On the stochastic bifurcations regarding random iterations of polynomials of the form z 2 + c n

In this paper, we consider random iterations of polynomial maps z 2 + c n where c n are complex-valued independent random variables following the uniform distribution on the closed disk with center c and radius r . The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter c = − 1, almost every random Julia set is totally disconnected with much smaller radial parameters r than expected. We also introduce several open questions worth discussing.


Introduction 1.Background
In this paper, we discuss the concrete and interesting random holomorphic dynamical systems with parameters c ∈ C and r ≥ 0.More precisely, we will denote f c (z) = z 2 + c throughout this paper and we consider random iterations of the form where c n (n = 1, 2, . . . ) are complex-valued independent random variables that follow the uniform distribution on the closed disk B(c, r) = {c ′ ∈ C : |c ′ −c| ≤ r} on the c-plane.The reader is referred to Remark 4.10 of [32].See Setting 4.1 for the rigorous setting.
The aim of this paper is twofold.First, we study the (dis)connectedness of random Julia sets and relate it to the bifurcation, and second, we investigate the bifurcation of our proposed random iterations and give quantitative estimates of bifurcation parameters.
In recent decades, there has been a rapid growth of studies on random dynamical systems.In deterministic autonomous dynamical systems, time-evolution rules are independent of time and homogeneous; however, they are dependent in random dynamical systems.Motivated by scientific demands, great efforts have been devoted to establishing fundamental results.For this field, refer to the textbook by Arnold [1].
Fornaess and Sibony first studied random dynamical systems of complex analytic maps [13].They proved that small "random perturbations" of a single map produce very stable dynamics on average.It is then natural to ask what happens when larger noise is added.This question is the central motivation of this paper.
Sumi expanded the results of Fornaess and Sibony by studying minimal sets and established foundations for the study of random holomorphic dynamical systems.He defined the mean stability and showed that the set of mean stable systems is open and dense in the space of all random dynamical systems having mild conditions [28].We will quickly review this theory in Subsection 2.2.For details, see [29,30].See also [31,32] for the author generalizing the setting from i.i.d. to non-i.i.d.setting.
These previous studies motivate us to study the bifurcation of (i.i.d.) random quadratic dynamical systems induced by the uniform distribution on B(c, r).More precisely, for a fixed parameter c ∈ C we can define the bifurcation radius r bif (c) as the parameter satisfying that the random dynamical system is mean stable if and only if r / ∈ {0, r bif (c)}.Compare this with Theorem 2.16 and Definition 2.17.
The dynamical systems of quadratic polynomials f c (z) = z 2 +c (c ∈ C) are extremely important for studying holomorphic dynamical systems.The celebrated Mandelbrot set is an iconic figure on the parameter plane, whose boundary is known as the deterministic bifurcation locus of quadratic polynomials.The Mandelbrot set M is the set of all parameters whose critical orbit is bounded: is the (autonomous) n-th iterate of a map f c .The condition f •n c (0) → ∞ is equivalent to the Julia set of f c being connected.See McMullen's book [20] and Milnor's book [22] for details.
Several authors have reported great studies on sample-wise dynamics for random iterations.For a fixed infinite sequence (c n ) ∞ n=1 , we consider the Julia set of compositions which is called a non-autonomous Julia set or a random Julia set.
Brück, Büger, and Reitz investigated the (dis)connectedness of random Julia sets [4].They found interesting examples that illustrated the difficulty of random iteration compared with deterministic iteration and established several conditions relating the noise amplitude to the connectedness of random Julia sets.See Remark 3.2 and [3,5].
Brück, Büger, and Reitz showed that if center c = 0 and radius r ≤ 1/4, then every random Julia set of [18], Lech and Zdunik proved that if c = 0 and r > 1/4, then almost every random Julia set is totally disconnected.For our viewpoint, this is equivalent to r bif (0) = 1/4.One motivation of this paper is to generalize this result for the cases where c = 0.

Main results
First, we consider a sufficient condition for the random Julia sets to be totally disconnected almost surely.
Main Result A (Theorem 3.7).In this staement, the distribution µ of c n needs not to be the uniform distribution on a disk.Under some assumption, if the critical point 0 tends to ∞ with probability one, then the random Julia set J ω is totally disconnected almost surely.
Second, we reveal the relationships between the bifurcation radius r bif and the connectedness of random Julia sets.More precisely, the following holds.
Main Result B (Theorem 4.6).Suppose that the interior of B(c, r) contains a superattracting parameter.Then the following four are equivalent.
(2) The random orbit of z = 0 does not tend to ∞ surely.
(3) The random Julia set is connected for every sample-path.
(4) The set of all total orbits of z = 0 is bounded in C. Also, the following four are equivalent.
(3 ′ ) The random Julia set is totally disconnected almost surely.
(4 ′ ) The set of all total orbits of z = 0 is not bounded in C.
Furthermore, either the former or the latter is valid.
In particular, the following dichotomy holds.
Corollary 1.1.Let c be a superattracting parameter.If r ≤ r bif (c), then every random Julia set is connected.If r > r bif (c), then almost every random Julia set is totally disconnected.
Corollary 1.1 is a generalization of the fact that the autonomous Julia set of a single map f c is either connected or totally disconnected according to c ∈ M or not.
Next, we show some properties and present some quantitative estimates for the bifurcation radius as follows.
(i) For every c ∈ C, we have r bif (c) ≤ dist(c, ∂M).It is worth noting that for c = −1 or c3 , statement (v) and (vi) illustrate a large gap r bif (c) ≪ dist(c, ∂M).Furthermore, we also have r bif (c + ǫ) ≪ dist(c + ǫ, ∂M) for a small ǫ ∈ C .The proofs of (v) and (vi) rely on parabolic implosion in a stochastic context.See Theorem 4. 19.
Combining (iv) with Corollary 1.1, we give a stronger result compared with the result by Lech and Zdunik.Compare Corollary 4.15 with Theorem 3.5.Moreover, we give a result for the case where c is far from 0.
• ϕ n on the iterated Julia set for every n ≥ 0. In particular, there exists a quasiconformal map ϕ which maps the autonomous Julia set J c onto the non-autonomous Julia set.

Structure of the paper
In Section 2, we discuss the elementary properties in the general form and define the bifurcation radius.Moreover, we quickly review Sumi's theory on minimal sets and mean stability in Subsection 2.2.In Section 3, we establish some tools to decide connectedness and totally disconnectedness.These are developed by Brück, Büger, and Reitz [4] and Lech and Zdunik [18].Besides, we show Main Result A relating bifurcation and connectedness.In Section 4, we prove Main Results B, C, and D. In Section 5, we discuss some open problems and generalization to other cases.

Preliminaries
In this section, we define the bifurcation radius and discuss the elementary properties required in subsequent sections.We consider minimal sets and mean stability, which appears in Subsection 2.2.They describe set-valued dynamics and we can apply them to quenched, or averaged, dynamics.The minimal sets provide the basis for our discussion and define the bifurcation radius, which is the main subject in this paper.

Setting and notations
In this section, we consider a slightly general setting.
Setting 2.1.Let µ be a Borel probability measure on the parameter plane C with compact support.Denote the support by supp µ.We define the probability measure P µ as the one-sided infinite product of µ, which is supported on Ω µ = ∞ n=1 supp µ with the Borel σ-algebra B µ .
The most simple and important example for µ is the normalized Lebesgue measure on B(c, r) = {c ′ ∈ C : |c ′ − c| ≤ r}, which means the uniform distribution.Another example is the Dirac measure at c, which can be regarded as a deterministic dynamical system.We analyze these examples in Section 4.
Our random dynamical systems are the stochastic process perturbed by i.i.d.noise in a sense.One can generalize the setting to a non-i.i.d.setting, which we do not pursue here.For details, the reader is referred to [31,32].
Regarding noise, a natural consideration may be a distribution µ with unbounded support, say the Gaussian distribution.However, in unbounded cases, there are no planar attractors which we intend to investigate.Thus, we assume that supp µ is compact.See [5] for the detail, and see also [11] for more wild phenomena for the distributions with unbounded supports.
The following is a well-known fact.See, for example, Section 6 of [12].
We now define the Julia set for every noise realization ω.
Definition 2.3.For each n ∈ N and infinite sequence of parameters ω n=1 ∈ C N , we define the non-autonomous Julia set or the random Julia set of ω by ω } ∞ n=1 is not normal on any neighborhood of z}.
If there exists c 0 ∈ C such that c n = c 0 for every n ∈ N, then for ω = (c n ) ∞ n=1 , the non-autonomous Julia set J ω is equal to the usual (autonomous) Julia set of f c 0 .
Analogously to the usual case, the non-autonomous Julia set is the common boundary of the basin at infinity and the filled Julia set defined as follows.
Definition 2.5.For each ω = (c n ) ∞ n=1 ∈ C N , we define the non-autonomous basin at infinity of ω by → ∞ as n → ∞}.We define the non-autonomous filled Julia set of ω by K ω = C \A ω .
Here, C = C ∪{∞} denotes the Riemann sphere, which is homeomorphic to the real two-dimensional sphere.We endow C with the spherical metric d.A polynomial map f : C → C can be extended analytically to f : For our convenience, we list the elementary properties of non-autonomous Julia sets, basins at infinity, and filled Julia sets.For R > 0, we denote by D R the open disk D R = {z ∈ C : |z| < R} on the dynamical plane.
Lemma 2.6.Let µ be a Borel probability measure on C with compact support.Then we have all the following.
(1) There exists R > 0 such that f (2) Take R > 0 as above.The non-autonomous basin at infinity is then the union of the increasing sequence of the open subsets; ) is a non-empty compact subset for every ω ∈ Ω µ .
(3) For ω ∈ Ω µ , we have For the proof, see [4].Note that the existence of such R is due to the compactness of supp µ.

Minimal sets, mean stability, and bifurcation
Consider the action of polynomial semigroups whose product is the composition of maps.Semigroup actions are related to random dynamical systems as set-valued dynamics.Here, we review the theory of polynomial semigroups, which enables us to define bifurcations.
Generally, one can consider rational semigroups, not polynomial.However, we focus on polynomial maps since we are interested in quadratic polynomials.The interested reader is referred to [16] and [25].
We start with the definition of polynomial semigroups.Denote by Poly the space of all polynomials of degree two or more endowed with the topology of uniform convergence on C. We say that G is a polynomial semigroup if G is a non-empty subset of Poly closed under mapping composition.Let Γ be a subset of Poly.We say that G is generated by Γ if for every g ∈ G, there exist n ∈ N and γ 1 , γ 2 , . . .
Definition 2.7.Let µ be a Borel probability measure on C with compact support.We define the polynomial semigroup G µ as the semigroup generated by the compact set Γ = {f c ∈ Poly : c ∈ supp µ}.Here f c (z) = z 2 + c is a quadratic polynomial, as we will assume throughout this paper.

Note that the map Poly
For every polynomial semigroup, we define the Julia set and Fatou set as follows.The following proposition connects the Julia set of polynomial semigroup and the random Julia sets.The idea of the proof can be found in [26].
Definition 2.8.For a polynomial semigroup G, define the Julia set by Proposition 2.9.For a Borel probability measure µ on C with compact support, we have J(G µ ) = ω∈Ωµ J ω .
Next, we define minimal sets which play a crucial role in studying random holomorphic dynamical systems.
is a minimal element of the set of all forward G-invariant, non-empty, and compact subsets of C with respect to the inclusion relation ⊂.
We can easily show that a non-empty compact set L ⊂ C is minimal if and only if for every z ∈ L, we have g∈G {g(z)} = L.
Trivially, the singleton {∞} is a minimal set of G for every polynomial semigroup G. Thus, we are interested mainly in planar minimal sets as defined below.
For example, if µ = δ c is the Dirac measure at c, then every periodic cycle of f c is a minimal set of G δc .Hence, there is an infinite number of planar minimal sets of G δc for every c ∈ C.
For the number of minimal sets, we have the following property.
Lemma 2.12.Let G 1 and G 2 be polynomial semigroups generated by Γ 1 and Γ 2 , re- It follows from Zorn's lemma that there exists a minimal set Definition 2.13.Let G be the polynomial semigroup generated by a subset Γ of Poly.
A minimal set L ⊂ C of G is said to be attracting if there exist N ∈ N and non-empty open subsets U and V of the Fatou set F (G) such that the following two conditions hold.
Here, A ⋐ B means that the closure A is a relatively compact subset of B.
One can easily show that if the polynomial semigroup G is generated by a compact subset, then the minimal set {∞} is attracting.By using the hyperbolic metric, we have the following characterization of attracting minimal sets.For the proof, see [29,Remark 3.5] or [32,Lemma 2.8].
Proposition 2.14.Let G be the polynomial semigroup generated by a compact subset Γ of Poly.A minimal set L ⊂ F (G) of G is attracting if and only if the following holds.There exist N ∈ N, 0 < α < 1, C > 0, and a neighborhood W of L such that for every k ∈ N, f 1 , f 2 , . . .f kN ∈ Γ, and z, z 0 ∈ W , the spherical metric satisfies Now, we present the classification theorem, which states that every minimal set either is attracting, intersects the Julia set, or intersects some rotation domain.Compare the following theorem with the classification theorem for autonomous iteration of a single map.For the latter classical theorem, see [22,Theorem 16.1] or [6, Theorem IV.2.1].
Theorem 2.15.Let G be the polynomial semigroup generated by a compact subset Γ of Poly.A minimal set L ⊂ C of G is either one of the following three.
For the proof, see [29,Lemma 3.8].The classification theorem leads to the following definitions.We say that a minimal set is J-touching if it is of type (ii) in Theorem 2.15.Besides, a minimal set is sub-rotative if it is of type (iii) in Theorem 2.15.
However, these two types of minimal sets are uncommon compared with the attracting minimal sets as stated by Theorem 2.16.Denote by H the set of all parameters c ∈ M such that f c is hyperbolic.• If c ∈ H, then G c,r has no planar minimal sets for every r > 0.
• If c ∈ H, then there exists r bif (c) > 0 such that the following holds.
-If 0 < r < r bif (c), then G c,r has exactly one planar minimal set and it is attracting.
-If r = r bif (c), then G c,r has exactly one planar minimal set and it is not attracting.
-If r bif (c) < r, then G c,r has no planar minimal sets.
Theorem 2.16 motivates us to define the bifurcation radius as follows.Note that our bifurcation occurs at most once because f c is degree 2, and in general, our bifurcation occurs at most d − 1 times if we work on polynomial maps of degree d.Definition 2.17.For every c ∈ C, we define the bifurcation radius r bif (c) ≥ 0 according to the following cases.If c / ∈ H, we define r bif (c) = 0.If c ∈ H, we define the bifurcation radius r bif (c) > 0 as the unique r > 0 such that G c,r has the non-attracting planar minimal set.
Equivalently, the bifurcation radius r bif (c) is the maximum value for r ≥ 0 such that G c,r has a planar minimal set, or equivalently, is the infimum value for r ≥ 0 such that G c,r has no planar minimal sets.
The rest of this subsection is devoted to emphasizing the importance of deciding the bifurcation parameters by presenting some interesting phenomena.See [28, Subsection 3.6] for the details.Definition 2.18.Let G be the polynomial semigroup generated by a compact subset Γ of Poly.We say that G is mean stable if all the minimal sets of G are attracting.By Theorem 2.16, for every c ∈ C and r ≥ 0, the polynomial semigroup G c,r is mean stable if and only if r / ∈ {0, r bif (c)}.Before describing the dynamics of a mean stable system, we define the function representing the probability of random orbits tending to an attractor.Definition 2.19.Let µ be a Borel probability measure on C with compact support.For every attracting minimal set L of G µ , we define the function T L,µ : C → [0, 1] by for every point z ∈ C, where d(w, L) = min w ′ ∈L d(w, w ′ ).
The following theorem reveals an interesting noise-induced order.That is, almost every orbit is attracted by some attracting minimal set and the probability T L,µ (z) tending to the attractor L depends continuously on initial point z if the system is mean stable.For the detail, see [28,Theorem 3.15].Definition 2.20.For a polynomial semigroup G, define the kernel Julia set by Theorem 2.21.Let µ be a Borel probability measure on C with compact support.If J ker (G) = ∅, then T {∞},µ is continuous on C .Moreover, the sum satisfies L T L,µ (z) = 1 for every z ∈ C where the sum runs over all attracting minimal sets L.
The following gives a useful sufficient condition for J ker (G) = ∅.
See [28,Lemma 5.34] for the proof.Notably, the continuity of T L,µ in Theorem 2.21 does not occur in deterministic dynamical systems.Namely, if µ is the Dirac measure at c, then the function T {∞},µ takes 1 on A(f c ) and 0 on K(f c ), where A(f c ) = A ω and K(f c ) = K ω for ω = (c, c, . . .).Thus, T {∞},µ is discontinuous on the autonomous Julia set J(f c ) = J ω .
Theorem 2.21 suggests that the dynamics of a mean stable system are similar to those of a hyperbolic map.In the latter case, every orbit on the Fatou set is contracted to an attracting cycle, while the complementary Julia set has zero areas.
The set of all hyperbolic rational maps is conjectured to be open and dense in the space of all rational maps.Even in the quadratic polynomial case, whether or not H = M remains unsolved.As is well-known, this conjecture can be affirmatively solved if one prove that the boundary ∂ M is locally connected.Meanwhile, it is still open whether or not the boundary of the Mandelbrot set has a positive area, although Shishikura showed that it has the full Hausdorff dimension.
Compared with the autonomous case, the following results show that we can solve random versions of the abovementioned conjectures.For the proof, see [32] and [29].
with the usual topology.
(ii) For all but countably many r ∈ [0, ∞), the set {c ∈ C : G c,r is not mean stable} has zero two-dimensional Lebesgue measure.
We also have the following tameness if noise is added.
Theorem 2.24.Let µ be a Borel probability measure on C with compact support.If J ker (G) = ∅, then the random Julia set J ω has zero two-dimensional Lebesgue measure for P µ -almost every ω ∈ Ω µ .
One may mistakenly consider that Theorem 2.24 is insignificant since it is difficult to construct autonomous Julia sets with positive areas.However, compared with the autonomous case, constructing the non-autonomous Julia set of positive areas is not very difficult.Jones et al. found that there exists ω = (c n ) ∞ n=1 with c n ∈ {0, 1/2, 1/4} for all n ∈ N such that J ω has positive areas.They use the parabolic fixed point 1/2 of the map f 1/4 .For the proof, see [7,Section 5.2].Developing the idea, Comerford proved the existence of measurable invariant sequences of line fields [10].
As presented above, the dynamics of mean stable systems are similar to those of hyperbolic polynomial maps.However, they cause interesting phenomena that cannot occur in deterministic dynamical systems, e.g., continuity of T L,µ , finiteness of the set of all minimal sets, etc.This motivates us to determine the bifurcation radius where the mean stability breaks.

Total disconnectedness
The systematic study of random iterations of the quadratic polynomial was done by Brück and Büger.In particular, Brück, Büger, and Reitz investigated the (dis)connectedness of the non-autonomous Julia sets in [4].Here, we present some conditions for the random Julia set to be connected or totally disconnected.
The following [4,Theorem 1.1] shows that the critical orbits determine connectedness of Julia sets.Compared with the autonomous case, we need to modify the statement suitably.
Then the non-autonomous Julia set J ω is connected if and only if the orbit {f We remark on the difficulty of the disconnected case compared with the autonomous case.Remark 3.2.For an autonomous case, the Julia set of a quadratic polynomial is either connected or totally disconnected [22, Problem 9-f].However, for a non-autonomous case, we can construct a Julia set that has more than one but finitely many connected components.For instance, if |c 1 | > 1 and c n = 0 for every n ≥ 2, then has exactly two connected components where ω = (c n ) ∞ n=1 .Similarly, we can construct various non-autonomous Julia sets with finitely many connected components.Furthermore, there exists ω = (c n ) ∞ n=1 such that |c n | < 2 for every n ∈ N and f (n) σ k ω (0) → ∞ as n → ∞ for every k ∈ N, but the filled Julia set K ω contains a segment (see [4,Example 4.4]).Thus, it is interesting to determine when the non-autonomous Julia set J ω is connected, has finitely many connected components, is totally disconnected, and so on.
For the case when µ is the normalized Lebesgue measure on the disk B(0, r), Brück, Büger, and Reitz showed that r = 1/4 is the critical value where connectedness drastically changes as we see below.If r ≤ 1/4, then the random Julia set J ω is connected for every ω ∈ Ω µ [4, Remark 1.2].Brück, Büger, and Reitz then questioned whether the random Julia set J ω is totally disconnected almost surely if r > 1/4.
Lech and Zdunik [18] showed that it is true.We pay attention to their idea for a while.
Definition 3.3.Let µ be a Borel probability measure on C with compact support.Let R > 0 be a large number as in Lemma 2.6 (1).
Note that the escape time is related to the value of the Green function where log + a = max{a, 0} for a ∈ R .By [18, Proposition 8], there exists C > 0 such that for every ω ∈ Ω µ and z ∈ A ω ∩ D R , we have Hence, typical fast escaping can be defined using the Green functions.For details, see [18].For other discussions on the Green function, see also [13,17,27].
The following theorem gives a sufficient condition of random Julia sets to be totally disconnected almost surely.For the proof, see [18,Theorem 11].
Theorem 3.4.Let µ be a Borel probability measure on C with compact support.If the critical point 0 is typically fast escaping, then the Julia set J ω is totally disconnected for P µ -almost every ω ∈ Ω µ .
Consider the case where µ is the normalized Lebesgue measure on the disk B(c, r).From above, the connectedness changes at r = 1/4 if c = 0. Since the value 1/4 is the distance from c = 0 to the boundary of the Mandelbrot set, one may conjecture that the transition from connected to disconnected Julia sets equals the distance from c to the boundary of the Mandelbrot set for any c ∈ M.However, Theorem 3.5 states that even if supp µ is contained in the closure of the main cardioid of the Mandelbrot set, it can happen that the random Julia set J ω is totally disconnected almost surely.
Before giving the sufficient condition of typical fast escaping, we show the following lemma.
Lemma 3.6.Let µ be a Borel probability measure on C with compact support.Define Proof.Since P µ is the product of the probability measure µ, Fubini's theorem implies for every z ∈ C .By induction, we have The following is the new and original result that gives a sufficient condition for typical fast escaping.Theorem 3.7.Let µ be a Borel probability measure on C with compact support.Suppose that T {∞},µ is continuous.If T {∞},µ (0) = 1, then the critical point 0 is typically fast escaping, and hence the random Julia set J ω is totally disconnected for P µ -almost every ω ∈ Ω µ .
Proof.The latter statement is a consequence of Theorem 3.4.Thus, it suffices to show that the critical point 0 is typically fast escaping.Let R > 0 be a large number as in Lemma 2.6 (1).Set E = {z ∈ C : T {∞},µ (z) = 1}, then E is compact since we assume that T {∞},µ is continuous.(This is the only place where the assumption of continuity is used.) For every z ∈ E, there exist We may and do assume that O z is precompact.
We now consider the open covering {O z } z∈E of E. Since E is compact, there exists a finite subcover.Namely, there exist ℓ ∈ N and ℓ open sets O j (j = 1, . . ., ℓ) such that E ⊂ ℓ j=1 O j .Moreover, for each j = 1, . . ., ℓ, there exist n j ∈ N and c j 1 , c j 2 , . . .c j nz ∈ supp µ such that for every z ′ ∈ O j , we have |f c j n j By Lemma 2.6 (1), we can assume that n j = N for every j = 1, . . ., ℓ.Since C ∋ c → f c ∈ Poly is continuous, for every n = 1, 2, . . ., N , there exists an open neighborhood U j n of c j n such that for every c n ∈ U j n and z ′ ∈ O j , we have ω (z)| ≤ R} for every z ∈ C.Then, by the construction above, we have P µ (Ω µ \ S N (z)) ≥ P µ ( N n=1 U j n × ∞ N +1 B(c, r)) for every j = 1, . . ., ℓ and z ∈ O j .Since c j n ∈ supp µ and U j n is its open neighborhood, we have µ(U j n ) > 0 for every j = 1, . . ., ℓ and n = 1, 2, . . ., N .Define α = min j=1,...,ℓ N n=1 µ(U j n ) > 0.Then, for every z ∈ E, we have P µ (S N (z)) ≤ 1 − α.
It follows from Fubini's theorem that ω (z)) = ∅ by Lemma 2.6.Thus, the integral on the complement of S k (z) is zero; Repeating this, we have P µ (S mN (0)) ≤ (1 − α) m for every m ∈ N .Thus, we can find a constant γ > 0 such that for every k ∈ N.This completes the proof.Corollary 3.8.Let µ be a Borel probability measure on C whose support is compact and satisfies int(supp µ) = ∅.If T {∞},µ (0) = 1, then the random Julia set J ω is totally disconnected for P µ -almost every ω ∈ Ω µ .
Proof.This is a consequence of Lemma 2.22 and Theorems 2.21 and 3.7.
Corollary 3.9.Let µ be a Borel probability measure on C whose support is compact.If G µ has no planar minimal sets, then the random Julia set J ω is totally disconnected for P µ -almost every ω ∈ Ω µ .
Proof.Since G µ has no planar minimal sets, the function satisfies T {∞},µ ≡ 1 by Theorem 2.21.Since the constant function is continuous, Theorem 3.7 implies the conclusion.
For instance, let µ = δ 0 /2 + δ c /2 be the convex combination of the Dirac measures at 0 and c with |c| This shows that G µ has no planar minimal sets, which implies that the random Julia set is totally disconnected P µ -almost surely.
In the next section, we apply Corollary 3.8 to the case where µ is the normalized Lebesgue measure on the disk B(c, r).

Main results on uniform noise process
Throughout this section, we consider the case where µ is the normalized Lebesgue measure on the disk B(c, r) as in Setting 4.1.
Remark 4.2.Setting 4.1 can be interpreted as an additive-noise stochastic process in the following way.Fix c ∈ C and r ≥ 0. On a probability space (Ω, B, P), take independent random variables ν n : Ω → C that follow the uniform distribution on B(0, r) for all n ∈ N. Fix z 0 ∈ C and define the random variable Z 0 : Ω → C by a constant map with value z 0 .Define the C-valued stochastic process {Z n } n≥0 on (Ω, B, P), inductively by Z n+1 = f c (Z n ) + ν n+1 for every n ≥ 0. Then the distribution of Z n are identical to that of the random variable f ω (z 0 ).This stochastic process describes the process driven by adding uniform noise independently to the deterministic dynamical systems of f c with the initial value z 0 .Here, the parameter r describes the size of noise that satisfies |ν n | ≤ r almost surely for every n ∈ N.
We investigate the bifurcation radius r bif (c) defined in Definition 2.17.Our final goal is to determine r bif (c) and we provide some partial results in this paper.

Relation between bifurcation and connectedness
In this subsection, we reveal the relation between bifurcation and connectedness.We start the discussion with the following corollary.
Corollary 4.3.Suppose that r > r bif (c).Then the random Julia set J ω is totally disconnected for P c,r -almost every ω.
Proof.If r > r bif (c), then T c,r ≡ 1 by Theorems 2.16 and 2.21.It follows from Corollary 3.8 that the random Julia set J ω is totally disconnected almost surely.
In the following, suppose that the interior of the support B(c, r) contains a superattracting parameter.
Proof.We show that for every z ∈ C, there exist The proof is divided into three cases.Denote by K(f c) the autonomous filled Julia set of the superattracting map This completes the proof.Proof.Fix any z ∈ L. First, we show g∈Gc,r {g(z)} ⊂ L. By definition, there exists g n ∈ G c,r such that g n (0) → z as n → ∞.Then, for every g ∈ G c,r , we have . Thus, we have g(z) ∈ L, which implies g∈Gc,r {g(z)} ⊂ L. Next, we show g∈Gc,r {g(z)} ⊃ L. For every open set U which intersects L, take g 1 ∈ G c,r such that g 1 (0) ∈ U. Since g 1 is continuous, there exists an open set U 0 such that 0 ∈ U 0 and g The following is the main theorem of this paper.Theorem 4.6.Suppose that the interior of B(c, r) contains a superattracting parameter c.Then the following four are equivalent.
(4) The set g∈Gc,r {g(0)} is a planar minimal set of G c,r .
In contraposition to this, the following four are equivalent.
(2 ′ ) For P c,r -almost every ω ∈ Ω c,r , the orbit f (3 ′ ) The random Julia set J ω is totally disconnected for P c,r -almost every ω ∈ Ω c,r .
Furthermore, either the former or the latter is valid.
Proof.It is obvious that (1) or (1 ′ ) is valid.First, we show the equivalence of the four from (1) to ( 4).
(3) ⇒ (4) : Suppose that the random Julia set J ω is connected for every ω ∈ Ω c,r .For a large R > 0 as in Lemma 2.6 (1), it follows from Theorem 3.1 that |f , n ∈ N, and k ∈ N. Thus, the set L = g∈Gc,r {g(0)} is bounded in C .By Lemma 4.5, the set L is a planar minimal set.
The argument above completes the proof of the equivalence from ( 1) to ( 4).Next, we show the equivalence of the four from (1 ′ ) to (4 ′ ).
( ( = 1.If r > 0, then the random Julia set J ω is totally disconnected P c,r -almost surely by Corollary 3.8.If r = 0, then it trivially follows that the autonomous Julia set is totally disconnected. ( ) holds, then there exists ω ∈ Ω c,r such that the random Julia set J ω is disconnected.By Theorem 3.1, there exists k ∈ N such that f Let R > 0 be large, as in Lemma 2.6 (1).If g∈Gc,r {g(0)} is not bounded, then there exists g ∈ G c,r such that |g(0)| > R. By definition of G c,r , there exist Proof.It is obvious that "only if" part is true.Suppose T c,r (0) = 0 and we show there does not exist ω ∈ Ω c,r such that f However, for every , contradicting the assumption T c,r (0) = 0.This shows that for every ω ∈ Ω c,r , we have |f

General properties
In this subsection, we give a trivial estimate of the bifurcation radius and show the continuity.The following is due to [4].
This completes the proof.
Denote by dist(c, ∂M) the Euclidean distance from the point c to the compact set ∂M.
Proof.From Lemma 4.8, if r > dist(c, ∂M), then the polynomial semigroup G c,r has no planar minimal sets.By Definition 2.17, r bif (c) is the infimum value for r ≥ 0 such that G c,r has no planar minimal sets.Hence, r bif (c) ≤ dist(c, ∂M).
The following is a part of Main Result C.

Inside the main cardioid
In this subsection, we present some estimates of r bif (c) when c is in the main cardioid.Recall that the main cardioid is the set of all parameters c ∈ C for which there exists λ ∈ C with |λ| < 1 such that c = λ/2 − λ 2 /4, and its boundary is contained in ∂M.
) diverges to ∞ as n → ∞.Thus, we have proved that there exist z 0 ∈ L and ω This contradicts that L is planar and forward invariant under G −ǫ,r .This completes the proof since r is an arbitrary number that satisfies r > 1/4 + ǫ − ǫ 2 .
Note that we can show that dist(−ǫ, This shows that the random Julia set J ω is totally disconnected for almost every ω even if B(c, r) ⊂ int M and B(c, r) ⊃ B(0, 1/4).This is stronger than Theorem 3.5 proved by Lech and Zdunik.

Outside the main cardioid
Recall that a parameter c is said to be superattracting if there exists p ∈ N such that f •p c (0) = 0.For example, c = 0 and −1 are superattracting parameters with p = 1 and 2, respectively.In this subsection, we show the nontrivial estimates of r bif (−1).
Next, we consider the upper bound of r bif (−1).The following theorem gives upper bounds, whose proof suggests a kind of parabolic implosion in the stochastic sense.
let z * denote the parabolic periodic point of g.Then there exists p ∈ N such thatg •p (z * ) = z * , and the multiplier of g •p at z * is 1.Thus, there exists a critical point z 0 of g such that g to be the identity map.In any case, we have g 0).Since the parabolic periodic point z * belongs to the autonomous Julia set J(g) of g and the Julia set is the boundary of the basin at infinity, perturbation of c N allows us to find a random orbit which escapes to the basin at infinity.Namely, there exists c ′ ∈ B(c N , ǫ) such that where A(g) denotes the basin at infinity for the autonomous iteration of g.Note that c ′ ∈ B(c, r) since c N ∈ B(c, r 0 ) and ǫ = r − r 0 .By definition, we have g •n (w ′ ) → ∞ as n → ∞.Thus, Lemma 4.4 gives that r bif (c) < r.This completes the proof since r is an arbitrary number that satisfies r > r 0 .Theorem 4.19 gives an alternative proof of r bif (0) ≤ 1/4 since f 1/4 has a parabolic fixed point at z = 1/2.
The following gives an algorithm to find an upper bound of the bifurcation radius.
Theorem 4.20.Suppose c is a superattracting parameter.Fix N ∈ N and k ∈ N, and let p i ∈ {0, 1, . . ., k − 1} for each i = 1, 2, . . ., N. Define ζ k = e 2πi/k and c i = c + ρζ p i k , where ρ is a complex variable.Let ∆(ρ) be the discriminant of the algebraic equation We have some comments on the upper bounds of r bif (−1).Remark 4.22.In Example 4.21, the numerical errors should be treated carefully because the discriminant of the equation with respect to z is a polynomial ∆(ρ) of degree 32, and each coefficient is a very large number in absolute value, i.e., about 10 19 -10 26 order.The author used validated numerics to estimate ρ 0 , which showed r bif (−1) ≤ |ρ 0 | ≤ 0.0399217.Since validated numerics gives values including mathematically strict error evaluation, this value 0.0399217 is a rigorous upper bound of r bif (−1).For more details on validated numerics, see [23] for example.
More precisely, we can show by using [8,Theorem 1.3] that the maximal dilatation K of the quasiconformally conjugacy {ϕ n } n≥0 depends on c and r but not on ω nor ω ′ .This is because the δ in the proof is independent of ω.Besides, we can show that the maximal dilatation K tends to 1 as r tends to 0.
Lastly, we give the following corollary.
Corollary 4.30.Suppose that the interior of B(c, r) contains a superattracting parameter c.Denote the autonomous Julia set by J c = J ω where ω = (c, c, . . .).If r < r bif (c), then for every ω ∈ B(c, r) N , there exists a map ϕ which maps J c onto the non-autonomous Julia set J ω quasiconformally.If r > r bif (c), then for P c,r -almost every ω, the autonomous Julia set J c is not homeomorphic to the non-autonomous Julia set J ω .
Proof.First suppose r < r bif (c).By Theorem 4.29, there exists a quasiconformal conjugacy {ϕ n } n≥0 between ω and ω for every ω ∈ B(c, r) N .By definition, the map ϕ 0 quasiconformally maps J c onto J ω .Next suppose r > r bif (c).Then by Theorem 4.6, the non-autonomous Julia set J ω is totally disconnected almost surely.Since the autonomous Julia set J c of superattracting parameter c is connected, these J c and J ω cannot be homeomorphic each other.
Question 5.2.What is the exact value of r bif (c) for a superattracting parameter c? How does it relate to other dynamical quantities?
Besides, it would be meaningful to investigate their asymptotic behavior.
Question 5.3.Let cp be a superattracting parameter with period p for every p ∈ N .How does r bif (c p ) decrease as p → ∞?
Our strategy to estimate r bif (−1) from above is to find multiple roots of some algebraic equation.This raises the following question.
Question 5.4.For every superattracting parameter c, is the bifurcation radius r bif (c) an algebraic number?At least, if c = 0, then r bif (0) = 1/4 is algebraic and is related to the parabolic parameter c = 1/4.
We are interested also in parameters which are not superattracting.Recall that Theorem 4.6 is concerned with the case where B(c, r) contains superattracting parameters.Question 5.5.Can we generalize Theorem 4.6 when B(c, r) does not contain superattracting parameters?Question 5.6.How does r bif (c) decrease as c → ∂ M? For example, within ǫ > 0, how does r bif (−3/4 + ǫ) decrease as ǫ → 0?

Other problems about geometric measure theory
One can consider the Hausdorff dimensions and other geometric properties of random Julia sets.For instance, the uniform perfectness and the Johnness was investigated.See Theorem 1.12 and Theorem 1.6 of [27].
Regarding dimensions, the following facts are known.Suppose c = 0 and 0 < r < 1/4 and let (P 0,r , Ω 0,r ) as in Setting 4.1.Denote by HD(J) the Hausdorff dimension of a set J.Then, there exists 1 < d 0,r < 2 such that HD(J ω ) = d 0,r for P 0,r -almost every ω ∈ Ω 0,r .See Theorems 8.8 and 8.10 of [19] by Mayer, Skorulski, and Urbański.Besides, denote by H d the d-dimensional Hausdorff measure and by P d the d-dimensional Packing measure, then H d 0,r (J ω ) = 0 and P d 0,r (J ω ) = ∞ for P 0,r -almost every ω ∈ Ω 0,r .See [19,Theorem 8.16].Note that this is different from the deterministic case, and hence almost every J ω is not bi-Lipschitz equivalent to any autonomous hyperbolic Julia sets.
Sumi kindly taught me that for every c ∈ C and r > 0, there exists a constant d c,r such that HD(J ω ) = d c,r for P c,r -almost every ω ∈ Ω c,r .Rugh showed that the dimension d c,r depends real-analytically on Re(c), Im(c), and r within |c| + r < 1/4.See [24] for the exact statements.This is the motivation for the following question.For other geometric properties, Brück showed that the random Julia set J ω is a quasi-circle for every ω ∈ Ω 0,r if r < 1/4.See [2,Corollary 4.5].
Theorem 4.29 generalize this result.However, the following questions are open.

(
vi) For the airplane parameter c3 ≈ −1.75487766, we have r bif ( c3 ) ≤ 0.0021.Here, dist(c, ∂M) denotes the Euclidean distance from c to the boundary of the Mandelbrot set ∂ M .Statement (iii) gives the examples of c satisfying equality r bif (c) = dist(c, ∂M).However, statement (iv), (v), and (vi) give the examples of c satisfying strict inequality r bif (c) < dist(c, ∂M).

Corollary 1 . 2 (
Corollaries 4.15 and 4.25).Almost every random Julia set is totally disconnected • if the central parameter −1/4 ≤ c < 0 and the noise amplitude r > 1/4 − c − c 2 , or • if the central parameter c = −1 + ǫ and the noise amplitude r > 0.0399 • • • + |ǫ|, or • if the central parameter c = c3 + ǫ and the noise amplitude r > 0.0021 + |ǫ|.This corollary gives new examples which satisfy that the random Julia set is totally disconnected almost surely even if B(c, r) ⊂ int M .Besides, Main Result B gives a quasiconformal conjugacy between Julia sets.Main Result D (Theorem 4.29 and Corollary 4.30).Suppose that the interior of B(c, r) contains a superattracting parameter c and suppose r < r bif (c).Then for every

Theorem 2 . 16 .
For every c ∈ C and r > 0, define the polynomial semigroup G c,r as the semigroup generated by {f c ′ ∈ Poly : |c ′ − c| ≤ r}.Here f c (z) = z 2 + c.Then G c,r has an attracting minimal set {∞}.Other than {∞}, possible planar minimal sets are described as follows.

Theorem 2 . 23 .
For every c ∈ C and r ≥ 0, define the polynomial semigroup G c,r as the semigroup generated by {f c ′ ∈ Poly : |c ′ − c| ≤ r}.Then we have all the following statements.

Setting 4 . 1 .
For fixed c ∈ C and r ≥ 0, we define the following Borel probability measure µ c,r on C with compact support.If r > 0, define µ c,r as the normalized Lebesgue measure on B(c, r) = {c ′ ∈ C : |c ′ − c| ≤ r}.If r = 0, define µ c,r as the Dirac measure at c.We define the probability space (Ω c,r , P c,r ) = (Ω µc,r , P µc,r ) as in Setting 2.1.Namely, denote by P c,r the one-sided infinite product of µ c,r supported on Ω c,r = ∞ n=1 B(c, r) with the Borel σ-algebra.Besides, let the polynomial semigroup G c,r be defined by the semigroup generated by {f c ′ ∈ Poly : c ′ ∈ B(c, r)} and define the function T c,r : C → [0, 1], abusing notation, by

Lemma 4 . 4 .
Suppose that the interior of B(c, r) contains a superattracting parameter c.If there exists ω ∈ Ω c,r such that f

(
), then by the No Wandering Domain Theorem, the orbit f •n c (z) converges to the attracting cycle of 0. By the assumption, we have 0 ∈ A ω .Since the non-autonomous basin A ω is open, there exists an open neighborhood D of 0 such that for every z ′ ∈ D, we have f

Lemma 4 . 5 .
Suppose that the interior of B(c, r) contains a superattracting parameter c.If L = g∈Gc,r {g(0)} is bounded in C, then L is a planar minimal set of G c,r .
r has no planar minimal sets by Theorem 2.16.It follows from Theorem 2.21 that T c,r (z) = 1 for every z ∈ C, and hence T c,r (0) = 1.

Question 5 . 7 .
How does the typical dimension d c,r depend on c and r?