Specification and thermodynamical properties for semigroups actions - PowerPoint PPT Presentation

Specification and thermodynamical properties for semigroups actions Paulo Varandas Federal University of Bahia (joint with F. Rodrigues - UFRGS) AMS/EMS/SPM Meeting - Porto, July 2015

Topological pressure for individual dynamics Classical results

Topological pressure for individual dynamics Classical results M compact metric space f : M ! M continuous ' : M ! R continuous potential The topological pressure 1 X e S n ' ( x ) } P top ( f , ' ) = lim " → 0 lim sup n log { sup n →∞ E n x ∈ E n • measures ’weighted complexity’ on the space of orbits • satisfies a variational principle Z � P top ( f , ' ) = sup h µ ( f ) + ' d µ µ ∈ M 1 ( f ) • (in some cases) measures exponential growth rate of weighted periodic points 1 X e S n ' ( x ) � � P top ( f , ' ) = lim sup n log n →∞ f n ( x )= x

Thermodynamical formalism for individual dynamics Classical results [Adler, Konheim, McAndrew 65’] Definition of topological entropy [Ruelle 68’] The pressure function � 7! P top ( f , ' + � ) is analytic [Bowen 71’] Specification ) Positive entropy [Ruelle 73’ Walters 75’] Variational principle for continuous maps [Parry 64’ Bowen 71’, 74’] Specification & expansiveness ) 9 ! equilibrium state µ ' for every H¨ older potential ' , obtained as weak ∗ -limit of 1 X e S n ' ( x ) � x X e S n ' ( x ) with Z n = Z n x ∈ Per n ( f ) x ∈ Per n ( f ) and 1 h top ( f ) = lim n log ] Per n ( f ) n →∞

Specification for individual dynamics

Specification for individual dynamics

Specification for individual dynamics

Specification for individual dynamics

Specification for individual dynamics

Specification for individual dynamics: precise definition A continuous map f : X ! X satisfies the specification property if for any � > 0 there exists an integer p ( � ) � 1 such that the following holds: for every k � 1, any points x 1 , . . . , x k , and any sequence of positive integers n 1 , . . . , n k and p 1 , . . . , p k with p i � p ( � ) there exists a point x in X such that ⇣ ⌘ f j ( x ) , f j ( x 1 )  � , 8 0  j  n 1 d and ⇣ ⌘ f j + n 1 + p 1 + ··· + n i − 1 + p i − 1 ( x ) , f j ( x i )  � d for every 2  i  k and 0  j  n i .

More general dynamical systems Motivated by the number of di ff erent applications the following classes of dynamical systems have been intensively studied: 1. Non-autonomous / sequential dynamical systems 2. Iterated function systems (IFS) 3. Group and semigroup actions

Sequential dynamical systems Non-autonomous (or sequential) dynamical systems F = ( f k ) k ≥ 1 F n = f n � · · · � f 2 � f 1 for n � 1 Some di ffi culties include: • non-stationarity (no common invariant measures!) • omega-limit sets are not necessarily invariant sets • ’periodic points’ defined by truncating dynamics ’Topological & probabilistic complexity’

Finitely generated (semi)groups ( G , � ) finitely generated (semi)group G 1 = { id , g 1 , g 2 , . . . , g m } generators & G = S n ∈ N 0 G n g 2 G n if and only if g = g i n . . . g i 2 g i 1 with g i j 2 G 1 (concatenations of at most n elements of G 1 ) ( G , � ) is a group ( G , � ) is a semigroup generators G 1 = { id , g ± 1 , g ± 2 , . . . , g ± m } generators G 1 = { id , g 1 , g 2 , . . . , g m } ( G n ) n ∈ N increasing family in G ( G n ) n ∈ N may be non-increasing d G ( h , g ) := | h − 1 g | distance no natural distance Notation: G ∗ 1 = G 1 \ { id } and G ∗ n = { g = g i n . . . g i 2 g i 1 with g i j 2 G ∗ 1 }

Continuous semigroup actions We say that T : G ⇥ X ! X is a continuous semigroup action on a topological space X if: 1. For every g 2 G the map g ⌘ T g : X ! X is continuous 2. ( gh ) x = g ( hx ) for every g , h 2 G and x 2 X The orbit of x 2 X is the set O T ( x ) = { gx : g 2 G } . x 2 X is ’periodic point’ (period n ) if g n ( x ) = x for some g n 2 G n Per ( G ) = S n ≥ 1 Per ( G n ) set of periodic orbits.

Continuous semigroup actions We say that T : G ⇥ X ! X is a continuous semigroup action on a topological space X if: 1. For every g 2 G the map g ⌘ T g : X ! X is continuous 2. ( gh ) x = g ( hx ) for every g , h 2 G and x 2 X The orbit of x 2 X is the set O T ( x ) = { gx : g 2 G } . x 2 X is ’periodic point’ (period n ) if g n ( x ) = x for some g n 2 G n Per ( G ) = S n ≥ 1 Per ( G n ) set of periodic orbits. Questions: i. Are there natural notions of complexity? ii. Can it be computed using periodic points/loops? iii. Does local complexity propagate?

Motivational example: geodesics and moving billiards table f : S 1 ! S 1 be smooth expanding map (Bowen-Series map) R ↵ : S 1 ! S 1 rotation angle ↵ G semigroup generated by G 1 = { id , f , R ↵ }

Coding: the semigroups G and T ( G )  C ( X , X ) Bijection Z + ⇥ Z 4 7! h g 1 ( x ) = R π 4 ( x ) , g 2 ( x ) = 4 x ( mod 1) i Non-injective Z 2 + 7! h g 1 ( x ) = 2 x ( mod 1) , g 2 ( x ) = 4 x ( mod 1) i

Coding: the semigroups G and T ( G )  C ( X , X ) Bijection F 2 (free group) 7! h g 1 , g 2 i Anosov di ff eos g 2 / 2 Z ( g 1 )

Topological pressure for (semi)group actions Some (di ff erent) notions and contributions: [Ruelle 73’] 9 > > [Ghys, Langevin, Walczak 88’] > > > > [Friedland 95’] > > > > [Bufetov 99’] Some of these notions require = [Lind, Schmidt 02’] abelianity or amenability > > [Bis 08’, 13’ ] > > > > [Ma, Wu 11’] > > > > [Miles, Ward 11’] ;

Di ff erent flavours [Ruelle 73’] Z d -expansive actions with (very strong) specification .

Di ff erent flavours [Ghys, Langevin, Walczak 88’] Entropy for pseudo-groups and foliations .

Di ff erent flavours [Bufetov 99’] Entropy free semigroup actions . . . .

Three concepts: topological pressure & entropy points & orbital specification I.1 Topological pressure: ⇣ 1 1 n X P n − 1 i =0 ϕ ( g i ( x )) o⌘ X P top (( G , G 1 ) , ' , E ) := lim ε → 0 lim sup n log sup e m n n →∞ F | g | = n x ∈ F supremum over all ( g , n , " )-separated sets F = F g , n , " ⇢ E ⇣ 1 1 ⌘ X h top (( G , G 1 ) , E ) := lim ε → 0 lim sup n log s ( g , n , " ) m n n →∞ | g | = n I.2 Entropy: 1 h (( G , G 1 ) , E ) = lim ε → 0 lim sup n log s ( n , " , E ) n →∞ where s ( n , " ) is maximal cardinality of ( n , " )-separated sets in E . Entropy taking the compact set E = X .

Three concepts: topological pressure & entropy points & orbital specification Simple illustration: g 1 : S 1 ! S 1 g 2 ( x ) = 2 x ( mod 1) g 2 : S 1 ! S 1 g 2 ( x ) = 3 x ( mod 1) g 3 : S 1 ! S 1 g 3 ( x ) = 5 x ( mod 1) I.1 Topological pressure: ⇣ 1 1 = log(10 ⌘ X h top (( G , G 1 ) , S 1 ) = lim ε → 0 lim sup n log s ( g , n , " ) 3 ) 3 n n →∞ | g | = n I.2 Entropy: 1 h (( G , G 1 ) , S 1 ) = lim n log s ( n , " , S 1 ) = log 3 ε → 0 lim sup n →∞

Three concepts: topological pressure & entropy points & orbital specification II.1 Entropy point x 0 2 X is an entropy point for h top (( G , G 1 ) , · ) if h top (( G , G 1 ) , U ) = h top (( G , G 1 ) , X ) for any open nhood U of x 0 II.2 Entropy point x 0 2 X is an entropy point for h (( G , G 1 ) , · ) if h (( G , G 1 ) , U ) = h (( G , G 1 ) , X ) for any open nhood U of x 0 Rmk: II.2 was introduced by [Bis 13’] which proved that the set of entropy points is non-emtpy provided X is compact.

Three concepts: topological pressure & entropy points & orbital specification III.1 Orbital specification

Three concepts: topological pressure & entropy points & orbital specification III.1 Orbital specification Rmk 1: Similar notion is studied on the space of push-forwards Rmk 2: Each element in G ∗ 1 must satisfy specification

Three concepts: topological pressure & entropy points & orbital specification III.2 Weak orbital specification

Three concepts: topological pressure & entropy points & orbital specification III.2 Weak orbital specification Rmk 3: Other notions of specification for semigroups / groups can be defined similarly (not needed for this talk!)

Three concepts: topological pressure & entropy points & orbital specification T : G ⇥ X ! X satisfies the weak orbital specification property if: for any " > 0 there exists p ( " ) > 0 so that for any p � p ( " ), there ] ˜ exists a set ˜ G p G p ⇢ G ∗ p satisfying lim p →∞ p = 1 for which: for any ] G ∗ h p j 2 ˜ G p j with p j � p ( " ), any points x 1 , . . . , x k 2 X , any natural numbers n 1 , . . . , n k and any concatenations g n j , j = g i nj , j . . . g i 2 , j g i 1 , j 2 G n j with 1  j  k there exists x 2 X so that d ( g ` , 1 ( x ) , g ` , 1 ( x 1 )) < " for every ` = 1 . . . n 1 and d ( g ` , j h p j − 1 . . . g n 2 , 2 h p 1 g n 1 , 1 ( x ) , g ` , j ( x j ) ) < " for every j = 2 . . . k and ` = 1 . . . n j .

Main Results