Quantitative equidistribution in non-archimedean and complex - PowerPoint PPT Presentation

Quantitative equidistribution in non-archimedean and complex dynamics Yˆ usuke Okuyama (Kyoto Inst. Tech., okuyama@kit.ac.jp) Complex and p -adic Dynamics; ICERN, Brown University 13 February, 2012

§ Berkovich projective line: Notation K : algebraically closed field, complete WRT a non-trivial absolute value | · | . (either non-archimedean or archimedean. e.g. C p , C u , C ) P 1 = P 1 ( K ) : (classical) projective line [ · , · ] : the normalized chordal distance on P 1 P 1 = P 1 ( K ) : Berkovich projective line, compactifying P 1 (Fact: For archimedean K , P 1 � P 1 ) H 1 : = P 1 \ P 1 : endowed with the hyperbolic metric ρ δ ( · , · ) can : the generalized Hsia kernel on P 1 WRT S can ∈ H 1 . 1

§ Gauss variational approach to dynamics A rational function of degree d > 1 f : P 1 → P 1 . (Fact: this extends to P 1 → P 1 , f ( H 1 ) = H 1 , conti, surj, open, discrete) ∃ 1 (non-degenerate homogeneous polynomial) lift of f F : K 2 → K 2 (upto × c ∈ K ∗ ), i.e. for canonical projection π : K 2 → P 1 ( K ) , π ◦ F = f ◦ π and the homogeneous resultant Res F does not vanish. 2

Def . The dynamical Green function on P 1 ∞ ( 1 ) 1 ∑ d j ( f j ) ∗ d log | F | − log | · | g F : = j = 0 (for ∀ c ∈ K ∗ , g cF = g F + (log | c | ) / ( d − 1) ). An upper semicontinuous F -kernel on P 1 Φ F ( z , w ) : = log δ ( z , w ) can − g F ( z ) − g F ( w ) . The F -energy of a Radon measure µ on P 1 (if exists) ∫ I F ( µ ) : = P 1 × P 1 Φ F ( z , w )d( µ × µ )( z , w ) . 3

The F -equilibrium energy of (the whole) P 1 V F : = sup { I F ( µ ); µ is a prob. Radon measure on P 1 } > −∞ . A possible definition of the canonical measure µ f is Thm . There is the unique solution of Gauss variational problem WRT external field g F . Concretely, ∃ 1 probability Radon measure µ f on P 1 s.t. I F ( µ f ) = V F . (Rem: µ f is independent of choices of F ) 4

§ Fekete configuration in dynamics Now we can be more canonical: the f -kernel on P 1 Φ f ( · , · ) : = Φ F ( · , · ) − V F , − Φ f is called the independent of choices of F . (Rem: Arakelov Green (kernel) function of f on P 1 ) Def . A sequence ( ν n ) of positive discrete measures on P 1 is f -asymptotically Fekete on P 1 if as n → ∞ , ν n ( P 1 ) ր ∞ , ( ν n × ν n )(diag P 1 ) = o ( ν n ( P 1 ) 2 ) , ∫ 1 Φ f d( ν n × ν n ) → 0 . ν n ( P 1 ) 2 P 1 × P 1 \ diag P 1 5

(Rem: this is an analogue of Gauss variational problem for positive discrete measures. Φ f ( S , S ) > 0 if S ∈ H 1 .) Def . The averaged pullback of a ∈ P 1 ∑ ( f n ) ∗ ( a ) : = deg w ( f n ) · ( a ) w ∈ f − n ( a ) ( ( a ) : the Dirac measure at a on P 1 ). The algebraic exceptional set of f (Rem: this is in P 1 ) ∪ E ( f ) : = { a ∈ P 1 ; # f − n ( a ) < ∞} . n ∈ N SAT ( f ) : superattracting periodic points of f 6

Def (main quantity) . For each a ∈ P 1 and each n ∈ N , ∫ E f ( n , a ) : = 1 Φ f d(( f n ) ∗ ( a ) × ( f n ) ∗ ( a )) d 2 n P 1 × P 1 \ diag 1 P ( ( f n ) ∗ ( a ) − µ f , ( f n ) ∗ ( a ) ) = − − µ f d n d n f (: the dyn version of Favre and Rivera-Letelier’s energy). (Fact) Then • For ∀ a ∈ P 1 \ E ( f ) , (( f n ) ∗ ( a )) is f -asymp Fekete on P 1 ⇔ lim n →∞ E f ( n , a ) = 0 . • For ∀ a ∈ E ( f ) , (( f n ) ∗ ( a )) is NEVER f -asymp Fekete on P 1 . 7

Another fundamental quantity Def . For ∀ a ∈ P 1 \ E ( f ) , η a , n = η a , n ( f ) : = w ∈ f − n ( a ) deg w ( f ) ∈ N . max Rem: if K has characteristic 0 , then  ≤ ( d 3 − 1) 1 / 3 ( a ∈ P 1 \ E ( f )) ,   η 1 / j   lim sup (1)  a , j = d ( a ∈ E ( f )) ,  j →∞    ( a ∈ P 1 \ SAT ( f )) , ≤ d 2 d − 2     sup η a , j (2)  = ∞ ( a ∈ SAT ( f )) .  j ∈ N   8

§ Main results: error estimates on Fekete Let f be a rat function on P 1 = P 1 ( K ) of degree d > 1 . Put C ( f ) : = { c ∈ P 1 ; f ′ ( c ) = 0 } , C ( f ) wan : = { c ∈ C ( f ); ( f n ( c )) is wandering under f } , CO( f ) wan : = { f n ( c ); c ∈ C ( f ) wan , n ∈ N } . (Rem: if f has char 0 , then ∑ c ∈ C ( f ) (deg c f − 1) = 2 d − 2 .) Thm 1 (principal estimates) . For ∀ a ∈ H 1 and ∀ n ∈ N , |E f ( n , a ) | ≤ Cd − n (3) for some C > 0 indep of n and loc bounded on a under ρ . 9

(cont.) If in addition K has char 0 , then there is C ′ > 0 s.t. for ∀ a ∈ P 1 and ∀ n ∈ N , n n − C ′ η a , j − C a − 1 1 1 ∑ ∑ ∑ η a , j d j log d n d n d n [ f j ( c ) , a ] c ∈ C ( f ) \ f − j ( a ) j = 1 j = 1 ≤E f ( n , a ) (4) n n + C ′ η a , j + C a ≤ − 1 1 1 ∑ ∑ ∑ d j log d n . d n d n [ f j ( c ) , a ] c ∈ C ( f ) \ f − j ( a ) j = 1 j = 1 Here the constant C a ≥ 0 , which is independent of n , vanishes if a ∈ P 1 \ CO( f ) wan . 10

Def . The classical omega limit set of each z 0 ∈ P 1 chordal ∩ { f n ( z 0 ); n ≥ N } ω ( z 0 ) = ω ( z 0 ) : = . N ∈ N A point z 0 ∈ P 1 is pre-recurrent if ∃ n 0 ∈ N , f n 0 ( z 0 ) ∈ ω ( z 0 ) . (The chordal open ball with center w ∈ P 1 and radius r > 0 B [ w, r ] : = { z ∈ P 1 ; [ z , w ] < r } ) Thm 1 estimates the non-Fekete locus E Fekete ( f ) : = { a ∈ P 1 ; (( f n ) ∗ ( a )) is not f -asymp Fekete on P 1 } 11

from above using ∩ ∪ ∪ B [ f j ( c ) , exp( − d j )] . E wan ( f ) : = N ∈ N j ≥ N c ∈ C ( f ) wan Thm 2. Suppose K has characteristic 0 . Then E ( f ) ⊂ E Fekete ( f ) ⊂ P 1 , E Fekete ( f ) \ E ( f ) ⊂ E wan ( f ) \ E ( f ) , and E wan ( f ) is of capacity 0 . ( finite Hyllengren meas for ( d j )) . Moreover, E Fekete ( f ) is G δ -dense in ω ( c ) for every pre- recurrent c ∈ C ( f ) wan . ( so, possibly E ( f ) � E Fekete ( f )) 12

§ Application: quantitative equidistribution / K Let f be a rational function on P 1 = P 1 ( K ) of degree d > 1 . Favre and Rivera-Letelier’s Cauchy-Schwarz inequality is Prop . For ∀ a ∈ P 1 , C 1 -test function ∀ φ on P 1 and ∀ n ∈ N , φ, ( f n ) ∗ ( a ) � �� � � � � � − µ f � � d n � � � �  � φ, φ � 1 / 2 √ ( a ∈ H 1 ) , |E f ( n , a ) |     ≤ √  C max { Lip( φ ) , � φ, φ � 1 / 2 } ( a ∈ P 1 ) . |E f ( n , a ) | + nd − n η a , n    Here C > 0 is independent of a ∈ P 1 , φ and n . 13

Theorem 1 establishes a quantitative equidistribution in terms of the proximity of wandering crit orbits to a ∈ P 1 . Thm 3 (Special case) . Suppose that K has characteristic 0 . Then there is C > 0 s.t. for ∀ a ∈ P 1 excluding E wan ( f ) of capacity 0 and ∀ n ∈ N large enough, |E f ( n , a ) | ≤ Cnd − n η a , n , (5) and there is C ′ > 0 s.t. for C 1 -test function ∀ φ on P 1 , ∀ a ∈ P 1 \ E wan ( f ) and ∀ n ∈ N large enough, φ, ( f n ) ∗ ( a ) � �� � √ � � � ≤ C ′ max { Lip( φ ) , � φ, φ � 1 / 2 } nd − n η a , n � � − µ f � � d n � � � ( Recall that sup n ∈ N η a , n ≤ d 2 d − 2 if in addition a � SAT ( f )) . 14

(cont.) On the other hand, for ∀ a 0 ∈ P 1 excluding chordal PC( f ) : = { f n ( c ); c ∈ C ( f ) , n ∈ N } , there are r 0 > 0 and N = N ( a 0 ) s.t. for ∀ a ∈ B [ a 0 , r 0 ] , C 1 -test function ∀ φ on P 1 , ∀ k > N , the same (but locally uniform) estimate φ, ( f n ) ∗ ( a ) √ � �� � � � � ≤ C ′ max { Lip( φ ) , � φ, φ � 1 / 2 } � � nd − n . − µ f � � d n � � � √ d − n ) estimate holds for ∀ a ∈ Rem . For K � C , the better O ( P 1 at which f is semihyp. (cf. D. Drasin and Ok, BLMS 2007 ). 15

§ Arithmetic application / global fields For a number field or a function field k , when f has its coefficients in k , the dynamics on algebraic points f : P 1 ( k ) → P 1 ( k ) , is also interesting. Fix a non-trivial absolute value u on k , and set K = C u . The dynamical Diophantine approximation (Silverman 1993, Szpiro and Tucker 2005): For ∀ a ∈ P 1 ( k ) \ E ( f ) and wandering ∀ z ∈ P 1 ( k ) , 1 d n log[ f n ( z ) , a ] u = 0 . lim n →∞ 16

Since ( E ( f ) ⊂ SAT ( f ) ⊂ ) C ( f ) ⊂ P 1 ( k ) , consequently E wan ( f ) u ∩ P 1 ( k ) ⊂ E ( f ) , (6) and Theorem 3 recovers (in a purely local manner) Favre and Rivera-Letelier’s arithmetic quantitative equidistribution: Under the above arithmetic setting, there is C > 0 s.t. for ∀ a ∈ P 1 ( k ) \ E ( f ) , C 1 -test function ∀ φ on P 1 ( C u ) and ∀ n ∈ N large enough, φ, ( f n ) ∗ ( a ) � �� � √ � � � ≤ C ′ max { Lip( φ ) , � φ, φ � 1 / 2 } � � nd − n η a , n . − µ f ,u � � d n � � � Rem . By Thm 2 with (6), also E Fekete ( f ) u ∩ P 1 ( k ) = E ( f ) . 17

§ In complex dynamics / C Let f be a rational function on P 1 ( C ) of degree > 1 . Q . When E ( f ) = E Fekete ( f ) ? Cor 1. If ∃ Cremer periodic points, Siegel disks or Herman rings of f , then E ( f ) � E Fekete ( f ) . If f is geometrically finite, then E Fekete ( f ) = E ( f ) . Rem . ∃ semihyperbolic real cubic polynomial f such that E Fekete ( f ) ∩ J ( f ) � ∅ , so E ( f ) � E Fekete ( f ) . 18