1. No category

Introduction
Categories
Entropy memento
Entropy and invariant measures of
weakly hyperbolic diffeomorphisms
Wandering Seminar, Wroclaw
Lecture 1
J´erˆome Buzzi (CNRS & Universit´e Paris-Sud)
http://jbuzzi.wordpress.com/
[email protected]
April 24, 2015
Exercices
Introduction
Categories
Entropy memento
Exercices
Entropy and invariant measures of diffeomorphisms
Entropy fundamental invariant (Kolmogorov 1958, Shannon 1948)
• counts the orbits
• classifies (Ornstein 1971, Adler-Weiss 1979)
Our goal:
Theorem (Boyle & B 2014)
Any C 1+ -diffeo of a compact surface is conjugate to a Markov
shift, ”up to zero entropy invariant probability measures”
Corollary
In particular, such diffeomorphisms with a unique mixing measure
maximizing the entropy are classified by their topological entropy
Introduction
Categories
Entropy memento
Exercices
Entropy and invariant measures of diffeomorphisms
Our route for the mini-course:
• positive entropy
hyperbolicity in the sense of Pesin
• converse: Katok’s horseshoe theorem
• symbolic model for non-uniformly hyperbolic dynamics:
Markov shifts (Gureviˇc)
• hyperbolicity on surfaces
Markov shift extension of Sarig
• Markov shifts have a universal property by Hochman
If time allows, study of the invariants:
• (non)existence of measures maximizing the entropy
• value of the entropy (continuity, local constancy, formula)
Introduction
Categories
Entropy memento
Plan for lecture 1
• Categories of dynamical systems
• Entropy memento
• Exercises
Exercices
Introduction
Categories
Entropy memento
Exercices
Categories of dynamical systems
Definition
dynamical system (d.s.) = isomorphism of a space up to relevant
identifications.
conjugacy of d.s. (X , S), (Y , T ) = isomorphism of spaces
ψ : X → Y that intertwines the maps: ψ ◦ S = T ◦ ψ
factor = epimorphism of spaces ψ : X → Y that intertwines the
maps.
• smooth: a compact C r manifold;
• topological : a compact metric space (t.d.s.);
• probability: a probability space up to subsets of zero measure
(p.d.s.);
• (standard) Borel: σ-algebra generated by topology defined
by complete distance;
• almost Borel: Borel up to subsets with zero measure wrt any
aperiodic ergodic invariant probability measure
Introduction
Categories
Entropy memento
Categories of dynamical systems
Our focus – interplay between these points of view:
• one can forget
• one can enrich, selecting from:
• P(S, X ) := { S-invariant Borel probability measures of X }
• Perg (S, X ) := {µ ∈ P(S, X ) : µ S-ergodic}
• P0erg (S, X ) := {µ ∈ Perg (S, X ) : µ({periodic points}) = 0}
Theorem (for t.d.s.)
• (Krylov-Bogolioubov) P(T ) and Perg (T ) are not empty
• (Dinaburg, Goodman)
htop (S) = sup h(S, µ) =
sup
µ∈P(S)
µ∈Perg (S)
h(S, µ)
(definitions are on the way...)
More importantly:smoothness constrains entropy of measures
Exercices
Introduction
Categories
Entropy memento
Exercices
Entropy Memento
Definition
The (Kolmogorov-Sinai) entropy of a p.d.s. is
h(T , µ) := supP h(T , µ, P)
Wn−1 −k h(T , µ, P) = limn→∞ n1 Hµ
P
k=0 T
P
Hµ (P) := A∈P −µ(A) log µ(A) (0 log 0 = 0)
The (Borel) entropy is h(T ) := supµ∈Perg (T ) h(T , µ)
Proposition
R
For µ ∈ P(S), Borel d.s., ∃ ergodic decomposition µ = µx dµ(x).
Then: x 7→ h(T , µx ) defined on a Borel set of total µ-measure and:
Z
h(T , µ) = h(T , µx ) dµ(x).
Introduction
Categories
Entropy memento
Entropy Memento
Theorem (Goodman-Dinaburg Variational principle)
For a t.d.s. (X , S), h(S) := supµ∈Perg (S) h(S, µ) is equal to the
topological entropy defined as:
htop (S) := lim→0 h(S, )
htop (S, ) := lim supn→∞ n1 log rS (X , , n)
where
S
rS (Y , , n) := min{#C ⊂ Y : Y ⊂ x∈C BS (x, , n)}
BS (x, , n) := {y ∈ X : ∀0 ≤ k < n d(S k y , S k x) < }.
One can replace lim inf n→∞ by lim supn→∞ .
Theorem (Katok entropy formula)
If (X , S) t.d.s., µ ∈ Perg (S), 0 < t < 1, then:
h(S, µ) = lim→0 ht (S, µ, )
ht (S, µ, ) := lim inf n→∞ n1 log inf µ(Y )>t rS (Y , , n)
One can replace lim inf n→∞ by lim supn→∞ .
Exercices
Introduction
Categories
Entropy memento
Exercices
Entropy Memento
Proof (Misiurewicz) of the variational principle:htop (S) = h(S)
Katok formula
h(S) ≤ htop (S)
Converse: measures equidistributed on minimal (, n)-covers
Remark. Limits of µ may have small entropy!
Definition
µ measure maximizing entropy (m.m.e.) if h(T , µ) = h(T )
Both existence and uniqueness can fail:
Theorem (Misiurewicz 1973)
∀1 ≤ r < ∞ ∃ f , g ∈ Diff r (T4 ):
• f has no m.m.e.
• g has infinitely many m.m.e. (trivial)
Question What about dimension 2?
Theorem (Newhouse 1988)
Any C ∞ -diffeomorphism of a compact manifold has some m.m.e.
Introduction
Categories
Entropy memento
Exercises
Facts
1. For (S, µ) p.d.s., n ∈ Z:
h(S n , µ) = |n| · h(S, µ)
2. If π : (S, µ) → (T , ν) is a factor of p.d.s. h(S, µ) ≥ h(T , ν)
Exercice
Prove them. Extend them to t.d.s. and Borel d.s.
(Ledrappier-Walters variational principle gives a formula for t.d.s.)
Box dimension of compact metric space:
dimB (X ) := lim→0 log N(X , )/|
S log |
where N(X , ) = min{#C : C ⊂ X and x∈C B(x, ) ⊃ X }
Exercice (Kushnirenko 1965)
X is a compact metric with dimB X < ∞
S : X → X Lipschitz
Then htop (S) ≤ dimB X · log+ (Lip(S))
(x + := max(x, 0))
Exercices
Introduction
Categories
Entropy memento
Exercices
Exercises
Exercice
Show htop = 0 for (1) isometries of compact metric spaces; (2)
homeomorphisms of the circle.
(ΣN , σN ) shift on {1, . . . , N}. Prove/recall: htop (ΣN ) := log N
A Misiurewicz (T , N)-horseshoe for f ∈ C 0 (I ):
open ∅ =
6 U1 , . . . , UN ⊂ I s.t.:
∀1 ≤ i 6= j ≤ N Ui ∩ Uj = ∅
∀1 ≤ i, j ≤ N int(f T (Ui )) ⊃ Uj
Exercice
T
Show K := n≥0 g −n (U1 ∪ . . . UN ) is compact g := f T -invariant
Use π : K → ΣN s.t. g n x ∈ Uπ(x)n to show htop (f ) ≥ T1 log N
Generic (residual, fat) = dense Gδ set
(Baire: stable by countable intersection in complete metric spaces)
Exercice (Continued)
Find a generic G ⊂ C 0 (I ) such that, for all g ∈ G, htop (g ) = ∞.