On The Minimal Number of Elements of Markov Partitions for Pseudo-Anosov Homeomorpisms

In the paper an estimation of the minimal number of elements of Markov partition for generalized pseudo-Anosov homeomorphism of closed non necessary orientable surface is given. It is formulated in terms of characteristic of invariant foliation of generalized pseudo-Anosov homeomorphism.


Introduction
Pseudo-Anosov homeomorphisms of orientable surfaces were introduced by W. Thurston (1988) in his research on J. Nielsen classification of surface homeomorphisms up to isotopy.Markov partitions are very useful tools for investigation of geometrical and dynamical properties of such homeomorphisms.The number of theirs elements is essential in particulary for combinatoric description of these partitions and for calculating the number of periodic points of homeomorphism and of entropy.It is well known that in particular case of hyperbolic homeomorphism of 2-torus there Markov partition onto two elements exists and of course there is no Markov partition with one element.We will consider the common case.
We begin with definitions.
We will mind rectangle [0, 1] × [0, 1] ⊂ R 2 with its partition onto intervals of horizontal lines under non-singular coordinate neighborhood with nonsingular foliation.Also, the singular coordinate neighborhood with d-pronged (d ∈ N, d 2) singularity will be the neighborhood of origin in R 2 fibered onto subsets of two types: 1) intervals of d rays issuing out of origin (singular leaves); 2) arcs of convex curves lying in sectors between these rays and asymptotic to them (in the case d = 1 these are arcs of parabolas with singular leaf as the mirror symmetry axis).We will say that d is the valency of singularity.
Definition 1 Let M be a closed surface, S ⊂ M its finite subset.Singular foliation of M with singularities in S is the family of linear connected subsets of M called leaves such that (1) M is the union of all leaves; (2) intersection of any two leaves either is empty, or is contained in S ; (3) for each x ∈ M \ S (correspondingly x ∈ S ) neighborhood U and its homeomorphism onto nonsingular (correspondingly singular of some valency d) coordinate neighborhood in R 2 which maps x to origin and any linear connected component of intersection with U of any leaf onto leaf of corresponding coordinate foliation exists.
Definition 2 The homeomorphism f : M → M is said to generalized pseudo-Anosov (GPA) if it preserves two transversal foliations W u , W s with common singularities expanding leaves of W u with the factor λ > 1 and contacting leaves of W s with the factor λ −1 .The number λ is called the dilatation of f .

Remarks.
1) Because there is no need in the smooth structure of surface and of its foliations we understand transversality in the following sense.Two arcs γ 1 , γ 2 are transversal if their intersection points are isolated and for each intersection point x exists neighborhood U ∋ x and its homeomorphism φ to R 2 which maps x to origin end parts of curves lying in U onto intervals of horizontal and vertical axis correspondingly.
2) Expanding and contracting in this definition are understood usually with respect to transversal invariant measures µ s , µ u .The transversal measure µ s is the family of Borel measures defined on the arcs of leaves of W u so that if two arcs are fiber homotopic (with respect to W u ), then their measures are equal.The transversal measure µ u is defined by swapping symbols s and u.So the property of foliations in the definition 1 means µ s (γ) = λµ s (γ) and µ u (γ) = λ −1 µ u (γ) for γ arc of leaf of W u (of W s correspondingly).
Definition 3 Let f : M → M be a generalized pseudo-Anosov homeomorphism.The rectangle is a closed set Π ⊂ M that is the image of the map φ : [0, 1] × [0, 1] → M so that it is one-to-one on the interior of square and maps each horizontal (vertical) interval onto arc of contracting (expanding) leaf.Let us denote by Π • image of interior of square.The images of horizontal (vertical) sides of this square are called contracting (expanding) sides of rectangle Π.
Definition 4 Markov partition for the generalized pseudo-Anosov homeomorphism f : Here ∂ s P and ∂ u P are unions of contracting (expanding) sides of all rectangles of P.
It is well known that for Anosov diffeomorphism of 2-torus (which essentially is the same as GPA-homeomorphism with no singularities) there exist Markov partition consisting of two rectangles.It is easy to see that there are no Markov partitions with one element for it.It is natural question on the minimal number of rectangles for arbitrary GPA-homeomorphism.In this paper we will establish en estimation from above for this number.We need two additional definitions to formulate the final result.
Theorem Let f is the generalized pseudo-Anosov homeomorphism of the closed surface.Let S = {s d : d ∈ N} is its singular type and m -the minimal period of periodic leaves of its contracting foliations.Then minimal number of elements of Markov partitions for f is To be sure that this estimate is valid for the Anosov diffeomorphism, it is natural to assume that the latter has a unique singularity of valence 2 in its arbitrary fixed point.Note that there are other reasons for such point of view on invariant foliations of Anosov diffeomorphisms.
Let us note also that because f maps singular leaves onto singular leaves and singular points to singular points of the same valency, the minimal number of periodic leaves may be estimated by m ≤ min Consequently, the minimal number of the elements of Markov partitions may be estimated be means of the singular type of GPA-homeomorphism.
The proof of the theorem consists of the explicit constructing of Markov partition with the number of elements equal to the value in right side of (1).

The Proof of the Theorem
In what follows is assumed that f : M → M is GPA-homeomorphism, W s , W u are its contracting and expanding foliations and S = {s d } its singular type.
To prove the theorem we consider two cases: 1) The periodic contracting leaf with minimal period M is singular; 2) The periodic contracting leaf with minimal period M is nonsingular.
In the proof we will use well known properties of invariant foliations of GPA-homeomorphisms whose proves can be found in (Fathi, 1979).These properties are valid for both foliations W s and W u .
1.There are no closed leaves and no leaves joining singular points.
2. Each singular leaf is dense in the surface.The same is true for the both linear connected components of every nonsingular leaf onto which arbitrary point divide it.
3. Each periodic leaf contain periodic point.
Let us begin with the proof in the first case Lemma 1.Let W be a periodic singular leaf of period m of the contracting foliation of f and W Then there exist arcs w k ⊂ W k with the following properties: 2) one endpoint of w k is singular point and another one belongs to expanding leaf beginning from some singular point; 3) for any singular point p whose arc of expanding leaf connects p with endpoint of some w k this arc do not intersect none of arcs w l (0 ≤ l < m) in their interior points.
Proof.Let p k be a singular periodic point belonging to the leaf W k .Note that some of these points may coincide (in the case that corresponding leaves coming out of the same singularity).Let x 0 p 0 be an arbitrary point of the leaf W 0 .Define points ) s an open arc of W k joining points p k and x k .Let p be some singular point (possibly one of p k or not) and arbitrary expanding leaf coming out of this point.Denote by y 0 the point of first intersection of this leaf with some of arcs (p k , x k ) s i.e. y 0 ∈ (p l , x l ) s for some l and the arc (p, y 0 ) u of this leaf (joining p and y 0 ) do not intersect other arcs (p k , x k ) s .Such point y 0 exists because each leaf of expanding and contracting foliation is dense in the surface.
Let us suppose that the point y 0 belongs to the arc (p 0 , x 0 ) s .In other case we can to renumber leaves W k and, correspondingly, points p k and x k so that f

Now we define points y
Consequently, the family arcs (p k , y k ) s satisfy conditions 1,2 of the Lemma 1 but possibly do not satisfy to the condition 3.
We change this family to obtain the family satisfying this condition too.For each k consider the set Y k which is the intersection with (p k , y k ] s (it is semi-open arc) all arcs ( f i p, y i ] u (0 ≤ i ≤ m − 1).Let q k be such point of the set Y k that the arc (p k , q k ) s does not contain other points of this set.Let w k := (p k , q k ) s .Then family of arcs w k satisfy to the conditions 2 and 3 and we need to prove that the condition 1 remains to be true.
To do this consider the sets Y k .First let us note that because (p, y Hence It follows that Y m \ {y m } = ∅ i.e. q m = y m .For k 1 we have In other case q 0 ∈ ( f i p, y i ) u for some i, 0 < i ≤ m − 1.Then f −1 (q 0 ) ∈ ( f −1 p, y −1 ) u .According to (2) the arc ( f −1 p, y i−1 ) u do not intersect the arc (p m−1 , y m−1 ) s .Hence the inclusion (p m−1 , f −1 (q 0 )) s ⊂ (p m−1 , y m−1 ) s = p m−1 , q m−1 s is false.Consequently f ( (p m−1 , q m−1 ) s ) ⊂ (p m−1 , y m−1 ) s = (p m−1 , q m−1 ) s as required.2 Lemma 2 If there exist the family of arcs w 0 , . . ., w m−1 enabling the properties 1-3 of lemma 1 then 1) there exists Markov partition P with ∂ s P = ∪ m k=1 w k ; 2) the number of elements of P is m + 1 2 ∑ d ds d .Proof.Let us denote by Γ s := ∪ m−2 k=0 w k .The property 1 of Lemma 2 implies that f (Γ s ) ⊂ Γ s .Let p k be the singular endpoint of w k .Let q k be another endpoint and of w k and p ′ k be the singular point arc [p ′ k , q k ] u of the expanding leaf that

Definition 5
The singular type of generalized pseudo-Anosov homeomorphism f is the sequence of S = S( f ) := {s d : d ∈ N} element s d of which is the number of d-pronged singularities of invariant foliations of f .Evidently, only finite number of elements of the sequence S( f ) are non-zero.Definition 6 Let us say that the family W = W of leaves of f -invariant foliation (either