dynamical Diophantine approximation

Dioph. Appro. Dynamical Dioph. Appro. in dynamical Diophantine approximation WANG Bao-Wei Huazhong University of Science and Technology Joint with Zhang Guo-Hua Central China Normal University 24-28 July 2017

Outline Dioph. Appro. Dynamical Dioph. Appro. 1 Diophantine approximation Introduction Mass transference principle 2 Dynamical Diophantine approximation Introduction

Dioph. Appro. Dynamical Dioph. Appro. Introduction Mass transference principle I. Diophantine approximation

Dioph. Appro. Dynamical Dioph. Appro. Introduction Mass transference principle Introduction

Dioph. Appro. Dynamical Dioph. Appro. Introduction Mass transference principle Classic Diophantine Approximation Main concerns : Distribution of rational numbers. Motivation : Density of rational numbers. Theorem Rational numbers are dense in R. Or equivalently, for any x R and ɛ > 0, x p/q < ɛ, for infinitely many rationals p/q. Remark : This is a qualitative result in nature with no quantitative information.

Dioph. Appro. Dynamical Dioph. Appro. Introduction Mass transference principle Quantitative results in classic Dioph. Appro. Theorem (Dirichlet s Thm) For any x R, x p/q < q 2, for infinitely many rationals p/q. Remark : q 2 can only improved to 1/ 5q 2 (Hurwitz s theorem). Main reason : bad points (eg. 1+ 5 2 = [1, 1,, ]).

Dioph. Appro. Dynamical Dioph. Appro. Introduction Mass transference principle Metric Diophantine approximation Main concerns. Instead of considering all points, consider the properties of generic points (in a sense of measure). Let ψ : N R +. The set of ψ-well approximable points : W (ψ) := { x [0, 1] : x p/q < ψ(q), i.o. (p, q) N 2, 0 p q }. How about the size of W (ψ) in the sense of a measure and fractal dimensions?

Dioph. Appro. Dynamical Dioph. Appro. Introduction Mass transference principle Size in measure Theorem (Khintchine s Thm) Let ψ : N R + be decreasing. Then L(W (ψ)) = 0 or 1 qψ(q) = or <. q 1 Multiplicative case : P. Gallagher {x R d : qx 1 qx d < ψ(q), i.o. q N} Linear forms : A. Groshev {x = (x 1,, x n ) R mn : qx 1 < ψ 1 (q),, qx n < ψ n (q), i.o. q N m }.

Dioph. Appro. Dynamical Dioph. Appro. Introduction Mass transference principle Size in Hausdorff measure Theorem (Jarnik-Bescotivch s Thm) Let ψ : N R + be decreasing and f a dimension function with f(r)/r being decease. Then H f (W (ψ)) = 0 or ( ) qf ψ(q) = or <. q 1 Linear forms : Bernik, Dickinson, Dodson, Levesley, Velani, Beresnevich, Bugeaud, et. al.

Dioph. Appro. Dynamical Dioph. Appro. Introduction Mass transference principle Mass transference principle

Dioph. Appro. Dynamical Dioph. Appro. Introduction Mass transference principle General results on dimensional theory : Mass transference principle Lebesgue measure statement = Hausdorff measure statement. Theorem (Beresnevich & Velani, Ann. Math. 06 ) Let x n R and ψ > 0 (n N). Let f be dimension function with f(r)/r decreasing. If { } x R : x B(x n, f(ψ(n))), i.o. n N is of full Lebesgue measure, then for any ball B, H f ( { x B : x B(xn, ψ(n)), i.o. n N }) = H f (B).

Direct consequence : Dioph. Appro. Dynamical Dioph. Appro. Introduction Mass transference principle Proof : (i). Khintchine s thm : Khintchine Thm = Jarnik Thm. ( ψ(q) ) qf = q n 1 { ( p = x [0, 1] : x B q, f( ψ(q) ) ), i.o. p/q} full Lebesgue. q (ii). Jarnik set : W (ψ) := { ( p x [0, 1] : x B q, ψ(q) ) }, i.o. p/q. q (iii) So, mass transference principle gives that H f (W (ψ)) =.

Dioph. Appro. Dynamical Dioph. Appro. Introduction Mass transference principle More words about Mass transference principle : as natural as it should be! Recall the definition of Hausdorff measure : { H f (E) = lim inf f( U i ) : E } U i, U i < δ, i 1. δ 0 i 1 i 1 Roughly speaking, the measure of B(x, r) in sense of H f the measure of B(x, f(r)) in sense of Lebesgue.

Dioph. Appro. Dynamical Dioph. Appro. Introduction Mass transference principle MTP : Mass transference principle. Roughly speaking : If L(lim sup A f n) full, then H f (lim sup A n ) full. Powerfulness : Most known results = MTP, since Lebesgue measure statements are already known.

II. Dynamical Diophantine approximation

Introduction

Dynamical Diophantine approximation Main concerns : Distribution of the orbits. Motivation : Density of orbits (in some sense). Let (X, d) be a separate complete metric space, T : X X, µ : a T -invariant finite Borel measure.

Density of orbits Theorem (Poincaré Recurrence Thm) For almost all x supp(µ) and ɛ > 0, T n x B(x, ɛ), i.o.. Theorem (Corollary of ergodic Thm) Assume that T is ergodic w. r. t. µ, for any y 0 supp µ and ɛ > 0, for µ a.s. x X, T n x B(y 0, ɛ), i.o.. for a µ-typical point x 0, for any y supp µ, lim inf n d(t n x 0, y) = 0. Qualitative properties Quantitative properties?

Quantitative studies How about the size of the set {x X : T n x B n, i.o. n N} where B n is decreasing in measure. We call the study on this dynamically defined limsup set as dynamical Diophantine approximation.

Dynamical Diophantine approximation What are sizes of the following dynamically defined limsup sets in the sense of measure (if µ is a measure on X)? Hausdorff dimension? Covering problem : C(T, x 0, ϕ) := { y : d(t n x 0, y) < ϕ(n), i.o. n N } Shrinking target problem : S(T, y 0, ϕ) := { x : d(t n x, y 0 ) < ϕ(n), i.o. n N } Quantitative recurrence : R(T, ϕ) := { x : d(t n x, x) < ϕ(n), i.o. n N }

Size in measure Covering problems Fan, Schmeling, Troubetzkoy, 13. T x = 2x(mod 1). µ φ, ν ψ : Gibbs measure. κ > ( φdν ψ ) 1, T n x t < n κ, i.o. n N, µ φ -a.e. x, ν ψ - a.e. t. L. Liao & S. Seuret, 13. General finite Markov maps. D. Kim & M. Fuchs, 15. Irrational rotation. a.e. x B(nα, ϕ(n)) i.o. iff n 1 ( qk+1 1 n=q k { min ϕ(n), q k α } ) =.

Shrinking target problems Chernov & Kleibock, 98. Under the condition that n,m N µ(t m A T n A n ) µ(a n ) µ(a m ) C n N = T n x A n infinitely often almost surely. Fayad, 05. µ(a n ). lim inf n n1/α d(t n x, y 0 ) <, µ-a.s. for any α > d µ (y 0 ).

Quantitative recurrence Boshernitzan, 93. If H α (X) is finite or σ-finte, then lim inf n n1/α d(t n x, x) <, µ-a.s. Barreira & Saussol, 01. lim inf n n1/α d(t n x, x) <, µ-a.s. for any α > d µ (x).

Size in dimension Covering problems Given an orbit {T n x 0 } n 1, how many points can be well approximated? { y : d(t n x 0, y) < ϕ(n), i.o. n N } Bugeaud 03, Schmeling & Troubetzkoy 03, Fan & Wu, 05, Liao & Rams 13, Kim-Rams-W, 16 : Irrational rotation : {y : nα y < ϕ(n), i.o. n N} Fan, Schmeling & Troubetzkoy 13, Liao & Seuret 13 : Expanding Markov systems. Järvenpää 2, Li, et al. 16 : random covering. W-Wu-Xu, 16. Triadic Cantor set.

Shrinking target problems : Given a point y 0, how many points x whose orbit can approximate y 0 with given speed? { } x X : T n x y < ϕ(n, x), i.o. n N. Hill & Velani 95, 97. T an expanding rational map on Riemann sphere and J its Julia sets. dim H { x J : T n x y 0 e Snf(x), i. o. } is given by the solution of the pressure function ( ) P T, s(log T + f) = 0.

Hill & Velani 99. Linear operator action on torus. Uranbski, 02. Infinite iterated function systems Levesley 98, Bugeaud, 04. Irrational rotation Stratmann & Uranbski, 02. Parabolic rational maps Fernàndez, Meliàn & Pestana 07. Finite expanding Markov systems Li-W-Wu-Xu, 13. Continued fraction systems Reeve, 11. General conformal iterated function systems. Shen-W, 13. beta expansions (Haudorff dimension). Coons-Hussain-W, 16. beta expansions (Hausdorff measure).

Quantitative recurrence How many points can return back to the initial one with given speed? { } x J : T n x x < e Snf(x), i. o.. Tan-W, 11. β-dynamical system : T x = βx(mod 1) dim H { x J : T n x x < e Snf(x), i. o. } is given by the solution of the pressure function P(T, s(f + log β)) = 0. Seuret-W, 15. Conformal iterated function systems. The dimension is { } inf s 0 : P(T, s(f + log T )) 0.

In the most known expanding systems, the dimension of the following set { } W := x X : T n (x) x or (y o ) < e Snf(x), i.o. n N is always given by the solution of a pressure function : P( t(log T + f)) = 0. It is high likely that there should be a general principle for the dimension of W. So we make the following conjecture : Conjecture : In expanding systems with some regular conditions, there should be dim H W = inf{t 0 : P( t(f + log T )) 0}.

Reason 1 In fact, the set W concerns the distributions of the periodic points or the inverse images of y o. So, we let χ n = {z n X : T n z n = z n } or χ n = {z n X : T n y = y o }. Then W is closely related to the following set { } W (f) : = x : x B(z n, e Sn(f+g)(zn) ), z n χ n, i.o. n N = N=1 n=n z n χ n B(z n, e Sn(f+log T While, in ergodic system, almost surely, one has X = N=1 n=n z n χ n B(z n, e Sn log T )(z n) ). (z n) ).

So, from X to W (f), it seems one shrinks balls to small balls in a dynamical way. B(z n, e Sn log T (z n) ) B(z n, e Sn(f+log T )(z n) )

Reason 2 Recall that the mass transference principle says that : If B(x n, f(r n )) is full in Lebesgue N=1 n=n then the shrunk limsup set B(x n, r n ) is full in Hausdorff. N=1 n=n full Lebesgue measure statement = full Hausdorff measure statement.

Reason 3 In many cases, the dimension of the phase space X can be given by the solution : P( t log T ) = 0. In many cases, the dimension of the set W (f) is given by the solution : P( t(f + log T )) = 0.

A general framework (X, T ) : be a topological dynamical system, X a compact metric space, T : X X continuous. Hypothesis : g C(X) with g 0 s.t. the following two are satisfied : (A). Dynamical Ubiquity. c 1 > 1 > c 2 > 0 such that for any y Y and n 1 (A1) Covering property. { B (z, c 1 e Sng(z)) } : z χ n covers X. (A2) Separation property. The balls are disjoint {B(z, c 2 e Sng(z) ) : z χ n }.

(B). Weak conformality. For any λ > 1, there exists 0 < b λ < 1 such that for any 0 < b b λ, (B1). Upper semi-conformality. ( ) T B(z, e g(z) b) B(T z, λb); (B2). Lower semi-conformality. B(T z, λ 1 b) T ( ) B(z, e g(z) b). Remark : More or less, g = log T.

Theorem (W-Zhang, 2016) Let (X, T ) be an exact topological dynamical system. Assume g satisfying the hypothesis (A-B). Then for any positive continuous function f, one has dim H X = inf{s : P( sg) 0}, dim H W (f) = inf{s : P( s(f + g)) 0} where P is the pressure function defined by 1 P(ϕ) := lim lim sup ɛ 0 n n log inf F :(n,ɛ)-spanning z F e Snϕ(z)

Applications 1. T x = 2x (mod 1). View it as a system on circle. Then clearly, it is an exact C 1 expanding topological dynamical system. χ n (y) = { ɛ1 2 + + ɛ n + y } 2 n : ɛ k {0, 1}, 1 k n (A). χ n is sufficiently well distributed and separated. χ n = 2 n, gap = 2 n = e Sng(x). (B). Conformality : λ = 1, b = 1/2. For any r < 1, T B(z, rb) = B(T z, 2rb).

Applications 2. T x = βx (mod 1) for β > 1. Expanding rational map acting on its Julia set. Conformal finitely expanding dynamical systems. Dynamical Diophantine approximation on triadic Cantor set. { } x C : T n x y < e Snf(x), i.o. n N.

Proof : I. Observations Under the hypothesis (A-B), one can observe that If x B(z, e Sng(z) ), then d(t i x, T i z) < c, for 1 i n(1 o(1)). So, T n y is a (n(1 o(1)), c)-spanning set. For any y X, T n y is (n(1 o(1)), c)-separated. Let N be the smallest integer s.t. T N B(z, e Sng(z) ) = X, then N = n(1 + o(1)), independent of y. So, one has for any y X, 1 P(ϕ) = lim n n log z T n y e Snϕ(z).

Proof : II. Cantor subset construction First level. Shrink the basic balls {B(z, c 2 e Sn 1 g(z) ) : z T n1 y 0 } to the desired ones {B(z, c 2 e Sn 1 (f+g)(z) ) : z T n1 y 0 }. Second level. For each B 1 = B(z, c 2 e Sn 1 (f+g)(z) ) in the first level, we will construct a local sub-level.

Assume that B(z, e Sn 1 (f+g)(z) ) B(z, e S N 1 g(z) ). Then T N1(1+o(1)) B(z, 1/2e Sn 1 (f+g)(z) ) = X. Lemma For each z 2 T n2 y 0, there exist z 2 T N1(1+o(1)) z, such that z 2 B(z, 1/2e Sn 1 (f+g)(z) ). Furthermore, by asking n 2 1, we can also have B(z 2, e S N 1 (1+o(1))+n 2 g(z) ) B(z, e Sn 1 (f+g)(z) ). Then shrink it to the desired ball consisting the second local level : {B(z 2, e S N 1 (1+o(1))+n 2 (f+g)(z) ) : z 2 T n2 y 0 }.

Mass distribution (1). Let B 1 = B(z 1, e Sn 1 (f+g)(z1) ) µ(b 1 ) = e tsn 1 (f+g)(z 1 ) z 1 T n 1 y o e tsn 1 (f+g)(z 1 ). (2). Let B 2 = B(z 2, e S N 1 (1+o(1))+n 2 (f+g)(z 2) ) be a ball in the second level and contained in B 1, where z 2 T N1(1+o(1)) z 2 for some z 2 T n2 y 0. µ(b 2 ) = µ(b 1 ) e tsn 2 (f+g)(z 2 ) z 2 T n 2 y o e tsn 2 (f+g)(z 2 ).

Thanks for your attention!