Fourier nonuniqueness sets for the hyperbola and the Perron-Frobenius operators
aa r X i v : . [ m a t h . C A ] S e p FOURIER NONUNIQUENESS SETS FOR THE HYPERBOLAAND THE PERRON-FROBENIUS OPERATORS
DEB KUMAR GIRI
Abstract.
Let Γ be a smooth curve or finite disjoint union of smooth curvesin the plane and Λ be any subset of the plane. Let X (Γ) be the space of allfinite complex-valued Borel measures in the plane which are supported on Γand are absolutely continuous with respect to the arc length measure on Γ . Let AC (Γ , Λ) = { µ ∈ X (Γ) : ˆ µ | Λ = 0 } , then we say that Λ is a Fourier uniquenessset for Γ or (Γ , Λ) is a Heisenberg uniqueness pair, if AC (Γ , Λ) = { } . Inparticular, let Γ be the hyperbola { ( x, y ) ∈ R : xy = 1 } and Λ β be the lattice-cross in R is defined by Λ β = ( Z × { } ) ∪ ( { } × β Z ) , where β is a positivereal. Then Canto-Mart´ın, Hedenmalm and Montes-Rodr´ıguez has shown thatthe space AC (Γ , Λ β ) is infinite-dimensional for β > . Further, they consideredthe branch Γ + = { ( x, y ) ∈ R : xy = 1 , x > } of the hyperbola xy = 1 andthe lattice-cross Λ γ = (2 Z × { } ) ∪ ( { } × γ Z ) , where γ is a positive real,and prove that AC (Γ + , Λ γ ) is infinite-dimensional for γ > . In this paper, we prove the following results:(a) For a rational perturbation of Λ β namely, Λ θβ = (( Z + { θ } ) × { } ) ∪ ( { } × β Z ) , where θ = 1 /p, for some p ∈ N , and β is a positive real, AC (cid:16) Γ , Λ θβ (cid:17) is infinite-dimensional whenever β > p. (b) For a rational perturbation of Λ γ namely, Λ θγ = ((2 Z + { θ } ) × { } ) ∪ ( { } × γ Z ) , where θ = 1 /q, for some q ∈ N , and γ is a positive real, AC (cid:0) Γ + , Λ θγ (cid:1) is infinite-dimensional whenever γ > q. Introduction
Heisenberg uniqueness pairs.
The uncertainty principle for Fourier trans-form states that a nonzero function and its Fourier transform both cannot betoo concentrated at the same time (for the details see [4, 14, 18]). The notionof Heisenberg uniqueness pair introduced recently by Hedenmalm and Montes-Rodr´ıguez as a version of this uncertainty principle. Further, the concept ofHeisenberg uniqueness pair has significant similarity with mutually annihilatingpairs of Borel measurable sets of positive measures. To describe this, consider apair of Borel measurable sets S , Σ ⊆ R . Then ( S , Σ) forms a mutually annihilat-ing pair if for any ϕ ∈ L ( R ) such that supp ϕ ⊂ S and whose Fourier transformˆ ϕ supported on Σ , implies ϕ is identically zero (for more details see [14]). Heisenberg uniqueness pair.
In [15], Hedenmalm and Montes-Rodr´ıguez proposesthe following: Let Γ be a smooth curve in R and Λ be a subset of R . Let X (Γ) be Date : September 22, 2020.2010
Mathematics Subject Classification.
Primary 42A10, 42B10; Secondary 35L10, 37A45.
Key words and phrases.
Ergodic theory, Fourier transform, Invariant measures, Klein-Gordon equation, Koopman operator, Perron-Frobenius operator. the space of all finite complex-valued Borel measures µ in R which are supportedon Γ and are absolutely continuous with respect to the arc length measure on Γ . For ( ξ, η ) ∈ R , the Fourier transform of µ is defined by(1.1) ˆ µ ( ξ, η ) = Z Γ e πi ( xξ + yη ) dµ ( x, y ) . Let AC (Γ , Λ) = { µ ∈ X (Γ) : ˆ µ | Λ = 0 } , then following [15], (Γ , Λ) is said tobe a
Heisenberg uniqueness pair (HUP) if AC (Γ , Λ) = { } . In this case, sinceΛ determine the measures µ ∈ X (Γ) , we say that Λ is a Fourier uniquenessset for Γ . The definition of Heisenberg uniqueness pair can be extended formore general measures but here we restrict our attention to those which areonly absolutely continuous. Heisenberg uniqueness pairs satisfy the followinginvariance properties:( inv -1) For any points u , v ∈ R , (cid:16) Γ + { u } , Λ + { v } (cid:17) is a HUP if and only if(Γ , Λ) is a HUP.( inv -2) (Γ , Λ) is a HUP if and only if (cid:16) T − (Γ) , T ∗ (Λ) (cid:17) is a HUP, where T : R → R be an invertible linear transform with adjoint T ∗ . The dual formulation. (Γ , Λ) is a HUP if and only if the subspace of all linearspan of the functions { e πi ( xξ + yη ) : ( ξ, η ) ∈ Λ } is weak-star dense in L ∞ (Γ) . Many examples of Heisenberg uniqueness pair have been obtained in the planeas well as in the higher dimensional Euclidean spaces. The Heisenberg unique-ness pairs (Γ , Λ) in which Γ is the union of any two parallel lines in the planewas investigated in [15]. In [2], Babot studied the HUP in which Γ is the certainsystem of three parallel lines. Later, the author has studied the HUP corre-sponding to a certain system of four parallel lines together with some algebraiccurves (see [10]). Further, the cases in which Γ is the union of a certain systemof finitely many parallel lines were studied in [3, 11]. The cases in which Γ isthe unit circle were independently investigated in [22, 23]. Later, Gonzal´ez Vieli(see [12]) generalized the cases for the circle to the higher dimension using theproperties of the Bessel functions J ( n +2 k − / ; k ∈ Z + . In [25], Srivastava studiedthe cases for the sphere in which Λ is the cone as well as the cone does notcontain in the zero sets of any homogeneous harmonic polynomial on R n . In [24],Sj¨olin investigated the HUP corresponding to the parabola while the author (see[7]) studied certain exponential surfaces and connect the notion of HUP to theEuclidean motion groups. The dynamical system approach was used in [20] tostudy the cases for hyperbola, polygon, ellipse, and graph of ϕ ( t ) = | t | α , when-ever α > . Later, Gr¨ochenig and Jaming [13] solved the cases corresponding tothe quadratic surface. In a recent article [8], the authors had extended the notionof Heisenberg uniqueness pair to the Heisenberg group. As a major developmentalong this direction, in [6, 15, 16, 17], the dynamics of Gauss-type maps, Ergodictheory, Klein-Gordon equation, and the Perron-Frobenius operators were usedto studying HUP which advanced the theory.In this paper, the problems (Theorem 1.7 and Theorem 1.13) we consider isinspired by the articles [6, 9, 15], and closely follows the methods of [6] with
OURIER NONUNIQUENESS SETS FOR THE HYPERBOLA 3 moderate modifications at appropriate places. We will follow definitions andnotation from [6] as much as possible.1.2.
Nonuniqueness sets for the hyperbola.
Let Γ = { ( x, y ) ∈ R : xy = 1 } be the hyperbola and µ ∈ X (Γ) , then there exists g ∈ L ( R , p /t dt ) suchthat for bounded and continuous function ϕ on R , Z Γ ϕ ( x, y ) dµ ( x, y ) = Z R \{ } ϕ ( t, /t ) g ( t ) p /t dt. In particular, for ( ξ, η ) ∈ R , the Fourier transform of µ can be expressed asˆ µ ( ξ, η ) = Z R \{ } e πi ( ξt + η/t ) f ( t ) dt, where f ( t ) := g ( t ) p /t ∈ L ( R ) . As a first known result on HUP, in [15], Hedenmalm and Montes-Rodr´ıguezhave studied that some lattice-cross in the plane is a
Fourier uniqueness set forthe hyperbola and have proved the following result.
Theorem 1.1. [15]
Let
Γ = { ( x, y ) ∈ R : xy = 1 } be the hyperbola and Λ β bethe lattice-cross Λ β := ( Z × { } ) ∪ ( { } × β Z ) , where β is a positive real. Then AC (Γ , Λ β ) = { } if and only if β ≤ . In [6], Canto-Mart´ın, Hedenmalm and Montes-Rodr´ıguez have studied thatsome lattice-cross in R , is a Fourier nonuniqueness set for the hyperbola andhave proved the following result.
Theorem 1.2. [6]
Let Γ be the hyperbola xy = 1 and Λ β be the lattice-cross Λ β := ( Z × { } ) ∪ ( { } × β Z ) , where β is a positive real. Then AC (Γ , Λ β ) isinfinite-dimensional for β > . Let M β be the subspace of all linear span of the functions { e n ( x ) := e πinx ; n ∈ Z } ∪ { e βn ( x ) := e πinβ/x ; n ∈ Z } in L ∞ ( R ) , where β is a positive real. Thecodimension of the weak-star closure of M β in L ∞ ( R ) is the dimension of itspre-annihilator space M ⊥ β := (cid:26) f ∈ L ( R ) : Z R f ( x ) e n ( x ) dx = Z R f ( x ) e βn ( x ) dx = 0 for all n ∈ Z (cid:27) . By dual formulation, Theorem 1.1 is equivalent to the following density result.
Theorem 1.3. [15]
The space M β is weak-star dense in L ∞ ( R ) if and only if < β ≤ . Similarly, by dual formulation, Theorem 1.2 is equivalent to the followingdensity result.
Theorem 1.4. [6] M ⊥ β is an infinite-dimensional subspace of L ( R ) for < β < ∞ . Next, we consider a rational perturbation of the lattice-cross Λ β , namely that(1.2) Λ θβ := (( Z + { θ } ) × { } ) ∪ ( { } × β Z ) , DEB KUMAR GIRI where θ = 1 /p, for some p ∈ N , and β is a positive real. Let Γ be the hyperbola xy = 1 , then the following result shows that (cid:0) Γ , Λ θβ (cid:1) is a Heisenberg uniquenesspair for 0 < β ≤ p. In other words, Λ := Λ θβ is a Fourier uniqueness set for thehyperbola Γ whenever β ≤ p. Theorem 1.5. [9] AC (cid:0) Γ , Λ θβ (cid:1) = { } if and only if < β ≤ p. Remark 1.6. (a) It is rather surprising that the condition on β depends on θ. The proof of Theorem 1.5 works along the same lines as in [15] but withmoderate modifications at appropriate places. The question is remains openfor irrational values of θ. (b) The notion of Heisenberg uniqueness pair may be extended for more generalfinite complex-valued Borel measures µ in R which are supported on Γ with-out assuming absolute continuity with respect to the arc length measure onΓ . But for Theorem 1.5, the measures µ must be absolutely continuous withrespect to the arc length measure on the hyperbola, without this assumption,Theorem 1.5 is not true.Next, we state a result of this paper which is a variant of Theorem 1.2. Theorem 1.7.
Let Γ be the hyperbola xy = 1 and Λ θβ be the lattice-cross definedin (1.2). Then AC (cid:0) Γ , Λ θβ (cid:1) is infinite-dimensional for β > p. Remark 1.8. (a) Theorem 1.7 asserts that Λ := Λ θβ is a Fourier nonuniquenessset for the hyperbola xy = 1 whenever β > p. The presence of θ showing upin the condition of β which is somewhat unexpected. The proof of Theorem1.7 works along the same lines as in [6] but with modifications at appropriateplaces.(b) As a corollary to Theorem 1.7, let Γ be the hyperbola xy = 1 and Λ ζβ bethe set Λ ζβ := (( Z + { ζ } ) × { } ) ∪ ( { } × β Z ) , where ζ = r/p, for some p ∈ N and r ∈ Z with gcd(p,r)=1 and β is a positive real. Then AC (cid:16) Γ , Λ ζβ (cid:17) isinfinite-dimensional for β > p. (c) Let Γ be the hyperbola xy = 1 , then any µ ∈ AC (cid:0) Γ , Λ θβ (cid:1) , u := ˆ µ is a solutionof the one-dimensional Klein-Gordon equation: ( ∂ ξ ∂ η + π ) u ( ξ, η ) = 0 in thesense of distributions. Theorem 1.7 says that for β > p, the solution spaceof the above partial differential equation is infinite-dimensional.Let F β be the subspace of all linear span of the functions e pn ( x ) , e βn ( x ); n ∈ Z in L ∞ ( R ) , where e pn ( x ) := e πi ( n +1 /p ) x and e βn ( x ) := e πinβ/x with p ∈ N and β is apositive real. The codimension of the weak-star closure of F β in L ∞ ( R ) is thedimension of its pre-annihilator space F ⊥ β . By dual formulation, Theorem 1.5 isequivalent to the following result.
Theorem 1.9. [9]
The space F β is weak-star dense in L ∞ ( R ) if and only if < β ≤ p. Similarly, Theorem 1.7 is equivalent to the following density result.
Theorem 1.10. F ⊥ β is an infinite-dimensional subspace of L ( R ) for p < β < ∞ . OURIER NONUNIQUENESS SETS FOR THE HYPERBOLA 5
Nonuniqueness sets for the branch of the hyperbola.
Let Γ + = { ( x, y ) ∈ R : xy = 1 , x > } be the branch of the hyperbola and µ ∈ X (Γ + ) , then there exists g ∈ L ( R + , p /t dt ) such that for bounded and continuousfunction ϕ on R , Z Γ + ϕ ( x, y ) dµ ( x, y ) = Z R + \{ } ϕ ( t, /t ) g ( t ) p /t dt. In particular, for ( ξ, η ) ∈ R , the Fourier transform of µ can be expressed asˆ µ ( ξ, η ) = Z R + \{ } e πi ( ξt + η/t ) f ( t ) dt, where f ( t ) := g ( t ) p /t ∈ L ( R + ) . In [6], Canto-Mart´ın, Hedenmalm and Montes-Rodr´ıguez have studied that somelattice-cross in R , is a Fourier nonuniqueness set for Γ + and have proved thefollowing result. Theorem 1.11. [6]
Let Γ + = { ( x, y ) ∈ R : xy = 1 , x > } be the branch ofthe hyperbola and Λ γ be the lattice-cross Λ γ := (2 Z × { } ) ∪ ( { } × γ Z ) , where γ is a positive real. Then AC (Γ + , Λ γ ) is infinite-dimensional for γ > . By dual formulation, Theorem 1.11 is equivalent to the following result.
Theorem 1.12. [6]
Let N γ be the subspace of all linear span of the functions { e n ( x ) := e πinx ; n ∈ Z } ∪ { e γn ( x ) := e πinγ/x ; n ∈ Z } in L ∞ ( R + ) , where γ isa positive real. Then the pre-annihilator space N ⊥ γ is infinite-dimensional for γ > . Next, we state a result of this paper which is a variant of Theorem 1.11.
Theorem 1.13.
Let Γ + = { ( x, y ) ∈ R : xy = 1 , x > } be the branch of thehyperbola and Λ θγ be the lattice-cross Λ θγ := ((2 Z + { θ } ) × { } ) ∪ ( { } × γ Z ) , where θ = 1 /q, q ∈ N , and γ is a positive real. Then AC (Γ + , Λ θγ ) is infinite-dimensional for γ > q. Remark 1.14. (a) Theorem 1.13 asserts that Λ := Λ θγ is a Fourier nonunique-ness set for the branch Γ + whenever γ > q. The presence of θ showing up inthe condition of γ which is somewhat unexpected. The proof of Theorem 1.3works along the same lines as in [6] but with modifications at appropriateplaces. The question is still open when θ is irrational.(b) Let Λ θγ be the lattice-cross ((2 Z + { θ } ) × { } ) ∪ ( { } × γ Z ) , where θ =1 /q, q ∈ N , and γ is a positive real. It seems likely, that AC (cid:0) Γ + , Λ θγ (cid:1) = { } if and only if γ < q, and for the critical case γ = q, AC (cid:0) Γ + , Λ θγ (cid:1) is one-dimensional in analogy with the results in [16]. The question is still open.By duality, Theorem 1.13 is equivalent to the following density result. Theorem 1.15.
Let K γ be the subspace of all linear span of the functions { e qn ( x ) := e πi ( n +1 /q ) x ; n ∈ Z } ∪ { e γn ( x ) := e πinγ/x ; n ∈ Z } in L ∞ ( R + ) , where q ∈ N and γ is a positive real. Then the pre-annihilator space K ⊥ γ is infinite-dimensional for γ > q. DEB KUMAR GIRI
The Perron-Frobenius operators.
In this section, we recall the defini-tions and notation related to the Perron-Frobenius operators associated with a C -smooth piecewise monotonic transform from ([6], Section 3) as far as possible.The spectral property of the Perron-Frobenius operators has played a significantrole in the HUP.1.4.1. Perron-Frobenius operators on bounded intervals.
Let I ⊂ R be a closed and bounded interval and m be the Lebesgue measuredefined on the σ -algebra of I. Following ([6], Definition 3.2), a measurable map τ : I → I is said to be a ”partially filling C -smooth piecewise monotonic trans-form” if there exists a countable collection of pairwise disjoint open intervals say, { I u } u ∈U , where U is the index set, such that the following holds:(i) m ( I \ S { I u : u ∈ U } ) = 0 , (ii) for any u ∈ U , the map τ u := τ | I u is strictly monotone and can be extendedto a C -smooth function on ¯ I u with τ ′ u = 0 on I ou , (iii) there exists a positive number say, δ such that m ( τ ( I u )) ≥ δ for all u ∈ U . Following ([6], Definition 3.1), τ is said to be a ”filling C -smooth piecewisemonotonic transform” , if the above conditions (i),(ii) holds for τ along with( iii ) ′ for every u ∈ U , the map τ u : ¯ I u → I is onto.Observe that, condition ( iii ) ′ is much stronger than condition ( iii ) . In theabove context, each I u is called a ”fundamental interval” and τ u is the corre-sponding ”branch” .The Koopman operator C τ : L ∞ ( I ) → L ∞ ( I ) corresponding to a measurablemap τ : I → I is defined by letting C τ [ ϕ ] = ϕ ◦ τ. The Perron-Frobenius operator P τ : L ( I ) → L ( I ) is the pre-dual adjoint (Ba-nach space dual) of C τ is given by(1.3) hP τ [ ψ ] , ϕ i I = h ψ, C τ [ ϕ ] i I , where ψ ∈ L ( I ) and ϕ ∈ L ∞ ( I ) . The operator P τ is linear and a norm contraction on L ( I ) , therefore, its spectrum σ ( P τ ) is contained in the closed unit disk ¯ D = { λ ∈ C : | λ | ≤ } . The spectral decomposition of Perron-Frobenius operators.Functions of bounded variation in one variable.
Let I ⊂ R be a closed boundedinterval. For any function h : I → C , the pointwise variation of h in I is definedby letting pV ( h, I ) := sup t ,...,t n ∈ I,t < ··· For any h ∈ BV( I ) , from ([1], Theorem 3.27) we get that the infimum in theexpression of eV ( h, I ) is achieved. Hence the space BV( I ) equipped with thenorm k h k BV = k h k L ( I ) + eV ( h, I ) , h ∈ BV( I ) , becomes a Banach space.Next, we state the spectral decomposition of Perron-Frobenius operators P τ associated to τ on I. Let ∂ ¯ D denote the boundary of the closed unit disk ¯ D andthe point spectrum of P τ is denoted by σ point ( P τ ) . Then the following spectraldecomposition for P τ is stated in ([6], Theorem C) which is a consequence ofthe Ionescu-Tulcea and Marinescu theorem (for details see [5, 19]). Recall thedefinition of U m♯ ; m ≥ m = 1 , we have U m♯ = U . Theorem A. Let τ : I → I is a partially filling C -smooth piecewise monotonictransform such that(i) [uniform expansiveness] there exists an integer m ≥ ǫ > | ( τ m ) ′ ( x ) | ≥ ǫ for all x ∈ S { I u : u ∈ U m♯ } , (ii) [second derivative condition] there exists M > | τ ′′ ( x ) | ≤ M | τ ′ ( x ) | for all x ∈ S { I u : u ∈ U } , then Λ τ := σ point ( P τ ) ∩ ∂ ¯ D is a finite set namely, Λ τ = { α , . . . , α s } and oneof the eigenvalues is 1, say α = 1 . Let E i denotes the eigenspace of P τ forthe eigenvalue α i , then E i is finite-dimensional and E i is contained in BV( I ) . Inaddition, P nτ [ h ] = s X i =1 α ni P τ,i [ h ] + Z nτ [ h ] , h ∈ L ( I ) , n = 1 , , . . . , where the operators P τ,i are projections onto E i , and the operator Z τ acts bound-edly on L ( I ) as well as on BV( I ) . Moreover, Z τ acting on BV( I ) has spectralradius < . Proof of Theorem 1.10 Dynamics of a Gauss-type map. A Gauss-type map. For t ∈ R , the expression { t } represent the uniquenumber in ( − , 1] such that t − { t } ∈ Z . We consider a Gauss-type map U onthe interval ( − p, p ]; p ∈ N which is defined by letting U ( x ) := ( p (cid:8) − px (cid:9) , x = 0 , x = 0 . For u ∈ Z ∗ = Z \{ } , the map U can be explicitly written as U ( x ) = p (2 u − p/x )whenever p u +1 < x ≤ p u − , and hence U : (cid:0) p u +1 , p u − (cid:3) → ( − p, p ] is one-to-oneand for x ∈ ( − p, p ] \ p Z +1 , the derivative of U is U ′ ( x ) = p x . For a continuous2 p -periodic function ϕ on R and a finite complex-valued Borel measure ν on( − p, p ] , the integral R ( − p,p ] ϕ ( x ) dν ( x ) is well-defined. The above integral makessense for all pseudo-continuous functions on ( − p, p ] . DEB KUMAR GIRI Note that for a pseudo-continuous function ϕ on ( − p, p ] , ϕ ◦ U is pseudo-continuous . Given λ ∈ C , a finite complex Borel measure ν on ( − p, p ] is ( U, λ )-invariant provided that Z ( − p,p ] ϕ ( U ( x )) dν ( x ) = λ Z ( − p,p ] ϕ ( x ) dν ( x )holds for all pseudo-continuous functions ϕ, that is, λν = ν ( { } ) δ + P u ∈ Z ∗ ν u , where δ t denote the point mass at t, and dν u ( x ) = dν (cid:16) p pu − x (cid:17) . It is easy to seethat, for | λ | > , there are no ( U, λ )-invariant measures except zero measure.In this work, we mainly study the properties the following map which is asso-ciated to the parameter β. For 0 < β < ∞ , the Gauss-type map U β : ( − p, p ] → ( − p, p ] is defined by letting(2.1) U β ( x ) := ( p (cid:8) − βx (cid:9) , x = 0 , x = 0 . For u ∈ Z ∗ = Z \{ } , on the interval (cid:0) β u +1 , β u − (cid:3) , the map U β can be expressed as U β ( x ) = p (2 u − β/x ) . In particular, U p = U. For λ ∈ ∂ ¯ D , a finite complex Borelmeasure ν on ( − p, p ] is ( U β , λ )-invariant provided that λν = ν ( { } ) δ + P u ∈ Z ∗ ν u , where dν u ( x ) = dν (cid:16) pβ pu − x (cid:17) . Dynamical properties of U β for β > p. Denote β := β/p > . In this section, we observe that U β is a partiallyfilling C -smooth piecewise monotonic transform for β > . We need to find aunique absolutely continuous invariant probability measure for U β having positivedensity.Let U := U β denote the index set which contain the points u ∈ Z ∗ so that theassociate fundamental interval is nonempty, that is, I u := (cid:18) β u + 1 , β u − (cid:19) ∩ ( − p, p ) = ∅ . If β is an odd integer, then { β } = 1 . In this case, it is easy to see that forall u ∈ U , where U be the set of all nonzero integers with | u | ≥ ( β + 1) , thefundamental intervals are given by(2.2) I u := (cid:18) β u + 1 , β u − (cid:19) so that U β ( I u ) = ( − p, p ) for all u ∈ U . Thus in this particular case, U β fulfillthe ”filing” condition for all u ∈ U . If β is not an odd integer, then − < { β } < . Denote u := ( β − { β } ) , then u ∈ Z with u ≥ . A simple calculation shows that for all u ∈ U \ {± u } , OURIER NONUNIQUENESS SETS FOR THE HYPERBOLA 9 where U be the set of all nonzero integers with | u | ≥ u , the fundamental intervalsare given by(2.3) I u := (cid:18) β u + 1 , β u − (cid:19) so that U β ( I u ) = ( − p, p ) for all u ∈ U \ {± u } . Thus in this case, U β fulfill the”filing” condition for all u ∈ U except two branches corresponding to {± u } . The edge fundamental intervals corresponding to ± u are explicitly given by(2.4) I u := (cid:18) β u + 1 , p (cid:19) , I − u := (cid:18) − p, − β u + 1 (cid:19) . In view of the above facts, we conclude that U β is a partially filling C -smoothpiecewise monotonic transform for β > . The map U β satisfy the ”uniform expansiveness” condition with m = 1 , be-cause of the derivative | U ′ β ( x ) | = pβx ≥ β > x ∈ { I u : u ∈ U } . Also, we have ”second derivative condition” due to | U ′′ β ( x ) | ≤ p | U ′ β ( x ) | for all x ∈ { I u : u ∈ U } . Our aim is to find a unique absolutely continuous invariantmeasure for U β which has a positive density. Since U β is a partially filling C -smooth piecewise monotonic transform , (see [6], p. 45, Remark 5.1(b)), thereexists a U β -invariant absolutely continuous probability measure, but it may notbe unique. Although, form ([6], p. 45, Remark 5.2) we can get the uniquenessfor filling C -smooth piecewise monotonic transform . Also, U β may not alwaysbe a Markov map, therefore, to get the uniqueness of the absolutely continuous U β -invariant measure, we have to prove the condition ( iii ) of Adler’s theoremwhich is stated in ([6], p. 46, Theorem D), for the details see [5, 21].2.1.3. The iterates of an interval. The following result will help to get the unique ergodic U β -invariant absolutelycontinuous probability measure with positive density. Lemma 2.1. ( p < β < ∞ ) Let J ⊂ [ − p, p ] be any nonempty open interval,then for sufficiently large positive integers, namely that n ≥ n , we have C p :=( − p, p ) ⊂ U nβ ( J ) . Proof. We want to show that U nβ ( J ) will cover the open interval C p for sufficientlylarge positive integers namely, n ≥ n . The proof will be carried out in thefollowing cases. 1. The case β is an odd integer. In this case β ≥ β / > . The fundamental intervals I u are given by(2.2) for all u ∈ U , where U be the set of all nonzero integers with | u | ≥ ( β +1) . Here, we want to show that C p ⊂ U nβ ( J ) for sufficiently large n. There are twopossibilities: (i). Suppose J contains one of the fundamental intervals say, I u for some u ∈ U , then it follows that C p = U β ( I u ) ⊂ U β ( J ) . (ii). If none of the fundamental intervals are contained in J , then we have thefollowing possibilities: (a). Suppose J is contained in one of the fundamental intervals namely, I u , u ∈U , then by uniform expansiveness condition of U β , we have m ( J ) ≥ β m ( J ) ≥ β m ( J ) , where J := U β ( J ) . (b). Suppose J has nonempty intersection with two neighbouring fundamentalintervals say I u , I u ′ and J is contained in the closure of I u ∪ I u ′ . In this case, oneof the sets J ∩ I u , J ∩ I u ′ say, J ∩ I u has length at least m ( J ) . It follows thatfor J := U β ( J ∩ I u ) with J ⊂ U β ( J ) , we have m ( J ) = m (cid:16) U β ( J ∩ I u ) (cid:17) ≥ β m ( J ∩ I u ) ≥ β m ( J ) . For both the cases (a) and (b) , there exist an interval J which is contained in U β ( J ) and m ( J ) ≥ β m ( J ) . Next, we consider J in place of J , and there-fore we get an interval namely, J such that m ( J ) ≥ β m ( J ) ≥ (cid:0) β (cid:1) m ( J ) . We repeat this process, and hence we get an increasing sequence of intervalsnamely, J , J , J , . . . such that m ( J l ) ≥ (cid:0) β (cid:1) l m ( J ) with J l ⊆ U lβ ( J ) . Since thelength of J l depends on l, after finitely many steps say, l = l we must stopthis process, because J l will contain one of the fundamental intervals, and hence C p ⊂ U β ( J l ) ⊆ U l +1 β ( J ) . 2. The case β is not an odd integer. Since { β } ∈ ( − , , it follows that 2 u − < β < u + 1 . The fundamentalintervals I u are given by (2.3) for all u ∈ U \ {± u } , where U be the set of allnonzero integers with | u | ≥ u . The edge fundamental intervals I u , I − u aregiven by (2.4). For every u ∈ U , the map U β is given by U β ( x ) = p (2 u − β/x )for all x ∈ I u . J is an edge fundamental interval. Assume that J = I − u , and the case J = I u is similar. In this case, we want to show that C p ⊂ U nβ ( J ) for sufficiently large n. We first observe that(2.5) U β ( J ) = U β ( I − u ) = (cid:16) p ( β − u ) , p (cid:17) ⊃ I ′ u := (cid:18) p ( β − u ) , β u + 1 (cid:19) . Then there are two possibilities: (i). If ( β − u ) ≤ β / (2 u + 3) , then we have I u +1 ⊂ I ′ u . Therefore, C p = U β ( I u +1 ) ⊂ U β ( I ′ u ) ⊂ U β ( I − u ) = U β ( J ) . (ii). If ( β − u ) > β / (2 u + 3) , then I ′ u ⊂ I u +1 and the point p ( β − u ) ∈ I u +1 . Next, we claim that for y ∈ I ′′ u := (cid:16) β u +3 , p ( β − u ) i ⊂ I u +1 , thereexists a positive number β ′ with β ≥ β ′ > , which depends only on β andclosest to the point 1 such that m (cid:16) U β ( I y ) (cid:17) ≥ β ′ m (cid:16) ( y, p ) (cid:17) , where I y := (cid:16) y, β u +1 (cid:17) . Since U β ( I y ) = (cid:16) p (2 u + 2 − β/y ) , (cid:17) , it is enough toshow that for y ∈ I ′′ u , (2.6) β ′ y + pβ/y ≥ p (2 u + 1) + pβ ′ . OURIER NONUNIQUENESS SETS FOR THE HYPERBOLA 11 By the change of variables y := py ′ , it is equivalent to show that for y ′ ∈ I ′′′ u := (cid:16) β u +3 , ( β − u ) i , we have(2.7) β ′ y ′ + β /y ′ ≥ u + 1 + β ′ . Now, (2.7) is same as ([6], p. 54, Equation 7.4). Thus we conclude that β ′ exists and sufficiently close to 1 so that the minimum exists in (2.6) at y := p ( β − u ) . Note that β > y > β / (2 u + 3) . In particular, wehave m (cid:16) ( U β ( y ) , p ) (cid:17) ≥ β ′ m (cid:16) ( y , p ) (cid:17) . If we consider J := U β ( J ) = ( y , p )and J := U β ( I ′ u ) = (cid:16) U β ( y ) , p (cid:17) , then J ⊂ U β ( J ) = U β ( J ) with m ( J ) ≥ β ′ m ( J ) . If U β ( y ) ≤ β/ (2 u + 3) , then I u +1 ⊂ J , and hence we are done,because C p = U β ( I u +1 ) ⊂ U β ( J ) = U β ( I ′ u ) ⊂ U β ( I − u ) = U β ( J ) . If U β ( y ) >β/ (2 u + 3) , then repeat the same argument to get a bigger interval say, J withright end point 1 so that I u +1 is contained in J . This completes proof of thecase J = I − u . J ⊂ [ − p, p ] is an arbitrary nonempty open interval. Recall that 2 u − < β < u + 1 and the edge fundamental intervals are givenby (2.4). Let the point x ∈ I u is given by(2.8) x := (2 u + 1) β u (2 u + 1) + β . A simple calculation gives pβ/x > . Since on the fundamental interval I u theGauss-type map is given by U β ( x ) = 2 u − β/x, the point x ∈ I u has the propertythat(2.9) U β ( I x ) = I − u and U β ( I − x ) = I u , where I x := (cid:18) β u + 1 , x (cid:19) and I − x := (cid:18) − x , − β u + 1 (cid:19) . Moreover, for x ∈ [ − x , x ] ∩ S { I u : u ∈ U } , we have U ′ β ( x ) ≥ pβ/x > . Therefore, if we write β ′′ := min n β , pβ x o , then β ′′ > . In this case, we showthat C p ⊂ U nβ ( J ) for sufficiently large n. There are two possibilities: (i). Suppose J contains one of the fundamental intervals say, I u for some u ∈U \ {± u } , then it directly follows that C p = U β ( I u ) ⊂ U β ( J ) . If J contains theedge fundamental intervals, then also we are done by the case above. (ii). If none of the fundamental intervals are contained in J , then we have thefollowing possibilities: (a). Suppose J ⊂ I u for some u ∈ U , then by uniform expansiveness conditionof U β , we get that m ( J ) ≥ β m ( J ) ≥ β ′′ m ( J ) , where J := U β ( J ) . (b). Suppose J has nonempty intersection with two neighbouring fundamentalintervals say I u , I u ′ and J is contained in the closure of I u ∪ I u ′ . There are twopossibilities: (b1). Assume that J ⊂ [ − x , x ] . In this case, one of the sets J ∩ I u , J ∩ I u ′ say, J ∩ I u has length at least m ( J ) . It follows that for J := U β ( J ∩ I u ) with J ⊂ U β ( J ) ,m ( J ) = m (cid:16) U β ( J ∩ I u ) (cid:17) ≥ pβx m ( J ∩ I u ) ≥ pβ x m ( J ) ≥ β ′′ m ( J ) . For both the cases (a) and (b1) , there exist an interval J which is contained in U β ( J ) with m ( J ) ≥ β ′′ m ( J ) . Next, we consider J in place of J , and thereforewe get a bigger interval namely, J such that m ( J ) ≥ β ′′ m ( J ) ≥ β ′′ m ( J ) . We repeat this process and hence we get an increasing sequence of intervalsnamely, J , J , J , . . . such that m ( J l ) ≥ β ′′ l m ( J ) with J l ⊆ U lβ ( J ) . Since thelength of J l depends on l, after finitely many steps say, l = l we must stopthis process, because J l will contain one of the fundamental intervals, and hence C p ⊂ U β ( J l ) ⊆ U l +1 β ( J ) . (b2). It only remains the case when J is not contained in [ − x , x ] . Then¯ I x ⊂ J ∩ I u or ¯ I − x ⊂ J ∩ I − u , and hence we are done, because in view of(2.9) we have one of the following:(b21). C p = U β ( I − u ) = U β ( I x ) ⊂ U β ( J ∩ I u ) ⊂ U β ( J ) , (b22). C p = U β ( I u ) = U β ( I − x ) ⊂ U β ( J ∩ I − u ) ⊂ U β ( J ) . This completes the proof of Lemma 2.1. (cid:3) Characterization of the pre-annihilator space F ⊥ β . Periodic and inverted periodic functions. Let L ∞ p ( R ) denote the space of all functions f ∈ L ∞ ( R ) such that the map x e − πix/p f ( x ) is 2-periodic. Then the weak-star closure in L ∞ ( R ) of thelinear span of the functions { e pn ( x ) := e πi ( n +1 /p ) x ; n ∈ Z } equals to L ∞ p ( R ) . Let L ∞ β ( R ) denote the space of all functions f ∈ L ∞ ( R ) such that the map x f ( β/x ) is 2-periodic. Then the weak-star closure in L ∞ ( R ) of the linearspan of the functions { e βn ( x ) := e πinβ/x ; n ∈ Z } equals to L ∞ β ( R ) . Observe that the functions in L ∞ p ( R ) are defined freely on [ − p, p ] , and due toperiodicity they are uniquely determined on R \ [ − p, p ] . Similarly, the functions in L ∞ β ( R ) are defined freely on R \ [ − β, β ] , and due to periodicity they are uniquelydetermined on [ − β, β ] . For E ⊆ R , the function χ E denote the characteristicfunction of E on R . This observations motivate to define the operators S p , T β asfollows.The operator S p : L ∞ ([ − p, p ]) → L ∞ ( R \ [ − p, p ]) is defined byS p [ ϕ ]( x ) = ϕ ( p { x/p } ) χ R \ [ − p,p ] ( x ) , where ϕ ∈ L ∞ ([ − p, p ]) . The operator T β : L ∞ ( R \ [ − β, β ]) → L ∞ ([ − β, β ]) isdefined by T β [ ψ ]( x ) = ψ (cid:18) β { β/x } (cid:19) χ [ − β,β ] \{ } ( x ) , OURIER NONUNIQUENESS SETS FOR THE HYPERBOLA 13 where ψ ∈ L ∞ ( R \ [ − β, β ]) . Now, in terms of the operators S p , T β the functionsspace L ∞ p ( R ) and L ∞ β ( R ) are given by L ∞ p ( R ) = n ϕ + S p [ ϕ ] : ϕ ∈ L ∞ ([ − p, p ]) o ,L ∞ β ( R ) = n ψ + T β [ ψ ] : ψ ∈ L ∞ ( R \ [ − β, β ]) o . The Perron-Frobenius operator. For p < β < ∞ , the Koopman operator C β : L ∞ ([ − p, p ]) → L ∞ ([ − p, p ]) associated to U β be the map C β [ ϕ ]( x ) = ϕ ◦ U β ( x ) , x ∈ [ − p, p ] . The predual adjoint of C β is the Perron-Frobenius operator P β : L ([ − p, p ]) → L ([ − p, p ]) associated to U β is given by P β [ h ]( x ) = X u ∈ Z ∗ pβ (2 pu − x ) h (cid:18) pβ pu − x (cid:19) , x ∈ [ − p, p ] . The operator P β is linear and a norm contraction on L ([ − p, p ]) . Thus the pointspectrum σ point ( P β ) of P β is contained in the closed unit disk ¯ D . Here, we usethe notation C β in place of C U β and P β for P U β . Next, we build up a connection between the operators S p , T β and C β , so thatwe can study the pre-annihilator space F ⊥ β via the properties of the Perron-Frobenius operators for p < β < ∞ . To do this, we need to define some restrictionoperators. For a measurable set E ⊆ R with m ( E ) > , we denote L s ( E ) theclosed subspace of L s ( R ) by extending the functions vanish on R \ E, where s = 1 , ∞ . For p < β < ∞ , consider the following restriction operators: R : L ∞ ( R \ [ − p, p ]) → L ∞ ( R \ [ − β, β ]) , R : L ∞ ([ − β, β ]) → L ∞ ([ − p, p ]) , R : L ∞ ([ − β, β ]) → L ∞ ([ − β, β ] \ [ − p, p ]) , R : L ∞ ( R \ [ − p, p ]) → L ∞ ([ − β, β ] \ [ − p, p ]) . Then the pre-dual adjoints (Banach space dual) are the maps R ∗ , R ∗ , R ∗ andR ∗ defined on the corresponding L -spaces. A simple calculation shows that C β = R T β R S p whenever β > p, and hence P β = S ∗ p R ∗ T ∗ β R ∗ . Proposition 2.2. For p < β < ∞ , suppose f ∈ L ( R ) such that f = f + f + f , where f ∈ L ([ − p, p ]) , f ∈ L ([ − β, β ] \ [ − p, p ]) , and f ∈ L ( R \ [ − β, β ]) . Then f ∈ F ⊥ β if and only if ( i ) ( I − P β ) f = S ∗ p ( − R ∗ + R ∗ T ∗ β R ∗ ) f , where I is theidentity operator on L ([ − p, p ]) , and ( ii ) f = − T ∗ β R ∗ f − T ∗ β R ∗ f . Proof. The proof of Proposition 2.2 works along the same lines as in the proofof ([6], p. 52, Proposition 6.1), hence omitted. (cid:3) Remark 2.3 (0 < β ≤ p ) . The composition operator T β S p : L ∞ ([ − p, p ]) → L ∞ ([ − p, p ]) is given byT β S p [ ϕ ]( x ) = ϕ (cid:18) p (cid:26) β { β/x } (cid:27) (cid:19) χ E β ( x ) , ϕ ∈ L ∞ ([ − p, p ]) , where β = β/p and E β = n x ∈ ( − β, β ] \ { } : β { β/x } ∈ R \ ( − , o , and theweighted Koopman operator C β : L ∞ ([ − p, p ]) → L ∞ ([ − p, p ]) associated to U β be the map C β [ ϕ ]( x ) = ϕ ◦ U β ( x ) χ [ − β,β ] ( x ) , where x ∈ R . The predual adjointof C β is the Perron-Frobenius operator P β : L ([ − p, p ]) → L ([ − p, p ]) is givenby P β [ h ]( x ) = P u ∈ Z ∗ pβ (2 pu − x ) h (cid:16) pβ pu − x (cid:17) . As the operator P β is linear and a normcontraction, the point spectrum σ point ( P β ) of P β is contained in ¯ D . Observe thatT β S p = C β . In [9], it has shown that λ ∈ ∂ ¯ D is not an eigenvalue of P β whenever0 < β ≤ p, which in turn implies that AC (cid:0) Γ , Λ θβ (cid:1) = { } for 0 < β ≤ . Exterior spectrum of P β , β > p and the proof of Theorem 1.10. Next, we study the exterior spectrum of the Perron-Frobenius operator P β for p < β < ∞ . We know from Theorem A that 1 is an eigenvalue of P β and theassociated eigenfunction is in BV([ − p, p ]) . The proof of Theorem 2.4 gives usthat one of the eigenfunctions for the eigenvalue 1 must be positive, and it canbe normalized by a suitable constant so that we get the positive density of anergodic U β -invariant absolutely continuous probability measure. Theorem 2.4. Let p < β < ∞ , then α = 1 is a simple eigenvalue of P β , and isthe only eigenvalue of P β contained in ∂ ¯ D . Moreover, the eigenfunctions for α =1 are nonzero scalar multiple of ̺ , where ̺ dm is the unique ergodic U β -invariantabsolutely continuous probability measure with ̺ > almost everywhere.Proof. Here, Lemma 2.1 will help to show that ̺ > (cid:3) For p < β < ∞ , the Gauss-type map ˜ U β : [ − β, β ] → [ − β, β ] is defined byletting ˜ U β ( x ) := ( p (cid:8) − βx (cid:9) , x = 0 , x = 0 . The Koopman operator ˜ C β : L ∞ ([ − β, β ]) → L ∞ ([ − β, β ]) associated to ˜ U β bethe map ˜ C β [ ϕ ]( x ) = ϕ ◦ ˜ U β ( x ) , x ∈ [ − β, β ] . Then the pre-dual adjoint of ˜ C β is the Perron-Frobenius operator ˜ P β : L ([ − β, β ]) → L ([ − β, β ]) associated to˜ U β . A simple calculation shows that ˜ C β = T β R S p R whenever β > p, and hence˜ P β = R ∗ S ∗ p R ∗ T ∗ β . It is easy to see that ˜ U β is a partially filling C -smooth piecewisemonotonic transform for β > p. Also, ˜ U β satisfy the uniform expansiveness condition with m = 2 , and second derivative condition . Therefore, it followsfrom Theorem A that ˜ P β maps BV([ − β, β ]) into BV([ − β, β ]) . Now, followingthe proof of ([6], Lemma 8.1), for any f ∈ BV([ − β, β ] \ [ − p, p ]) , we have(2.10) − S ∗ p R ∗ f + S ∗ p R ∗ T ∗ β R ∗ f ∈ BV([ − p, p ]) . To complete the proof of Theorem 1.10, we prove the following result. Althoughthe proof of Theorem 2.5 follows the same lines as in ([6], Theorem 8.2), we writeit here for the sake of completeness. OURIER NONUNIQUENESS SETS FOR THE HYPERBOLA 15 Theorem 2.5. For p < β < ∞ , there exits a bounded linear operator B : BV ([ − β, β ] \ [ − p, p ]) → L ( R ) such that the range of B is infinite-dimensional, and containedin F ⊥ β . Moreover, the range of B is contained in the weighted L -space L ( R , ω ) , where ω ( x ) = 1 + x . Proof. We actually prove more precise statement, namely that, there exits abounded linear operator B : BV([ − β, β ] \ [ − p, p ]) → L ( R ) such that B f ( x ) = f ( x ) a.e. x ∈ [ − β, β ] \ [ − p, p ] , and for all f ∈ BV([ − β, β ] \ [ − p, p ]) . Moreover, B has infinite-dimensional range which is contained in F ⊥ β . In view of Theorem A and Theorem 2.4, we have the following spectral de-composition for the Perron-Frobenius operator P β associated to U β :(2.11) P nβ [ h ] = {h h, φ i [ − p,p ] } ̺ + Z nβ [ h ] , n = 1 , , . . . , where h ∈ L ([ − p, p ]) , and φ ∈ L ∞ ([ − p, p ]) such that h ̺ , φ i [ − p,p ] = 1 . Also, ̺ is the positive density of the ergodic U β -invariant absolutely continuous prob-ability measure on [ − p, p ] , and we have ̺ ∈ BV([ − p, p ]) , in addition, we cannormalized by a suitable constant so that h ̺ , i [ − p,p ] = 1 . Moreover, Z β actson BV([ − p, p ]) and its spectral radius smaller than 1 . In particular, we have Z β [ ̺ ] = 0 , because ̺ is invariant under P β . Observe that, Z nβ [ h ] → n → ∞ . Next, we claim that φ is the constant function 1 almost everywhere on [ − p, p ] . To see this, let h ∈ BV([ − p, p ]) , then by (2.11) we infer that h h, i [ − p,p ] = h h, C nβ [1] i [ − p,p ] = hP nβ [ h ] , i [ − p,p ] (2.12) = h h, φ i [ − p,p ] h ̺ , i [ − p,p ] + hZ nβ [ h ] , i [ − p,p ] = h h, φ i [ − p,p ] + hZ nβ [ h ] , i [ − p,p ] → h h, φ i [ − p,p ] , as n → ∞ . It is well known that BV([ − p, p ]) is dense in L ([ − p, p ]) . Thus from (2.12) getthe claim. Further, as soon as φ = 1 , (2.11) can be rewrite as(2.13) P nβ [ h ] = {h h, i [ − p,p ] } ̺ + Z nβ [ h ] , n = 1 , , . . . , where h ∈ L ([ − p, p ]) , and from (2.12) we get that(2.14) hZ nβ [ h ] , i [ − p,p ] = 0; h ∈ L ([ − p, p ]) , n = 1 , , . . . . Now, we are in a position to construct an extension operator B from BV([ − β, β ] \ [ − p, p ]) onto L ( R ) . To do so, pick an arbitrary f ∈ BV([ − β, β ] \ [ − p, p ]) , thenfrom (2.10), we know that − S ∗ p R ∗ f + S ∗ p R ∗ T ∗ β R ∗ f ∈ BV([ − p, p ]) . Since Z β actson BV([ − p, p ]) has spectral radius smaller than 1 , I − Z β is invertible and hence,we define the operator B by letting(2.15) B f := ( I − Z β ) − ( − S ∗ p R ∗ + S ∗ p R ∗ T ∗ β R ∗ ) f ∈ BV([ − p, p ]) . A simple calculation gives h− S ∗ p R ∗ f + S ∗ p R ∗ T ∗ β R ∗ f , i [ − p,p ] = 0 . If we denote f := B f , then by (2.14) and (2.15),(2.16) h f , i [ − p,p ] = (cid:10)(cid:0) I − Z β (cid:1) [ f ] , (cid:11) [ − p,p ] = 0 . Next, we define the operator B on BV([ − β, β ] \ [ − p, p ]) by letting(2.17) B f := − T ∗ β R ∗ f − T ∗ β R ∗ f ∈ L ( R \ [ − β, β ]) . We write f := B f . Now, we define the operator B : BV([ − β, β ] \ [ − p, p ]) → L ( R ) by letting B f := f + f + f ∈ L ( R ) , with the understanding that each f k ; k = 1 , , R by con-sidering zero outside of their domain of definition. Then the bounded andlinear operator B is clearly an extension operator, in the sense that for all f ∈ BV([ − β, β ] \ [ − p, p ]) , B f ( x ) = f ( x ); a.e. x ∈ [ − β, β ] \ [ − p, p ] . Observe that the range of B is infinite-dimensional. Next, we claim that therange of B is contained in F ⊥ β . Actually, we have to verify the conditions ( i ) and( ii ) of the Proposition 2.2 for the functions f k ; k = 1 , , . In view of (2.16), from(2.13) we get that P nβ [ f ] = Z nβ [ f ]; n = 1 , , . . . . Thus, from (2.15) and (2.17)we have the conditions ( i ) and ( ii ) of the Proposition 2.2.It follows from the proof of (Proposition 8.3, [6]) that the range of B is con-tained in the weighted L -space L ( R , ω ) , where the weight ω ( x ) = 1 + x . Thiscompletes the proof of Theorem 2.5. (cid:3) Proof of Theorem 1.15 Dynamics of a Gauss-type map. A Gauss-type map. For t ∈ R , the expression { t } is the unique numberin [0 , 1) such that t − { t } ∈ Z . For 0 < γ < ∞ , consider the Gauss-type map V γ on the interval [0 , q ) . The map V γ : [0 , q ) → [0 , q ) is defined by letting V γ ( x ) = ( q (cid:8) γx (cid:9) , x = 00 x = 0 . Note that, for v ∈ N , the map V γ can be expressed as V γ ( x ) = q (cid:0) γx − v (cid:1) whenever γv +1 < x ≤ γv , and hence V γ : (cid:0) γv +1 , γv (cid:3) → [0 , q ) is one-to-one.3.1.2. Dynamical properties of Gauss-type map for γ > q. Denote γ := γ/q which is > . In this section, we observe that V γ is a partiallyfilling C -smooth piecewise monotonic transform for γ > . We need to find aunique absolutely continuous invariant probability measure for V γ having positivedensity.Let V := V γ denote the index set which contain the points v ∈ N so that theassociate fundamental interval is nonempty, that is, J v := (cid:18) γv + 1 , γv (cid:19) ∩ (0 , q ) = ∅ . If γ is an integer, then it is easy to see that for all v ∈ V , where V be theset of all nonzero positive integers with v ≥ γ , the fundamental intervals aregiven by J v := (cid:0) γv +1 , γv (cid:1) , v ∈ V so that V γ ( J v ) = (0 , q ) for all v ∈ V . Thus in OURIER NONUNIQUENESS SETS FOR THE HYPERBOLA 17 this particular case, V γ fulfill the ”filing” condition for all v ∈ V . If γ is notan integer, write v := γ − { γ } ≥ , then a simple calculation shows that J v := (cid:0) γv +1 , γv (cid:1) , v ∈ V \ { v } , where V be the set of all positive integers with v ≥ v . Observe that V γ ( J v ) = (0 , q ) for all v ∈ V \ { v } , that is, in this case, V γ fulfill the ”filing” condition for all v ∈ V except one branch correspondingto { v } . In view of the above facts, we conclude that V γ is a partially filling C -smooth piecewise monotonic transform for γ > . The map V γ satisfy the ”uniform expansiveness” condition with m = 1 , be-cause of the derivative | V ′ γ ( x ) | = qγx ≥ γ > x ∈ { J v : v ∈ V} . Also, we have ”second derivative condition” due to | V ′′ γ ( x ) | ≤ q | V ′ γ ( x ) | for all x ∈ { J v : v ∈ V} . We aim to find a unique absolutely continuous invariantprobability measure corresponding to V γ which has a positive density. Since V γ is a partially filling C -smooth piecewise monotonic transform , there exists a V γ -invariant absolutely continuous probability measure, but it may not be unique.Although, form ([6], p. 45, Remark 5.2) we can get the uniqueness for filling C -smooth piecewise monotonic transform . Also, V γ may not always be a Markovmap, therefore, to get the uniqueness of the absolutely continuous V γ -invariantmeasure, we have to prove the condition ( iii ) of Adler’s theorem which is statedin ([6], p. 46, Theorem D) for the details see [5, 21].3.2. Characterization of the pre-annihilator space K ⊥ γ . Periodic and inverted periodic functions. Let L ∞ q ( R + ) denote the space of all functions f ∈ L ∞ ( R + ) such that the map x e − πix/q f ( x ) is 1-periodic. Then the weak-star closure in L ∞ ( R + ) of thelinear span of the functions { e qn ( x ) := e πi ( n +1 /q ) x ; n ∈ Z } equals to L ∞ q ( R + ) . Let L ∞ γ ( R + ) denote the space of all functions f ∈ L ∞ ( R + ) such that the map x f ( γ/x ) is 1-periodic. Then the weak-star closure in L ∞ ( R + ) of the linearspan of the functions { e γn ( x ) := e πinγ/x ; n ∈ Z } equals to L ∞ γ ( R + ) . Observe thatthe functions in L ∞ q ( R + ) are defined freely on [0 , q ] , and because of periodicitythey are uniquely determined on R + \ [0 , q ] . Similarly, the functions in L ∞ γ ( R + ) aredefined freely on R + \ [0 , γ ] , and due to periodicity they are uniquely determinedon [0 , γ ] . The operator O q : L ∞ ([0 , q ]) → L ∞ ( R + \ [0 , q ]) is defined by(3.1) O q [ ϕ ]( x ) = ϕ ( q { x/q } ) χ R + \ [0 ,q ] ( x ) , where ϕ ∈ L ∞ ([0 , q ]) . The operator T γ : L ∞ ( R + \ [0 , γ ]) → L ∞ ([0 , γ ]) is defined by(3.2) T γ [ ψ ]( x ) = ψ (cid:18) γ { γ/x } (cid:19) χ [0 ,γ ] \{ } ( x ) , where ψ ∈ L ∞ ( R + \ [0 , γ ]) . In view of the above facts, we get that L ∞ q ( R + ) = { ϕ + O q [ ϕ ] : ϕ ∈ L ∞ ([0 , q ]) } and L ∞ γ ( R + ) = { ψ + T γ [ ψ ] : ψ ∈ L ∞ ( R + \ [0 , γ ]) } . The Perron-Frobenius operators. For q ≤ γ < ∞ , the Koopman operator C γ : L ∞ ([0 , q ]) → L ∞ ([0 , q ]) associated to U γ be the map C γ [ ϕ ]( x ) = ϕ ◦ U γ ( x ) , where ϕ ∈ L ∞ ([0 , q ]) . The predual adjoint of C γ is the Perron-Frobenius operator P γ : L ([0 , q ]) → L ([0 , q ]) given by(3.3) P γ [ h ]( x ) = ∞ X v =1 qγ ( qv + x ) h (cid:18) qγqv + x (cid:19) . The operator P γ is linear and a norm contraction on L ([0 , q ]) . Thus the pointspectrum σ point ( P γ ) of P γ is contained in ¯ D . For q < γ < ∞ , consider thefollowing restriction operators: R : L ∞ ( R + \ [0 , q ]) → L ∞ ( R + \ [0 , γ ])R : L ∞ ([0 , γ ]) → L ∞ ([0 , q ])R : L ∞ ([0 , γ ]) → L ∞ ([0 , γ ] \ [0 , q ])R : L ∞ ( R + \ [0 , q ]) → L ∞ ([0 , γ ] \ [0 , q ])The corresponding pre-dual adjoints are the maps R ∗ , R ∗ , R ∗ and R ∗ respectively.As γ > q, a simple calculation shows that C γ = R T γ R O p and hence P γ =O ∗ q R ∗ T ∗ γ R ∗ . Proposition 3.1. For q < γ < ∞ , suppose f ∈ L ( R + ) such that f = f + f + f , where f ∈ L ([0 , q ]) , f ∈ L ([0 , γ ] \ [0 , q ]) , and f ∈ L ( R + \ [0 , γ ]) . Then f ∈ K ⊥ γ if and only if ( i ) ( I − P γ ) f = O ∗ q ( − R ∗ + R ∗ T ∗ γ R ∗ ) f , where I is the identityoperator on L ([0 , q ]) , and ( ii ) f = − T ∗ γ R ∗ f − T ∗ γ R ∗ f . Proof. The proof of Proposition 3.1 works along the same lines as in the proofof ([6], p. 52, Proposition 6.1), hence omitted. (cid:3) Remark 3.2. For 0 < γ < q, in analogy with the results in [16] it seems likelythat, the Perron-Frobenius operator P γ has no eigenfunction corresponding tothe eigenvalue 1 . For the critical case γ = q, P γ has one-dimensional eigenspacecorresponding to the eigenvalue 1 . This in turn implies that AC (cid:0) Γ + , Λ θγ (cid:1) = { } for 0 < γ < q, and for γ = q, AC (cid:0) Γ + , Λ θγ (cid:1) is one-dimensional. This problem isopen.3.3. Proof of Theorem 1.15. The proof of Theorem 1.15 directly follows fromthe proof of Theorem 3.3. Theorem 3.3. For q < γ < ∞ , there exits a bounded linear operator B + : BV ([0 , γ ] \ [0 , q ]) → L ( R + ) such that the range of B + is infinite-dimensional, and containedin K ⊥ γ . Proof. We actually prove more precise statement, namely that, there exits abounded linear operator B + : BV([0 , γ ] \ [0 , q ]) → L ( R + ) such that B + f ( x ) = f ( x ) a.e. x ∈ [0 , γ ] \ [0 , q ] , and for all f ∈ BV([0 , γ ] \ [0 , q ]) . Moreover, B + hasinfinite-dimensional range which is contained in K ⊥ γ . The proof of Theorem 3.3works along a similar path as in Theorem 2.5, hence omitted. (cid:3) OURIER NONUNIQUENESS SETS FOR THE HYPERBOLA 19 Acknowledgements. The author wishes to thank E. K. Narayanan, RamaRawat, R. K. Srivastava, and Sundaram Thangavelu for several valuable sug-gestions during the preparation of this manuscript. The author gratefully ac-knowledges the support provided by NBHM post-doctoral fellowship from theDepartment of Atomic Energy (DAE), Government of India. The author wassupported by the Department of Mathematics, IISc Bangalore, India. References [1] L. Ambrosio, N. Fusco, and D. Pallara Functions of bounded variation and free dis-continuity problems, Oxford Mathematical Monographs, Oxford University Press, NewYork, 2000. MR1857292[2] D. B. Babot, Heisenberg uniqueness pairs in the plane, Three parallel lines, Proc. Amer.Math. Soc. 141 (2013), no. 11, 3899-3904.[3] S. Bagchi, Heisenberg uniqueness pairs corresponding to a finite number of parallel lines, Adv. Math. 325 (2018), 814-823.[4] M. Benedicks, On Fourier transforms of functions supported on sets of finite Lebesguemeasure, J. Math. Anal. Appl. 106 (1985), no. 1, 180-183.[5] A. Boyarsky and P. ´Gora, Laws of Chaos. Invariant Measures and Dynamical Systemin One Dimension, Probab. Appl., Birkh¨auser Boston (1997). MR 1461536[6] F. Canto-Mart´ın, H. Hedenmalm, and A. Montes-Rodr´ıguez, Perron-Frobenius oper-ators and the Klein-Gordon equation, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 1,31-66.[7] A. Chattopadhyay, S. Ghosh, D. K. Giri, and R. K. Srivastava, Heisenberg uniquenesspairs on the Euclidean spaces and the motion group, C. R. Math. Acad. Sci. Paris 358(2020), no. 3, 365-377.[8] S. Ghosh and R. K. Srivastava, Heisenberg uniqueness pairs for the Fourier transformon the Heisenberg group, (2020), arXiv:1810.06390.[9] D. K. Giri and R. Rawat, Heisenberg uniqueness pairs for the hyperbola, Bull. Lond.Math. Soc., doi: 10.1112/blms.12391 (to appear)[10] D. K. Giri and R. K. Srivastava, Heisenberg uniqueness pairs for some algebraic curvesin the plane, Adv. Math. 310 (2017), 993-1016.[11] D. K. Giri and R. K. Srivastava, Heisenberg uniqueness pairs for the finitely manyparallel lines with an irregular gap, (submitted).[12] F. J. Gonzal´ez Vieli, A uniqueness result for the Fourier transform of measures on thesphere, Bull. Aust. Math. Soc. 86 (2012), 78-82.[13] K. Gr¨ochenig and P. Jaming, The Cram´er-Wold theorem on quadratic surfaces andHeisenberg uniqueness pairs, J. Inst. Math. Jussieu 19 (2020), no. 1, 117-135.[14] V. Havin and B. J¨oricke, The Uncertainty Principle in Harmonic Analysis, Ergebnisseder Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas(3)], 28. Springer-Verlag, Berlin, 1994.[15] H. Hedenmalm and A. Montes-Rodr´ıguez, Heisenberg uniqueness pairs and the Klein-Gordon equation, Ann. of Math. (2) 173 (2011), no. 3, 1507-1527.[16] H. Hedenmalm and A. Montes-Rodr´ıguez, The Klein-Gordon equation, the Hilbert trans-form, and dynamics of Gauss-type maps, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 6,1703-1757.[17] H. Hedenmalm and A. Montes-Rodr´ıguez, The Klein-Gordon equation, the Hilbert trans-form, and Gauss-type maps: H ∞ approximation, J. Anal. Math. (2020) (to appear).[18] W. Heisenberg, ¨Uber den anschaulichen Inhalt der quantentheoretischen Kinematik undMechanik, Z. Physik 43 (1927), 172-198. JFM 53.0853.05. [19] M. Iosifescu and S. Grigorescu, Dependence with Complete Connections and its Appli-cations, Cambridge Tracts in math, 96, Cambridge Univ. Press, Cambridge (1990). MR1070097[20] P. Jaming and K. Kellay, A dynamical system approach to Heisenberg uniqueness pairs, J. Anal. Math. 134 (2018), no. 1, 273-301.[21] A. Lasota and J. A. Yorke Existence of invariant measures for piecewise monotonictransformations, Trans. Amer. Math. Soc. 186 (1973), 481-488.[22] N. Lev, Uniqueness theorem for Fourier transform, Bull. Sci. Math. 135 (2011), 134-140.[23] P. Sj¨olin, Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin, Bull.Sci. Math. 135 (2011), 125-133.[24] P. Sj¨olin, Heisenberg uniqueness pairs for the parabola, J. Fourier Anal. Appl. 19 (2013),410-416.[25] R. K. Srivastava, Non-harmonic cones are Heisenberg uniqueness pairs for the Fouriertransform on R n , J. Fourier Anal. Appl. 24 (2018), no. 6, 1425-1437. Deb Kumar Giri, Department of Mathematics, Indian Institute of Science,Bangalore-560012, India. E-mail address ::