Main idea and Generalizations. Main idea of the proof: Using Markov partitions reduce to a subshift of finite type and establish the theorem there. Then push everything back to the manifold using the semi-conjugacy. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 6/20
Main idea and Generalizations. Main idea of the proof: Using Markov partitions reduce to a subshift of finite type and establish the theorem there. Then push everything back to the manifold using the semi-conjugacy. Several generalizations of the above theorem exists. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 6/20
Main idea and Generalizations. Main idea of the proof: Using Markov partitions reduce to a subshift of finite type and establish the theorem there. Then push everything back to the manifold using the semi-conjugacy. Several generalizations of the above theorem exists. Highlight: Anosov flows. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 6/20
Main idea and Generalizations. Main idea of the proof: Using Markov partitions reduce to a subshift of finite type and establish the theorem there. Then push everything back to the manifold using the semi-conjugacy. Several generalizations of the above theorem exists. Highlight: Anosov flows. Some hyperbolicity is usually required, either for f of for the potential (see for example Y. Lima’s course next week). P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 6/20
Main idea and Generalizations. Main idea of the proof: Using Markov partitions reduce to a subshift of finite type and establish the theorem there. Then push everything back to the manifold using the semi-conjugacy. Several generalizations of the above theorem exists. Highlight: Anosov flows. Some hyperbolicity is usually required, either for f of for the potential (see for example Y. Lima’s course next week). We discuss now one important example where the available methods fail. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 6/20
Diagonal actions (on locally homogeneous spaces). P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 7/20
Diagonal actions (on locally homogeneous spaces). In the course of Mohammadi it’s been discussed diagonal actions on quotients of SL ( n , R ). P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 7/20
Diagonal actions (on locally homogeneous spaces). In the course of Mohammadi it’s been discussed diagonal actions on quotients of SL ( n , R ). G = Sl (3 , R ) . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 7/20
Diagonal actions (on locally homogeneous spaces). In the course of Mohammadi it’s been discussed diagonal actions on quotients of SL ( n , R ). G = Sl (3 , R ) . A = { diag ( e a , e b , e c ) : a + b + c = 0 } < G P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 7/20
Diagonal actions (on locally homogeneous spaces). In the course of Mohammadi it’s been discussed diagonal actions on quotients of SL ( n , R ). G = Sl (3 , R ) . A = { diag ( e a , e b , e c ) : a + b + c = 0 } < G Γ < G co-compact torsion free lattice. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 7/20
Diagonal actions (on locally homogeneous spaces). In the course of Mohammadi it’s been discussed diagonal actions on quotients of SL ( n , R ). G = Sl (3 , R ) . A = { diag ( e a , e b , e c ) : a + b + c = 0 } < G Γ < G co-compact torsion free lattice. α : A � M = Sl (3 , R ) / Γ P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 7/20
Diagonal actions (on locally homogeneous spaces). In the course of Mohammadi it’s been discussed diagonal actions on quotients of SL ( n , R ). G = Sl (3 , R ) . A = { diag ( e a , e b , e c ) : a + b + c = 0 } < G Γ < G co-compact torsion free lattice. α : A � M = Sl (3 , R ) / Γ is an Anosov action P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 7/20
Diagonal actions (on locally homogeneous spaces). In the course of Mohammadi it’s been discussed diagonal actions on quotients of SL ( n , R ). G = Sl (3 , R ) . A = { diag ( e a , e b , e c ) : a + b + c = 0 } < G Γ < G co-compact torsion free lattice. α : A � M = Sl (3 , R ) / Γ is an Anosov action, meaning there exists an element f = α ( g ) having an invariant splitting TM = E s ⊕ E c ⊕ E u P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 7/20
Diagonal actions (on locally homogeneous spaces). In the course of Mohammadi it’s been discussed diagonal actions on quotients of SL ( n , R ). G = Sl (3 , R ) . A = { diag ( e a , e b , e c ) : a + b + c = 0 } < G Γ < G co-compact torsion free lattice. α : A � M = Sl (3 , R ) / Γ is an Anosov action, meaning there exists an element f = α ( g ) having an invariant splitting TM = E s ⊕ E c ⊕ E u such that (for some metric) df | E u expansion, df | E s contraction and E c is tangent to the orbits of the action. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 7/20
Center isometries. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 8/20
Center isometries. The metric in M can be chosen so that f acts as an isometry on the orbit foliation. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 8/20
Center isometries. The metric in M can be chosen so that f acts as an isometry on the orbit foliation. Hence, f is a center isometry (the definition is nearly evident). P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 8/20
Center isometries. The metric in M can be chosen so that f acts as an isometry on the orbit foliation. Hence, f is a center isometry (the definition is nearly evident). Remark: no hyperbolicity whatsoever along the center. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 8/20
Center isometries. The metric in M can be chosen so that f acts as an isometry on the orbit foliation. Hence, f is a center isometry (the definition is nearly evident). Remark: no hyperbolicity whatsoever along the center. For a center isometry all bundles E s , E u , E c , E cs = E c ⊕ E s , E cu = E c ⊕ E u are integrable to f -invariant foliations W ∗ . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 8/20
Main Theorems. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 9/20
Main Theorems. f : M → M center isometry of class C 2 s.t P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 9/20
Main Theorems. f : M → M center isometry of class C 2 s.t W s , W u are minimal. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 9/20
Main Theorems. f : M → M center isometry of class C 2 s.t W s , W u are minimal. ϕ : M → R H¨ older potential. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 9/20
Main Theorems. f : M → M center isometry of class C 2 s.t W s , W u are minimal. ϕ : M → R H¨ older potential. Theorem A [P.C., F. Rodriguez-Hertz] There exist µ ϕ ∈ P f ( M ) and families of measures µ u = { µ u x } x ∈ M , µ s = { µ s x } x ∈ M , µ cu = { µ cu x } x ∈ M , µ cs = { µ cs x } x ∈ M satisfying the following. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 9/20
Main Theorems. f : M → M center isometry of class C 2 s.t W s , W u are minimal. ϕ : M → R H¨ older potential. Theorem A [P.C., F. Rodriguez-Hertz] There exist µ ϕ ∈ P f ( M ) and families of measures µ u = { µ u x } x ∈ M , µ s = { µ s x } x ∈ M , µ cu = { µ cu x } x ∈ M , µ cs = { µ cs x } x ∈ M satisfying the following. 1. The probability µ ϕ is an equilibrium state for the potential ϕ . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 9/20
Main Theorems. f : M → M center isometry of class C 2 s.t W s , W u are minimal. ϕ : M → R H¨ older potential. Theorem A [P.C., F. Rodriguez-Hertz] There exist µ ϕ ∈ P f ( M ) and families of measures µ u = { µ u x } x ∈ M , µ s = { µ s x } x ∈ M , µ cu = { µ cu x } x ∈ M , µ cs = { µ cs x } x ∈ M satisfying the following. 1. The probability µ ϕ is an equilibrium state for the potential ϕ . 2. For every x ∈ M the measure µ σ , σ ∈ { u , s , cu , cs } is a Radon measure on W σ ( x ) which is positive on relatively open sets, and y ∈ W σ ( x ) implies µ σ x = µ σ y . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 9/20
Cont. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 10/20
Cont. 3. If ξ is a measurable partition that refines the partition by unstable (stable) leaves then the conditionals ( µ ϕ ) ξ x of µ ϕ are equivalent to µ u x (resp. µ s x ) for µ ϕ − a . e . ( x ). P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 10/20
Cont. 3. If ξ is a measurable partition that refines the partition by unstable (stable) leaves then the conditionals ( µ ϕ ) ξ x of µ ϕ are equivalent to µ u x (resp. µ s x ) for µ ϕ − a . e . ( x ). 4. For every ǫ > 0 sufficiently small, for every x ∈ M the measure µ ϕ | D ( x ; ǫ ) has product structure with respect to the pair µ u x , µ cs x , i.e. its equivalent to µ u x × µ cs x . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 10/20
Cont. 3. If ξ is a measurable partition that refines the partition by unstable (stable) leaves then the conditionals ( µ ϕ ) ξ x of µ ϕ are equivalent to µ u x (resp. µ s x ) for µ ϕ − a . e . ( x ). 4. For every ǫ > 0 sufficiently small, for every x ∈ M the measure µ ϕ | D ( x ; ǫ ) has product structure with respect to the pair µ u x , µ cs x , i.e. its equivalent to µ u x × µ cs x . 5. Given ǫ > 0 there exist a ( ǫ ) , b ( ǫ ) > 0 such that if U ( x , ǫ, n ) = { y ∈ W u ( x , ǫ ) : d ( f j x , f j y ) < ǫ, j = 0 , . . . , n − 1 } then a ( ǫ ) ≤ µ u x ( U ( x , ǫ, n ))) e S n ϕ ( x ) − nP top ( ϕ ) ≤ b ( ǫ ) . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 10/20
Bernoulli property and uniqueness. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 11/20
Bernoulli property and uniqueness. Theorem B [P.C., F. Rodriguez-Hertz] Under certain technical conditions the system ( f , µ ϕ ) is metrically isomorphic to a Bernoulli shift. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 11/20
Bernoulli property and uniqueness. Theorem B [P.C., F. Rodriguez-Hertz] Under certain technical conditions the system ( f , µ ϕ ) is metrically isomorphic to a Bernoulli shift. Theorem C [P.C., F. Rodriguez-Hertz] If either dimE s , dimE u = 1 , or P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 11/20
Bernoulli property and uniqueness. Theorem B [P.C., F. Rodriguez-Hertz] Under certain technical conditions the system ( f , µ ϕ ) is metrically isomorphic to a Bernoulli shift. Theorem C [P.C., F. Rodriguez-Hertz] If either dimE s , dimE u = 1 , or f is an ergodic automorphisms of T N (no repeated eigenvalues), P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 11/20
Bernoulli property and uniqueness. Theorem B [P.C., F. Rodriguez-Hertz] Under certain technical conditions the system ( f , µ ϕ ) is metrically isomorphic to a Bernoulli shift. Theorem C [P.C., F. Rodriguez-Hertz] If either dimE s , dimE u = 1 , or f is an ergodic automorphisms of T N (no repeated eigenvalues), the equilibrium state µ ϕ is unique. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 11/20
Bernoulli property and uniqueness. Theorem B [P.C., F. Rodriguez-Hertz] Under certain technical conditions the system ( f , µ ϕ ) is metrically isomorphic to a Bernoulli shift. Theorem C [P.C., F. Rodriguez-Hertz] If either dimE s , dimE u = 1 , or f is an ergodic automorphisms of T N (no repeated eigenvalues), the equilibrium state µ ϕ is unique. Work in progress: Uniqueness also holds in the homogeneous examples (Weyl Chambers’ flow). P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 11/20
SRB measures. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 12/20
SRB measures. In the general setting, an SRB measure is a invariant measure whose unstable conditionals are absolutely continuous with respect to Lebesgue. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 12/20
SRB measures. In the general setting, an SRB measure is a invariant measure whose unstable conditionals are absolutely continuous with respect to Lebesgue. We keep working with center isometries. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 12/20
SRB measures. In the general setting, an SRB measure is a invariant measure whose unstable conditionals are absolutely continuous with respect to Lebesgue. We keep working with center isometries. • SRB measures exist (Sinai-Pesin). P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 12/20
SRB measures. In the general setting, an SRB measure is a invariant measure whose unstable conditionals are absolutely continuous with respect to Lebesgue. We keep working with center isometries. • SRB measures exist (Sinai-Pesin). • Ledrappier-Young: µ is an SRB if and only if � log J u ( x ) d µ ( x ) J u ( x ) = det ( df | E u h µ ( f ) = x ) P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 12/20
SRB measures. In the general setting, an SRB measure is a invariant measure whose unstable conditionals are absolutely continuous with respect to Lebesgue. We keep working with center isometries. • SRB measures exist (Sinai-Pesin). • Ledrappier-Young: µ is an SRB if and only if � log J u ( x ) d µ ( x ) J u ( x ) = det ( df | E u h µ ( f ) = x ) Implicit: Unstable manifolds coincide with Pesin’s unstable manifolds. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 12/20
Eq. states are determined by their unstable conditionals. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 13/20
Eq. states are determined by their unstable conditionals. Theorem D [P.C., F. Rodriguez-Hertz] If µ is an equilibrium state for ϕ then µ has conditionals along unstables equivalent to µ u . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 13/20
Eq. states are determined by their unstable conditionals. Theorem D [P.C., F. Rodriguez-Hertz] If µ is an equilibrium state for ϕ then µ has conditionals along unstables equivalent to µ u . Similarly it has conditionals along stables equivalent to µ s . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 13/20
Eq. states are determined by their unstable conditionals. Theorem D [P.C., F. Rodriguez-Hertz] If µ is an equilibrium state for ϕ then µ has conditionals along unstables equivalent to µ u . Similarly it has conditionals along stables equivalent to µ s . Conversely if µ ∈ P f ( M ) has unstable conditionals absolutely continuous wrt µ u , then µ is an equilibrium state for ϕ . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 13/20
Eq. states are determined by their unstable conditionals. Theorem D [P.C., F. Rodriguez-Hertz] If µ is an equilibrium state for ϕ then µ has conditionals along unstables equivalent to µ u . Similarly it has conditionals along stables equivalent to µ s . Conversely if µ ∈ P f ( M ) has unstable conditionals absolutely continuous wrt µ u , then µ is an equilibrium state for ϕ . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 13/20
Eq. states are determined by their unstable conditionals. Theorem D [P.C., F. Rodriguez-Hertz] If µ is an equilibrium state for ϕ then µ has conditionals along unstables equivalent to µ u . Similarly it has conditionals along stables equivalent to µ s . Conversely if µ ∈ P f ( M ) has unstable conditionals absolutely continuous wrt µ u , then µ is an equilibrium state for ϕ . The families µ u , µ s provide the reference measures to which one can compare. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 13/20
Sketch of the proof: Thm D. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 14/20
Sketch of the proof: Thm D. Definition We call a measurable partition ξ a SPLY partition if P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 14/20
Sketch of the proof: Thm D. Definition We call a measurable partition ξ a SPLY partition if f ξ < ξ . ξ subordinated to W u µ -a.e. x the atom ξ ( x ) contains a neighbourhood of x inside W u ( x ). P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 14/20
Sketch of the proof: Thm D. Definition We call a measurable partition ξ a SPLY partition if f ξ < ξ . ξ subordinated to W u µ -a.e. x the atom ξ ( x ) contains a neighbourhood of x inside W u ( x ). SPLY partitions exist: (Sinai, Pesin - Ledrappier, Strelcyn). P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 14/20
Sketch of the proof: Thm D. Definition We call a measurable partition ξ a SPLY partition if f ξ < ξ . ξ subordinated to W u µ -a.e. x the atom ξ ( x ) contains a neighbourhood of x inside W u ( x ). SPLY partitions exist: (Sinai, Pesin - Ledrappier, Strelcyn). We fix one SPLY partition and take m ∈ P f ( M ) equilibrium state for ϕ ; P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 14/20
Sketch of the proof: Thm D. Definition We call a measurable partition ξ a SPLY partition if f ξ < ξ . ξ subordinated to W u µ -a.e. x the atom ξ ( x ) contains a neighbourhood of x inside W u ( x ). SPLY partitions exist: (Sinai, Pesin - Ledrappier, Strelcyn). We fix one SPLY partition and take m ∈ P f ( M ) equilibrium state for ϕ ; denote m x = unstable conditional of m on ξ ( x ). P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 14/20
’Change of variables’ for µ u P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 15/20
’Change of variables’ for µ u Theorem For every x ∈ M 1. µ σ fx = e P top ( ϕ ) − ϕ f ∗ µ σ σ ∈ { u , cu } . x 2. µ σ fx = e ϕ − P top ( ϕ ) f ∗ µ σ σ ∈ { s , cs } . x P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 15/20
Cont. proof P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 16/20
Cont. proof If m x << µ u x m − a . e . ( x ) we can write dm x = ρ d µ u x where ρ is measurable on M . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 16/20
Cont. proof If m x << µ u x m − a . e . ( x ) we can write dm x = ρ d µ u x where ρ is measurable on M . One can then show ρ ( x ) ρ ( f − 1 x ) e P − ϕ ( f − 1 x ) x �→ is constant on the atoms of ξ . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 16/20
Cont. proof If m x << µ u x m − a . e . ( x ) we can write dm x = ρ d µ u x where ρ is measurable on M . One can then show ρ ( x ) ρ ( f − 1 x ) e P − ϕ ( f − 1 x ) x �→ is constant on the atoms of ξ . From here one deduces that ∞ e ϕ ◦ f − k ( y ) y ∈ ξ ( x ) ⇒ ρ ( y ) � ρ ( x ) = e ϕ ◦ f − k ( x ) = ∆ x ( y ) . k =1 P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 16/20
Cont. proof P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 17/20
Cont. proof ⇒ ρ (provided is defined) should have the form ρ ( y ) = ∆ x ( y ) L ( x ) , y ∈ ξ ( x ) ξ ( x ) ∆ x ( y ) d µ u with L ( x ) = � x ( y ). P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 17/20
Cont. proof ⇒ ρ (provided is defined) should have the form ρ ( y ) = ∆ x ( y ) L ( x ) , y ∈ ξ ( x ) ξ ( x ) ∆ x ( y ) d µ u with L ( x ) = � x ( y ). Define a measure ν by requiring ν = m on B ξ and such its conditionals on ξ are given by d ν x = ρ d µ x . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 17/20
Cont. proof ⇒ ρ (provided is defined) should have the form ρ ( y ) = ∆ x ( y ) L ( x ) , y ∈ ξ ( x ) ξ ( x ) ∆ x ( y ) d µ u with L ( x ) = � x ( y ). Define a measure ν by requiring ν = m on B ξ and such its conditionals on ξ are given by d ν x = ρ d µ x . We want to prove m = ν . P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 17/20
Cont. proof ⇒ ρ (provided is defined) should have the form ρ ( y ) = ∆ x ( y ) L ( x ) , y ∈ ξ ( x ) ξ ( x ) ∆ x ( y ) d µ u with L ( x ) = � x ( y ). Define a measure ν by requiring ν = m on B ξ and such its conditionals on ξ are given by d ν x = ρ d µ x . We want to prove m = ν . It is enough to show m = ν on every B f − n ξ , n ≥ 0. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 17/20
Cont. proof ⇒ ρ (provided is defined) should have the form ρ ( y ) = ∆ x ( y ) L ( x ) , y ∈ ξ ( x ) ξ ( x ) ∆ x ( y ) d µ u with L ( x ) = � x ( y ). Define a measure ν by requiring ν = m on B ξ and such its conditionals on ξ are given by d ν x = ρ d µ x . We want to prove m = ν . It is enough to show m = ν on every B f − n ξ , n ≥ 0. An induction argument shows that after proving m = ν on B f − 1 ξ we are done. P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 17/20
Cont. proof q ( x ) = ν x ( f − 1 ξ ( x )) ⇒ P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 18/20
Cont. proof q ( x ) = ν x ( f − 1 ξ ( x )) ⇒ e P − ϕ ( y ) 1 � ∆ x ( f − 1 fy ) e P − ϕ ( f − 1 fy ) d µ u q ( x ) = x L ( x ) f − 1 ( ξ ( fx )) 1 fx ( z ) = L ( fx ) � L ( x ) e ϕ ( x ) − P ≤ 1 ∆ x ( f − 1 z ) e ϕ ( f − 1 z ) − P d µ u = L ( x ) ξ ( fx ) P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 18/20
Cont. proof q ( x ) = ν x ( f − 1 ξ ( x )) ⇒ e P − ϕ ( y ) 1 � ∆ x ( f − 1 fy ) e P − ϕ ( f − 1 fy ) d µ u q ( x ) = x L ( x ) f − 1 ( ξ ( fx )) 1 fx ( z ) = L ( fx ) � L ( x ) e ϕ ( x ) − P ≤ 1 ∆ x ( f − 1 z ) e ϕ ( f − 1 z ) − P d µ u = L ( x ) ξ ( fx ) L ( fx ) L ( x ) ≤ e ϕ ( x ) − P ∈ L 1 ( M , m ) P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 18/20
Cont. proof q ( x ) = ν x ( f − 1 ξ ( x )) ⇒ e P − ϕ ( y ) 1 � ∆ x ( f − 1 fy ) e P − ϕ ( f − 1 fy ) d µ u q ( x ) = x L ( x ) f − 1 ( ξ ( fx )) 1 fx ( z ) = L ( fx ) � L ( x ) e ϕ ( x ) − P ≤ 1 ∆ x ( f − 1 z ) e ϕ ( f − 1 z ) − P d µ u = L ( x ) ξ ( fx ) L ( fx ) L ( x ) ≤ e ϕ ( x ) − P ∈ L 1 ( M , m ) � log L ◦ f � � dm = 0 ⇒ − log q ( x ) dm ( x ) = P − ϕ dm . L P. Carrasco, ICMC-USP visiting ICTP Equilibrium States are determined by their unstable conditionals. 18/20
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