ENTROPY COMPUTATION ON THE UNIT DISC OF A MEROMORPHIC MAP

We propose a new definition of entropy based on both topological and metric entropy for the meromorphic maps. The entropy is then computed on the unit disc of a meromorphic map, which is called the extended Blaschke function, and is a nonlinear extension of the normalized Lorentz transformation. We find that the defined entropy is computable and observe several interested results, such as maximal entropy, entropy overshoot due to topological transition, entropy reduction to zero, and scaling invariance in conjunction with parameter space.


Introduction
The definitions of entropy of a dynamical system, namely, a discrete map or a continuous flow are different from those contemporary ones in physics. Mathematically entropy represents the measures of dynamical complexity as the system evolves with time or in space. For rigor and soundness of the theorems and proofs, we normally explore the maps and flows in conjunction with differentiable structures, such as differentiable manifolds. The key concepts of the efforts include invariant measures on the sets under specific partitions. A common approach for defining a partition is accordingly based on the differentiable structures. For examples, we can read the symbol of ϵ → 0, namely, number of partition becoming infinite in the definitions of topological entropy, Kotak's entropy, Brin-Kotak entropy, Romagnoli's entropy, Ornstein-Weiss type entropy, Modified Bowen entropy, Newhouse's local entropy, Modified Misiurewicz's entropy, etc. A comprehensive review and update are in [4] and the references therein.
From the perspectives of the recent efforts toward bridging the definitions between mathematics and physics [3], and the limitation of numerical analysis, we face new challenges. The first challenge is that we need to handle singularities of non-smooth structures in the physical world, and secondly the computing hardware can only provide finite resolutions. In 2002, Milnor raised several questions in an article entitled "Is Entropy Effectively Computable" [5] as follows: • Is there an effective procedure to carry out the computation in a reasonable time?
• Topological entropy does not always depend continuously on parameters.
• Are both upper and lower bound of the defined entropy effectively computable?
The topological entropy of a map F can be described as the supremum over all finite F -invariant sets in conjunction with a partition of a bounded region containing a probability measure which is invariant under F [1]. The metric or measuretheoretic entropy introduced by Kolmogorov and Sinai in 1959 can be described as the supremum in conjunction with a set of partitions of a phase space, where the orbits stepping forward with a probability measure. Other definitions also face the same questions raised by Milnor. Some ideas, such as variational principle, were introduced to develop structures and computing procedures for answering these questions. However, general and solid examples for the structures with singularities are still under intensive researches.
In this paper, we construct a meromorphic map, which is a nonlinear extension of the normalized Lorentz transformation and define a new version of entropy based on the concepts of ratio of preimage of divergent and convergent partitions, namely, Julia sets and Fatou sets. This approach allows us to simplify the procedure of computing entropy in the momentum space instead of computing with time and in phase space.

Maps
Given two inertial frames with different velocities, u and v, the observed velocity, u, from v-frame is as follows: We set c 2 = 1 and then multiply a phase connection, exp(iϕ(u)), to the normalized complex form of the equation (2.1) based on gauge transformation as follows: We further define a generalized complex function in conjunction with phase connection, exp(iϕ(u)), as follows: and C i has the following forms: where z is a complex variable representing the velocity u, a i is a parameter representing velocity v,ā i is the complex conjugate of a complex number a i , and m is an integer representing the functional order. The term g i (z) is a function assigned to ∑ p 2pπiz with p as an integer.
The function f B (z, m) is called an extended Blaschke function (EBF) [2]. The extended Blaschke equation (EBE) is defined as follows: For m = 1 case, we have the normalized Lorentz transformation. For m > 1 cases, we have nonlinear extension of Lorentz transformation. The convergent sets of EBF represent the stable sets of normalized momentum of the particles interacting nonlinearly with others in an ensemble.

Original and Mapped Domains
A domain can be the entire complex plane, C ∞ , or a set of complex numbers, such as z = x + yi , with (x 2 + y 2 ) 1/2 ≤ R and R is a real number. For solving the EBE, a function f will be iterated as: where n is a positive integer indicating the number of iteration. The function operates on a domain, which is called the original domain. The sets of f n (z) is called the mapped domain. In the figures of this paper, the regions in black color represent Fatou sets containing the convergent points of the concerned functions or equations and the white (i.e., blank) regions correspond to Julia sets, the complementary sets of Fatou sets on C ∞ . The original domain is analogical to the preimage of the holomorphic maps.

Parameter Space
In order to characterize and to classify the original and the mapped domains, we define a set of parameters, which is the parameter space, for specifying the concerned domains. The parameter space includes five parameters: 1) z, 2) a, 3) exp(g i (z)), 4) m, and 5) iteration. In the context of this paper, we use the set {z, a, exp(g i (z)), m, iteration} to represent the parameter space. For example, {a}, is one of the subsets of the parameter space.

Conformal Mapping and Fractals
On the complex plane, the convergent domains of the functions form fractal patterns with limited-layered structures, which demonstrate skip-symmetry, symmetry broken, chaos, and degeneracy in conjunction with parameter space [6].  Figure 2 shows two types of the fractal patterns of the original domains. These patterns are plotted on the different scales. In order to show the figures more observable, we intend to reverse the color tone of the Fatou and Julia sets, namely, the sets in the black color are the Julia sets in the Figure 2.
Conformal mapping and fractal patterns directly verify that the numerical results are not generated by the embedded algorithms in the computer architecture.

Layered Ring Structure
The Fatou sets of the original domains form ring structures with the fractal patterns. The smaller rings are inside the larger rings. For m > 2 cases, there exist five layers of rings as shown in Figure 3. The ring on scale of 10 −2 , Fig. 3(b), is topologically similar to that on scale of 10 5 as shown in Fig. 3(e). We called this observation as skip symmetry. The value of a will determine the size and number of layers of the rings [5].

Topological Transition of Parameter-dependent Domains
The parameters in the parameter space determine the topological structures of the domains. In this section, we examine two cases related to the parameter {a}. We call this observation as nonlinear-to-linear degeneracy [7]. As the degeneracy continues to evolve, the layered-ring structure for all m > 1

Continuous to Discrete Transition
As the value of {a} is further approaching to unity, we observe the continuous and connected Fatou sets transform into discrete Cantor-like sets. Figure 5(a) shows that the sets in Fig. 4(h) further transform into a unit disc with the shrinking fractal patterns on the scale of 10 −7 at the vicinity of z = 1. Figure 5(b) shows the continuous and connected Fatou sets transform into Cantor-like sets when 1 − a ∼ 10 −17 . This interesting property may potentially bridge the theories of relativity and quantum mechanics [7] .

Entropy Definition
From the computation perspective, we define a new version of entropy in order to explore Milnors questions.
Definition 4.1. Let X be a subset of C ∞ , a topological partition of X is dividing X into a group of subsets such that the individual subset has no intersection with one another. where P (X j ) and P (X) are the σ-algebra measures on a given topological partition of X j and X, respectively.
We propose these definitions with the assumption that there are conditionally invariant properties between P (X) and P (Y ) of the meromorphic map in this study.

Entropy Computation
Since the layered ring structure of the original domains of EBF(f B ) demonstrates a property of parameter-dependent area growth, we limit the computation of the normalized entropy on the unit disc, as shown in Fig. 4(c) and 4(d). Further exploration on the entropy related to area growth will be addressed in a future publication. To compute the entropy, we firstly analyze the sensitivity of entropy values to the topological partition and parameter {iteration}. Then we continue to analyze the dependency of entropy values on the parameters {a} and {m}.     Figure 8 shows the computed normalized entropy values versus logarithm of base 10 of the functional orders with a = 0.5. There are three regions: linear, overshoot, and nonlinear regions and two characteristic functional orders: Maximal Entropy and Stability Edge. The latter order shows the entropy value suddenly drops to a value close to zero, namely, the Julia sets diminish.

Entropy vs. Functional Order
In the linear region, the topological patterns of the Fatou sets are shown as in Fig. 4(a). In the overshoot region, the Fatou sets diminish. This region is directly related to the linear-to-nonlinear degeneracy as described in Section 3.1. In the nonlinear region, the Julia sets will grow to a peak value at specific functional order depending of parameter {a}. This peak value is called the Maximal Entropy. The normalized entropy will then decline as the functional order increases and suddenly drops to a value close to zero at a second functional order indicated as Stability Edge in Fig. 8. Beyond the Stability Edge, the Julia sets diminish. Figure 9 shows the computed entropy values versus logarithm of base 10 of the functional orders with a = 0.1, 0.5, 0.9, 0.99, 0.999, and 1, respectively. In the figure, we observe the following interested properties: a) for all individual {a} value except a = 1, there exists one maximal value and one stability edge excluding the overshoot region, b) there exists an invariant scaling factor of the functional order between maximal entropy and stability edge. c) for a = 1 case, the entropy values are computed after the continuous-to-discrete transition occurs as described in Section 3.2.

Remarks
This paper proposes a new version of entropy definition based on both topological and metric entropy. We define the normalized entropy as the ratio of topological partition of Julia sets, where the map divergent, over the topological partition of total bounded region, which is the unit disc in this paper. Based on the definition, we perform computation on the entropy and find this special case: • For a reasonable resolution based on the topological partition, we are able to obtain a known computing error for the computed values.
• For a reasonable iteration, such as iteration = 20, in this study, we are able to obtain the entropy values with a definite accuracy in a reasonable computing time.
• There are parameter-dependant transitions, such as the continuous-to-discrete transition, wherein the normalized entropy does not depend continuously on • We are able to compute both upper and lower bounds of the normalized entropy.
In addition, we have potential to bridging the entropy definitions between mathematics and physics since the stable sets of normalized momentum space representing the allowable momentum values in a normalized ensemble. In this study, we can argue that the max-entropy, where the system demonstrates highest complexity, corresponding to the definition of minimal entropy in physics and the minimal entropy at stability edge, where the system becomes stable and randomized, corresponding to the maximal entropy in physics.

Acknowledgement
The author appreciates the opportunity to participate the conference entitled "Frontiers of Complex dynamics" in Banff Center, Alberta, Canada, Feb. 21-25, 2011 and would like to dedicate this paper to John Milnors 80 th birthday.