On the dimension of diagonally a ffi ne self-a ffi ne sets and - PowerPoint PPT Presentation

On the dimension of diagonally a ffi ne self-a ffi ne sets and overlaps. K´ aroly Simon Department of Stochastics Institute of Mathematics Budapest University of Technolugy and Economics www.math.bme.hu/˜simonk Joint work with Bal´ azs B´ ar´ any and Michal Rams ICERM February 2016 B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 1 / 43

Co-authors B´ ar´ any Bal´ azs Univ. of Warwick/ TU Budapest (left) Michal Rams, Warsaw IMPAN (right) B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 2 / 43

Introduction Introduction 1 Abstract Notation An old result of B´ ar´ any Bal´ azs Hochman-condition Results 2 Theorem A Theorem B Proposition C Theorem D Feng-Hu Theorem about Ledrappier-Young formula 3 Notation The self-similar case Diagonally self-a ffi ne case Stu ff what I am going to have no time for 4 B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 3 / 43

Introduction Abstract Abstract In this talk we consider diagonally a ffi ne, planar IFS Φ = { S i ( x , y ) = ( – i x + t i , 1 , — i y + t i , 2 ) } m i =1 . Combining the techniques of Hochman and Feng-Hu we compute the Hausdor ff dimension of the self-a ffi ne attractor and measures and we give an upper bound for the dimension of the exceptional set of parameters. B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 4 / 43

Introduction Notation Definitions Φ = { S i ( x , y ) = ( – i x + t i , 1 , — i y + t i , 2 ) } m (1) i =1 , where 0 < | – i | , | — i | < 1, and we assume that S i ([0 , 1] 2 ) µ [0 , 1] 2 . We call a Borel probability measure µ self-a ffi ne if it is compactly supported with support Λ and there exists a p = ( p 1 , . . . , p m ) probability vector such that m p i µ ¶ S ≠ 1 (2) µ = ÿ . i i =1 B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 5 / 43

Introduction Notation Lyapunov exponent The entropy and the Lyapunov exponents of µ : m h µ := ≠ ÿ p i log p i , i =1 and m m ‰ – := ≠ p i log | – i | , ‰ — := ≠ (3) ÿ ÿ p i log | — i | . i =1 i =1 B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 6 / 43

Introduction Notation Lyapunov exponents: 0 < χ 1 ≤ χ 2 ≤ χ 3 ≤ χ 4 Λ µ ( t ) slope= χ 1 slope= χ 3 ) t ( µ slope= χ 2 Λ − h µ D ( µ ) t 1 2 3 4 Figure: Definition of the Laypunov dimension D ( µ ) B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 7 / 43

Introduction Notation Lyapunov dimension by formula If (4) k := max { i : 0 < h ‹ ≠ ‰ 1 ( ‹ ) ≠ · · · ≠ ‰ i ( ‹ ) } Æ d ≠ 1 , then we define the Lyapunov dimension of ‹ : D ( ‹ ) := k + h ‹ ≠ ‰ 1 ( ‹ ) ≠ · · · ≠ ‰ k ( ‹ ) ; ‰ k +1 ( ‹ ) If h ‹ ≠ ‰ 1 ( ‹ ) + · · · ≠ ‰ d ( ‹ ) > 0 then we define h ‹ D ( ‹ ) := d · ‰ 1 ( ‹ ) + · · · + ‰ d ( ‹ ) , where h ‹ is the entropy of the measure ‹ . B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 8 / 43

Introduction Notation Lyapunov dimension on the plane h ‹ Y if h ‹ Æ ‰ 1 ( ‹ ); _ ‰ 1 ( ‹ ) , _ _ _ _ _ _ _ 1 + h ‹ ≠ ‰ 1 ( ‹ ) _ _ ] if ‰ 1 ( ‹ ) < h ‹ Æ ‰ 1 + ‰ 2 ( ‹ ); D ( ‹ ) := , ‰ 2 ( ‹ ) _ _ _ h ‹ _ _ _ 2 · ‰ 1 ( ‹ ) + ‰ 2 ( ‹ ) , if h ‹ > ‰ 1 ( ‹ ) + ‰ 2 ( ‹ ). _ _ _ _ [ What does this mean exactly? See it in a special case: B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 9 / 43

Introduction Notation Lyapunov dimension in a very special case f 3 ( Q ) 1 Q ξ λ Let ‹ be the self-a ffi ne measure which corresponds f 2 ( Q ) to p 1 = p 2 = p 3 = 1 3 . ξ ‰ 1 = ≠ log ⁄ < ≠ log › = ‰ 2 λ f 1 ( Q ) Clearly, h ‹ = log 3 ξ 0 λ 1 log 3 Y if ⁄ < 1 3 ; _ ≠ log ⁄ , _ _ _ ] D ( ‹ ) = log 3 ≠ log ⁄ , if ⁄ > 1 _ 3 ; _ _ _ ≠ log › [ B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 10 / 43

Introduction An old result of B´ ar´ any Bal´ azs An old result First we consider an old result due to Bal´ azs B´ ar´ any. Notation s – = dim B proj x Λ , s — := dim B proj y Λ . d – and d — are the solutions of the equations: m m i – d β ≠ s β i — d α ≠ s α — s β – s α ÿ = 1 and ÿ = 1 i i i =1 i =1 B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 11 / 43

Introduction An old result of B´ ar´ any Bal´ azs B´ ar´ any’s Theorem Theorem 1.1 (B´ ar´ any (2011)) W.L.G. we may assume that S i ([0 , 1] 2 ) µ [0 , 1] 2 . Assume that { S i } m i =1 satisfies Strong Separation Condition: S i ([0 , 1] 2 ) fl S j ([0 , 1] 2 ) = ÿ . (5) Then (6) dim B Λ = max { d – , d — } . B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 12 / 43

Introduction Hochman-condition Hochman-condition We say that an IFS G = { f i ( x ) } i œ S of similarities on the real line satisfies the Hochman-condition if there exists an Á > 0 such that for every n > 0 min { ∆ ( ı, ä ) : ı, ä œ S n , ı ” = ä } > Á n , where Y f Õ ı (0) ” = f Õ Œ ä (0) ] ∆ ( ı, ä ) = | f ı (0) ≠ f ä (0) | f Õ ı (0) = f Õ ä (0) . [ B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 13 / 43

Introduction Hochman-condition Examples when Hochman Condition holds If the parameters of the IFS G = { f i ( x ) = r i x + t i } i œ S of similarities are algebraic, i.e. both t i and r i are algebraic numbers , then either the Hochman-condition holds or there is a complete overlap, that is, there exist n Ø 1, and ı ” = ä œ S n such that f ı (0) = f ä (0) . B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 14 / 43

Introduction Hochman-condition Hochman Theorem Suppose that an IFS Ψ = { r i x + t i } m i =1 , | r i | < 1 of contracting similarities on the real line satisfies the Hochman-condition. Let P := { p 1 , . . . , p m } N . Then for the measure µ = P ¶ Π ≠ 1 , 1 , h µ I J dim H µ = min , ‰ where h µ is the entropy and ‰ is the Lyapunov exponent: M ÿ p i log r 1 h µ = ≠ ÿ p i log p i and ‰ = ≠ i =1 . B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 15 / 43

Introduction Hochman-condition Hausdor ff dimension of a measure Here we recall the Hausdor ff dimension of a probability measure µ , dim H µ = inf { dim H A : µ ( A ) = 1 } log µ ( B r ( x )) = ess sup lim inf , log r µ ≥ x r æ 0+ B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 16 / 43

Introduction Hochman-condition Families of self-similar IFS Let I µ R be a compact parameter interval and m Ø 2. For every parameter t œ I given a self ≠ similar IFS on the line: Φ t := { Ï i , t ( x ) = r i ( t ) · ( x ≠ a i ( t )) } m i =1 , where r i : I æ ( ≠ 1 , 1) \ { 0 } and a i : I æ R are real analytic functions. Let Π t be the natural projection from Σ := { 1 , . . . , m } N to the attractor Λ t of Φ t . B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 17 / 43

Introduction Hochman-condition Families of self-similar IFS (cont.) For every probability vector p := ( p 1 , . . . , p m ) the associated self-similar measure is ‹ p , t := ( Π t ) ú ( p N ) . Its similarity dimension is defined by m i =1 p i log p i q dim S ( ‹ p , t ) := m i =1 p i log r i ( t ) q The similarity dimension of Λ t is the solution s ( t ) of r s ( t ) ( t ) + · · · + r s ( t ) m ( t ) = 1 . 1 B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 18 / 43

Introduction Hochman-condition Families of self-similar IFS (cont.) t œ I is exceptional We say that a parameter if either dim H Λ t < min { 1 , s ( t ) } or there exists a probability vector p := ( p 1 , . . . , p m ) such that dim H ( ‹ p , t ) < min { 1 , dim S ( ‹ p , t ) } . B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 19 / 43

Introduction Hochman-condition Families of self-similar IFS (cont.) Theorem 1.2 (Hochman) Assume that for i , j œ Σ = { 1 , . . . , m } N we have if Π t ( i ) = Π t ( j ) holds for all t œ I then i = j . Then both the Hausdor ff and the packing dimension of the set of exceptional parameters are equal to 0 . B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 20 / 43

Results Introduction 1 Abstract Notation An old result of B´ ar´ any Bal´ azs Hochman-condition Results 2 Theorem A Theorem B Proposition C Theorem D Feng-Hu Theorem about Ledrappier-Young formula 3 Notation The self-similar case Diagonally self-a ffi ne case Stu ff what I am going to have no time for 4 B. B´ ar´ any, M. Rams, K. Simon Diagonally Self-a ffi ne sets and measures February 19, 2016 21 / 43