Lower bounds on directional complexity for irrational triangle billiards
aa r X i v : . [ m a t h . D S ] N ov Lower bounds on directional complexity forirrational triangle billiards
Dmitri ScheglovUniversity of OklahomaNovember 7, 2018
Abstract
We provide explicit lower bounds on directional complexity for a class ofirrational triangle billiards for a full measure F σ -set of directions. This paper is a continuation of our paper [7], where we provided an explicitversion of the theorem by Galperin, Kruger and Troubeczkoy on the splittingof a thin parallel beam of triangle billiard trajectories. In paper [7] we gavean explicit upper bound on the splitting time in terms of a particular numbertheoretic function of angles F αβ ( ǫ ).We also described some class of angles α, β for which the function F αβ ( ǫ ) iscorrectly defined and we conjectured that F αβ ( ǫ ) is in fact defined for any pairof irrational numbers α, β and moreover is uniformly bounded by some universalfunction F ( ǫ ), namely F αβ ( ǫ ) ≤ F ( ǫ ).Here we continue to exploit this function F αβ ( ǫ ) and in particular we providean explicit lower bound on the important dynamical characteristic of a billiard- its directional complexity.The complexity of a polygonal billiard is a dynamical characteristic whichroughly speaking measures the growth of combinatorial types of different or-bits. Namely one enumerates the sides of the k -gon by symbols 1 , , . . . , k andthen associates a word from this alphabet to a billiard trajectory of length n reading the sides of the polygon which it hits.The complexity function p ( n ) is a total number of different words of length n obtained this way.There is also an analogous definition of the directional complexity function. Weintroduce a restriction on the trajectories, namely we consider the trajectoriesstarting from a fixed side of a polygon under a particular angle θ . And then1e again count the number of different words of the length n . The resultingfunction is called a directional complexity in the direction θ . We will denote itas p θ ( n ) or sometimes just p ( n ) if it does not lead to ambiguity.By trivial reasons complexity function can not grow faster than exponentiallyand A. Katok [4] proved that for any polygon the complexity function in factgrows subexponentially, but his estimate is not explicit. It is still the bestknown upper bound on the complexity growth and it is a difficult open problemto provide any explicit subexponential upper estimate on p ( n ).S.Troubeczkoy found a quadratic lower bound for p ( n ) in case of any polygon.[8]In case of a rational polygon, meaning that the angles are rational multiples of π , the billiard is essentially equivalent to the geodesic flow on the flat compactsurface with a finite number of singularities. Then one can use Teichmullertheory to investigate the billiard dynamics.In this case using the results of H.Masur on the growth for quadratic differentials[6], J.Cassaigne, P. Hubert, S. Troubeczkoy [1] proved that there are constants c and c such that c n < p ( n ) < c n .For a directional complexity p θ ( n ) of any polygon with k sides there is a poly-nomial upper bound by E.Gutkin and S. Troubeczkoy [3]: p θ ( n ) ≤ krn (cid:0) n (cid:1) s ,where r is the least common denominator of the rational angles of P and s is anumber of distinct irrational angles of p .Note that this upper boundary is quite universal, namely it is basically inde-pendent on the angles of the polygon and on the direction.The natural question is then to give the lower bound on the directional com-plexity. This question however is more delicate and the current paper is devotedto the lower bound estimates on directional complexity for irrational triangles. First of all let us consider a simple example of a biliard in the square andhorizontal direction θ . By ” horizontal ” we mean the direction parallel to oneof the square sides. In this direction p ( n ) = 1 which means there is no growthat all. The same picture happens in any rational direction in the square, forwhich p ( n ) is bounded.Intuitively this happens because any orbit in a rational direction is periodic andso the whole billiard flow in such a direction just splits into a finite number of”periodic beams” without producing any complexity.On the other side the complexity in any irrational direction is closely connectedto the so-called ” Sturmian sequences” and certainly grows.2n order to understand the nature of the complexity growth we make the folowingwell-known observation. Namely, the function p θ ( n ) increases at those timemoments n when the thin beam of parallel trajectories of a given combinatorics,starting in a given direction, hits a polygon vertex.At this time moment one ” half” of the beam hits one side of the polygonand another ” half” of the beam hits another side and so we have at least twodifferent words of length n + 1 which implies p θ ( n + 1) ≥ p θ ( n ) + 1.This mechanism of complexity growth however fails in general if the direction θ is periodic. In this case there exist thin ” periodic beams” which never splitand do not produce complexity. The before mentioned example of a rationaldirection in the square billiard perfectly demonstrates this fenomenon.It is theoretically possible that for some irrational polygons there exist periodicdirections where not all the orbits are periodic. In the other words so-called” partially - periodic directions” may exist. However the analysis of periodictrajectories in irrational polygons so far is very difficult problem and it is noteven known if periodic orbits always exist.On the other side even if almost all the orbits in a given direction θ are periodic,it could possibly happen that there is an infinite number of periodic beams,which would imply the growth of p θ ( n ). But again, for generic polygons there areyet no tools, allowing one to effectively analyse the behaviour and distributionof periodic orbits.Having this said it is natural to consider a direction θ which is not periodic, inthe other words, there are no periodic orbits in θ -direction. A priory there isstill a possibility that the thin parallel beam of trajectories never splits, howeverthis possibility is excluded by the very nice result of Galperin, Kruger andTroubeczkoy[2], which can be reformulated as follows: Theorem ( Galperin, Kruger, Troubeczkoy).
In any polygonal billiardany non-parallel beam of trajectories splits.We would like to remark that the theorem above in the original paper wasformulated slightly differently but the given formulation easily follows.Having in mind this result the approximate scheme on the estimating p n ( θ )would approximately look as follows.We divide a side of the polygon on m equal pieces of length 1 /m and start aparallel beam from each piece. If there is a uniform splitting time time T ( m )for each beam, then obviously p ( T ( m )) ≥ m . The numbers T ( m ) form a veryspecial increasing sequence, so in order to estimate p ( n ) we find a maximal m such that T ( m ) ≤ n , which provides a lower bound p ( n ) > m .However we would like to explain the reader several important obstacles whichwe must overcome during this process. 3irst of all even if the beam B ( ǫ ) is not periodic, it can be very close to aperiodic beam. Namely if θ is a beam direction and α is a direction of a fixedperiodic beam, than if ǫ is small enough and the middle point of the base of thebeam B ( ǫ ) coincides with a middle point of periodic beam, then as the value φ = | θ − α | −→ n ( ǫ ) −→ ∞ .Informally speaking this example shows that the periodic orbits are ” the ob-stacles” to the uniform splitting.And the second issue to keep in mind is the following. Assume that our beam B θ ( ǫ ) is in the reasonable sense ”far” from periodic orbits. The precise meaningof the word ”far” will be explained little later. We then have to provide anexplicit bound for the splitting time n ( ǫ ) in the other words an effective versionof the splitting theorem by Galperin, Kruger and Troubeczkoy.The main result of the paper [7] was a dichotomy, briefly formulated as follows:either beam B ( ǫ ) contains a periodic orbit inside or its length is uniformlybounded by some function M ( ǫ ). In the other words ” parallel ǫ - beams, notcontaining periodic orbits can not be too long”.Having this dichotomy we may assume that if the beam B ( ǫ ) does not splitin time M ( ǫ ) then it has a periodic orbit inside. For any fixed direction θ weintroduce a sequence of functions φ ( N ) = min | θ − α | , where α runs through allperiodic directions, corresponding to the directions of the length less than N .Using some delicate analysis we estimate the splitting time for a beam whichpossibly contains a periodic orbit inside, in terms of the sequence φ n .We would like to note that this result is independent on the fact whether ornot there exist periodic orbits inside a triangle. If in some triangle there are noperiodic orbits at all ( which presumably never happens), then the correspondingterms in the explicit formulas for the splitting time would vanish.This way we obtain the explicit formulas for any non-periodic directions θ interms of the sequence φ n . However these lower bounds are in a sense ” notexplicit enough”, as we basically have no information about the sequence φ n .However these explicit formulas combined with some simple measure-theoreticarguments allow us to provide a lower bound on the p θ ( n ) for typical directions θ as we are able to say that typical direction ” stays away” from periodic directionswith a prescribed distance.Now it is time to turn into formal definitions.
4n this section we borrow several definitions from the paper [7] in order to keepthe current paper self-contained and provide the results needed for the proof ofthe main theorem.First of all we will remind the so-called Katok-Zemlyakov construction[5], whichis a main technical tool in our analysis of the billiard trajectories. Consider abilliard trajectory inside a polygon and in particular a moment when it hits thepolygon side. Instead of reflecting the trajectory, we reflect a polygon, using the” optic law”, that the angle of reflection coincides with the angle of incidence.From the point of view of the observer inside the polygon, looking at the side asin the mirror, the billiard trajectory instead of reflection goes further ” behindthe mirror”. Then we continue the process indefinitely many times.As a result, instead of the piecewise linear billiard trajectory in a fixed polygon,we have a fixed straight line and a sequence of reflected polygons along theline. In the picture below we show the result of the one-time application ofthe Katok-Zemlyakov construction to the triangle. This way we obtain a niceshape, which we call a
Kite . ▲▲▲▲▲▲▲▲✓✓✓✓✓❙❙❙❙❙☞☞☞☞☞☞☞☞ Pic. 1.
KiteThe next picture represents Katok-Zemlyakov construction, applied several timesto a kite. ❈❈❈❈❈❈❈❈✡✡✡❏❏❏✄✄✄✄✄✄✄(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✦✦✦✦ ✲❤❤❤❤❇❇❇❇❇❇❇☎☎☎☎☎☎☎☎◗◗◗◗◗◗ ❅❅❅✔✔✔✔
Pic.2.
Katok-Zemlyakov construction applied to a kite.We will also need the notion of the parallel beam of the trajectories and weborrow corresponding definition from [1].5 efinition 1.
An ( ǫ, T ) - beam is a set of parallel segments, corresponding tothe application of the Katok-Zemlyakov construction along some direction andfrom some base point, where ǫ is a width of the beam and T is the length of themaximal parallel segment.The left interval on the kite side, transversal to the beam direction is calleda base segment or base of a beam and the right interval is called an end segmentor end of the beam.Note, that by the definition beam does not have any kite vertices inside, as theKatok-Zemlyakov is undefined on when the trajectory hits a vertex.We will usually denote ( ǫ, T ) - beam as B ( ǫ, T ).Below one may see a picture, which provides a geometric intuition behind thenotion of ( ǫ, T ) - beam. ❈❈❈❈❈❈❈❈✡✡✡❏❏❏✄✄✄✄✄✄✄(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✦✦✦✦ ✲❤❤❤❤❇❇❇❇❇❇❇☎☎☎☎☎☎☎☎◗◗◗◗◗◗ ❅❅❅✔✔✔✔ ✄✄✄✄✄✄✄✄✲✜✜✜✜✜✜✜✧✧✧ ❊❊❊❊ Pic.3. ( ǫ, T ) - beam of parallel trajectories.We also remind the following well-known inequality, which will be used later. Lemma 1.
Let S be a billiard trajectory, L ( S ) be its geometric length and N ( S )be its combinatorial length, meaning the number of reflections in the Katok-Zemlyakov construction. Then there exist positive constants, c , C , dependingonly on the kite, such that: cN ( S ) ≤ L ( S ) ≤ CN ( S ).The proof of Lemma 1. is elementary and we will not reproduce it here.From here and further we will denote as C any large enough or small enoughconstant, depending on the context.We will also use the following simplifying convention from [1], which would allowus to avoid too complicated expressions: Convention.
Any function defined on the positive integers f : Z + → R is bydefault extended to a function f : R + → R by the rule f ( x ) = f ([ x ] + 1), where x ∈ R + \ Z + .The next several definitions are borrowed from [1] in order to formulate adichotomy theorem. We introduce them in order to keep our exposition self -contained. 6 efinition 1. Consider a finite subset S ⊂ ∆ of a segment ∆ of the circle. S is called a relative ǫ -net if after linear ”blowing up” of the segment ∆ to thelength 1, S becomes an ǫ -net in the standard sence. Definition 2.
Let 0 < α, β < π . A finite sequence x , . . . , x n ∈ S is called αβ -connected if for any i : 1 ≤ n − x i +1 − x i = ± α or x i +1 − x i = ± β .For any finite set of points S ⊂ S let | S | denote its cardinality. We then havethe following definition. Fix a pair of numbers α , β as above. The net -function F αβ : (0,1) → Z + is defined as follows. Definition 3. F αβ ( ǫ ) = min { n ∈ Z + | ∀ αβ -connected S ⊂ S , | S | > n ∃ S ⊂ S , S − relative ǫ -net } Informally speaking F αβ ( ǫ ) is a minimal cardinality of αβ - connected sequencewhich guarantees that it contains a relative ǫ -net. Definition 4.
Let α, β be rationally independent. Then N αβ ( k ) = min {h nα + mβ i| for all | n | + | m | ≤ k |} , where h x i is a distance from x to the closest integer.The function F αβ ( ǫ ) defined above turns out to be extremely important for ourpurposes. In fact our approach works precisely for all angles α, β for which F αβ is defined. In the paper [1] we proved the existence and gave an explicitupper estimate on the function F αβ for all angles α, β for which β is a fixedirrational number and α/β allows very fast approximation by rational numbers.This result shows that the class of angles, for which F αβ is explicitely defined isquite non-trivial.In the paper [7] we conjecture that F αβ ( ǫ ) is defined for all α, β and moreover,that it is uniformly bounded from above for all α, β , namely F αβ ( ǫ ) ≤ F ( ǫ ).The proof of this number-theoretic conjecture seems to be quite important forunderstanding the triangle billiard dynamics. In particular due to the result ofthe current paper it would automatically give the explicite lower bounds on thetypical directional complexity for all triangle billiards.Even if the number-theoretic conjecture is not true for all pairs α, β its proof forsome particular pair of angles α, β implies the lower bound on the directionalcomplexity for that particular triangle.We now formulate a theorem from [7] which will serve as a key tool in ourapproach. Theorem (Effective dichotomy).
Let α , β be a pair of rationally indepen-dent numbers such that F αβ ( ǫ ) is a correctly defined function for positive ǫ . And7et K αβ be the kite of diameter 1 with angles α , β . Let B( ǫ , M ) be a parallelbeam.Let us also introduce the following notations: P αβ ( ǫ ) = ǫ F αβ (cid:0)(cid:0) ǫ (cid:1) ǫ (cid:1) Then either M ≤ M ( ǫ ) = CN αβ ( P αβ ( ǫ )) or B contains a periodic trajectory inside,starting from the base. Here C is a constant, depending only on the kite.One more definition precisely defines the speed of approximation of a givendirection by periodic. Definition 7.
Fix a kite K on the plane, choose a side of K and a direction θ .As a tangent bundle to K is naturally trivialized, we may think of any directionas of the point on the circle S . Then we define a sequence φ ( n ) as follows: φ ( n ) = min {| θ − α |} , where α runs through all the periodic directions of thecombinatorial length less than n .Here by periodic direction α we mean a direction such that there exists at leastone periodic orbit starting from the given side in the direction α .We should notice that the set of directions α from the definition above is fi-nite and so the minimum is taken over a finite number of values. It happensbecause any periodic direction is uniquely defined by the combinatorics of acorresponding periodic trajectory which is an easy well-known observation.In the hypothetical case when there are no periodic trajectories at all, we put φ ( n ) = 1 identically. We now have all the tools to provide the lower bound for a directional complexitygrowth. First we prove a useful theorem, which estimates the splitting time ofa thin beam of width ǫ in the direction θ . Theorem.
Let α, β be a pair of kite angles, such that the function F αβ ( ǫ ) iscorrectly defined. Let us also fix a side of the kite I , a non-periodic direction θ and ǫ small enough. Let φ ( n ) be the sequence, defined above and B θ ( ǫ, T ) be aparallel beam in the direction θ .Note that non-periodicity of θ implies that all the terms φ ( n ) are positive.Then T ≤ T ( ǫ ) = max { M ( ǫ ) , Cφ ( C · M ( ǫ )) } , C >
Proof.
Let us assume that
T > T ( ǫ ), which implies T > M ( ǫ ). By thetheorem 1 there is a periodic trajectory S of the length L ( S ) inside the beam B = B θ ( ǫ, M ( ǫ )), starting from its base. It implies that there is a subbeam B θ = B θ ( ǫ, L ( ǫ )) with the same base, which has parallel base and end kites andcontains a small parallel beam of periodic orbits close to S .8e are going to take a closer look on the trajectory S and estimate how theendpoints of S are located inside B . ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✲✲✲✲ ✲✲✲ L ( ǫ ) ǫ α δL ( S ) Pic.4.
The beam B θ = B θ ( ǫ, L ( ǫ )) and a beam of periodic trajectories inside.As we see on the picture there is a horizontal sheer on the length δ of the beambase. δ is exactly the difference between ǫ and the width of the periodic beam,located inside B θ ( ǫ, L ( ǫ )).Let us now estimate the sheer δ from below.As the beam length is L ( ǫ ) and α is exactly the angle between the periodicdirection and θ then by definition of the sequence φ ( n ) we have α ≥ φ ( CL ( ǫ )).As for small enough α we have sin ( α ) ≈ α then from the picture 4 we have δ ≥ Csin ( α ) L ( ǫ ) ≥ c · sin ( φ ( C · L ( ǫ ))) L ( ǫ ) ≥ cφ ( C · L ( ǫ )) L ( ǫ ) ≥ cφ ( C · M ( ǫ )) L ( ǫ ).We now take a new beam B θ which base is a union of the base and end of thebeam B θ ( ǫ, L ( ǫ )). The width of B θ is ǫ + δ which is clear from the picture 5.The picture 5 shows how the part of the end of the beam B θ ( ǫ, L ( ǫ )) is attached9o its base. ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✲✲✲✲ ✲✲✲ L ( ǫ ) ǫ α δ ✲✲✲✲ δ L ( S ) Pic.5.
The second beam B θ of the width ǫ + δ .Assume first that the extended beam B θ does not split at the time L ( ǫ ).In this case we still have a periodic trajectory S inside the extended beamwith angle α to the direction θ . It implies that the previous argument can beapplied to the extended beam B θ ( ǫ, L ( ǫ )) and we construct a third beam B θ and contiinue the process.Note that each time the width of a new beam is greater then the width of theprevious on δ .Let us assume that this process terminates at n -th step, meaning that the n -thextended beam splits. As each of the extended beams has length less than L ( ǫ ) itimplies that the splitting time T ( ǫ ) of the original beam satisfies T ( ǫ ) ≤ nL ( ǫ ).Since the perimeter of the kite is 1 and each time the beam width increases by δ , it implies δn ≤ T ( ǫ ) ≤ C/φ ( M ( ǫ )).10s the computations above were maid under assumption that L ( ǫ ) ≤ M ( ǫ ) thenin general for arbitrary beam B θ ( ǫ, T ) we have T ≤ max ( M ( ǫ ) , /φ ( C · M ( ǫ ))).Let us now estimate the splitting time in the generic direction. First of all weremind an easy upper estimate on the number P ( n ) of periodic beams of lengthless than n . As each such a beam is uniquely determined by its coding then thenumber of the periodic beams of length precisely n is at most 4 n . And sinse wecount all the trajectories of the smaller length, we have: P ( n ) ≤ . . . + 4 n ≤ n +1 We now take a small enough ρ > U ( n, ρ ) = { θ ∈ S : | θ − α | < ρ − n − n − } where α runs through all periodic directions of lengthless then n .As the number of periodic directions of length less then n is bounded by 4 n +1 we have that the Lebesguse measure of our set Leb ( U ( n, ρ )) ≤ ρ − n .Let us denote U ( ρ ) = S n U ( n, ρ )Summing up these inequalities for all n we have Leb ( U ( ρ )) ≤ ρ .Now for any direction θ in the complement of U ( ρ ) we have by definition φ ( n ) ≥ ρ − n − n − which implies that there is a large enough constant C dependingon ρ that the splitting time of the parallel beam in the direction θ satisfies T ( ǫ ) ≤ C exp( C · M ( ǫ )) for large enough C Each U ( ρ ) is an open set so the complement V ( ρ ) is closed. Let us now take asequence ρ n = 1 /n and consider V = S n V ( ρ n ).The set V belongs to the class F σ which is a countable union of closed sets andfrom considerations above easily follows:1) V ∈ F σ V has a full Lebesgue measure.3) For any direction θ ∈ V we have T ( ǫ ) ≤ C exp( C · M ( ǫ )) for large enough C Now let us turn to the directional complexity estimate. We first remind the ideain more details.Let us consider the billiard side I and let all the the points of I move in thedirection θ . Without loss of generality let us assume that the length of I equalsto 1. We fix a positive integer m on divide I onto m equal intervals of the length1 /m .Each of these intervals splits at the time T (1 /m ), which in particular impliesthat p θ ( T (1 /m )) ≥ m .Now we define the following function N θ ( n ) = max { m | T (1 /m ) ≤ n } . As p θ ( n )is increasing function, the inequality above implies our main result, the lowerbound on the directional complexity: p θ ( n ) ≥ N θ ( n ).Let us finally summarize our observations to the following concluding theorem.11 heorem 2. Lower bound on directional complexity. Let α, β be a pair of rationally independent irrational numbers, such that thefunction F αβ ( ǫ ) is correctly defined.Let R ( m ) = N αβ (cid:0) mF αβ (cid:0)(cid:0) m (cid:1) m (cid:1)(cid:1) and let L ( n ) = max { m ∈ Z + | R ( m ) ≥ C/ ln( n ) } , where C is a big enough constantdepending on K and θ .Then for a kite K with angles α, β and for any direction θ from F σ -set of a fullmeasure, the directional complexity p θ ( n ) satisfies: p θ ( n ) ≥ L ( n ) 12 eferenceseferences