States 2016
Measure entropy and the variational principle For a flow F = ( f t ) t ∈ R let M ( f t ) be the set of f t -invariant Borel probability measures and M ( F ) = � t ∈ R M ( f t ) be the set of flow invariant Borel probability measures. For µ ∈ M ( F ) the measure theoretic entropy of F for µ is h µ ( F ) = h µ ( f 1 ). ( Variational Principle ) h top ( F ) = sup µ ∈M ( F ) h µ ( F ) Todd FIsher (BYU) Equilibrium States 2016
Topological pressure ϕ : X → R a continuous function. We will refer to this as a potential function or observable. Λ span ( ϕ ; T , ǫ ) = � T �� 0 ϕ ( f t x ) dt | E ⊂ X is ( T , ǫ )-spanning � inf x ∈ E e 1 P ( ϕ, F ) = lim T log Λ span ( ϕ ; T , ǫ ) ǫ → 0 lim sup T →∞ Remark: When ϕ ≡ 0, P ( ϕ, F ) = h top ( F ) Todd FIsher (BYU) Equilibrium States 2016
The variational principle for pressure Let µ ∈ M ( F ). Then � P µ ( ϕ, F ) = h µ ( F ) + ϕ d µ ( Variational Principle for Pressure ) P ( ϕ, F ) = sup P µ ( ϕ, F ) µ ∈M ( F ) µ ∈ M ( F ) is an equilibrium state for ϕ if P µ ( ϕ, F ) = P ( ϕ, F ). If the flow is C ∞ there is an equilibrium state for any continuous potential (Newhouse: upper semi continuity of µ �→ h µ , and the set of measures is compact) Todd FIsher (BYU) Equilibrium States 2016
Geodesic flow M compact Riemannian manifold with negative sectional curvatures. v Todd FIsher (BYU) Equilibrium States 2016
Geodesic flow M compact Riemannian manifold with negative sectional curvatures. v f (v) t Let T 1 M be the unit tangent bundle. f t : T 1 M → T 1 M geodesic flow Todd FIsher (BYU) Equilibrium States 2016
Properties of geodesic flows for negative curvature The flow is Anosov ( TT 1 M = E s ⊕ E c ⊕ E u ) where the splitting is given by the flow direction ( E c ) and the tangent spaces to the stable and unstable horospheres ( E s and E u ), and the flow is volume preserving (Liouville measure) v cu W (v) cs W (v ) Todd FIsher (BYU) Equilibrium States 2016
Properties of geodesic flows for negative curvature The flow is Anosov ( TT 1 M = E s ⊕ E c ⊕ E u ) where the splitting is given by the flow direction ( E c ) and the tangent spaces to the stable and unstable horospheres ( E s and E u ), and the flow is volume preserving (Liouville measure) v cu W (v) cs W (v ) Bowen: any H¨ older continuous potential ϕ has a unique equilibrium state, and the geometric potential ϕ u = − lim t → 0 1 t log Jac ( Df t | E u ) has a unique equilibrium state Todd FIsher (BYU) Equilibrium States 2016
Outline Introduction 1 Surfaces with nonpositive curvature 2 Climenhaga-Thompson program 3 Decomposition for a surface of nonpositive curvature 4 Todd FIsher (BYU) Equilibrium States 2016
Nonpositive curvature M compact manifold with nonpositive curvature f t : T 1 M → T 1 M geodesic flow. There are still foliations globally defined that are similar to stable and unstable foliations. Two types of geodesic: regular: horospheres in M do not make second order contact singular: horospheres in M make second order contact (stable and unstable foliations are tangent) Todd FIsher (BYU) Equilibrium States 2016
Nonpositive curvature M compact manifold with nonpositive curvature f t : T 1 M → T 1 M geodesic flow. There are still foliations globally defined that are similar to stable and unstable foliations. Two types of geodesic: regular: horospheres in M do not make second order contact singular: horospheres in M make second order contact (stable and unstable foliations are tangent) For a surface: regular = geodesic passes through negative curvature singular = curvature is zero everywhere along the geodesic Todd FIsher (BYU) Equilibrium States 2016
Rank of the manifold A Jacobi field along a geodesic γ is a vector field along γ that satisfied the equation J ′′ ( t ) + R ( J ( t ) , ˙ γ ( t ))˙ γ ( t ) = 0 The rank of a vector v ∈ T 1 M is the dimension of the space of parallel Jacobi vector fields on the geodesic γ v : R → M . The rank of M is the minimum rank over all vectors in T 1 M (always at least 1). Standing Assumption: M is a compact rank 1 manifold with nonpositive curvature. (Rules out manifolds such as the torus.) Todd FIsher (BYU) Equilibrium States 2016
Example ———– singular geodesic ———– regular geodesic Todd FIsher (BYU) Equilibrium States 2016
Previous result Reg = set of vectors in T 1 M whose geodesics are regular, and Sing = set of vectors in T 1 M whose geodesics are singular. T 1 M = Reg ∪ Sing Reg is open and dense, and geodesic flow is ergodic on Reg. Open problem: What is Liouville measure of Reg? Todd FIsher (BYU) Equilibrium States 2016
Previous result Reg = set of vectors in T 1 M whose geodesics are regular, and Sing = set of vectors in T 1 M whose geodesics are singular. T 1 M = Reg ∪ Sing Reg is open and dense, and geodesic flow is ergodic on Reg. Open problem: What is Liouville measure of Reg? (Knieper, 98) There is a unique measure of maximal entropy. It is supported on Reg. For surfaces h top ( Sing , F ) = 0. Generally, h top ( Sing , F ) < h top ( F ) Todd FIsher (BYU) Equilibrium States 2016
Main results Assume: M is a compact rank 1 manifold with nonpositive curvature and Sing � = ∅ . The next results follow from a general result that is stated later. Todd FIsher (BYU) Equilibrium States 2016
Main results Assume: M is a compact rank 1 manifold with nonpositive curvature and Sing � = ∅ . The next results follow from a general result that is stated later. Theorem (Burns, Climenhaga, F, Thompson) If ϕ : T 1 M → (0 , h top ( F ) − h top ( Sing )) is H¨ older continuous, then ϕ has a unique equilibrium state. Todd FIsher (BYU) Equilibrium States 2016
Main results Assume: M is a compact rank 1 manifold with nonpositive curvature and Sing � = ∅ . The next results follow from a general result that is stated later. Theorem (Burns, Climenhaga, F, Thompson) If ϕ : T 1 M → (0 , h top ( F ) − h top ( Sing )) is H¨ older continuous, then ϕ has a unique equilibrium state. Theorem (Burns, Climenhaga, F, Thompson) If h top ( Sing ) = 0 , then q ϕ u has a unique equilibrium state for each q < 1 and the graph of q �→ P ( q ϕ u ) has a tangent line for all q < 1 . Todd FIsher (BYU) Equilibrium States 2016
Main results Assume: M is a compact rank 1 manifold with nonpositive curvature and Sing � = ∅ . The next results follow from a general result that is stated later. Theorem (Burns, Climenhaga, F, Thompson) If ϕ : T 1 M → (0 , h top ( F ) − h top ( Sing )) is H¨ older continuous, then ϕ has a unique equilibrium state. Theorem (Burns, Climenhaga, F, Thompson) If h top ( Sing ) = 0 , then q ϕ u has a unique equilibrium state for each q < 1 and the graph of q �→ P ( q ϕ u ) has a tangent line for all q < 1 . Theorem (Burns, Climenhaga, F, Thompson) There is q 0 > 0 such that if q ∈ ( − q 0 , q 0 ) , then the potential q ϕ u has a unique equilibrium state. Todd FIsher (BYU) Equilibrium States 2016
P ( q ϕ u ) for a surface with Sing � = ∅ The graph of q �→ P ( − q ϕ u ) has a corner at (1 , 0) created by µ Liou and measures supported on Sing. Todd FIsher (BYU) Equilibrium States 2016
Outline Introduction 1 Surfaces with nonpositive curvature 2 Climenhaga-Thompson program 3 Decomposition for a surface of nonpositive curvature 4 Todd FIsher (BYU) Equilibrium States 2016
Bowen’s result There is a unique equilibrium state for ϕ if F is expansive: For every c > 0 there is some ǫ > 0 such that d ( f t x , f s ( t ) y ) < ǫ for all t ∈ R all x , y ∈ R and a continuous function s : R → R with s (0) = 0 if y = f γ x for some γ ∈ [ − c , c ]. F has specification: next slide ϕ has the Bowen property: there exist K > 0 and ǫ > 0 such that for any T > 0, if d ( f t x , f t y ) < ǫ for 0 ≤ t ≤ T , then � T � T � � � � ϕ ( f t x ) dt − ϕ ( f t y ) dt � < K . � � � 0 0 Todd FIsher (BYU) Equilibrium States 2016
Recommend
More recommend
