Finiteness of matrix equilibrium states Jairo Bochi (Pontifical - PowerPoint PPT Presentation

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Finiteness of matrix equilibrium states Jairo Bochi (Pontifical Catholic University of Chile) Webminar New Trends in Lyapunov exponents July 7, 2020 https://cemapre.iseg.ulisboa.pt/events/event.php?id=197

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Main reference This talk is based on the paper: J. B., Ian D. Morris. Equilibrium states of generalised singular value potentials and applications to affine iterated function systems. Geometric and Functional Analysis , 28 (2018), no. 4, pp. 995–1028. http://dx.doi.org/10.1007/s00039-018-0447-x

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Subadditive thermodynamical formalism

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Subadditive pressure Given: a compact metric space X ; a continuous map T : X → X ; a sequence F = ( f n ) n ≥ 1 of continuous functions f n : X → [ − ∞ , + ∞ ) which is subadditive : f n + m ≤ f n ◦ T m + f m . Define the (topological) pressure : 1 � e f n ( x ) . P ( F ) := lim ϵ → 0 limsup sup log n n → ∞ E ⊆ X x ∈ E ( n,ϵ ) -separated

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Subadditive ergodic theorem � � M T ( X ) := T -invariant probability measures . A Borel set B ⊆ X has full probability if μ ( B ) = 1 , ∀ μ ∈ M T ( X ) . Kingman’68: If F = ( f n ) is a (say, continuous) subadd. seq. then the asymptotic average f n ( x ) ¯ f ( x ) := lim exists for all x in a full probability set. n → ∞ n Furthermore, for all μ ∈ M T , � f n � ¯ f dμ = lim dμ . n → ∞ n ���� inf

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Subadditive variational principle Subadditive variational principle : If ¯ f is the asymptotic avg. of the subadd. seq. F , then: � � � ¯ (if h top ( T ) < ∞). P ( F ) = sup h μ ( T ) + f dμ μ ∈ M T ( X ) (Cao–Feng–Huang’08; related work by Falconer’88, Barreira’96, Kaënmäki’04, Mummert’06. ) An equilibrium state is an invariant measure μ that attains the sup. As in the classical (additive) setting, (ergodic) equilibrium states exist provided the metric entropy is upper-semicontinuous (e.g. T expansive or C ∞ ).

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends An easy remark: If ( f n ) and ( g n ) are two subadditive sequences, then we can construct a third one by: � � h n ( x ) := max f n ( x ) , g n ( x ) . Asymptotic average: � ¯ � ¯ f ( x ) , ¯ h ( x ) = max g ( x ) . If μ is ergodic, then � �� � � ¯ ¯ ¯ hdμ = max f dμ, gdμ . So: {erg. equil. states for ( h n ) } ⊆ {erg. equil. states for ( f n ) } ∪ {erg. equil. states for ( g n ) } .

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Locally constant case Consider: ( X, T ) = (Σ N , σ ) = one-sided full shift on the alphabet {1 , . . . , N }. a subadditive sequence F = ( f n ) s.t. each f n is constant on the cylinders of depth n . Equivalently, for every word w f n | [ w ] ≡ log Φ( w ) , where Φ: Σ ∗ N → [ 0 , + ∞ ) is a submultiplicative potential , i.e., a function on the set Σ ∗ N of words s.t. ∀ w , v ∈ Σ ∗ Φ( wv ) ≤ Φ( w )Φ( v ) . N , Rem.: Even if the subadd. seq. ( f n ) (for the shift) is not loc. const., we can still define a submult. potential Φ( w ) := expsup [ w ] f | w | .

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends In the locally constant situation f n | [ w ] = log Φ( w ) , the pressure has a simpler expression: 1 � P (Φ) = lim log Φ( w ) n → ∞ n w ∈ Σ ∗ N | w | = n That is, � Φ( w ) = e nP (Φ)+ o ( n ) w ∈ Σ ∗ N | w | = n

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Semi- and Quasi-multiplicativity Let Φ: Σ ∗ N → [ 0 , + ∞ ) be a submultiplicative potential. Φ is semimultiplicative if there exists c ∈ ( 0 , 1 ] such that for every pair of words w , v , Φ( wv ) ≥ c Φ( w )Φ( v ) . Example: Norm potential under 1-domination hypothesis. Φ is quasimultiplicative if there exists c ∈ ( 0 , 1 ] and ℓ ∈ N such that for every pair of words w , v there exists a word u of length | u | ≤ ℓ such that: Φ( wuv ) ≥ c Φ( w )Φ( v ) . Example: Norm potential under irreducibility hypothesis (more about this later).

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Consequences of quasimultiplicativity Theorem (Feng’11) Every quasimultiplicative potential Φ has a unique equilibrium state μ, which is ergodic, and satisfies Gibbs inequalities : there exists C > 0 such that for every cylinder [ w ] ⊆ Σ N , C − 1 Φ( w ) e −| w | P (Φ) ≤ μ ([ w ]) ≤ C Φ( w ) e −| w | P (Φ) . In particular, μ has full support.

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Singular values, Lyapunov exponents

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Singular values A linear map L : R d → R d has singular values s 1 ( L ) ≥ · · · ≥ s d ( L ) . So s 1 ( L ) = � L � , s d ( L ) = “co-norm”. Ellipsoid L ( S d − 1 ) . d k ) , Λ k L : Λ k R d → Λ k R d . Exterior powers: Λ k R d = R ( � Λ k ( L ) � = biggest expansion rate of k -volume = s 1 ( L ) · · · s k ( L ) . Submultiplicativity: � Λ k ( L 1 L 2 ) � ≤ � Λ k ( L 1 ) � � Λ k ( L 2 ) � .

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Lyapunov exponents Given a continuous matrix cocycle A : X → Mat ( d × d ) , we form the products: A ( n ) ( x ) := A ( T n − 1 x ) · · · A ( Tx ) A ( x ) . For all x on a full probability set, the Lyapunov exponents 1 log s k ( A ( n ) ( x )) λ 1 ( x ) ≥ · · · ≥ λ d ( x ) , λ k ( x ) := lim n → ∞ n exist. Indeed, for each k , λ 1 + · · · + λ k is the asymptotic average of the following subadditive sequence: k � f n , k ( x ) := log � Λ k ( A ( n ) ( x )) � = log s i ( A ( n ) ( x )) . i = 1

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Singular value potential Given an ordered vector α = ( α 1 , . . . , α d ) ∈ R d � with α 1 ≥ · · · ≥ α d , the function d � s i ( L ) α i α : Mat ( d × d ) → [ 0 , + ∞ ) , α ( L ) := φ � φ � i = 1 α ( L 1 L 2 ) ≤ φ � is submultiplicative: φ � α ( L 1 ) φ � α ( L 2 ) . Indeed:   d − 1 � � Λ i L � α i − α i + 1 � Λ d L � α d α ( L ) = φ �   � �� � � �� � i = 1 submult. multiplicative

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Singular value pressure Given the cocycle ( T, A ) and a ordered vector α = ( α 1 , . . . , α d ) ∈ R d � consider the subadditive sequence � d � s α i α ( A ( n ) ( x )) , α ( x ) := log φ � α := f n , � where φ � i i = 1 The corresponding pressure is: � � � P ( A , � ( α 1 λ 1 + · · · + α d λ d ) d μ α ) = sup h μ ( T ) + μ ∈ M T ( X ) (provided h top ( T ) < ∞).

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Some subjects that we won’t go into: Continuity of the singular value pressure: See Feng–Shmerkin’14, Morris’16, Cao–Pesin–Zhao’19. Differentiability of the singular value pressure. Multifractal analysis of Lyapunov exponents: Given a linear cocycle, how big (in terms of topological entropy) is the set of points with a given Lyapunov spectrum? See Feng–Huang’10. Transfer operators and applications: See Guivarc’h–LePage’04, Piraino (ETDS, to appear).

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Applications to dimension theory

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Contracting IFS Consider an IFS (iterated function system) specified by contractions T 1 , . . . , T N : R d → R d . Its attractor is the unique nonempty compact set Λ ⊆ R d such that N � Λ = T i (Λ) . i = 1 In fact, Λ consists of all limits A non-conformal Sier- i → ∞ T i n ◦ · · · ◦ T i 1 ( x ) . lim pi´ nski gasket. If the contractions T i are affine then Λ is called a self-affine set .

Opening Subadd. therm. form. Lyapunov exp. Dim. Finiteness New trends Dimension estimate Let Λ ⊆ R d be a self-affine set. Let A be the loc. const. cocycle induced by the linear parts of the contractions. The unique root s of the “Bowen-like” equation � � P ( A, � α ( s )) = 0 where � α ( s ) := 1 , . . . , 1 , s − ⌊ s ⌋ , 0 , . . . , 0 � �� � ⌊ s ⌋ is called affinity dimension of the IFS. Theorem (Falconer’88) dim H (Λ) ≤ dim aff ( T 1 , . . . , T N ) . Rem.: A corresponding equilibrium states on Σ N project to measures on Λ which are natural candidates for measure of maximal dimension (Käenmäki’04).