self-overlays and shape of the julia set of a rational map · d. edwards and h. hastings, cech and...

Semi-flows, shape and overlays Spherical multipliers Plotting algorithms Spherical algorithm Futher work References

Self-overlays and shape of the Julia set of a rational map

L. J. Hernandez Paricio and M. T. Rivas Rodrıguez

University of La Rioja

XV Meeting on Computer Algebra and Applications (Encuentro deAlgebra Computacional y Aplicaciones), 22-24 June 2016

(L. J. Hernandez and M. T. Rivas) EACA 2016 22-24 June 2016 1 / 57


Table of Contents

1 Semi-flows, shape and overlays

2 Spherical multipliers

3 Plotting algorithms

4 Spherical algorithm

5 Futher work



Discrete metric semi-flows

Let (X , d) be a metric space with metric d .Given a continuous map h : X → X , the pair (X , h) is called a discretesemi-flow and the triple (X , d , h) is said to be a metric discretesemi-flow.

Given an integer n ≥ 0, hn denotes the n-th composition h ◦ · · · ◦ h andh0 = idX .

Let X = (X , h) be a discrete semi-flow:

A point x ∈ X is said to be a fixed point if, for all n ∈ N, hn(x) = x ; x issaid to be a periodic point if there exists n ∈ N, n 6= 0, such thathn(x) = x and x is said to be a q-cyclic point if hq(x) = x andhk(x) 6= x , 1 ≤ k < q.



Shape theory

Given a discrete semi-flow (X , f ), a point x0 ∈ X is said to be a Fatoupoint if there is an open neighborhood U at x0 verifying that for everyε > 0 there is nε such that for every x , y ∈ U and for every n ≥ nε,d(f n(x), f n(y)) < ε.

Denote F (f ) the open subset of all the Fatou points x0 of X . It is saidthat F (f ) is the Fatou set of f .

The Julia set J(f ) is defined to be the closed subset J(f ) = X \ F (f ).



Shape theory

Let Top denote the category of topological spaces and its localizationHo(Top) is the category obtained by inverting all the weak equivalences.

Let pro-C denote the category of inverse systems (or projective systems) ofa category C. In particular, we consider the categories pro-Top andpro-Ho(Top).

Given a manifold M and a compact subspace J ⊂ M the shape of J canbe defined as the isomorphism class in pro-Ho(Top) of the inverse systemof the neighborhoods at J induced by the topology of M.

It is interesting to remark that the shape only depends on J and it doesnot depend on the ‘ambient manifold’ M.



Shape theory

For more information about shape and strong shape theory we refer thereader to:

D. Edwards and H. Hastings, Cech and Steenrod homotopy theories withapplications to Geometric Topology, Springer Lect. Notes in Math., 542,(1976).

S. Mardesic and J. Segal, Shape Theory –The Inverse Limit Approach,North-Holland, 1982.

T. Porter, Cech and Steenrod homotopy and the Quigley exact couple inStrong Shape and Proper Homotopy Theory, J. Pure Apl. Alg. 24 (1983)303–312.



Overlays

The usual notion of covering of a space p : X → Y assumes that Y is alocally connected space. A map is a covering if there is an open coveringU of Y such that (i) if U ∈ U then U is connected, (ii) for each U ∈ Uthere is an index set IU and and a homeomorphism p−1(U) ∼=

∑i∈IU Ui

verifying that p|Ui : Ui → U is a homeomorphism.

When one removes the condition of local connectivity, it can appear thefollowing problem: given a continuous map p : X → Y and a nonconnected open subset U of Y , p−1(U) could be decomposed in twodifferent ways p−1(U) ∼=

∑Ui∼=

∑Vj verifying that p|Ui : Ui → U,

p|Vj : Vj → U are homeomorphisms.

Ralph H. Fox introduced the following notion: An overlay structure forp : X → Y is given by taking a collection of decompositionsp−1(U) ∼=

∑i∈IU Ui where U runs over an open covering U of Y satisfying

the following property: If for U,V ∈ U , p−1(U) ∼=∑

i∈IU Ui and

p−1(V ) ∼=∑

j∈IV Vj and Ui ∩ Vj1 6= ∅ and Ui ∩ Vj2 6= ∅ , then j1 = j2.



Overlays

For the notion of overlay, its origins and some relations to shape theory:

R. H. Fox, On Shape, Fund. Math. 74 (1972) 47–71.

R. H. Fox, Shape theory and covering spaces, Springer Lect. Notes inMath., 375, (1974) 71–90.

S. Mardesic and V. Matijevic, Classifying overlay structures of topologicalspaces, Topology and its Applications, 113, (2001)1–3.

L. J. Hernandez and V. Matijevic, Fundamental groups and finite sheetedcoverings, J. Pure Appl. Algebra, 214, (2010) 281–296.



Notation

S2 = {(r1, r2, r3) ∈ R3 | r 21 + r 2

2 + r 23 = 1} (the unit 2-sphere)

C = C ∪ {∞} (the Alexandroff compactification)

P1(C) (the complex projective line)

θ : S2 → C (the stereographic projection)

θ : P1(C)→ C (the change from homogenous to absolutecoordinates)

Θ: P1(C)→ S2, Θ = (θ)−1θ

Let x and y be functions from P1(C) to C with domains:Dom x = {[z : t] ∈ P1(C) | t 6= 0} and Dom y = {[z : t] ∈ P1(C) | z 6= 0}x([z : t]) = z/t and y([z : t]) = t/z .



Normaized homogeneous coordinates

Given a point [z : t] ∈ P1(C), the coordinates (z , t) are called thehomogeneous coordinates of the point.The normalized homogeneous coordinates for any point in P1(C), whichare given as follows:

[z : t] =

{[z/t : 1] if |t| ≥ |z |,[1 : t/z ] if |t| < |z |.

where |t| and |z | represent the absolute value (or modulus) of the complexnumbers t and z , respectively.



Spherical multiplier

Given a rational map ϕ ∈ C(z), we can extend in the usual way thefunction ϕ to a map ϕ+ : C→ C.We denote by f : P1(C)→ P1(C) and g : S2 → S2 the correspondingmaps given by f = θ−1ϕ+θ and g = θ−1ϕ+θ.Since S2 has a canonical Riemannian structure, if g : S2 → S2 is a rationalfunction, one has that for a given point p ∈ S2, there is an induced lineartransformation Tp(g) : Tp(S2)→ Tg(p)(S2) on the euclidean tangentspaces at p and g(p) of the Riemannian manifold S2. Then, the sphericalmultiplier of a rational function g : S2 → S2 at a point p ∈ S2 is given by

sm(g)(p) = ‖Tp(g)‖.

Note that f : P1(C)→ P1(C) and g : S2 → S2 are related by the equationΘ−1gΘ = f .




It is interesting to remark that if we translate the Riemannian metric fromS2 to P1(C), the translated metric agrees up to a positive scalar with theFubini-Study metric on P1(C).

The bijection Θ: P1(C)→ S2 permits us to define a similar version of thespherical multiplier sm(f ) : P1(C)→ R given by

sm(f )([z : t]) = ‖T[z:t](f )‖.

It is clear from the definitions that for [z : t] ∈ P1(C) or p ∈ S2:

sm(f )([z : t]) = sm(g)(Θ([z : t])), sm(g)(p) = sm(f )(Θ−1(p)).

Note that sm(f ) : P1(C)→ R (sm(g)) is a continuous map such thatsm(f ) ≥ 0 (sm(g) ≥ 0).




Any rational map f : P1(C)→ P1(C) can be represented by a pair ofbivariate polynomials:

f ([z : t]) = [A(z , t) : B(z , t)]

The coordinate representations of f with respect to x , y are given by

xfx−1(z) = A(z , 1)/B(z , 1), yfx−1(z) = B(z , 1)/A(z , 1),

(1)

xfy−1(t) = A(1, t)/B(1, t), yfy−1(t) = B(1, t)/A(1, t),

so that we can consider its corresponding derivatives:

(xfx−1)′(z) =d(A(z , 1)/B(z , 1))

dz, (yfx−1)′(z) =

d(B(z , 1)/A(z , 1))

dz,

(2)

(xfy−1)′(t) =d(A(1, t)/B(1, t))

dt, (yfy−1)′(t) =

d(B(1, t)/A(1, t))

dt.




Proposition

Given a rational map f : P1(C)→ P1(C) induced by homogeneouspolynomials A(z , t),B(z , t) and a point [z : t] ∈ P1(C), if we denote[z ′ : t ′] = [A(z , t) : B(z , t)], then

sm(f )([z : t]) =

(|t|2+|z|2)|t′|2(|t′|2+|z ′|2)|t|2 |(xfx−1)′(z/t)|, if |t| ≥ |z | and |t ′| ≥ |z ′|,(|t|2+|z|2)|z ′|2(|t′|2+|z ′|2)|t|2 |(yfx−1)′(z/t)|, if |t| ≥ |z | and |t ′| < |z ′|,(|t|2+|z|2)|t′|2(|t′|2+|z ′|2)|z|2 |(xfy−1)′(t/z)|, if |t| < |z | and |t ′| ≥ |z ′|,(|t|2+|z|2)|z ′|2(|z ′|2+|z ′|2)|z|2 |(yfy−1)′(t/z)|, if |t| < |z | and |t ′| < |z ′|.

(3)




In the case that we are working with normalized homogeneous coordinates(that is, [z : t] such that either t = 1 and |z | ≤ 1 or |t| < 1 and z = 1) wehave the following formula:

sm(f )([z : t]) =

1+zz

1+z ′z ′|d(A(z,1)/B(z,1))

dz |, if t = 1, t ′ = 1,1+zz1+t′ t′

|d(B(z,1)/A(z,1))dz |, if t = 1, z ′ = 1,

1+tt1+z ′z ′

|d(A(1,t)/B(1,t))dt |, if z = 1, t ′ = 1,

1+tt1+t′ t′

|d(B(1,t)/A(1,t))dt |, if z = 1, z ′ = 1.

(4)

Moreover, if [z : t] is a fixed point ([z ′ : t ′] = [z : t]) and we takenormalized homogeneous coordinates, the formula (4) is given by

sm(f )([z : t]) =

{|d(A(z,1)/B(z,1))

dz |, if t = 1, t ′ = 1,

|d(B(1,t)/A(1,t))dt |, if z = 1, z ′ = 1.

(5)




Theorem

Given a rational map f : P1(C)→ P1(C) and the induced sphericalmultiplier map sm(f ) : P1(C)→ R, then we have:

(i) sm(f ) : P1(C)→ R is a continuous map and sm(f ) ≥ 0,

(ii) sm(f )2 : P1(C)→ R is a differentiable map,

(iii) If p ∈ P1(C) is a fixed point, it follows that

sm(f )(p) = |(θf θ−1)′(θ(p))|.

Proof.

(i) and (ii): in formula (3), every function is differentiable except for themodule function which is continuous. However the square of the modulefunction is also a differentiable function.(iii) It follows from equation (5).



(Super) Attracting, indifferent and repelling cyclic points

Let a ∈ C be a fixed point of ϕ+ : C→ C. If a ∈ C the multiplier of ϕ+

at a is given by m(ϕ+)(a) = ϕ′(a). If a =∞, the multiplier m(ϕ+)(a) ofϕ+ at a =∞ is given by the derivative of 1/ϕ(1/z) at z = 0. A fixedpoint a is said to be super-attracting , attracting , indifferent orrepelling if |m(ϕ+)(a)| = 0,|m(ϕ+)(a)| < 1, |m(ϕ+)(a)| = 1 or|m(ϕ+)(a)| > 1, respectively.

Corollary

Let f : P1(C)→ P1(C) be a rational function. Then, a fixed point p of fis super-attracting, attracting, indifferent or repelling if and only ifsm(f )(p) is zero, lower than 1, equal to 1 or greater than 1, respectively.

Remark

Similar results can be obtained for a rational map g : S2 → S2 and thecorresponding version of spherical multiplier sm(g) : S2 → R.



(Super) Attracting, indifferent and repelling cyclic points

Given f : P1(C)→ P1(C) a rational map and a point p ∈ P1(C), it is saidthat p is a critical point if Tp(f ) is a singular linear map (similarly forS2). We denote by C (f ) the set of critical points of f .

Proposition

Let f : P1(C)→ P1(C) be a rational map, then the following propertiesare equivalent:

C (f ) ∩ J(f ) = ∅,s = min{sm(f )(z)|z ∈ J(f )} > 0,



Our approach to the Julia set of a rational function is based in differenttypes of algorithms:

We can compute the repelling q-cyclic points, 1 ≤ q ≤ n (the Juliaset is the closure of the set of repelling cyclic points).

We can compute the basins of non-repelling q-cycles for 1 ≤ q ≤ n(the Julia set is the boundary of the these basins).

In this talk, we also present a new procedure that looks for points p ∈ S2

such that sm(gq)(p) > 1.



Computing repelling cycles

For rational functions of degree ≥ 2, we have the following description ofJulia set:

Theorem

Let g : S2 → S2 be a rational map. If degree(g) ≥ 2, then J(g) is theclosure of repelling cyclic points of g .

For a proof of this result we refer the reader to Theorem 6.9.2. ofA. F. Beardon, Iteration of rational functions, Springer-Verlag, New York,2000.




Given a rational function g , this algorithm runs through the followingsteps:

(1) it represents g (via Θ) by using a pair of bivariate homogeneouspolynomials;(2) it computes this type of representation of gq for 1 ≤ q ≤ n;(3) it calculates the fixed points of gq;(4) it computes the spherical multiplier of gq at these fixed points;(5) it removes the fixed points with multiplier ≤ 1 and(6) it plots the resulting set of repelling fixed points of gq for 1 ≤ q ≤ n.

Its implementation in Mathematica is given by the function:

SphericalPlotRepellingCycles[{P, Q}, n]{P,Q}, the numerator and denominator of the rational function,

n, the upper bound




Example ϕ(z) = 2z/− (1 + 3z2),P(z) = 2z ,Q(z) = −(1 + 3z2)

n = 4 n = 5 n = 6




The graphic output corresponding to three executions of the Mathematicafunction related to the values of the second argument n ∈ {4, 5, 6}. In thecorresponding plot each q has an associated color; for example, for q = 6we have the green color which correspond to repelling 6-cyclic points; forq = 5, the red color is for 5-cyclic points, et cetera.

It is interesting to note that g has 3 repelling fixed points. Since thedegree of g is 2, this implies that g does not have neither attracting norsuper-attracting fixed points.

In this case, one can see that the repelling q-cyclic points are contained ina great circle of the unit sphere and therefore this suggests that thecorresponding Julia set is this great circle.




For this example, other representations of J(g) are the following:

Table : Spherical plots of J(g) where g = θ−1ϕ+θ and ϕ(z) = 2z−(1+3z2) . On the

left side, J(g) is the boundary of the basin of attracting 2-cyclic points . On theright side, a small neighborhood of J(g) has been constructed as the union ofsmall 2-spherical 2-cubes .



Spherical plots of basins of non-repelling cyclic points

Iterated subdivision on the 2-sphere: Γ0∗, Γ

1∗, Γ

2∗.

This algorithm takes a point at each 2-cube and analyses if the sequence

(p, g(p), g 2(p), g 3(p), . . . , gk−1(p), gk(p))

converges to a non repelling cycle.

L. J. Hernandez, M. Maranon and M. T. Rivas, Plotting basins of endpoints of rational maps with Sage, Tbilisi Math. J., vol. 5 (2) (2012),71–99 .




These ideas has been implemented in the function:

SphericalPlotNonRepellingCyclesBasins[P, Q, untiln, subdivision]

P,Q are the numerator and denominator of a rational function.

untiln is a positive integer. For instance, if untiln=3, we can obtain aspherical plot with the basin of non-repelling 1-cyclic points (fixed points),2-cyclic points and 3-cyclic points.

subdivision is again a non negative integer which denotes the number ofconsecutive subdivisions of the standard cubic subdivision of the sphere.For instance, for subdivision=2, this second subdivision has 6× 42 = 96spherical quadrilaterals.




Let us illustrate this algorithm with an example. Take the iterationfunction

ϕ(z) =P(z)

Q(z), P(z) = (2i/3)(i +

√3)z + ((1− i

√3)/6)z4, Q(z) = 1, (6)

untiln=3, subdivision=8.




9

2-sphere and other algorithms to complement our dynamical study. We cancomplement these algorithms with the ones given by Varona [11] for plottingthe basins of attraction of different iterative processes in the complex plane.

3.1 Algorithm 1: Spherical plots of basins of fixed and non-repelling cyclicpoints

p = 1{{0., {1, 0}}}

p = 2 Null { }

p = 3{0, {1., 1}}{5.74 ∗ 10−14, {−0.5 + 0.866025i, 1}}{5.74 ∗ 10−14, {−0.5 − 0.866025i, 1}}

Table 1 Output coming from our first algorithm. We can see the basins of p-cycles for1 ≤ p ≤ 3 for the iteration function given in (3.1).

With the target to finding an end point associated to a point x ∈ S2, therational map f must be iterated to obtain a finite sequence

(x, f(x), f2(x), f3(x), . . . , fk−1(x), fk(x)).

In this context, we remind that a maximum number of iterations l must beconsidered and a certain precision c1 must be prefixed to determine when tostop the iterative process while programming the function which returns suchsequence. That is why we shall always work with sequences in which k ≤ l.Now we explain how our first algorithm works. It is divided into several sub-algorithms in the following way.

One of the basic sub-algorithms is based on the notion of Cauchy sequencein order to stop the iterative process. After each iteration, there will be twopossible cases:

1) If the chordal distance from fk−1(x) to fk(x) is lower than 10−c1 , thentake as output the list [fk(x), k]; otherwise, case 2) is applied.

2) If k < l, a new iteration is done and case 1) is applied again; otherwise (ifk = l), then the output [f l(x), l] is taken.

Let us suppose that we know some fixed points {x1, x2, . . . , xm+1} of arational function f .

q = 1{{0., {1, 0}}}

q = 2 Null { }

q = 3{9.46228 ∗ 10−15, {1., 1}}{5.74839 ∗ 10−14, {−0.5 + 0.866025i , 1}}{5.74839 ∗ 10−14, {−0.5− 0.866025i , 1}}




In the first row, we can see the basins of q-cycles for 1 ≤ q ≤ 3. Thecorresponding Julia set J(g) is the boundary of the basin of theinfinite point which is the same the boundaty of the basis of thesuper-atracting 3-cycle.

The second row contains two approaching neigborhoods of the Juliaset J(g) contructed with a plotting algorithm based on the sphericalmultiplier map sm(g) : S2 → R.




Main problem:

This type of algorithms have to deal with rational functions of high degreeand this fact increases the execution time.



Plotting algorithms based on spherical multipliers

Given a point p ∈ S2 and a rational function g : S2 → S2, we cancompute the spherical multiplier by the formula:

‖Tp(gk)‖ = ‖Tp(g)‖‖Tg(p)(g)‖ · · · ‖Tgk−1(p)(g)‖

or equivalently,

sm(gk)(p) = (sm(g)(p)) (sm(g)(g(p))) · · · (sm(g)(gk−1(p)))

Using the formula above, we can compute the spherical multiplier of gk ata point p ∈ S2 without computing rational functions of higher degree.Note that if p is a repelling q-cyclic point, then sm(gq)(p) > 1 and thisin-equality holds in a small neighborhood at p. Using this property of thespherical multiplier and subdivisions of the canonical cubic structure of the2-sphere we can find neighborhoods of the Julia set J(g) for expansiverational maps.




Definition

Given a rational map g : S2 → S2, we say that g is expanding on J(g) ifthere is c, λ ∈ R, c > 0, λ > 1 such that sm(gn)(p) > cλn, 1 ≤ n ∈ N, forevery p ∈ J(g).

Note that for n = 1, sm(g)(p) > cλ > 0, then C (g) ∩ J(g) = ∅.

Theorem

Let g : S2 → S2 be a rational map and suppose that degree(g) ≥ 2. If theforward orbit of any critical point accumulates at a (super)attracting cycleof g , then g is expanding on J(g).

For a proof of this fact, we refer the reader to Theorem 9.7.5. ofA. F. Beardon, Iteration of Rational Functions, Graduate Texts inMathematics, Springer-Verlag, New York, 1991.




DenoteMr = {p ∈ S2|sm(g r )(p) > 1}

If g is expanding, then there is r0 such that J(g) ⊂ Mr for r ≥ r0 andJ(g) ⊂ ⋂∞

r0Mr .

If the forward orbit of any critical point of g accumulates at a(super)attracting cycle of g , then J(g) =

⋂∞r0

Mr .

Corollary

Let g : S2 → S2 be a rational map and suppose that degree(g) ≥ 2.Suppose that the forward orbit of any critical point accumulates at a(super)attracting cycle of g . If we take Ns =

⋂r0+sk=r0

Mr , thenN0 ⊃ N1 ⊃ N2 ⊃ · · · is a resolution of J(g).




We have developed the implemented Mathematica function:

SphericalPlotNeigborhoodsRepellingCycles[P, Q, ntimes, subdivision]

For each n, s ∈ N this function computes a set of points Pns suchthat if p ∈ Pns , then sm(gn)(p) > 1

Let g : S2 → S2 be a rational map (degree(g) ≥ 2) such that the forwardorbit of any critical point accumulates at a (super)attracting cycle of g .Then, given a resolution {Ni} of J(g), there are increasing subsequencesni , si such that the sets Pni si ⊂ Ni verify that:

for every p ∈ J(g) there a sequence of points pi ∈ Pni si such thatpi → p.

Note that that a neighborhood of Pni si can be considered as anapproaching of Ni .




Taking into account the limitations of any computational system (the sizeof plotted point, the size of a pixel in any monitor, the memory of acomputer), the better possible visual description of J(g) is given adetermined set of points Pni si and taking more subdivisions or moreiterations do not improve the visualization of J(g).



Overlay structure of g|J(g)

Since J(g) is invariant (g(J(g)) = J(g)), we can prove that ifN0 ⊃ N1 ⊃ N2 ⊃ · · · is a resolution of J(g), then there a subsequencer1 < r2 < · · · ri < · · · such that g(Nri+1) ⊂ Nri .Taking suitable neighborhoods, we obtain:

Theorem

Let g : S2 → S2 be a rational map of degree d ≥ 2 and suppose thatC (g) ∩ J(g) = ∅. Then, there is a resolution {Ni} of J(g) such thatJ(g) ⊂ Int(Ni ), Ni is a planar surface with boundary, g(Ni+1) ⊂ Ni andthere is n0 such that g |Ni

: Ni → g(Ni ) is a covering map for i ≥ n0.Moreover, g |J(g) : J(g)→ J(g) has an induced d-overlay structure.

Note that if N is planar surface with boundary, then N has the homotopytype of a union of wedges of circles.



Overlay structure of g|J(g)

Note that in the following commutative diagram

J(g)

g |J(g)

��

// Ni+1

g |Ni+1

��

// Ni

g |Ni��

J(g) // Ni// Ni−1

the map g |J(g) : J(g)→ J(g) is obtained as an inverse limit of mapswhich are homotopy equivalent to covering maps.

Example. Computing J(g) with the spherical algorithm for the rationalmap g : S2 → S2 induced by

ϕ(z) = 2z/− (1 + 3z2),P(z) = 2z ,Q(z) = −(1 + 3z2)



The spherical algorithm applied to ϕ(z) = 2z/− (1 + 3z2)

sq 6 7 8 9

1

2

3(L. J. Hernandez and M. T. Rivas) EACA 2016 22-24 June 2016 38 / 57



sq 6 7 8 9

4

5

6(L. J. Hernandez and M. T. Rivas) EACA 2016 22-24 June 2016 39 / 57



The different colors are assigned depending on the values of thespherical multiplier; the black color corresponds to the region wherethe spherical multiplier is ≤ 1

This pictures suggest the overlay structure of g |J(g)

In this case the functions (sm(gq))2 are Morse functions defined on aplanar surface with boundary



Non expanding rational maps

The following example suggests that the algorithms based on sphericalmultipliers can be applied in more general settings.

Consider the rational map:

ϕ(z) = z2 + i

The critical point of this map is z = 0 and its forward orbit is0, i ,−1 + i ,−i ,−1 + i , · · · converges to the 2-cycle −1 + i ,−i . In thiscase sm(g)(i) = ((1 + 1)/(1 +

√2))2 = 4/(1 +

√2) > 1 and

sm(g)(−1 + i) = ((1 +√

2)/2)2√

2 = 2 +√

2 > 1. Then the critical pointis in the basin of a repelling 2-cycle and it is in J(g).

In this case, the map g defined on a neigborhood at J(g) has thestructure of a branched covering. Then, a notion of branched overlayhave to be considered to study the structure of the map g |J(g).



Non expanding rational maps

The first and second images contain two approaching neigborhoods of the Julia

set J(g) constructed with a plotting algorithm based on the spherical multiplier

map sm(g) : S2 → R, where g = θ−1ϕ+θ and ϕ(z) = z2 + i . The third image

has been extracted from the book of Beardon.



Julia Language

In order to obtain a better performance, Julia Language has the requiredproperties to develop these family of algorithms.

The language Julia name comes from the important French mathematicianGaston Maurice Julia (1893 – 1978), who discovered fractals associated tosome iterative processes. His works were popularized by Frenchmathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals areclosely related.



Julia Language

Some characterisctics:

It is a high-level, high-performance dynamic programming languagefor technical computing.

It naturally has many of the mathematical and statistical librariesfound in any high performance environment.

It is also very extensible: There is a built-in package manager for theaddition of new external libraries and packages.



Julia Language

Julia is built for speed. For classical algorithms:

fib (Fibonaccy sequence),

mandel (drawing the Mandelbrot set and associated Julia sets),

pars int (Express an integer in different bases),

π sum (Summation of a power series),

printfd (Printing to a file descriptor),

quicksort (Sorting of random numbers using quicksort),

rand mat mul (Multiplication of random matrices),

rand mat stat(Statistics on a random matrix).

We show some comparisons from the Julia Language website:



Julia Language

homesourcedownloadsdocspackagesblogcommunitylearningteachingpublicationsGSoCjuliacon

JuliaCon 2016June 21st - 25th

Massachusetts Institute of Technology, Cambridge, Massachusettstickets on sale now | more information

Google Summer of CodeGet paid to work on a project of your choice over the summer!

ideas list | how to apply

Figure: benchmark times relative to C (smaller is better, C performance = 1.0).

C and Fortran compiled with gcc 5.1.1. C timing is the best timing from all optimization levels (-O0through -O3). C, Fortran and Julia use OpenBLAS v0.2.14. The Python implementations ofrand_mat_stat and rand_mat_mul use NumPy (v1.9.2) functions; the rest are pure Python

implementations. Plot created with Gadfly and IJulia from this notebook.

Fortran Go Java JavaScript Julia Lua Mathematica Matlab Octave Python R

mandelparse_intpi_sumprintfdquicksortrand_mat_mulrand_mat_stat

benchmark

10-1

100

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Julia Benchmarks http://julialang.org/benchmarks/

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Julia Language

More information about Julia:

Julia run fast because of its LLVM-based just-in-time (JIT) compiler,which is designed for a high performance environment.

Julia is also designed for cloud computing and parallelism as itprovides a number of key building blocks for distributed computation.

The LLVM compiler infrastructure project (formerly Low Level VirtualMachine) is a “collection of modular and reusable compiler and toolchaintechnologies” used for developing compiler front ends and back ends.Languages with compilers that use LLVM include ActionScript, Ada, C#,Common Lisp, D, Fortran, OpenGL Shading Language, Haskell, Javabytecode, Julia, Lua, Objective-C, Python, R, Ruby, Rust, Scala, andSwift.



Implementations in Julia Language

Julia packages for Julia sets:

We have developed some packages devoted to give a global visualizationof the Julia set of a rational map using two rectangular plots.Plotting Basins of Univariate Rational Functions with Julia (User manualof the package PBURF03.jl),https://github.com/luisjavierhernandez/GVBURF03.jl


https://github.com/luisjavierhernandez/GVBURF03.jl



Next work will be:

(i) To develop 3D Julia Packages to plot the basins of fixed point and itsboundary in the usual unit sphere contained in the 3-dimensional euclideanspace.(ii) To develop 3D Julia Packages to plot the basins of non-repelling cyclesin the usual unit sphere contained in the 3-dimensional euclidean space.



The shape of the Julia set and measures of basins

The developed algorithms also provide a system of cubic complexes withdifferent resolutions that can be used to compute:

shape invariants of the Julia set,

representations of shape groups and classifications of the overlaystructures of Julia sets.



The shape of the Julia set and measures of basins

Thank you very much

for your attention!!



References

A. F. Beardon, Iteration of rational functions, Springer-Verlag, New York,2000.

D. Edwards and H. Hastings, Cech and Steenrod homotopy theories withapplications to Geometric Topology, Springer Lect. Notes in Math., 542,(1976).

A. Eskin and M. Mirzakhani, Invariant and stationary measures for theSL2(R) action on moduli space, preprint 2013; arXiv:1302.3320.

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution,and orbit closures for the SL2(R) action on moduli space, preprint, 2013;arXiv:1305.3015

G. Fubini, Sulle metriche definite da una forme Hermitiana, Atti Istit.Veneto, 63 (1904) 502–513.



References

R. H. Fox, Covering spaces with singularities. In: Algebraic Geometry andTopology. A Symposium in honor of S. Lefschetz (Fox et al., eds.),243–257, Princeton Univ. Press, 1957.

R. H. Fox, On Shape, Fund. Math. 74 (1972) 47–71.

R. H. Fox, Shape theory and covering spaces, Springer Lect. Notes inMath., 375, (1974) 71–90.

J. M. Garcıa-Calcines, L. J. Hernandez and M. T. Rivas, Limit and endfunctors of dynamical systems via exterior spaces, Bull. Belg. Math.Soc.–Simon Stevin, 20 (2013), 937–959.

J. M. Garcıa-Calcines, L. J. Hernandez and M. T. Rivas, A completionconstruction for continuous dynamical systems, Topol. Methods NonlinearAnal., 44 (2), (2014) 497–526.



References

J. M. Garcıa-Calcines, J. M. Gutierrez. L. J. Hernandez and M. T. Rivas,Graphical Representations for the Homogeneous Bivariate Newton’sMethod, Appl. Math. Comput., 269, 988–1006 (2015). DOI:10.1016/j.amc.2015.07.102

J. M. Gutierrez. L. J. Hernandez, A.A. Magrenan and M. T. Rivas,Measures of the basins of attracting n-cycles for the relaxed Newton’smethod, to appear, Springer, 2016.

J. M. Gutierrez, L. J. Hernandez-Paricio, M. Maranon-Grandes and M. T.Rivas-Rodrıguez, Influence of the multiplicity of the roots on the basins ofattraction of Newton’s method, Numer. Algorithms, vol. 66 (3) (2014),431–455.

L. J. Hernandez, Bivariate Newton-Raphson method and toroidalattraction basins, Numer. Algorithms, 71 (2), 349-38 (2016). DOI:10.1007/s11075-015-9996-3



References

L. J. Hernandez, Plotting Basins of Univariate Rational Functions withJulia (User manual of the package PBURF03.jl),https://www.researchgate.net/profile/Luis_Hernandez_

Paricio/contributions,https://github.com/luisjavierhernandez/GVBURF03.jl

L. J. Hernandez, M. Maranon and M. T. Rivas, Plotting basins of endpoints of rational maps with Sage, Tbilisi Math. J, 5 (2) (2012), 71–99.

L. J. Hernandez and V. Matijevic, Fundamental groups and finite sheetedcoverings, J. Pure Appl. Algebra, 214, (2010) 281–296.

S. Mardesic and J. Segal, Shape Theory –The Inverse Limit Approach,North-Holland, 1982.

S. Mardesic and V. Matijevic, Classifying overlay structures of topologicalspaces, Topology and its Applications, 113, (2001)1–3.


https://www.researchgate.net/profile/Luis_Hernandez_Paricio/contributions

https://www.researchgate.net/profile/Luis_Hernandez_Paricio/contributions

https://github.com/luisjavierhernandez/GVBURF03.jl


References

T. Porter, Cech and Steenrod homotopy and the Quigley exact couple instrong shape and proper homotopy Theory, J. Pure Apl. Alg. 24 (1983)303–312.

E. Study, Kurzeste Wege im komplexen Gebiet, Math. Ann., 60, (1905)321–378


self-overlays and shape of the julia set of a rational map · d. edwards and h. hastings, cech and...

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