Diabolical Entropy Neil Dobbs Nicolae Mihalache June 2016 - PowerPoint PPT Presentation

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Diabolical Entropy Neil Dobbs Nicolae Mihalache June 2016 Parameter Problems in Analytic Dynamics

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Topological Entropy and the Quadratic Family f a : x �→ x 2 + a , a ∈ A = [ − 2 , 1 / 4 ] • h ( a ) := lim n →∞ 1 n log # f − n a ( 0 ) . • h ( a ) exists, a �→ h ( a ) is continuous and monotone. [MS, DHS, MT, BvS] • λ ( a ) := lim n →∞ 1 n log | Df n a ( a ) | • If λ ( a 0 ) < 0 , a 0 is hyperbolic, there is a periodic attractor, a �→ h ( a ) is locally constant at a 0 . [LPS] • the hyperbolic set Hyp is open and dense. [G ´ S, Ly] • for almost every a ∈ A , λ ( a ) exists, λ ( a ) � = 0 . [AM,Ly] • for pos. measure set of parameters, λ ( a ) > 0 . [J, BC, AM]

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Topological Entropy and the Quadratic Family f a : x �→ x 2 + a , a ∈ A = [ − 2 , 1 / 4 ] • h ( a ) := lim n →∞ 1 n log # f − n a ( 0 ) . • h ( a ) exists, a �→ h ( a ) is continuous and monotone. [MS, DHS, MT, BvS] • λ ( a ) := lim n →∞ 1 n log | Df n a ( a ) | • If λ ( a 0 ) < 0 , a 0 is hyperbolic, there is a periodic attractor, a �→ h ( a ) is locally constant at a 0 . [LPS] • the hyperbolic set Hyp is open and dense. [G ´ S, Ly] • for almost every a ∈ A , λ ( a ) exists, λ ( a ) � = 0 . [AM,Ly] • for pos. measure set of parameters, λ ( a ) > 0 . [J, BC, AM]

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Regularity of Toplogical Entropy WR ( a ) : lim δ ց 0 lim inf 1 a ( a ) | <δ, j ≤ n log | f j ( a ) | = 0 n log � | f j • Tsujii’s weak regularity condition : "does not return too close, too soon, too often" • W := { a : λ ( a ) exists and λ ( a ) > 0 and WR ( a ) } • full measure in Hyp c [AM,Ly,L,T] • A NLC = { a : { a } = h − 1 ( h ( a )) } – positive measure set Theorem (D, Mihalache) Suppose a ∈ W and { a } = h − 1 ( h ( a )) . Then log | h ( a + t ) − h ( a ) | = h ( a ) lim λ ( a ) . log t t → 0

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Regularity of Toplogical Entropy WR ( a ) : lim δ ց 0 lim inf 1 a ( a ) | <δ, j ≤ n log | f j ( a ) | = 0 n log � | f j • Tsujii’s weak regularity condition : "does not return too close, too soon, too often" • W := { a : λ ( a ) exists and λ ( a ) > 0 and WR ( a ) } • full measure in Hyp c [AM,Ly,L,T] • A NLC = { a : { a } = h − 1 ( h ( a )) } – positive measure set Theorem (D, Mihalache) Suppose a ∈ W and { a } = h − 1 ( h ( a )) . Then log | h ( a + t ) − h ( a ) | = h ( a ) lim λ ( a ) . log t t → 0

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Visible measures of maximal entropy µ acip : absolutely continuous invariant probability measure µ max : measure of maximal entropy � Lyapunov exponent χ ( µ ) := log | Df | d µ. Theorem (D, Mihalache) Let g be a real-analytic unimodal map with non-degenerate critical point. Then µ max = µ acip if and only if g is analytically conjugate to x �→ x 2 − 2 . [Shub-Sullivan, Martens de Melo] Expanding maps : abs cns conjugacy upgrades to smooth/analytic conjugacy. [D] expanding induced map : upgrades to smooth conjugacy on an interval... implies pre-Chebyshev. Analytic conjugacy between renormalised map and x �→ x 2 − 2 , contradiction if renormalised (too many critical points).

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Visible measures of maximal entropy µ acip : absolutely continuous invariant probability measure µ max : measure of maximal entropy � Lyapunov exponent χ ( µ ) := log | Df | d µ. Theorem (D, Mihalache) Let g be a real-analytic unimodal map with non-degenerate critical point. Then µ max = µ acip if and only if g is analytically conjugate to x �→ x 2 − 2 . [Shub-Sullivan, Martens de Melo] Expanding maps : abs cns conjugacy upgrades to smooth/analytic conjugacy. [D] expanding induced map : upgrades to smooth conjugacy on an interval... implies pre-Chebyshev. Analytic conjugacy between renormalised map and x �→ x 2 − 2 , contradiction if renormalised (too many critical points).

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Escalier du diable Corollary a � = − 2 implies h ( a ) h ( a ) acip ) > 1 , max ) < 1 . χ ( µ a χ ( µ a For a � = − 2 χ ( µ acip ) = h ( µ a acip ) < h ( µ a max ) = h ( a ) < χ ( µ a max ) . Corollary Moreover, h ′ ( a ) = 0 almost everywhere. λ ( a ) = χ ( µ a acip ) = h ( µ a For almost every a ∈ W , acip ) [AM]. Thus, almost everywhere, Hölder exponent > 1 . Uniformity ?

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Escalier du diable Corollary a � = − 2 implies h ( a ) h ( a ) acip ) > 1 , max ) < 1 . χ ( µ a χ ( µ a For a � = − 2 χ ( µ acip ) = h ( µ a acip ) < h ( µ a max ) = h ( a ) < χ ( µ a max ) . Corollary Moreover, h ′ ( a ) = 0 almost everywhere. λ ( a ) = χ ( µ a acip ) = h ( µ a For almost every a ∈ W , acip ) [AM]. Thus, almost everywhere, Hölder exponent > 1 . Uniformity ?

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Uniformity Theorem (Misiurewicz Szlenk, Raith, D Todd, D Mihalache) s �→ g s continuous family of S-unimodal maps, each with positive topological entropy. Then s �→ µ g s s �→ χ ( µ g s s �→ h top ( g s ) , max , max ) are continuous. Pressure P s ( t ) = sup µ h ( µ ) − t � log | Dg s | d µ • pressure is analytic on a nbd of zero [based on DT] • pressure functions converge on a nbd of zero [DT] • slope of pressure at zero is − χ ( µ g s max ) • Therefore s �→ χ ( µ g s max ) is continuous.

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Uniformity II Lemma If g k → g 0 , S-unimodal, and h top ( g k ) /χ ( µ g k acip ) → 1 . Then µ g 0 max = µ g 0 acip . Lemma In a neighbourhood of a F (Feigenbaum), there exists ε > 0 with h ( a ) h ( a ) max ) < 1 − ε, acip ) > 1 + ε. χ ( µ a χ ( µ a . • Take a sequence a n converging to a F , f a n is ( m n + 1 ) times Feigenbaum renormalisable. • subsequence of (rescaled) m n -renormalised maps converge to some S-unimodal map g [Sullivan] • by 2nd Theorem, µ max � = µ acip .

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Uniformity III Summing up : Theorem Given ε > 0 , there exists δ > 0 for which • for all a ∈ ( − 2 + ε, a F ) , if µ a acip exists then h ( a ) /χ ( µ a acip ) > 1 + δ • for all a ∈ ( − 2 + ε, a F ) , h ( a ) /χ ( µ a max ) < 1 − δ. Recall first theorem : Suppose a ∈ W and { a } = h − 1 ( h ( a )) . Then log | h ( a + t ) − h ( a ) | = h ( a ) λ ( a ) . lim log t t → 0

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Uniformity and Dimension λ ( a ) = χ ( µ a X ε := { a ∈ A NLC ∩ W : a > − 2 + ε, acip ) } λ ( a ) = χ ( µ a Y ε := { a ∈ A NLC ∩ W : a > − 2 + ε, max ) } . log | h ( a + t ) − h ( a ) | = h ( a ) λ ( a ) . lim log t t → 0 Theorem dim H ( h ( X ε )) < 1 , dim H ( Y ε ) < 1 . • ∪ ε X ε has full measure in A NLC [Avila Moreira Lyubich Levin Tsujii...] • ∪ ε h ( Y ε ) has full measure in [ 0 , log 2 ] . [Bruin Sands]

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Proof of main theorem log | h ( a + t ) − h ( a ) | = h ( a ) λ ( a ) . lim log t t → 0 • h monotone, cns, suffices to prove for t n with log t n / log t n + 1 aribtrarily close to 1 . • Tent map T b : x �→ 1 − b | x | , turning point at 0 , entropy log b • ξ n ( a ) = f n φ n ( b ) = T n a ( a ) , b ( 1 ) 1 • n log | D ξ n ( a + t ) | ≈ λ ( a ) for a subsequence of n , for a neighbourhood which gets mapped to the large scale 1 • n log | D φ n ( b ) | ≈ log b 0 = h ( a ) on corresponding nbds • Use conjugacy with tent map to measure change of entropy • log | t | ≈ − n λ ( a ) , log | h ( a + t ) − h ( a ) | ≈ − nh ( a )

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 Tsujii’s Lemma Lemma Suppose a 0 ∈ W . Let δ > 0 . There exist r 0 > 0 , m ≥ 1 , a sequence ( k n ) n ≥ 0 and decreasing neighbourhoods ω n ∋ a 0 for which k n + 1 • ≤ 1 + δ k n • ξ k n ( ω n ) ⊃ B ( ξ k n ( a 0 ) , r 0 ) • ξ j has bounded distortion on ω n for j = m , m + 1 , . . . , k n 1 • k n | D ξ k n ( a ) | ≈ λ ( a ) . Requires Weak Regularity, Transversality [L], Collet-Eckmann.

Entropy Visible MME Uniformity Main proof Iceland 2 — England 1 HAPPY BIRTHDAY Sebastian