Thermodynamic formalism for systems withoverlaps and applications in number theory

Eugen Mihailescu

Institute of Mathematics of the Romanian Academy, Bucharest

Thermodynamic formalism - Applications to geometry andnumber theory, Bremen, July 2017

I. First, we look at dimension results for smooth endomorphismsf : M → M hyperbolic on locally maximal sets of saddle typeΛ.This is based on joint work with Bernd Stratmann.We want to estimate the stable dimension at x ∈ Λ, i.e

δs(x) := HD(W sr (x) ∩ Λ)

May exist infinitely many unstable manifolds through a givenpoint. In fact W u

r (x) depends on the whole prehistoryx = (x , x−1, x−2, . . .) ∈ Λ, where x−i ∈ Λ and f (x−i ) = x−i+1, fori ≥ 1.

Define the preimage counting function as:

∆(x) := Card(f −1(x) ∩ Λ), x ∈ Λ

Falconer studied fractals with overlaps obtained from linearcontractions Ti (x) = λix , i = 1, ..., ` in Rn, with 0 < |λi | < 1 and∑

1≤i≤`|λi | < 1

He showed that the Hausdorff dimension of the invariant set of{Ti + ai , : 1 ≤ i ≤ `} is equal to s, for Lebesgue a.e(a1, . . . , a`) ∈ R× ...× R, where s is the similarity dimension, i.e.the solution of ∑

1≤i≤`|λi |s = 1

However, this result fails if∑

1≤i≤` |λi | ≥ 1, as observed by Edgar.Thus fractals originating from overlapping constructions can haveHausdorff dimension less than their similarity dimension.

Define the stable potential Φs on Λ by

Φs(x) := log |Dfs(x)|, x ∈ Λ

Theorem (M., Stratmann)

Let f : M → M be a C2-endomorphism which is c-hyperbolic on abasic set Λ of f and for which there exists a continuous functionω : Λ→ R such that ∆(x) ≥ ω(x), for all x ∈ Λ. It then follows

δs(x) ≤ tω,

where tω is the unique zero of

t 7→ P(tΦs − logω)

As an application of the above Theorem we obtain:

Corollary (M., S.)

If in addition we have that the minimal value of ∆ on Λ is equal tod , and that there exists a point x ∈ Λ at which δs is equal to theunique zero td of

t 7→ P(tΦs − log d),

then ∆ = d on an open dense subset of Λ and δs(y) = td , ∀y ∈ Λ.

This applies in particular when d = 1 (no overlap). Then, thereexists an open dense set where f has precisely one preimage in Λ.So, f is “almost” a homeomorphism on Λ.

For self-similar sets K , by Schief, if σ is the similarity dimension ofK and Hσ(K ) > 0, then K satisfies Strong Open Set Condition.

To obtain ”almost injectivity” on Λ, we only require δs(x) = t1, i.ethe zero of t → P(tΦs), for some x ∈ Λ; we do not requireHt1(W s

r (x) ∩ Λ) > 0. In our case t1 is the analogue of similaritydimension in the stable direction.

II. Another class of endomorphisms are skew product Smaleendomorphisms over countable shifts of finite type.Joint work with M. Urbanski.

First we want to prove exact dimensionality for certain measures.Exact dimensionality was studied in many various settings (Young,Barreira, Pesin, Schmeling, Feng, Hu, etc).

Let E be a countable and A : E × E → {0, 1} be a matrix. A finiteor countable word ω is called A-admissible iff Aab = 1 for anyconsecutive elements a, b of ω.A is called finitely irreducible if ∃ a finite set F of finiteA-admissible words, s.t ∀a, b ∈ E , ∃γ ∈ F s.t aγb is A-admissible.Given β > 0 define,

dβ((ωn)+∞0 , (τn)∞0 ) = exp(−βmax{n ≥ −1 : (0 ≤ k ≤ n), ωk = τk}),

with e−∞ = 0. Denote the unilateral shift

E+A = {(ωn)+∞0 : ∀n∈N Aωnωn+1 = 1}

E+A is a closed in EN, with metric dβ from EN. For everyω = ω0ω1 . . . ωn−1, put |ω| = n, the length of ω and the cylinder

[ω] = {τ ∈ E+A : ∀(0≤j≤n−1) : τj = ωj}.

Let ψ : E+A → R continuous, then the topological pressure P(ψ) is,

P(ψ) := limn→∞

1

nlog

∑|ω|=n

exp(sup(Snψ|[ω]))

The limit exists, log∑|ω|=n exp(sup(Snψ|[ω])), n ∈ N, sub-additive.

ψ : E+A → R is summable if∑

e∈Eexp(sup(ψ|[e])) < +∞.

A shift-invariant µ on E+A is a Gibbs state of ψ if ∃ C ≥ 1,P ∈ R,

C−1 ≤ µ([ω])

exp(Snψ(τ)− Pn)≤ C

for n ≥ 1, ω, τ ∈ [ω]. If ψ has a Gibbs state, then P = P(ψ).Mauldin and Urbanski showed for every locally Holder continuoussummable ψ : E+

A → R, ∃ a unique Gibbs state µψ, ergodic.

Variational Principle for One-Sided Shifts and locally Holdersummable potentials,

sup{hµ(σ) +

∫E+A

ψdµ, µ ◦ σ−1 = µ,

∫ψdµ > −∞} = P(ψ)

Moreover P(ψ) = hµψ(σ) +∫E+Aψdµψ, and µψ is the only measure

where the supremum is attained.

Thermodynamic formalism on two-sided shifts.Again E countable and A : E × E → {0, 1} a finitely irreduciblematrix, and

EA = {(ωn)+∞−∞ : ∀n∈Z Aωnωn+1 = 1}.

For ω ∈ EA and −∞ ≤ m ≤ n ≤ +∞, set

ω|nm = ωmωm+1 . . . ωn.

Let E ∗A be the set of A-admissible finite words. For τ ∈ E ∗, set

[τ ]nm = {ω ∈ EA : ω|nm = τ}

The family of cylinders from m to n is Cnm.

Let ψ : EA → R continuous and,

P(ψ) := limn→∞

1

nlog

∑ω∈Cn−1

0

exp(sup(Snψ|[ω]))

ψ1 and ψ2 are cohomologous in a class G if and only if ∃u ∈ G s.t

ψ2 − ψ1 = u − u ◦ σ

A function u− u ◦ σ is called a coboundary in G . Any functions onEA, cohomologous in C (EA), the class of bounded functions onEA, have the same topological pressure and set of Gibbs measures.

ψ : EA → R is called past-independent if ∀τ ∈ C+∞0 and

ω, ρ ∈ [τ ], we have ψ(ω) = ψ(τ).

We showed that every locally Holder continuous ψ : EA → R iscohomologous to a past-independent locally Holder continuousψ+ : EA → R in the class HB of bounded Holder cont functions.ψ : EA → R is called summable on EA iff∑

e∈Eexp(sup(ψ|[e])) < +∞

For every locally Holder continuous summable ψ : EA → R, ∃unique Gibbs state µψ on EA, and µψ is ergodic.

Moreover, we have:

sup{hµ(σ) +

∫EA

ψdµ,with µ ◦ σ−1 = µ,

∫ψdµ > −∞} = P(ψ)

=hµψ(σ) +

∫EA

ψdµψ,

and µψ is the only measure at which sup is attained.

DefinitionLet (Y , d) be a complete bounded metric space, and ∀ω ∈ E+

A aset Yω ⊂ Y and a continuous injective Tω : Yω → Yσ(ω). Define

Y :=⋃ω∈E+

A

{ω} × Yω ⊂ E+A × Y .

Define T : Y → Y by

T (ω, y) = (σ(ω),Tω(y)).

(Y ,T ) is called a skew product Smale endomorphism if T isunif contracting on fibers, i.e ∃γ > 1 s.t ∀ω ∈ E+

A and y1, y2 ∈ Yω,

d(Tω(y2),Tω(y1)) ≤ γ−1d(y2, y1)

For τ ∈ EA(−n,+∞),

T nτ := Tτ |+∞−1

◦ Tτ |+∞−2◦ . . . ◦ Tτ |+∞−n

: Yτ → Yτ |+∞0

Then for τ ∈ EA, define

T nτ := T n

τ |+∞−n= Tτ |+∞−1

◦ Tτ |+∞−2◦ . . . ◦ Tτ |+∞−n

: Yτ |+∞−n→ Yτ |+∞0

(T nτ (Yτ |+∞−n

))∞n=0 are descending sets, and

diam(T nτ (Yτ |+∞−n

)) ≤ γ−ndiam(Y ). Since (Y , d) is complete, let

π2(τ) := ∩∞n=1Tnτ (Yτ |+∞−n

)

So, π2 : EA → Y , and define

π : EA → E+A × Y , π(τ) = (τ |+∞0 , π2(τ))

Let π1 : EA → E+A , π1(τ) = τ |+∞0 , and for ω ∈ E+

A define thestable Smale fibers,

Jω := π2([ω]) ∈ Y

The global invariant set

J := π(EA) =⋃ω∈E+

A

{ω} × Jω ⊂ E+A × Y ,

is called the Smale space induced by the Smale system T .

For each τ ∈ EA, π2(τ) ∈ Y τ |+∞0, so Jω ⊂ Y ω, for every ω ∈ E+

A .Since all maps Tω : Yω → Yσ(ω) are Lipschitz continuous with aLipschitz constant γ−1, they extend uniquely to Lipschitz mapsfrom Y ω to Y σ(ω). Then, ∀ω ∈ E+

A ,

Tω(Jω) ⊂ Jσ(ω), and⋃

e∈E ,Aeω0=1

Teω(Jeω) = Jω, and

T ◦ π = π ◦ σ,

Assume now more about Yω and Tω : Yω → Yσ(ω), ω ∈ E+A :

(a) Yω is closed bounded in Rd , with some d ≥ 1 s.tInt(Yω) = Yω.

(b) Each Tω : Yω → Yσ(ω) extends to a C 1 conformal embeddingfrom Y ∗ω to Y ∗σ(ω), where Y ∗ω is a bounded connected open

subset of Rd containing Yω.

(c) contraction in every fiber with constant γ > 1.

(d) (Bounded Distortion Property 1) ∃ constants α > 0 andH > 0 such that for all y , z ∈ Y ∗ω ,∣∣ log |T ′ω(y)| − log |T ′ω(z)|

∣∣ ≤ H||y − z ||α.

(e) EA 3 τ 7−→ log |T ′τ (π2(ω))| ∈ R is Holder continuous.

(f) (Open Set Condition) For every ω ∈ E+A and all a, b ∈ E with

Aaω0 = Abω0 = 1 and a 6= b,

Taω(Int(Yaω)) ∩ Tbω(Int(Ybω)) = ∅

(g) (Boundary Condition) ∃ measurable δ : E+A → (0,∞) s.t for

all ω ∈ E+A ,

Jω ∩ (Yω \ B(Y cω , δ(ω)) 6= ∅

We call a system with (a)-(g) conditions, a conformal skewproduct Smale endomorphism.Consider now the measurable partition of EA,

P− = {[ω|+∞0 ] : ω ∈ EA} = {[ω] : ω ∈ E+A }.

If µ is a Borel measure on EA, let

{µω : ω ∈ E+A }

be the canonical system of conditional measures induced by P−and µ (Rokhlin).Recall the projection:

π1 : EA → E+A , π1(τ) = τ |∞0 , τ ∈ EA

Then {µω, ω ∈ E+A } is determined by the fact that, ∀g ∈ L1(µ),∫

EA

g dµ =

∫E+A

∫[ω]

g d µω d(µ ◦ π−11 )(ω)

Let T : Y → Y a conformal skew product Smale endomorphism. Ifµ is a σ-invariant measure on EA, then its Lyapunov exponent,

χµ(σ) :=−∫EA

log∣∣∣T ′τ |+∞0 (π2(τ))

∣∣∣ dµ(τ)

= −∫E+A

∫[ω]

log∣∣T ′ω(π2(τ))

∣∣ d µω(τ) dm(ω),

where m = π1∗µ is the canonical projection of µ onto E+A .

Now, if µ is an invariant measure on EA, take the conditionalmeasures µω on [ω], for ω ∈ E+

A .

Recall π2 : EA → Y , and

Jω = π2([ω]), for ω ∈ E+A

Theorem (Mihailescu, Urbanski)

Let T : Y → Y be a Holder conformal skew product Smaleendomorphism, and let ψ : EA → R be a Holder continuoussummable potential.

Then, for mψ-a.e ω ∈ E+A , the measure µωψ ◦ π

−12 is exact

dimensional on Jω, and

HD(µωψ ◦ π−12 ) = limr→0

logµωψ ◦ π−12 (B, r))

log r=

hµψ(σ)

χµψ(σ),

for mψ-a.e. ω ∈ E+A and µωψ ◦ π

−12 -a.e. z ∈ Jω (equivalently for

µψ ◦ π−1-a.e. (ω, z) ∈ J).

Now want to enlarge the class of skew products to skew productsover countable-to-1 endomorphisms.

Let a skew product F : X ×Y → X ×Y , where X ,Y are completebounded metric spaces, Y ⊂ Rd for some d ≥ 1, and

F (x , y) = (f (x), g(x , y)),

where gx(y) defined as

Y 3 y 7−→ g(x , y),

is injective and continuous for all y ∈ Y .Assume f : X → X expanding and countable-to-1, and thedynamics of f is modeled by a 1-sided Markov shift on a countablealphabet E with A finitely irreducible, i.e ∃ a surjective Holdercontinuous coding,

π : E+A → X , π ◦ σ = f ◦ π

Given F , take for ω ∈ E+A , Tω : Y → Y ,

Tω(y) = g(π(ω), y).

Assume (a)–(g) hold for Tω. If Y = E+A × Y , we obtain a

conformal skew product Smale endomorphism

T : Y → Y , T (ω, y) = (σ(ω),Tω(y))

Jω = π2([ω]) is the set of points⋂n≥1

Tτ−1ω ◦ Tτ−2τ−1ω ◦ . . . ◦ Tτ−n...τ−1ω(Y )

The space of full prehistories is X (the natural extension, or inverselimit) of (f ,X ). Consider

Jx = {⋂n≥1

gx−1 ◦ gx−2 ◦ . . . ◦ gx−n(Y ), for x = (x , x−1, x−2, . . .) ∈ X}

If η = (η0, η1, . . .) ∈ E+A with π(η) = x , then for η−1 with

η−1η ∈ E+A , π(η−1η) = x ′−1 where x ′−1 is 1-preimage of x . Hence,

Jx =⋃

ω∈E+A ,π(ω)=x

Jω

Denote the fibered limit sets for T , and F by:

J =⋃ω∈E+

A

{ω} × Jω ⊂ E+A × Y , and J(X ) =

⋃x∈X{x} × Jx ⊂ X × Y

We have the Holder continuous projection

πJ : J → J(X ), πJ(ω, y) = (π(ω), y),

Given F , the symbolic lift of F is

F : E+A × Y → E+

A × Y , F (ω, y) =(σ(ω)), g(π(ω), y)

),

The Lyapunov exponent for an F -invariant measure µ on J(X ), is

χµ(F ) =

∫J(X )

log |g ′x(y)| dµ(x , y)

Theorem (M., U.)

Let F : X × Y → X × Y , F (x , y) = (f (x), g(x , y)) be as above,so there exists a surjective Holder continuous coding π : E+

A → X

with π ◦ σ = f ◦ π. Assume F : E+A × Y → E+

A × Y is a Holderconformal skew product Smale endomorphism, and φ : J(X )→ Ris locally Holder continuous s.t∑

e∈Eexp(sup(φ|π([e])×Y )) <∞

Then, for µφ ◦ π−11 -a.e x ∈ X , the conditional measure µxφ is exactdimensional on Jx , and for µφ-a.e (x , y) ∈ J(X ),

limr→0

logµxφ(B(y , r))

log r=

hµφ(F )

χµφ(F )

For X ⊂ Rd ,Y ⊂ Rm, and a skew product F as above, let anF -invariant probability µ on X × Y . Denote ν = µ ◦ π−11 .

Theorem (M., U.)

In the above setting, let ν = µ ◦ π−11 , and assume that for ν-a.ex ∈ X the conditional measure µx is exact dimensional with itspointwise dimension equal to a fixed value α. Assume also that theprojection measure ν is exact dimensional on X , with its pointwisedimension equal to γ.Then, µ is exact dimensional globally on X × Y , and for µ-a.e(x , y) ∈ X × Y ,

limr→0

logµ(B((x , y), r))

log r= α + γ

Definition (Expanding Markov-Renyi maps)

Let I an interval, I = ∪n≥0In, where In are closed intervals withmutually disjoint interiors. f : I → I is called an EMR map if:a) f is C2 on ∪n≥0int(In).b) ∃ a uniformly expanding iterate of f , i.e ∃K > 1 and m apositive integer, s.t

|(f m)′(x)| ≥ K > 1,∀x ∈ ∪n≥0int(In)

c) f is Markov, i.e for any n ≥ 0, f |int(In) is a homeomorphismfrom IntIn to the interior of a union of some Ij ’s, j ≥ 0.d) f satisfies Renyi condition, i.e ∃K ′ > 0 s.t

supn

supx ,y ,z∈In

|f ′′(x)||f ′(y)| · |f ′(z)|

≤ K ′ <∞

For an EMR map f : I → I , ∃ coding,

π : NN → I , π((k1, k2, . . .)) = ∩n≥0

f −n(Ikn)

Every point x which never hits the boundary of some interval Inunder an iterate of f , has a unique such coding, i.e there exists aunique (k1, k2, . . .) ∈ NN with π((k1, k2, . . .)) = x . Thus,π : E+

A → X is injective outside a countable set.

The continued fractions (Gauss) map f1 : [0, 1]→ [0, 1],

f1(x) =1

x− [

1

x] =

{1

x

}, x 6= 0, and f1(0) = 0

is an example of an EMR map.

Theorem (M., U.)

Let an EMR map f : I → I , an open bounded set Y ⊂ Rd , and aHolder conformal skew product endomorphism over f ,F : I × Y → I × Y . Consider φ : J(I )→ R a locally Holdercontinuous potential, s.t∑

e∈Nexp(sup(φ|π([e])×Y )) <∞

Then, for (π1∗µφ)-a.e x ∈ I , the conditional measure µxφ is exactdimensional on Jx , and

limr→0

logµxφ(B(y , r))

log r=

hµφ(F )

χµφ(F ),

for µxφ-a.e y ∈ Jx ; hence, equivalently for µφ-a.e (x , y) ∈ J(I ).

Diophantine Approximation, and the Doeblin-LenstraConjecture.

We apply the above results, towards the conjecture of Doeblin andLenstra.Consider an irrational number x ∈ [0, 1],

x = [a1, a2, . . .] =1

a1 + 1a2+

1

...+ 1an+...

,

where ai ≥ 1, i ≥ 1. The continued fraction transformation is

T : [0, 1]→ [0, 1], T (x) = {1

x}, x 6= 0, and T (0) = 0

By truncating the representation at n, we obtain a rational numberpnqn

, where pn, qn ≥ 1, n ≥ 1, (pn, qn) = 1,

pnqn

= [a1, . . . , an]

Denote by

Θn = Θn(x) = |x − pnqn| · q2n, n ≥ 1

Also,Tn := T n(x) = [an+1, an+2, . . .], n ≥ 1,

andVn := [an, . . . , a1], n ≥ 1

Now, ∀n ≥ 1,

Vn =qn−1qn

, Θn−1 =Vn

1 + TnVn, Θn =

Tn

1 + TnVn

For ([0, 1),T ), the natural extension is given by the map on[0, 1)× [0, 1) :

T (x , y) = (Tx ,1

a1(x) + y), (x , y) ∈ [0, 1)2

Hence, T (x , 0) = (Tx , 1a1(x)

), and T 2(x , 0) = (T 2(x), 1a2(x)+

1a1(x)

).

By induction, ∀n ≥ 1,

T n(x , 0) = (Tn, [an, . . . , a1]) = (Tn,Vn)

The approximation coefficients Θn were the object of an importantConjecture by Doeblin, reformulated by Lenstra, namely that forLebesgue-a.e x ∈ [0, 1] and all z ∈ [0, 1],

limn→∞

Card{1 ≤ k ≤ n,Θn(x) ≤ z}n

exists, and equals

F (z) =z

log 2, z ∈ [0,

1

2], and F (z) =

1

log 2(1−z+log 2z), z ∈ [

1

2, 1]

The conjecture was solved by Bosma, Jager and Wiedijk. Theyneeded the natural extension ([0, 1)2, T , µG ), of ([0, 1),T ) withGauss measure µG .

Consider now for s > 12 , the geometric potential

φs(x) = −s log |T ′(x)|, x ∈ [0, 1)

Then φs has an equilibrium measure µs on [0, 1). Let also

ψs(x , y) = φs(x), (x , y) ∈ [0, 1)2,

and µs the equilibrium measure of ψs w.r.t T on [0, 1)2. Then weshowed µs is exact dimensional on [0, 1)× [0, 1).

Question: When x is in some set of Lebesgue measure zero, whatcan be said about the asymptotic frequencies at which theapproximation coefficients of x come close to arbitrary values?

Theorem (Mihailescu, Urbanski)

For µs -a.e x ∈ [0, 1), all z , z ′ ∈ [0, 1), and r , r ′ > 0,

limn→∞

Card{k, 0 ≤ k ≤ n − 1, (Tk ,Vk) ∈ B(z , r)× B(z ′, r ′)

}n

=

= µs(B(z , r)× B(z ′, r ′))

The Lyapunov exponent of µs is

λ(µs) :=

∫I

log |T ′|(x) dµs(x)

Denote by λ0 the Lyapunov exponent of the Gauss measure, i.eλ0 =

∫log |T ′|dµG = π2

6 log 2 . Then, from Pollicott-Weiss, for

λ ∈ [λ0,∞), ∃s = s(λ) > 12 and an uncountable dense set:

Λs = {x ∈ [0, 1), limn→∞

1

nlog |x − pn(x)

qn(x)| = −λ}, and

λ(µs) = λ, HD(Λs) =hµs (T )

λ

We want to show that, for x ∈ Λs , the asymptotic frequency ofΘk(x) being r -close to some z , and Θk−1(x) being r -close to z ′,is comparable to r δ(µs)±ε.

Theorem (M., U.)

For any λ ∈ [λ0,∞), there exists s > 12 and a set Λs ⊂ [0, 1) with

HD(Λs) =hµs (T )λ , such that ∀ε > 0, x ∈ Λs , and µs -a.e pair

(z , z ′) ∈ [0, 1)2, there exists r(x , z , z ′) > 0, so that for0 < r < r(x , z , z ′, ε), we have:

limn→∞

Card{k ≤ n − 1,

(Θk(x),Θk−1(x)

)∈ B( z

1+zz ′ , r)× B( z ′

1+zz ′ , r)}

n

∈(r δ(µs)−ε, r δ(µs)+ε

),

where

δ(µs) = HD(µs) =hµs (T )

λ+

hµs (T )

2∫[0,1)2 log(a1(x) + y) d µs(x , y)

Beta-trasformations.For β > 1, the β-map is Tβ : [0, 1)→ [0, 1),

Tβ(x) = βx (mod 1),

and the β-expansion of x is x =∞∑k=1

dkβk , dk = [βT k−1

β (x)], k ≥ 1,

and dk ∈ {0, 1, . . . , [β]}. A sequence (d1, d2, . . .) is admissible if ∃x ∈ [0, 1) with β-expansion

∑k≥1

dkβk .

Tβ does not preserve Lebesgue measure λ, but has a uniquemeasure νβ = hβdλ, equivalent to λ and Tβ-invariant.

For general β > 1, Tβ cannot be coded directly with a subshift offinite type, but the induced map to [0, 1)2 of the natural extensionTβ, is coded by a shift on countably many symbols, as shown byDajani, Kraaikamp and Solomyak.

Theorem (M., U.)

Let β > 1 and Tβ : [0, 1)→ [0, 1),Tβ(x) = βx(mod 1). Letφ : [0, 1)2 → R a Holder continuous with∑

n≥1exp(supφ|In×[0,1)) <∞,

where I = {In, n ≥ 1} are given in the natural extensionexpression, and µφ its equilibrium state w.r.t the induced mapTβ,[0,1)2 . Denote by S the natural extension of the GLS(I)

transformation, and Z0 = [0, 1)2.

Then, for µφ ◦ π−11 -a.e x ∈ [0, 1), the conditional measure µxφ of µφis exact dimensional on [0, 1), and for µxφ-a.e y ∈ [0, 1),

HD(µxφ) = limr→0

logµxφ(B(y , r))

log r=

hµφ,Z0 (S)

χµφ,Z0 (S)