jd445/arixv_year_clean_txt · Datasets at Hugging Face

text stringlengths 0 603k
Thus the orbit of EQU is “captured” by the periodic orbit of EQU.
Case EQU.
Disjoint Periodic Sinks.
Again there are disjoint neighborhoods EQU and EQU, but in this case the first return map carries each EQU into itself, say EQU and EQU.
Figure 6.
Schematic diagrams for maps representing the centers of the four distinct classes of hyperbolic components.
Each critical point is indicated by a heavy dot, and each arrow is labeled by a corresponding number of iterations.
(Compare 4 and Appendix B.)
In all four cases, the corresponding configuration in the EQU -parameter plane has a unique well defined “center” point EQU, characterized by the property that the first return map EQU carries critical points to critical points.
Thus this center map EQU has the Thurston property of being post critically finite.
In fact EQU has the sharper property that the orbits of both critical points are finite, and eventually superattracting.
It follows from Thurston’s theory that this center point EQU is uniquely determined by its “kneading invariants”, or in other words by the mutual ordering of the various points on the two critical orbits.
(Compare as well as the analogous discussion for quadratic maps in.)
Furthermore, any ordering which can occur for a post critically finite continuous map with two critical points can actually occur for a cubic polynomial map.
Case EQU is exceptional, and occurs only in one region, which has center point EQU corresponding to the map EQU.
In Cases EQU we will see rw-swa.2.ps pixels 800 by 800 scaled 225 Figure 7.
Detail of Figure 3 showing a “swallow configuration” centered at the point EQU.
For the cubic map associated with this central point, the two critical points EQU satisfy EQU.
Hence both lie on a common orbit of period 4.
(Region: EQU.)
that the corresponding point of the real EQU -parameter plane is associated respectively to a swallow configuration, to an arch configuration, or to a product configuration.
(Compare Figures 7, 11, 13.)
There are two qualifications: If this configuration is immediatelyadjacent and subordinated to another larger configuration, then it will be highly deformed.
Furthermore, along the EQU -axis the pictures in the EQU -plane are rather strange; and one should rather work with the EQU or EQU -plane, as in Figures 4, 5.
In each of these cases EQU, the first return map from EQU to itself canbe approximated by a map which is quadratic on each component.
Hence we canconstruct a simplified prototypical model for this kind of behavior by replacing eachinterval EQU by a copy EQU of the entire real line, and by replacing the smooth map EQU, which has one critical point in each component, by a componentwise quadratic map EQU from the disjoint union EQU to itself.
rb.3.ps pixels 800 by 800 scaled 225 Figure 8.
The prototype swallow configuration in the EQU -parameter plane, associated with the family of biquadratic maps EQU from the real line to itself.
(Region: EQU.)
First consider the case of a swallow configuration, as illustrated in Figure 7.
The prototypical model in this case is obtained by replacing these two intervals by disjoint copies of the real line, with parameters EQU and EQU respectively, and by replacing the first return map by the quadratic map which interchanges the two components of the disjoint union EQU.
Here EQU and EQU are real parameters.
Thus we obtain a universal swallow configuration in the EQU -parameterplane, as illustrated in Figure 8.
(Compare Ringland and Schell.)
The central tear drop shaped body of this swallow corresponds to the connectedness locus for this family, consisting of those biquadratic maps for which both critical orbits remain bounded.
On the other hand, the wings and tails correspond to maps for which only one critical orbit is bounded.
It is interesting to note that this same swallow configuration seems to occur in a quite different context, where there are no critical points at all.
Consider the two-parameter family of Hénon maps.
These are quadratic diffeomorphisms of the plane which can be written for example as with constant Jacobian determinant EQU.
A picture of those parameter pairs EQU for which there exists an attracting periodic orbit typically exhibits quite similar swallow shaped configurations.
( Compare.)
For example such a region is shown in Figure 9, corresponding to an attracting orbit of period 5.
This phenomenon can be explained intuitively as follows.
If EQU is small, then the dynamics of the two-dimensional Hénon map is quite similar to the dynamics of the one-dimensional map EQU.
In particular, the Hénon map can be closely approximated locally by a linear map, except at points near the axis EQU, where the second derivative plays an essential role.
Similarly, the dynamics of a composition of two Hénon maps with small determinant resembles the dynamics of a composition of two one-dimensional quadratic maps.
Now consider a periodic orbit for some Hénon map.
If this orbit is to be attracting, then it must contain at least one point which is close to the axis EQU.
If exactly two points of the orbit are close to EQU, then the dynamics will resemble that for a composition of two quadratic maps.
Hence, in this case, as we vary the parameters we obtain a swallow shaped configuration within the Hénon parameter plane.
henpar.3.ps pixels 640 by 480 scaled 225 Figure 9.
A swallow configuration in the Hénon parameter plane.
A location EQU is colored white if a random search of initial conditions found an attracting orbit of low period for the quadratic diffeomorphism EQU ; and grey indicates that only high periods or chaotic behavior were found.
In the black area to the lower right, no bounded orbits were found.
The graininess in the picture is presumably due to the random nature of the algorithm used.
(Region: EQU.)
The swallow configuration of Figures 7, 8, 9 should not be confused with the somewhat similar configuration shown in Figure 4, which can perhaps be described as a “pointed swallow”.
This pointed configuration also plays a role in many dynamical systems.
Here is a well known example.
(I am indebted to communications from S. Ushiki and T.
Consider the two-parameter family of circle maps These are diffeomorphisms for EQU, but have two critical points for larger values of EQU.
The corresponding picture in the EQU -parameter plane, shown in Figure 10, contains one immersed copy of the configuration of Figure 4 corresponding to each rational rotation number.
Each of these configurations terminate in a “tongue” which reaches down to the corresponding rational point on the EQU axis.
These are known as Arnold tongues.
arn-to.3a.ps pixels 960 by 960 scaled 225 Figure 10.
Arnold tongues ending in “pointed-swallow” configurations for the family of circle maps EQU.
(Region: EQU in the EQU -parameter plane.)
wh-arch.3.1.ps pixels 800 by 800 scaled 225 Figure 11.
Detail of Figure 3 showing an arch configuration.
For the cubic map correspondingto the center point EQU, the two critical points EQU satisfy EQU.
(Region: EQU.)
arch.ps pixels 1152 by 900 scaled 225 Figure 12.
The prototype arch configuration in the EQU -plane.
Here we consider the orbit of the point EQU under the map EQU.
Dark grey indicates that both EQU and EQU have chaotic orbits, while white means that both escape to infinity.
(Region: EQU.)
Next let us consider the “arch configuration”, as illustrated in Figure 11\.
Recall that a point of the cubic parameter plane belongs to an arch configuration if there are disjoint neighborhoods EQU and EQU as above so that some iterate of EQU maps EQU into EQU, and some iterate maps EQU into itself, but so that every forward image of EQU or EQU is disjoint from EQU.
In this case, the universal configuration, as illustrated in Figure 12, is obtained by studying a quadratic map from EQU to itself depending on two parameters EQU and EQU as follows.
We map a point EQU in the first copy of EQU to the point EQU in the second copy, so that the critical point maps to EQU, and we map the second copy of EQU to itself by EQU.
The real connectedness locus in this prototypical case consists of all pairs EQU with EQU and EQU.
rw-det3.1.ps pixels 480 by 480 scaled 225 Figure 13.
Detail of Figure 3 showing a product configuration.
For the map corresponding to the center point EQU, there are two superattracting periodic orbits with periods 3 and 4 respectively.
(Region: EQU.)
qp.4.ps pixels 800 by 800 scaled 225 Figure 14.
The prototype product configuration in the EQU -parameter plane.
Finally consider the product configuration of Figure 13.
We say that a point of the cubic parameter plane belongs to a product configuration if there are disjoint neighborhoods EQU and EQU as above so that some iterate of EQU maps EQU into itself and some iterate maps EQU into itself, but no forward image of either one of the EQU intersects the other.
In this case, the universal model is obtained by taking two disjoint real lines, say with parameters EQU and EQU respectively, and looking at independent quadratic maps EQU.
The “real connectedness locus” for this two-parameter family, that is the set of parameter pairs for which both critical points have bounded orbits, is evidently equal to the square EQU in the EQU -plane, as illustrated in Figure 14.
According to Jakobson, the set set of parameter pairs for which both critical orbits are chaotic (indicated by dark grey in the figure) has positive measure.
A classical conjecture, not yet proved, asserts that this set is totally disconnected.
Thus it seems natural to make the corresponding conjecture for the cubic parameter plane of Figure 3, that the set of parameter pairs for which both critical orbits are chaotic is a totally disconnected set of positive measure.
Some further discussion of these shapes, and other related ones, will be given in 4, which discusses the corresponding four cases for complex cubic maps, and in Appendix B.
One useful tool for studying real polynomial mappings EQU of degree EQU is provided by the topological entropy EQU of EQU considered as a map from the compact interval EQU to itself.
According to Rothschild and, this can be computed as where EQU is the number of points along the real axis at which the derivative EQU changes sign.
In the cubic case, a more practical algorithm for computing EQU has recently been described by.
The entropy EQU varies continuously as EQU varies through polynomials of fixed degree.
Furthermore EQU takes a constant value, equal to the logarithm of an algebraic integer, throughout each hyperbolic component.

Thus the orbit of EQU is “captured” by the periodic orbit of EQU.

Case EQU.

Disjoint Periodic Sinks.

Again there are disjoint neighborhoods EQU and EQU, but in this case the first return map carries each EQU into itself, say EQU and EQU.

Figure 6.

Schematic diagrams for maps representing the centers of the four distinct classes of hyperbolic components.

Each critical point is indicated by a heavy dot, and each arrow is labeled by a corresponding number of iterations.

(Compare 4 and Appendix B.)

In all four cases, the corresponding configuration in the EQU -parameter plane has a unique well defined “center” point EQU, characterized by the property that the first return map EQU carries critical points to critical points.

Thus this center map EQU has the Thurston property of being post critically finite.

In fact EQU has the sharper property that the orbits of both critical points are finite, and eventually superattracting.

It follows from Thurston’s theory that this center point EQU is uniquely determined by its “kneading invariants”, or in other words by the mutual ordering of the various points on the two critical orbits.

(Compare as well as the analogous discussion for quadratic maps in.)

Furthermore, any ordering which can occur for a post critically finite continuous map with two critical points can actually occur for a cubic polynomial map.

Case EQU is exceptional, and occurs only in one region, which has center point EQU corresponding to the map EQU.

In Cases EQU we will see rw-swa.2.ps pixels 800 by 800 scaled 225 Figure 7.

Detail of Figure 3 showing a “swallow configuration” centered at the point EQU.

For the cubic map associated with this central point, the two critical points EQU satisfy EQU.

Hence both lie on a common orbit of period 4.

(Region: EQU.)

that the corresponding point of the real EQU -parameter plane is associated respectively to a swallow configuration, to an arch configuration, or to a product configuration.

(Compare Figures 7, 11, 13.)

There are two qualifications: If this configuration is immediatelyadjacent and subordinated to another larger configuration, then it will be highly deformed.

Furthermore, along the EQU -axis the pictures in the EQU -plane are rather strange; and one should rather work with the EQU or EQU -plane, as in Figures 4, 5.

In each of these cases EQU, the first return map from EQU to itself canbe approximated by a map which is quadratic on each component.

Hence we canconstruct a simplified prototypical model for this kind of behavior by replacing eachinterval EQU by a copy EQU of the entire real line, and by replacing the smooth map EQU, which has one critical point in each component, by a componentwise quadratic map EQU from the disjoint union EQU to itself.

rb.3.ps pixels 800 by 800 scaled 225 Figure 8.

The prototype swallow configuration in the EQU -parameter plane, associated with the family of biquadratic maps EQU from the real line to itself.

(Region: EQU.)

First consider the case of a swallow configuration, as illustrated in Figure 7.

The prototypical model in this case is obtained by replacing these two intervals by disjoint copies of the real line, with parameters EQU and EQU respectively, and by replacing the first return map by the quadratic map which interchanges the two components of the disjoint union EQU.

Here EQU and EQU are real parameters.

Thus we obtain a universal swallow configuration in the EQU -parameterplane, as illustrated in Figure 8.

(Compare Ringland and Schell.)

The central tear drop shaped body of this swallow corresponds to the connectedness locus for this family, consisting of those biquadratic maps for which both critical orbits remain bounded.

On the other hand, the wings and tails correspond to maps for which only one critical orbit is bounded.

It is interesting to note that this same swallow configuration seems to occur in a quite different context, where there are no critical points at all.

Consider the two-parameter family of Hénon maps.

These are quadratic diffeomorphisms of the plane which can be written for example as with constant Jacobian determinant EQU.

A picture of those parameter pairs EQU for which there exists an attracting periodic orbit typically exhibits quite similar swallow shaped configurations.

( Compare.)

For example such a region is shown in Figure 9, corresponding to an attracting orbit of period 5.

This phenomenon can be explained intuitively as follows.

If EQU is small, then the dynamics of the two-dimensional Hénon map is quite similar to the dynamics of the one-dimensional map EQU.

In particular, the Hénon map can be closely approximated locally by a linear map, except at points near the axis EQU, where the second derivative plays an essential role.

Similarly, the dynamics of a composition of two Hénon maps with small determinant resembles the dynamics of a composition of two one-dimensional quadratic maps.

Now consider a periodic orbit for some Hénon map.

If this orbit is to be attracting, then it must contain at least one point which is close to the axis EQU.

If exactly two points of the orbit are close to EQU, then the dynamics will resemble that for a composition of two quadratic maps.

Hence, in this case, as we vary the parameters we obtain a swallow shaped configuration within the Hénon parameter plane.

henpar.3.ps pixels 640 by 480 scaled 225 Figure 9.

A swallow configuration in the Hénon parameter plane.

A location EQU is colored white if a random search of initial conditions found an attracting orbit of low period for the quadratic diffeomorphism EQU ; and grey indicates that only high periods or chaotic behavior were found.

In the black area to the lower right, no bounded orbits were found.

The graininess in the picture is presumably due to the random nature of the algorithm used.

(Region: EQU.)

The swallow configuration of Figures 7, 8, 9 should not be confused with the somewhat similar configuration shown in Figure 4, which can perhaps be described as a “pointed swallow”.

This pointed configuration also plays a role in many dynamical systems.

Here is a well known example.

(I am indebted to communications from S. Ushiki and T.

Consider the two-parameter family of circle maps These are diffeomorphisms for EQU, but have two critical points for larger values of EQU.

The corresponding picture in the EQU -parameter plane, shown in Figure 10, contains one immersed copy of the configuration of Figure 4 corresponding to each rational rotation number.

Each of these configurations terminate in a “tongue” which reaches down to the corresponding rational point on the EQU axis.

These are known as Arnold tongues.

arn-to.3a.ps pixels 960 by 960 scaled 225 Figure 10.

Arnold tongues ending in “pointed-swallow” configurations for the family of circle maps EQU.

(Region: EQU in the EQU -parameter plane.)

wh-arch.3.1.ps pixels 800 by 800 scaled 225 Figure 11.

Detail of Figure 3 showing an arch configuration.

For the cubic map correspondingto the center point EQU, the two critical points EQU satisfy EQU.

(Region: EQU.)

arch.ps pixels 1152 by 900 scaled 225 Figure 12.

The prototype arch configuration in the EQU -plane.

Here we consider the orbit of the point EQU under the map EQU.

Dark grey indicates that both EQU and EQU have chaotic orbits, while white means that both escape to infinity.

(Region: EQU.)

Next let us consider the “arch configuration”, as illustrated in Figure 11\.

Recall that a point of the cubic parameter plane belongs to an arch configuration if there are disjoint neighborhoods EQU and EQU as above so that some iterate of EQU maps EQU into EQU, and some iterate maps EQU into itself, but so that every forward image of EQU or EQU is disjoint from EQU.

In this case, the universal configuration, as illustrated in Figure 12, is obtained by studying a quadratic map from EQU to itself depending on two parameters EQU and EQU as follows.

We map a point EQU in the first copy of EQU to the point EQU in the second copy, so that the critical point maps to EQU, and we map the second copy of EQU to itself by EQU.

The real connectedness locus in this prototypical case consists of all pairs EQU with EQU and EQU.

rw-det3.1.ps pixels 480 by 480 scaled 225 Figure 13.

Detail of Figure 3 showing a product configuration.

For the map corresponding to the center point EQU, there are two superattracting periodic orbits with periods 3 and 4 respectively.

(Region: EQU.)

qp.4.ps pixels 800 by 800 scaled 225 Figure 14.

The prototype product configuration in the EQU -parameter plane.

Finally consider the product configuration of Figure 13.

We say that a point of the cubic parameter plane belongs to a product configuration if there are disjoint neighborhoods EQU and EQU as above so that some iterate of EQU maps EQU into itself and some iterate maps EQU into itself, but no forward image of either one of the EQU intersects the other.

In this case, the universal model is obtained by taking two disjoint real lines, say with parameters EQU and EQU respectively, and looking at independent quadratic maps EQU.

The “real connectedness locus” for this two-parameter family, that is the set of parameter pairs for which both critical points have bounded orbits, is evidently equal to the square EQU in the EQU -plane, as illustrated in Figure 14.

According to Jakobson, the set set of parameter pairs for which both critical orbits are chaotic (indicated by dark grey in the figure) has positive measure.

A classical conjecture, not yet proved, asserts that this set is totally disconnected.

Thus it seems natural to make the corresponding conjecture for the cubic parameter plane of Figure 3, that the set of parameter pairs for which both critical orbits are chaotic is a totally disconnected set of positive measure.

Some further discussion of these shapes, and other related ones, will be given in 4, which discusses the corresponding four cases for complex cubic maps, and in Appendix B.

One useful tool for studying real polynomial mappings EQU of degree EQU is provided by the topological entropy EQU of EQU considered as a map from the compact interval EQU to itself.

According to Rothschild and, this can be computed as where EQU is the number of points along the real axis at which the derivative EQU changes sign.

In the cubic case, a more practical algorithm for computing EQU has recently been described by.

The entropy EQU varies continuously as EQU varies through polynomials of fixed degree.

Furthermore EQU takes a constant value, equal to the logarithm of an algebraic integer, throughout each hyperbolic component.