# Filtering Reconfigurable Intelligent Computational Surface for RF Spectrum
Purification

Kaining Wang, Bo Yang, Zhiwen Yu, Xuelin Cao, Mérouane Debbah, and Chau Yuen
K. Wang, B. Yang, and Z. Yu are with the School of Computer Science,
Northwestern Polytechnical University, Xi’an, Shaanxi, 710129, China. X. Cao
is with the School of Cyber Engineering, Xidian University, Xi’an, Shaanxi,
710071, China. M. Debbah is with the Center for 6G Technology, Khalifa
University of Science and Technology, P O Box 127788, Abu Dhabi, United Arab
Emirates. C. Yuen is with the School of Electrical and Electronics
Engineering, Nanyang Technological University, Singapore.

###### Abstract

The increasing demand for communication is degrading the electromagnetic (EM)
transmission environment due to severe EM interference, significantly reducing
the efficiency of the radio frequency (RF) spectrum. Metasurfaces, a promising
technology for controlling desired EM waves, have recently received
significant attention from both academia and industry. However, the potential
impact of out-of-band signals has been largely overlooked, leading to RF
spectrum pollution and degradation of wireless transmissions. To address this
issue, we propose a novel surface structure called the Filtering
Reconfigurable Intelligent Computational Surface (FRICS). We introduce two
types of FRICS structures: one that dynamically reflects resonance band
signals through a tunable spatial filter while absorbing out-of-band signals
using metamaterials and the other one that dynamically amplifies in-band
signals using computational metamaterials while reflecting out-of-band
signals. To evaluate the performance of FRICS, we implement it in device-to-
device (D2D) communication and vehicular-to-everything (V2X) scenarios. The
experiments demonstrate the superiority of FRICS in signal-to-interference-
noise ratio (SINR) and energy efficiency (EE). Finally, we discuss the
critical challenges faced and promising techniques for implementing FRICS in
future wireless systems.

###### Index Terms:

FRICS, metamaterial, spectrum purification, interference cancellation.

##  I Introduction

Metasurface is a two-dimensional planar structure composed of meta-atoms
arranged in a specific pattern, typically with a thickness smaller than the
wavelength of electromagnetic (EM) waves. This structure exhibits unique EM
properties, offering flexible and efficient control over various
characteristics of EM waves, such as polarization, amplitude and phase. In
this context, reconfigurable intelligent metasurfaces (RISs) have recently
emerged as a representative technology capable of managing wireless signal
propagation and enhancing communication quality [1]. For example, in scenarios
where users experience dead zones, implementing RISs, especially at the
periphery of a cell, can create propagation paths that enable beamforming,
thereby improving signal reception [2].

Although RISs are advancing as a technology for creating intelligent wireless
RF environments, there remains a critical need to enhance their capability to
adapt to complex radio frequency (RF) communication environments where
multiple signals (e.g., the desired signals and interfering signals) overlap
through passive reflection [3]. Addressing this challenge, Yang et al.
proposed an innovative approach integrating task-oriented computing and
communication functions using computational metamaterials [4]. They introduced
a novel metasurface structure known as reconfigurable intelligent
computational surface (RICS), designed to enable dynamic tunable signal
reflection and amplification via computational metamaterials. However, in
complex RF environments with significant interference, both RICS and
traditional RISs struggle to effectively mitigate interferences. For instance,
when signals out-of-band interact with a metasurface, they often undergo
unpredictable and uncontrolled reflection. This unmanaged reflection sometimes
will exacerbate interference issues, impacting receiver efficiency and causing
severe spectrum pollution, ultimately degrading overall communication system
performance.

Various methods have been explored to achieve interference cancellation in
wireless communications. In passive RISs, such as those discussed by Tang et
al. [5], interference elimination is achieved by utilizing signal reflection
from the RISs. However, passive RISs can only degrade interference from the
front side of the metasurface and are ineffective against interference from
the back side. To address this limitation, STAR-RIS proposed by the authors in
[6] divides the incident signal into two components, allowing for both
reflection and refraction. This capability enables interference elimination
from both the front and back sides through phase shift optimization.
Nonetheless, these methods often result in significant signal attenuation
after passing through the metasurface, thereby reducing the effectiveness of
interference cancellation. In contrast, active RISs, as discussed in [7],
integrate amplifier components directly onto the metasurface. This design
enhances the amplitude of the incident signal, thereby increasing signal power
and maximizing interference cancellation at the receiving end. However, the
incorporation of amplifiers increases energy consumption, which can impact the
overall energy efficiency of the system.

To tackle the challenges previously discussed, we propose a novel filtering
reconfigurable intelligent computational surface, called FRICS, which can
leverage absorbing metamaterials to effectively absorb interference and
computational metamaterials to amplify the desired signal. The four types of
metasurfaces are compared in Table I. We observe that passive RISs leverage
reflection to cancel interference and STAR-RIS enhances cancellation by
combining reflection and refraction, active RISs offer better interference
cancellation through signal amplification despite potential drawbacks in
energy efficiency. In contrast with them, the innovative configuration enables
FRICS to achieve comprehensive interference elimination while simultaneously
reducing energy consumption. Compared to existing interference cancellation
methods, FRICS enhances both the signal-to-interference-plus-noise ratio
(SINR) and energy efficiency (EE) of the system.

TABLE I: Comparison of different metasurfaces designs

| filtering | frequency adjustment | Simultaneous reflection and transmission | signal amplification | power consumption | interference cancellation capability
---|---|---|---|---|---|---
Passive RIS[5] |  2715 |  2715 |  2715 |  2715 | middle | low
STAR-RIS[6] |  2715 |  2715 |  1F5F8 |  2715 | middle | low
Active RIS [7] |  2715 |  2715 |  2715 |  1F5F8 | high | middle
proposed FRICS |  1F5F8 |  1F5F8 |  1F5F8 |  1F5F8 | low | high

Figure 1: The structure of FRICS. It consists of a spatial filtering layer, an
intermediate layer made of metamaterials, and a control layer. Design A on the
left includes a bandstop filter layer plus an absorbing material layer. Design
B on the right includes a bandpass filter layer plus a computational
metamaterial layer.

##  II Fundamentals of FRICS

As shown in Fig. 1, our proposed FRICS comprises three layers: a spatial
filtering layer, a metamaterial intermediate layer, and a control layer. The
control layer includes a circuit board hosting an intelligent controller
responsible for adjusting the tunable parameters of the spatial filtering
layer. This adjustment is commonly facilitated by a field-programmable gate
array (FPGA). In the subsequent sections, we will detail the structure of the
spatial filtering layer and explore its collaborative design with the
metamaterial intermediate layer to achieve RF spectrum purification in typical
application scenarios.

###  II-A Spatial Filtering Layer

Based on the generalized Huygens’ principle and the theory of metamaterial
equivalent circuit models [8], a metasurface can be conceptually represented
as a series connection of capacitive grids and inductive grids, thereby
forming a series LC resonant circuit. This equivalence arises because the
metal resonators within the metasurface structure can be analogously
considered inductors. In contrast, the gaps or spaces between these resonators
can be analogously treated as capacitors in an electrical circuit model.

In the equivalent LC resonant circuit, the capacitive reactance of the
equivalent capacitance decreases with increasing frequency, while the
inductive reactance of the equivalent inductance increases. This behavior
impacts the performance of ring-shaped elements in the circuit, which exhibit
a relatively high shunt impedance, thus behaving similarly to an open circuit.
Consequently, EM waves can effectively pass through the FRICS at both higher
and lower frequencies. However, the transmission performance of the FRICS is
weakest in the intermediate frequency range, resulting in a bandstop
characteristic, as illustrated in Fig. 2(a). At intermediate frequencies, the
combination of the decreasing capacitive reactance and the increasing
inductive reactance creates conditions that inhibit the efficient transmission
of EM waves, leading to a notable dip in performance.

On the other hand, as illustrated in Fig. 2(b), aperture-shaped elements
exhibit the opposite behavior to ring-shaped elements, resulting in a bandpass
characteristic. This means that EM waves can pass through the structure more
effectively at certain intermediate frequencies, rather than being blocked.
The unit structure of the spatial filtering layer of Design B is depicted on
the left side of Fig. 2(b). In this figure, the yellow color represents the
substrate, the green color represents the metal patches, and the blue color
represents the variable capacitor diodes. These diodes can adjust their
capacitance values, which in turn influences the transmission characteristics
of the spatial filtering layer. The right figure in Fig. 2(b) shows the
transmission coefficient of the spatial filtering layer with different
capacitance values of the variable capacitor diodes. By varying the
capacitance, the transmission properties of the filter can be tuned, allowing
for the selective passage of certain frequency ranges while blocking others.
This tunability is key to achieving the desired bandpass characteristic,
enabling the filter to efficiently manage the passage of EM waves based on
their frequency.

(a)

(b)

Figure 2: The left shows the signal propagation of the FRICS, and the right
shows the CST simulation results of the spatial filtering layer, where the
design A of FRICS is in (a), and the design B of FRICS is in (b).

In practical scenarios, the frequencies of the desired signal and the
interfering signal can vary due to the communication environment. To address
this, we employ variable capacitors, commonly referred to as tuning diodes, to
dynamically adjust the operating frequency range of the FRICS by controlling
the bias voltage. This allows the metasurface to be integrated into the
spatial filtering layer to achieve frequency tuning. Specifically, when a
reverse bias voltage is applied to the variable capacitor, its capacitance
changes accordingly. In this paper, we utilize the variable capacitor
SMV2019-079LF as an example [9], in which the capacitance changes with the
reverse bias voltage. For example, when the bias voltage of the diode changes
from 0V to -19V, the capacitance of the diode changes from 2.31pF to 0.24pF.
Consequently, the tunable control layer can dynamically adjust the reflection
frequency range of the FRICS by controlling the reverse bias voltage of each
unit using a FPGA. We further explore the CST simulations and we vary the
values of the RLC series circuit to simulate the electrical characteristics of
the variable capacitor under different bias voltages. By implementing this
dynamic tuning capability, the FRICS can effectively adapt to varying signal
frequencies, enhancing its performance in diverse communication environments.

###  II-B Metamaterial Intermediate Layer

To fully leverage the potential of the metamaterial intermediate layer, it is
crucial to appropriately design the surface structure. This can be achieved by
using absorbing materials to absorb interference signals or employing
computational materials to amplify desired signals. Specifically, as a novel
type of material that can significantly weaken or absorb EM wave energy,
absorber metamaterials can help to reduce EM interference [10]. Additionally,
absorber metamaterials possess characteristics such as being lightweight,
corrosion-resistant, and temperature-resistant, making them suitable for
deployment in wireless communication environments. According to the material
compositions, the common absorber materials can be classified into carbon-
based absorber materials, iron-based absorber materials, ceramic-based
absorber materials, and chiral materials. In our design, iron oxide material
is implemented as the intermediate layer due to its high magnetic
permeability, high resistivity, and good impedance matching characteristics.
For example, $\rm\ce{BaCo_{0.9}Si_{0.95}Fe_{10.1}O_{19}}$ exhibits an
absorption rate of up to 99% within the frequency range of 11-13 GHz [11]. On
the other hand, computational metamaterials refer to a specially designed
block of metasurface that allows mathematical operations on the incident
waves, such as spatial differentiation, integration, and convolution [4]. To
be specific, a computational metamaterial can achieve amplification of the
incident signal that impings the FRICS. Therefore, leveraging the properties
of the computational metamaterial can enhance the signal-to-noise ratio by
amplifying the desired signal while avoiding additional energy consumption.

To sum up, by incorporating absorbing materials, the intermediate layer can
effectively mitigate unwanted/interfering signals, thereby enhancing the
overall signal quality. Meanwhile, using computational materials can help to
amplify the desired signals, improving their strength and clarity. This dual
capability of absorption and amplification makes the metamaterial intermediate
layer a vital component in the FRICS design, enabling it to adapt to various
signal environments and maintain efficient communication. Therefore, the
intermediate layer’s design should be tailored to the specific requirements of
the application, ensuring optimal performance of the FRICS system.

###  II-C FRICS Architecture Design

According to the configuration of the spatial filtering layer and the
metamaterial intermediate layer, FRICS has two designs, which are described as
follows.

  * •

FRICS Design A: bandstop filtering layer + absorbing intermediate layer. In
this design, the spatial filtering layer achieves a bandstop filtering effect
while an absorber material is used as the metamaterial intermediate layer. As
simulated using CST in the right of Fig. 2(a), the transmission coefficient
exhibits a notable decrease within the frequency range of 16 GHz to 18 GHz,
indicating reflective characteristics. In the remaining frequency range, the
transmission coefficient remains relatively unchanged, demonstrating
transmission characteristics. Therefore, as shown in the left figure of Fig.
2(a), the in-band signal ($f_{2}$) can be reflected and the out-of-band
signals ($f_{1}$ and $f_{3}$) can be passed through the filtering layer. Then
absorbing material can absorb these signals to achieve interference
cancellation.

  * •

FRICS Design B: bandpass filtering layer + computational intermediate layer.
In this design, the spatial filtering layer achieves a bandpass filtering
effect while the computational metamaterial is used as the intermediate layer.
As illustrated in Fig. 2(b), there is a notable decrease in the reflection
coefficient within the frequency range of 17 GHz to 19 GHz, indicating
transmission characteristics. In the out-of-band range, the reflection
coefficient remains relatively unchanged, demonstrating reflective
characteristics. Consequently, as illustrated in the left figure of Fig. 2(b),
the desired signal ($f_{2}$) passes through the spatial filtering layer and
then can be amplified with the aid of computational metamaterials. Meanwhile,
the out-of-band signals ($f_{1}$ and $f_{3}$) can be reflected through the
filtering layer.

Based on the two designs of FRICS, for the communication scenario where the
desired signal and interfering signal coexist, the proposed FRICS can
effectively separate the desired signals from the interfering signals.
Meanwhile, the unwanted interfering signals can be absorbed and the desired
signal can be amplified by configuring the design of FRICS appropriately,
thereby improving energy efficiency.

##  III Applications and Performance Evaluation

Based on the two designs introduced above, we validate the superiority of
FRICS in two typical scenarios.

###  III-A Application Scenarios

(a)

(b)

Figure 3: In (a), the application scenario of FRICS with Design A is shown on
the left, wherein the outdoor interference signal at the frequency $f_{1}$ or
$f_{3}$ is filtered and absorbed via FRICS. Concurrently, the signal $f_{2}$
is reflected to enhance wireless link quality. In contrast,
passive/active/star-RIS can only compensate for interference signals by
adjusting the reflection phase, as shown on the left of (b). The typical
application scenario of FRICS with Design B is depicted on the right of (a),
where the desired signal from the vehicle to the roadside unit (RSU) via
$f_{2}$ can be filtered and amplified by FRICS. Meanwhile, the signals $f_{1}$
and $f_{3}$ are reflected to enhance the V2V transmission. For comparison,
passive/active/star-RIS just reflects/refracted the desired signals to improve
the receiving quality, as shown in the right of (b).

####  III-A1 Interference signal filtering and absorbing

As illustrated in Fig. 3(a), we examine a device-to-device (D2D) communication
scenario for efficient transmission, where a D2D indoor communication pair
operating at the frequency $f_{1}$ and may experience significant co-frequency
interference from outdoor cellular communications. To eliminate the co-
frequency interference from outdoor cellular communication, the FRICS is
implemented on the exterior wall of the building. This setup allows the
incoming interfering signal at $f_{1}$ or $f_{3}$ to pass through the spatial
filtering layer of the FRICS. Once through this layer, the signal is absorbed
by the metamaterial intermediate layer, which consists of absorbing materials
designed to mitigate interference. At the same time, the signal between the
users and the base station (BS) at $f_{2}$ will be reflected to enhance the
communication quality. It is observed that the implementation of FRICS plays a
crucial role in maintaining high-quality indoor communication by mitigating
external interferences.

####  III-A2 Desired signal filtering and amplification

As illustrated in Fig. 3(b), we consider a vehicular-to-everything (V2X)
scenario, where a vehicle is communicating with a roadside unit (RSU) at
frequency $f_{2}$. Concurrently, a vehicle-to-vehicle (V2V) pair transmits at
frequencies $f_{1}$ or $f_{3}$. To improve the SINR for the RSU, when the
uplink signal from the vehicle impinges the FRICS, the signal is allowed to go
through the filtering layer and then amplified through computational
metamaterial. This amplification ensures that the uplink signal is strong and
clear when it reaches the RSU, thereby boosting the intended signal’s strength
and thus enhancing communication reliability and efficiency. At the same time,
V2V communication is occurring. The FRICS can reflect the V2V communication
signals, thereby enhancing their communication quality. By reflecting these
signals, the FRICS helps to maintain robust and uninterrupted V2V
communication, which is crucial for safety and coordination between vehicles.

(a)

(b)

(c)

(d)

Figure 4: The comparison of SINR and EE for the receiving device in an indoor
environment after passing through FRICS (operating with Design A), passive
RIS, active RIS, and star RIS (with simultaneous transmission and reflection
mode) is shown in (a) and (b), respectively. The comparison of SNR for the RSU
and the V2V communication after passing through FRICS (operating with Design
B), passive RIS, active RIS, and star RIS (with full transmission mode) is
presented in (c) and (d), respectively.

###  III-B Simulation Results and Discussions

The configuration parameters for the spatial filtering layer are shown in
Table II. To demonstrate the benefits of FRICS, we evaluated energy efficiency
with the bandwidth set to $1$ MHz. To accurately compare the energy
consumption of different metasurfaces designs with interference mitigation
when using variable capacitors, the power consumption of the RIS reflection
units is $0$, the power consumption of the driver generator is $250$ mW, the
power consumption of the bias voltage amplifier is $180$ mW, and the FPGA
power consumption is $1.5$ W [12].

####  III-B1 Simulation results of Design A

TABLE II: Experimental Parameters of FRICS Parameters |  Values |  Parameters |  Values
---|---|---|---
Substrate side length |  6mm |  Patch width |  1mm
Substrate thickness |  0.38mm |  Pore size |  1mm
Substrate material |  FR-4 |  Patch material |  Copper
diode capacitance |  0.24-2.31pF |  Patch thickness |  0.038mm

In this scenario, we compare the SINR and EE of our proposed FRICS with the
latest RIS models (passive RIS, and active RIS and star RIS) in terms of
interference mitigation. The D2D transmission power ranges from 15 dBm to 39
dBm. For the communication simulation scenarios, we assume a D2D communication
distance of 10 meters and a cellular communication distance of 20 meters.

As illustrated in Fig. 4(a), a comparison is presented between the SINR after
interference mitigation utilizing FRICS, passive RIS, active RIS, and star RIS
at the D2D receiver indoor. It is observed that FRICS achieves the highest
SINR due to its ability to completely absorb the interfering signals. This
superior performance can be attributed to the efficient absorption
capabilities of the metamaterial intermediate layer, which effectively
eliminates interference. In comparison, passive RIS experiences a greater
degree of signal attenuation when eliminating interference, which ultimately
affects its ability to eliminate interference signals. The star RIS, which can
simultaneously transmit and reflect signals, even experiences a significant
reduction in reflected signals compared to passive RIS, which affects its
interference mitigation performance. Active RIS, on the other hand, amplifies
the signals by incorporating additional amplifying circuits, resulting in
stronger signals reaching the receiving end and effectively mitigating
interference, thus achieving a higher SINR.

Fig. 4(b) illustrates the energy efficiency of FRICS for interference
cancellation. As the D2D transmit power increases, the SINR at the D2D
communication receiver can be improved, resulting in a gradual increase in the
energy efficiency for interference cancellation. Notably, the energy
efficiency of FRICS remains the highest due to its ability to effectively
cancel interference without introducing additional energy, resulting in near-
zero power consumption for the driving generator. Given the related power
consumption parameters, the energy efficiency of amplifying signals for active
RIS (with the amplification factor of 2 and each reflection unit consumes 150
mW of energy) is higher than the others.

####  III-B2 Simulation results of Design B

In this scenario, we evaluate the SNR achieved at the RSU when the
transmission power of the vehicle ranges from 15 dBm to 39 dBm. We assume that
the gain of star RIS and FRICS is 2.

Fig. 4(c) illustrates a comparison between the FRICS and the benchmark RISs
designs in terms of SNR for V2X communication, with the transmission power of
the vehicle ranging from 15 to 39 dBm. We observe that for the RSU, the FRICS
achieves the highest SNR performance. This is because the computational
metamaterials amplify the desired signal. In contrast, passive RIS has no
signal amplification capability, resulting in the worst performance.
Similarly, Fig. 4(d) illustrates that only FRICS can extend the performance of
V2X by amplifying in-band signal while improving the quality of V2V
communication via out-of-band signal reflection.

To sum up, we observe from Fig. 4(a)-(d) that the proposed FRICS demonstrates
superior interference mitigation capabilities without the need for additional
energy input, achieving the highest SINR/SNR among the compared benchmark
schemes. Its ability to absorb interfering signals and amplify the desired
signals completely provides a significant advantage in maintaining high
communication quality and energy efficiency.

##  IV Challenges and Future Directions

The introduction of the FRICS designs has significant implications for future
wireless communication systems, particularly in the context of upcoming sixth-
generation (6G) networks with a more complicated RF communication environment.
In the following, we outline the challenges and potential directions, which
aim at paving the way for more feasible solutions to achieving RF spectrum
purification in future 6G networks.

###  IV-A FRICS Design for Energy Harvesting

In this paper, we have utilized absorbing materials as the intermediate layer
to achieve interference cancellation. However, we believe that the
metamaterial intermediate layer holds even greater potential. By implementing
energy-conversion metamaterials, such as photovoltaic conversion
metamaterials, as the intermediate layer, we can achieve energy harvesting
when the filtered signals, including interference signals, impinge on the
intermediate layer [13]. This approach enhances the deployment flexibility of
the metasurfaces by enabling the collection of energy from various sources,
such as solar power or wireless signals. It eliminates the need for additional
power sources, allowing metasurface deployment on various object surfaces.
This capability can significantly extend the sustainability of the
metasurfaces, making them more adaptable to diverse environments and
applications.

Looking ahead, there are opportunities to explore different types of
metasurfaces to achieve additional functionalities. For example, time-
targeting metasurfaces can dynamically adjust their properties based on
temporal changes, enabling precise control over when certain signals are
absorbed or amplified. By integrating such advanced functionalities, future
metasurfaces can offer even greater capabilities and versatility, paving the
way for innovative applications in wireless communication, energy harvesting,
and beyond.

###  IV-B FRICS Design for Phase Modulation

We have observed that although FRICS can effectively eliminate interference
signals, controlling the phase of the reflected signals remains a challenge
since precise phase control is crucial for several metasurfaces-based
applications such as beamforming. Recent works have achieved filtering while
adjusting the phase [14]. However, their multi-layer structures and additional
amplifiers consume significant energy, which violates the principle of
metasurface design. Moreover, they ignore the issue of same-frequency
interference, which is more serious in practice. Therefore, it holds promise
to investigate how to directly modulate the phase of FRICS.

Furthermore, when we consider the coexistence of the FRICS and existing RISs
within the same physical space, the functionalities provided by FRICS can
complement the limitations of traditional RISs. For instance, when the
metamaterial intermediate layer of FRICS employs optoelectronic conversion
materials, the energy obtained by the FRICS can not only sustain its operation
but also serve as an energy supplement for other RISs. Meanwhile, the
beamforming capability of the RISs can compensate for the limitations of FRICS
by enabling phase adjustment.

###  IV-C Interplay between FRICS Design and ISAC

Integrated sensing and communications (ISAC) has emerged as a crucial
technology in the context of 6G aiming to achieve both perception and
communication [15]. Traditional approaches to integrated sensing often
prioritize either communication-centric or perception-centric designs, making
it challenging to achieve a unified ISAC system. However, the proposed FRICS
holds the potential to address this challenge by filtering signals in
different frequency bands, allowing the reflected signals to be used for
communication and the filtered signals for sensing. For instance, the signal
filtered to the intermediate layer of FRICS can directly serve as information
for environmental sensing, such as humidity, pollutant density, and more.

Furthermore, to effectively suppress interferences using the metamaterial
intermediate layer of FRICS, the transmission coefficient of FRICS should be
configured appropriately based on the frequency of signals. To obtain this
frequency information, ISAC may play a crucial role in detecting the frequency
of the signals, e.g., with the aid of deep learning methods [3]. This dual
functionality enables FRICS to enhance the capabilities of ISAC systems,
providing a seamless integration of communication and sensing tasks within a
single framework.

###  IV-D Low-Energy and Low-Complexity Transceiver Design

In traditional wireless communication systems, the receiver’s antenna captures
wireless signals, which then require filtering to extract the desired signal.
However, these conventional designs often result in increased receiver
complexity and higher energy consumption. In contrast, the proposed FRICS
offers several advantageous characteristics, including low energy consumption
and tunability. By integrating FRICS into receiver designs, it mitigates
receiver complexity and reduces energy consumption significantly. Moreover,
FRICS facilitates dynamic adjustment of the filtering frequency, which aligns
with principles of green and low-carbon design. This capability not only
provides enhanced flexibility but also holds the potential to enhance
communication performance.

If we go one step further and suppose that the receiver is surrounded by
multiple FRICS unis, by assigning different operating frequencies to each
directional FRICS, we can selectively ensure that the desired signal is
received while filtering out interfering signals. This approach leverages the
directional capabilities of FRICS to achieve significant improvements in
signal reception quality, interference management, and overall communication
performance.

##  V Conclusion

This paper investigated a novel filtering RICS (FRICS) structure and
introduced two different designs. Design A uses a tunable spatial filter to
dynamically reflect signals within the resonance band while absorbing out-of-
band signals. Design B enables the transmission and amplification of the
desired signals within the resonant band while reflecting out-of-band signals.
Furthermore, this paper explores two typical application scenarios where FRICS
can be employed and evaluates its performance in these contexts. Finally, we
outline several challenges and opportunities, highlighting promising
directions for future research in this field.

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$\partial_{+}N$, and a pseudo-Anosov flow $\varphi$ on $M$ positively
transverse to $W$.

  2. (b)

The surfaces $W,Y\subset M$ are taut and lie over the face of the Thurston
norm ball determined by the Euler class of $\varphi$.

  3. (c)

Further, the pullback semiflow $\varphi_{N}$ on $N$ has the following
properties:

    1. (i)

$\varphi_{N}$ is compatible with the sutured structure of $N$ in that it
points out of $N$ along $\partial_{+}N$ and into $N$ along $\partial_{-}N$

    2. (ii)

“The semiflow is a product away from $H$”: the flowlines of $\varphi_{N}$
lying in $S\times I$ and $T\times I$ are all of the form $x\times I$ for some
$x\in S\cup T$.

    3. (iii)

There are exactly 2 closed orbits of $\varphi_{N}$ lying in $H$, $\gamma_{1}$
and $\gamma_{2}$. They are both freely homotopic to the core of $H$ and also
have the property that $\gamma_{1}$ is freely homotopic to $\gamma_{2}$ with
the orientation reversed.

    4. (iv)

Every other flowline of $\varphi_{N}$ in $H$ is either a properly embedded
interval or has one end on $\partial_{\pm}N$ and accumulates on $\gamma_{1}$
or $\gamma_{2}$ in the forward or backward direction.

    5. (v)

There exists a periodic orbit $\gamma_{Z}$ of $\varphi$ which pulls back to a
single oriented flowline of $\varphi_{N}$ in $H$ from $A_{\text{front}}$ to
$A_{\text{back}}$.

The orbits $\gamma_{1}$ and $\gamma_{2}$, as well as parts of their stable and
unstable leaves, are shown in the center image of Figure 17; the orbit
$\gamma_{Z}$ is shown in the righthand image.

We remark that the flow $\varphi$ in the above lemma is constructed using the
Gabai–Mosher construction of pseudo-Anosov flows transverse to sutured
hierarchies in closed oriented atoroidal 3-manifolds, which is unpublished.
However, Mosher laid much of the foundation in [Mos96], and Landry and Tsang
are expositing the rest in detail ([LT23] and work in progress).

Building on Lemma 6.6, we have:

######  Theorem 6.7.

There exists a closed 3-manifold $M$, a transitive pseudo-Anosov flow
$\varphi$ on $M$, and a taut surface $Z\subset M$ such that $[Z]$ pairs
nonnegatively with each closed orbit of $\varphi$ but is not almost transverse
to $\varphi$ up to isotopy.

###### Proof.

Let $M$ be as in Lemma 6.6; we will use the same notation as in that lemma.
Note that since $W$ and $Y$ are taut and lie over the same face of the
Thurston norm ball, the surface $Z=W\cup Y$ is taut. Note that $Z$ is not
almost transverse to $\varphi$ up to isotopy because this would imply the same
for $Y$, which is impossible: by Lemma 6.6(c)(v), $Y$ has pairing $-1$ with
the orbit $\gamma_{Z}$.

We now claim that for every closed orbit $\gamma$ of $\varphi$, $[Z]$ pairs
nonnegatively with $[\gamma]$; this will complete the proof.

If $\gamma$ is equal to either $\gamma_{1}$ or $\gamma_{2}$ then $\gamma$ has
null pairing with $Y$, since the $\gamma_{i}$’s are homotopic to the core of
$H$, which can be homotoped into $Y$. Hence we may assume that $\gamma$ is any
other closed orbit of $\varphi$. In this case, $\gamma$ pulls back to finitely
many oriented segments $\gamma_{1},\dots,\gamma_{n}$ in $N$ beginning on
$\partial_{-}N$ and ending on $\partial_{+}N$. Fix $1\leq i\leq n$ and
consider $\gamma_{i}$. By Lemma 6.6(c)(ii) $\gamma_{i}$ either lies in $(S\cup
T)\times I$ or in $H$.

Suppose $\gamma_{i}\subset(S\cup T)\times I$. Then $\gamma_{i}$ contributes
only positive intersections to the algebraic intersection number of $Z$ and
$\gamma$ by Lemma 6.6(c)(ii).

Next, suppose $\gamma_{i}\subset H$. The absolute value of the algebraic
intersection of $\gamma_{i}$ and $Y$ is at most 1 since $Y\cap H$ separates
$H$, while the algebraic intersection of $\gamma_{i}$ with $W$ is equal to
$1$. It follows that $\gamma_{i}$ contributes nonnegatively to the algebraic
intersection of $Z$ and $\gamma$. Summing over $i=1,\dots,n$, we see that
$[Z]$ and $[\gamma]$ pair nonnegatively as claimed. ∎

One can modify the example above to give a manifold $M$, flow $\varphi$, and
surface $Z$ as in the statement of the theorem with $Z$ connected. In the
construction, replace the surface $S\cup T$ with a connected surface $R$ with
6 boundary components as indicated on the left of Figure 19. The same methods
that Mosher uses to justify Lemma 6.6 can be used to glue the inward and
outward pointing boundaries of $N=H\cup(R\times I)$ to produce a hyperbolic
manifold together with a pseudo-Anosov flow $\varphi$ satisfying all the
properties in Lemma 6.6 (c).

Now cap off $R\times\\{\frac{1}{2}\\}$ inside $H$ as shown on the right of
Figure 19 to form a connected surface $Z$; Mosher’s arguments apply equally
well to show that $Z$ is taut. The orbit $\gamma_{Z}$ furnished by Lemma
6.6(c) passes through the annulus in $Z$ running between $a_{1}$ and $a_{4}$
negatively and exactly once, up to an isotopy of this annulus supported in
$H$.

It remains to show that the negative intersection point of $\gamma_{Z}$ with
$Z$ cannot be removed by isotopy. To see this, let $\Sigma$ be the image in
$M$ of the outward pointing boundary of $N$. Let $\widetilde{\gamma}_{Z}$ be a
lift of $\gamma_{Z}$ to the universal cover $\widetilde{M}$ of $M$. Then
$\gamma_{Z}$ passes through a bi-infinite sequence
$(\widetilde{\Sigma}_{n})_{n\in\mathbb{Z}}$ of distinct lifts of $\Sigma$ to
$\widetilde{M}$.

In the region of $\widetilde{M}$ between $\widetilde{\Sigma}_{0}$ and
$\widetilde{\Sigma}_{1}$, $\widetilde{\gamma}_{Z}$ has a negative point of
intersection with a lift $\widetilde{Z}$ of $Z$. Let $\widetilde{N}$ be the
universal cover of $N$ obtained as the component of the preimage of $N\subset
M$ in $\widetilde{M}$ containing $\widetilde{\Sigma}_{0}$ and
$\widetilde{\Sigma}_{1}$ in its boundary. Since $\widetilde{N}$ is assembled
by gluing together copies of the universal covers of $H$ and $R\times I$, we
can see that $\widetilde{Z}$ separates $\widetilde{\Sigma}_{0}$ and
$\widetilde{\Sigma}_{1}$ by observing that $Z$ separates the positive and
negative boundaries of $N$ within each of $H$ and $R\times I$. Moreover, the
intersection of $\widetilde{Z}$ with any copy of the universal cover of $H$ or
$R\times I$ is connected; together, this implies that $\widetilde{Z}$ meets
$\widetilde{\gamma}_{Z}$ in a single negative point of intersection. We
conclude that the negative intersection between $Z$ and $\gamma_{Z}$ cannot be
removed by an isotopy and the example is complete. Note that in this example
we could have replaced $\gamma_{Z}$ by any periodic orbit whose pullback to
$N$ contains a segment from $A_{\text{front}}$ to $A_{\text{back}}$; there are
many of these, since the periodic orbits of $\varphi$ are dense.

Figure 19. Modifying the example of Theorem 6.7 so that the surface $Z$ in the
statement is connected.

##  7\. Shadows of transverse surfaces

In Section 8 we will show that almost transverse position is essentially
unique. The key to this is understanding the projections to the orbit space of
lifts of compact connected surfaces transverse to an almost pseudo-Anosov
flow. This is the goal of the present section. In the case where the ambient
manifold has no boundary, this was essentially done by Fenley in [Fen09,
Section 4].

First we set some notations and terminology.

###  7.1. Notation and conventions

Let $\mathcal{O}^{s}$ and $\mathcal{O}^{u}$ denote the stable and unstable
foliations, respectively, of the orbit space $\mathcal{O}$ of an almost
pseudo-Anosov flow $\psi$. For $p\in\mathcal{O}$, let $\mathcal{O}^{s}(p)$
denote the leaf of $\mathcal{O}^{s}$ through $p$, and similarly for
$\mathcal{O}^{u}(p)$. Recall that $\Theta\colon\widetilde{M}\to\mathcal{O}$ is
the projection to the flow space from the universal cover of $M$. If
$p\in\mathcal{O}$, we set $\gamma(p):=\Theta^{-1}(p)$ to be the orbit of
$\widetilde{\psi}$ corresponding to $p$.

A quadrant of $p\in\mathcal{O}$ is a component of
$\mathcal{O}\\!\bbslash\\!(\mathcal{O}^{s}(p)\cup\mathcal{O}^{u}(p))$. Points
in $\partial\mathcal{O}$, as well as those connected to $\partial\mathcal{O}$
by singular leaves of either $\mathcal{O}^{s}$ or $\mathcal{O}^{u}$, have
infinitely many quadrants; all other points have $2n$ quadrants for some
$n\geq 2$. Two quadrants are adjacent if they meet along a noncompact half-
leaf.

A stable or unstable slice leaf is a properly embedded copy of $\mathbb{R}$
contained in a leaf of $\mathcal{O}^{s}$ or $\mathcal{O}^{u}$ respectively. A
stable or unstable boundary slice ray is a ray embedded in $\mathcal{O}$
meeting $\partial\mathcal{O}$ exactly at its basepoint, obtained as the union
of finitely many blown segments with a noncompact half-leaf of
$\mathcal{O}^{s}$ or $\mathcal{O}^{u}$, respectively.

We observe that each component of $\partial\mathcal{O}$ is a slice leaf of
both $\mathcal{O}^{s}$ and $\mathcal{O}^{u}$. Each slice leaf which is not an
entire component of $\partial\mathcal{O}$ separates $\mathcal{O}$ into either
$2$ or $3$ components. The $3$ component case arises when a slice leaf meets
$\partial\mathcal{O}$ in finitely many, necessarily contiguous, blown
segments. Similarly, a boundary slice ray separates $\mathcal{O}$ into $2$
components.

Now suppose $S$ is a properly embedded compact surface transverse to $\psi$.
Lift $S$ to $\widetilde{S}\subset\widetilde{M}$, and let
$\Omega=\Theta(\widetilde{S})\subset\mathcal{O}$. (Here $\Omega$ stands for
_ombre_ , or shadow.) Note that the restriction of $\Theta$ to $\widetilde{S}$
is a homeomorphism onto $\Omega$ since $\widetilde{S}$ separates
$\widetilde{M}$ and orbits intersect $\widetilde{S}$ positively. If $a$ is a
blown annulus of $\psi$ and $S\cap a$ is nonempty, then it is either $(1)$ a
collection of nonseparating arcs or $(2)$ a collection of circles homotopic to
the core of $a$. If $\widetilde{a}$ is a lift of $a$ to $\widetilde{M}$ that
intersects $\widetilde{S}$, we see that $\Theta(\widetilde{a})\cap\Omega$ is
equal to the entire blown segment $\Theta(\widetilde{a})$ in case $(1)$, and
equal to $\Theta(\widetilde{a})$ minus its endpoints in case $(2)$. If $a$ is
not contained in the boundary of $M$ and its intersection with $S$ is a
(possibly empty) collection of arcs, then $a$ can be collapsed to obtain a new
almost pseudo-Anosov flow transverse to $A$ with fewer blown annuli.

Hence for simplicity we assume that if $S$ intersects a blown annulus in a
collection of arcs, then that annulus lives in $\partial M$; we say $S$ has
the clean annulus property with respect to $\psi$. Note that assuming $S$ has
the clean annulus property is slightly weaker than $\psi$ being a minimal
blowup with respect to $S$, because we are allowing the existence of blown
annuli not contained in $\partial M$ which do not meet $S$. The advantage of
this relaxation is that it allows us to individually consider the components
of a disconnected surface that is minimally transverse to $\psi$. Such a
surface may not be componentwise minimally transverse; however, each component
has the clean annulus property.

Without the clean annulus property, the entire discussion in this section goes
through with minor modifications.

###  7.2. Executive summary

The main results of this section are fairly easily stated and we do so now.
Their proofs appear in Section 7.3.

As before, let $\psi$ be an almost pseudo-Anosov flow in a compact 3-manifold
$M$ which is transverse to a compact, connected, properly embedded surface
$S$. Let $\widetilde{S}$ be a lift of $S$ to the universal cover
$\widetilde{M}$ of $M$. Let $\Omega=\Theta(\widetilde{S})\subset\mathcal{O}$;
this set is connected and disjoint from its frontier
$\operatorname{fr}(\Omega)$ because it is open.

######  Proposition 7.1 (Shadows of transverse surfaces).

The frontier of $\Omega=\Theta(\widetilde{S})$ is a disjoint union of
stable/unstable slice leaves and boundary slice rays, each isolated from the
others and bounding $\Omega$ to one side.

This is an immediate consequence of the more technical Proposition 7.7.

A blowup tree is defined to be either a maximal connected union of blown
segments or a singular point not contained in a blown segment.

######  Proposition 7.2.

Let $\psi$ be an almost pseudo-Anosov flow on a compact manifold $M$ and let
$S$ be a properly embedded compact surface transverse to $\psi$. Let
$\widetilde{S}$ be a lift of $S$ to $\widetilde{M}$ and let
$\Omega=\Theta(\widetilde{S})$ be its image in the flow space. Let $T$ be a
blowup tree that intersects $\operatorname{cl}(\Omega)\subset\mathcal{O}$.

If $T\cap\Omega$ meets the interior of a blown segment, then either:

  1. (a)

$T\cap\Omega=\operatorname{int}(s)$ for a blown segment $s$ that meets two
nonadjacent quadrants,

  2. (b)

$T\cap\Omega=\operatorname{int}(s)$ for a blown segment
$s\subset\partial\mathcal{O}$, or

  3. (c)

$T$ is an entire component of $\partial\mathcal{O}$ contained in $\Omega$.

If $T$ is disjoint from $\Omega$, then:

  1. (d)

$T\cap\operatorname{fr}(\Omega)$ is either a single point or an arc comprised
of finitely many blown segments. Moreover, $\Omega$ meets precisely two
quadrants of $T$, and these quadrants are adjacent.

Figure 20. _Upper chart:_ simple examples of the local pictures described in
cases (a)-(d) of Proposition 7.2, with examples showing possible interaction
with $\partial\mathcal{O}$ in the second row. The set $\Omega$ is indicated in
green. Cases (b) and (c) only concern boundary segments, which is why two
spaces are empty. _Lower chart_ : pictures of the corresponding pieces of the
surface (green) in $M$.

See the top chart in Figure 20 for a visual companion to Proposition 7.2.

######  Remark 7.2.

The situations described Proposition 7.2 translate to straightforward
statements about $S$ in $M$. Recall that $S$ has the clean annulus property.
Thus $S$ intersects each blown annulus in a collection of closed curves or not
at all, unless the blown annulus lies in $\partial M$, in which case $S$ may
intersect a blown annulus $a$ in a family of arcs connecting the two boundary
components of $a$. From this one sees that cases (a)-(c) in Proposition 7.2
correspond to the following situations in $M$:

  1. (a)

$S$ passes through a blown annulus that lies in the interior of $M$.

  2. (b)

A boundary component of $S$ lies interior to a blown annulus $a\subset\partial
M$.

  3. (c)

A boundary component of $S$ meets each blown annulus that lies on the boundary
component of $M$ corresponding to the given component of
$\partial\mathcal{O}$.

For the final case, note that the two quadrants in (d) of Proposition 7.2
share a single periodic half-leaf of $\mathcal{O}^{u}$ or $\mathcal{O}^{s}$,
corresponding to a half-leaf $\widetilde{\lambda}$ of $\widetilde{W}^{u}$ or
$\widetilde{W}^{s}$. By analyzing the intersection of $\langle
g\rangle\cdot\widetilde{S}$ with $\widetilde{\lambda}$, where $\langle
g\rangle\leq\pi_{1}(M)$ is the stabilizer of $\widetilde{\lambda}$, one can
show without too much work that $S$ intersects the projection $\lambda$ of
$\widetilde{\lambda}$ to $M$ in a circle. (This also follows from the proof of
Proposition 8.7). Thus we have:

  1. (d)

There is a periodic half-leaf $\lambda$ of $W^{u}$ or $W^{s}$ such that
$S\cap\lambda$ contains a circle.

The corresponding pictures are shown in the lower chart of Figure 20.

###  7.3. Shadows of transverse surfaces: details

Note that $\widetilde{S}$ separates $\widetilde{M}$. If $B$ is a subset of
$\mathcal{O}$, we say $B$ lies above or lies below $\widetilde{S}$ if
$\Theta^{-1}(B)$ lies entirely above or below $\widetilde{S}$ in
$\widetilde{M}$, respectively (with respect to the coorientation of
$\widetilde{S}$).

The following is a basic observation about asynchronously proximal orbits.

######  Lemma 7.3.

Let $S$ be a compact, connected surface in an irreducible 3-manifold $M$ which
is transverse to a flow $\varphi$, and let $\widetilde{S}$ be a lift of $S$ to
$\widetilde{M}$. Suppose that the orbits passing through $a,b\in\widetilde{M}$
are asynchronously proximal. Then for all $t\gg 0$,
$\widetilde{\varphi}_{t}(a)$ and $\widetilde{\varphi}_{t}(b)$ lie on the same
side of $\widetilde{S}$. The same is true for all $t\ll 0$ if $a$ and $b$ are
backward asynchronously proximal.

###### Proof.

We shall prove the forward statement; the backward one is similar. Let $N$ be
an $\varepsilon$-tubular neighborhood of $S$ for some $\varepsilon>0$ such
that $N$ can be identified with $S\times I$ where the $I$-fibers are orbit
segments of $\varphi$. Let $\widetilde{N}$ be the lift of $N$ containing
$\widetilde{S}$. If $\widetilde{\varphi}_{t}(a)$ and
$\widetilde{\varphi}_{t}(b)$ lie on different sides of $\widetilde{S}$ for
large $t$, then they will be at least $2\varepsilon$ apart for large $t$,
hence are not asynchronously proximal. ∎

Let $\ell$ be a periodic half-leaf based at the periodic point
$x\in\mathcal{O}$. We say that $\ell$ is locally expanding at $x$ if the local
first return map to a transverse disk in $M$ through a point along the closed
orbit corresponding to $x$ expands along the periodic leaf corresponding to
$\ell$. Note that any blown segment is locally expanding at one of its
endpoints and locally contracting (similarly defined) at its other. We also
observe that a periodic half-leaf which is not a blown segment is locally
expanding/contracting depending on whether it belongs to $\mathcal{O}^{u}$ or
$\mathcal{O}^{s}$, respectively, and that the half-leaves terminating at a
periodic point alternate between locally expanding and locally contracting in
the circular order at that point. Each periodic half-leaf has at least one
endpoint and we endow it with a unique orientation defined by requiring half-
leaves to be oriented toward locally contracting endpoints and away from
locally expanding endpoints.

######  Lemma 7.4.

Let $\psi$ be an almost pseudo-Anosov flow on a compact manifold $M$ and let
$S$ be a properly embedded compact surface transverse to $\psi$. Let
$\widetilde{S}$ be a lift of $S$ to $\widetilde{M}$ and let
$\Omega=\Theta(\widetilde{S})$ be its image in the flow space. Let $s$ be a
blown segment that intersects $\Omega$.

If $s$ is not a subset of $\partial\mathcal{O}$, then:

  1. (a)

$\operatorname{int}(s)\subset\Omega$ and $\partial
s\subset\operatorname{fr}(\Omega)$.

If $s\subset\partial\mathcal{O}$ then either:

  1. (b)

$\operatorname{int}(s)\subset\Omega$ and $\partial
s\subset\operatorname{fr}(\Omega)$, or

  2. (c)

the component of $\partial\mathcal{O}$ containing $s$ lies entirely in
$\Omega$.

In cases (a) and (b), the locally expanding and contracting endpoints of $s$
lie under and over $\widetilde{S}$, respectively.

###### Proof.

Let $A$ be a blown annulus. If $S\cap A$ is a family of circles, then each
circle must be cooriented toward the attracting boundary component of
$\partial A$. Choosing a lift $\widetilde{A}$ of $A$ that intersects
$\widetilde{S}$, we see that $\widetilde{S}\cap\widetilde{A}$ is a line
intersecting every orbit in $\widetilde{A}$, cooriented toward the attracting
side of $\widetilde{A}$. Letting $s=\Theta(\widetilde{A})$, we see that
$\operatorname{int}(s)\subset\Omega$ and $\partial
s\subset\operatorname{fr}(\Omega)$, with the locally expanding and locally
contracting endpoints of $s$ lying below and above $\widetilde{S}$,
respectively. This describes situations (a) and (b). Now suppose that $A$ lies
in a component $T$ of $\partial M$ and $S\cap A$ is a family of intervals.
Then each component of $S\cap T$ intersects each blown annulus in $T$ in a
family of intervals. Lifting to $A$ to $\widetilde{A}\subset\widetilde{M}$, we
see the entire component of $\partial\mathcal{O}$ containing $\widetilde{A}$
is contained in $\Omega$. This is case (c).

Since $S$ has the clean annulus property there are no other types of
nontrivial intersections between $S$ and blown annuli, so the proof is
complete. ∎

######  Lemma 7.5 (Local frontier).

Let $x\in\operatorname{fr}(\Omega)$ lie below $\widetilde{S}$. If
$x\notin\partial\mathcal{O}$, then there is an open interval
$I\subset\mathcal{O}^{s}(x)$ such that $x\in I$ and
$I\subset\operatorname{fr}(\Omega)$. If $x\in\partial\mathcal{O}$, then the
same is true unless $x$ is a limit of points in
$\Omega\cap\partial\mathcal{O}$, in which case $I$ can be taken to be a half-
closed interval with boundary point $x$ such that
$I-\\{x\\}\subset\mathcal{O}\smallsetminus\partial\mathcal{O}$.

If $x$ lies above $\widetilde{S}$, the same is true after replacing below with
above and $\mathcal{O}^{s}$ with $\mathcal{O}^{u}$.

Figure 21. Left: the local picture around a point $x$ in
$\operatorname{fr}(\Omega)$ in the setting case 1 of the proof of Lemma 7.5.
Center and right: the local pictures around $x$ in cases 2 and 3 of the same
proof, respectively. In the righthand picture the vertical (in the page) line
through $x$ lies in $\partial\mathcal{O}$, while in the center and left
pictures it may or may not lie in $\partial\mathcal{O}$.

###### Proof.

We will assume that $x$ lies below $\widetilde{S}$; the proof in the other
case is symmetric.

Let $D$ be a disk in $\widetilde{M}$ transverse to $\widetilde{\psi}$
intersecting the lifted orbit $\gamma(x)=\Theta^{-1}(x)$ in a point. Because
$\gamma(x)$ is disjoint from $\widetilde{S}$, we can choose $D$ small enough
so that it is also disjoint from $\widetilde{S}$.

Let $N(x)$ be a neighborhood of $x$ contained in $\Theta(D)$ and small enough
so that it contains no singular points or endpoints of blown segments, except
possibly $x$.

Case 1: $x$ is not an endpoint of a blown segment that meets $\Omega$.

Let $x^{\prime}$ be any point in $N(x)\cap\mathcal{O}^{s}(x)$. Since $x$ and
$x^{\prime}$ are not separated by a singular point by the definition of
$N(x)$, $\gamma(x^{\prime})$ is asynchronously proximal to $\gamma(x)$ (Lemma
3.2). Since $\gamma(x)$ lies below $\Omega$, Lemma 7.3 implies that
$\gamma(x^{\prime})$ lies below $\Omega$ so $x^{\prime}\notin\Omega$. This
shows that $N(x)\cap\mathcal{O}^{s}(x)$ is disjoint from $\Omega$ and lies
below $\widetilde{S}$.

Now take a sequence of points $(x_{n})$ in $N(x)\cap\Omega$ converging to $x$
such that each $x_{i}$ lies in a single complementary component $Q$ of
$\mathcal{O}^{s}(x)$. Let $d_{i}$ be the intersection of $\gamma(x_{i})$ with
$D$. Since $D$ lies entirely below $\widetilde{S}$ and all the points in
$W^{s}(d_{i})\cap D$ are asynchronously proximal, Lemma 7.3 gives that all
points in $W^{s}(d_{i})\cap D$ intersect $\widetilde{S}$ in forward time.
Hence $\mathcal{O}^{s}(x_{i})\cap N(x)\subset\Omega$. Letting $i\to\infty$
these segments limit on $\partial Q\cap N(x)$, so $\partial Q\cap
N(x)\subset\operatorname{cl}(\Omega)$. Since $\partial
Q\subset\mathcal{O}^{s}(x)$ , $\partial Q\cap
N(x)\subset\operatorname{fr}(\Omega)$ and $\partial Q\cap N(x)$ lies below
$\widetilde{S}$. The local picture is shown on the left of Figure 21.

Case 2: $x$ is an endpoint of a blown segment $s$ meeting $\Omega$, and
$\operatorname{int}s\cap\partial\mathcal{O}=\varnothing$.

By Lemma 7.4, $\operatorname{int}(s)\subset\Omega$ and $s$ is locally
expanding at $x$ since $x$ lies below $\widetilde{S}$.

Let $Q_{1}$ and $Q_{2}$ be the two complementary components of
$\mathcal{O}^{s}(x)$ which meet along $s$. Let $\ell_{1}$ and $\ell_{2}$ be
the two periodic locally contracting half-leaves at $x$ adjacent to $s$,
indexed so that $\ell_{i}\subset\partial Q_{i}$. As in case 1, Lemma 7.3
implies that $\Omega$ is disjoint from both $\ell_{1}\cap N(x)$ and
$\ell_{2}\cap N(x)$, and that $\gamma(x^{\prime})$ lies below $\widetilde{S}$
for all $x^{\prime}\in\ell_{1}\cup\ell_{2}$.

Since $\operatorname{int}(s)\subset\Omega$ and $\Omega$ is open we can choose
a sequence of points $(x_{n})$ in $\Omega$ converging to $x$ so that each
$x_{i}$ lies in, say, $Q_{1}\cap N(x)$. As in case 1,
$\mathcal{O}^{s}(x_{i})\cap N(x)\subset\Omega$; these segments accumulate on
$(\ell_{1}\cup s)\cap N(x)$ so $\ell_{1}\cap N(x)\subset\operatorname{fr}(S)$.
We can argue the same way in $Q_{2}$ to complete the proof of this case. The
local picture is as shown in the center of Figure 21.

Case 3: $x$ is in the boundary of a blown segment
$s\subset\partial\mathcal{O}$ intersecting $\Omega$.

The argument for this case is the same as that of case 2, except now $s$ is
adjacent to only a single complementary component of $\mathcal{O}^{s}(p)$
since it lies in $\partial\mathcal{O}$. This produces a half-closed interval
$I$ in $\mathcal{O}^{s}(x)\cap\operatorname{fr}(\Omega)$, with boundary point
$x$. Every point in $I-\\{x\\}$ must lie in $\operatorname{int}(\mathcal{O})$
because $x$ is a 3-pronged singularity and $s\subset\partial\mathcal{O}$. The
local picture is shown on the right of Figure 21. ∎

The next lemma turns the previous local statement into a global one.

######  Lemma 7.6.

Let $x\in\operatorname{fr}(\Omega)$. If $x$ lies below $\widetilde{S}$, then
$x$ is contained in a unique slice leaf or boundary slice ray contained in
$\operatorname{fr}(\Omega)\cap\mathcal{O}^{s}(x)$, which lies below
$\widetilde{S}$. The symmetric statement is true if $x$ lies above
$\widetilde{S}$, with $\mathcal{O}^{u}$ replacing $\mathcal{O}^{s}$.

###### Proof.

Let $\ell$ be the component of
$\mathcal{O}^{s}(x)\smallsetminus(\Omega\cap\partial\mathcal{O})$ containing
$x$ and let $\lambda$ be the component of $\ell\cap\operatorname{fr}(\Omega)$
that contains $x$. We will show that $\lambda$ lies below $\widetilde{S}$ and
is a slice leaf if $\ell=\mathcal{O}^{s}(x)$ and a boundary slice ray
otherwise.

Observe that $\ell$ is a closed, connected subspace of $\mathcal{O}^{s}(x)$
and hence a properly embedded tree in $\mathcal{O}$ with at most one degree
$1$ vertex on $\partial\mathcal{O}$ (since a leaf meets at most one component
of $\partial\mathcal{O}$). By Lemma 7.5, $\lambda$ contains an interval around
each of its points; this interval is open for points in
$\operatorname{int}(\mathcal{O})$ and half open otherwise. We also have that
$\lambda$ is closed as a subspace of $\ell$ since $\operatorname{fr}(\Omega)$
is closed. Hence, $\lambda$ is a properly embedded subtree of
$\mathcal{O}^{s}(x)$ with at most one degree $1$ vertex on
$\partial\mathcal{O}$ (exactly in the case when
$\ell\subsetneq\mathcal{O}^{s}(x)$). Moreover, $\lambda$ lies below
$\widetilde{S}$: it is connected and disjoint from $\Omega$, contains $x$
which lies below $\widetilde{S}$, and points lying below $\widetilde{S}$
cannot be accumulated by points lying above $\widetilde{S}$ because “not lying
above $\widetilde{S}$” is an open condition.

It remains to show that $\lambda$ is a line or half line. Otherwise, it has a
vertex $v$ of degree $n\geq 3$, but only one of the complementary regions of
$\lambda$ incident to $v$ contains $\Omega$, contradicting the fact that
$\lambda\subset\operatorname{fr}(\Omega)$. ∎

The following proposition establishes Proposition 7.1 and more.

######  Proposition 7.7.

Let $\ell$ be a component of $\operatorname{fr}(\Omega)$. Then the following
hold:

  1. (a)

$\ell$ is a slice leaf or boundary slice ray of $\mathcal{O}^{s}$ or
$\mathcal{O}^{u}$, and $\ell$ is stable (unstable) if and only if $\ell$ lies
below (above) $\widetilde{S}$.

  2. (b)

$\ell$ has a neighborhood disjoint from all other components of
$\operatorname{fr}(\Omega)$.

  3. (c)

if $\ell$ contains periodic points, then there is a unique one $p$ with the
property that there is a half-leaf $\lambda$ terminating at $p$ that
intersects $\Omega$. Moreover, $\lambda$ is the unique such half-leaf, and
$\lambda$ is locally expanding or contracting at $p$ depending on whether
$\ell$ lies below or above $\widetilde{S}$, respectively.

###### Proof.

Let $x\in\operatorname{fr}(\Omega)$ and suppose without loss of generality
that $x$ lies below $\widetilde{S}$. Using Lemma 7.6 we can find a boundary
slice leaf in $\mathcal{O}^{s}(x)\cap\operatorname{fr}(\Omega)$, or a slice
ray in $\mathcal{O}^{s}(x)\cap\operatorname{fr}(\Omega)$ terminating in a
point accumulated by points in $\partial\mathcal{O}\cap\Omega$.

Let $\lambda$ be this slice leaf or boundary slice ray. Lemma 7.6 also tells
us that $\lambda$ lies below $\widetilde{S}$. Note that $\Omega$ is connected,
so it only accumulates on $\lambda$ from one side of $\lambda$. This implies
that $\lambda$ is a maximal connected subset of $\mathcal{O}^{s}(x)$ contained
in $\operatorname{fr}(\Omega)$.

Let $\ell_{1}$ and $\ell_{2}$ be two slice leaves or boundary slice rays in
$\operatorname{fr}(\Omega)$. We claim $\ell_{1}\cap\ell_{2}=\varnothing$. If
$\ell_{1}\cap\ell_{2}$ is nonempty, it could consist of a single point or some
union of half-leaves. In either case, $\ell_{1}\cup\ell_{2}$ separates
$\mathcal{O}$ into a number of components, none of which contains all of
$\ell_{1}\cup\ell_{2}$ in its frontier. Since $\Omega$ lies in a single
component of $\ell_{1}\cup\ell_{2}$, this is a contradiction.

It follows that $\operatorname{fr}(\Omega)$ is a disjoint union of slice
leaves or boundary slice rays. Let $C$ be a component of
$\operatorname{fr}(\Omega)$. If $C$ consists of more than one slice leaves or
boundary slice rays then it would have nonempty interior, a contradiction. We
conclude that $C$ is a single slice leaf or boundary slice ray. If the
component contains any point lying under $\widetilde{S}$, we have seen that it
is stable and lies entirely under $\widetilde{S}$. Conversely suppose there is
a stable component $\ell^{s}$ containing a point $p$ lying over
$\widetilde{S}$. Then by Lemma 7.6 there is an unstable slice leaf $\ell^{u}$
passing through $x$ contained in $\operatorname{fr}(\Omega)$, contradicting
that $\operatorname{fr}(\Omega)$ is a disjoint union of slice leaves or
boundary slice rays. This proves (a).

As noted above, $\Omega$ must lie to one side of each slice or slice ray in
$\operatorname{fr}(\Omega)$, and we know now these are precisely the
components of $\operatorname{fr}(\Omega)$. This implies that boundary
components do not accumulate on each other, proving (b).

Turning our attention to (c), suppose that $p$ is a periodic point in $\ell$
lying below $\Omega$. If $p$ is the boundary point of blown segment
intersecting $\Omega$, we have already seen in the proof of Lemma 7.5 (cases 2
and 3) that this blown segment is locally expanding at $p$ and is the only
half-leaf terminating at $p$ that intersects $\Omega$. Otherwise we are in
case 1 of the proof of Lemma 7.5, and there is an open interval
$I\subset\operatorname{fr}(\Omega)$ containing $x$ that is the limit of
segments of leaves of $\mathcal{O}^{s}$, accumulating on $I$ from the $\Omega$
side. In this situation there can be at most one half-leaf terminating at $p$
lying on the $\Omega$ side of $I$. If there is such a half-leaf it is
transverse to $\mathcal{O}^{s}$, and hence is an unstable half-leaf which is
not a blown segment. In particular it is locally expanding at $p$.

Next we prove there is at most one periodic point $p$ in $\ell$ which is an
endpoint of a periodic half-leaf intersecting $\Omega$. If $p\in\ell$ is a
periodic point incident to two half-leaves in $\ell$ which are locally
contracting at $p$, then there must be at least one (and hence only one by
above) locally expanding half-leaf emanating from $p$ into $\Omega$. This is
because the half-leaves meeting $p$ alternate between locally expanding and
contracting. Call such a point $p$ a _sink_ of $\ell$. If $\ell$ is a slice
leaf then the two noncompact half-leaves in $\ell$ are stable, and hence
oriented toward their endpoints. This forces the existence of at least one
sink in $\ell$. If $\ell$ is a boundary slice ray containing no sinks, then
all its half-leaves must be consistently oriented, so the half-leaf of $\ell$
meeting $\partial\mathcal{O}$ is locally contracting at its endpoint $q$.
There is then a blown segment in $\partial\mathcal{O}\cap\Omega$ meeting $q$
which is locally expanding at $q$.

Now suppose that $\ell$ contains at least two periodic points which are
endpoints of periodic half-leaves intersecting $\Omega$. Choose two such
points $p$ and $p^{\prime}$ which are adjacent along $\ell$ and are the
endpoints of two such half-leaves, $r$ and $r^{\prime}$ respectively. By
above, $r$ and $r^{\prime}$ are both locally expanding. This means that the
union of $r$, $r^{\prime}$, and the segment of $\ell$ between $p$ and
$p^{\prime}$ is not consistently oriented. Since this is part of the boundary
of a quadrant, we have a contradiction (see Figure 8 and related discussion in
Section 4.2). This proves (c) when $\ell$ lies below $\widetilde{S}$, and the
other case is symmetric. ∎

###### Proof of Proposition 7.2.

Suppose $T$ is a blowup tree containing a blown segment $s$ that intersects
$\Omega$. By Lemma 7.4, it is either the case that (1)
$\operatorname{int}(s)\subset\Omega$ and $\partial
s\subset\operatorname{fr}(\Omega)$, or (2) $s$ is contained in a component
$\ell$ of $\partial\mathcal{O}$ that lies entirely in $\Omega$. In case (1),
Proposition 7.7 implies that $T\cap\Omega$ equals $\operatorname{int}(s)$. In
case (2) we need only check that there are no blown segments in $\Omega$
incident to $\ell$, but this is true since $S$ has the clean annulus property.
Hence $T\cap\Omega=\ell$, completing the analysis when $T$ intersects
$\Omega$.

To finish the proof, suppose $T$ is disjoint from $\Omega$ and suppose $p\in
T\cap\operatorname{fr}(\Omega)$. Suppose without loss of generality that $p$
lies below $\widetilde{S}$. By Proposition 7.7, $p$ is contained in $\ell$
where $\ell$ is a slice leaf or boundary slice ray of $\mathcal{O}^{s}$, and
there is a single periodic point in $\ell$ from which a half-leaf $h$
(necessarily locally expanding) emanates into $\Omega$. Since $T$ and $\Omega$
are disjoint, $h$ is not a blown segment. Hence
$T\cap\operatorname{fr}(\Omega)$ consists precisely of the blown segments
contained in $\ell$. Since the two quadrants of $T$ intersecting $\Omega$ meet
along $h$, which is not a blown segment, they are adjacent. ∎

The following corollary will be used in the next section. A _regular open set_
is one that is the interior of its closure.

######  Corollary 7.8.

$\Omega$ is regular.

###### Proof.

First note that
$\operatorname{fr}(\Omega)=\operatorname{fr}(\operatorname{cl}(\Omega))$.
Indeed, the fact that each boundary component bounds $\Omega$ to only one side
gives that
$\operatorname{fr}(\Omega)\subset\operatorname{fr}(\operatorname{cl}(\Omega))$,
and the reverse inclusion follows from the openness of $\Omega$. Now
$\Omega=\operatorname{int}(\Omega)=\operatorname{cl}(\Omega)-\operatorname{fr}(\Omega)=\operatorname{cl}(\Omega)-\operatorname{fr}(\operatorname{cl}(\Omega))=\operatorname{int}(\operatorname{cl}(\Omega))$.
∎

##  8\. Uniqueness of almost transverse position

The following results, which are the goals of this section, comprise our
uniqueness statement from D.

######  Theorem 8.1.

Let $\varphi$ be a pseudo-Anosov flow on $M$ and let $S_{1}$ and $S_{2}$ be
isotopic properly embedded surfaces which are minimally transverse to
generalized dynamic blow ups $\varphi_{1}^{\sharp}$ and
$\varphi_{2}^{\sharp}$, respectively. Then $\varphi_{1}^{\sharp}$ and
$\varphi_{2}^{\sharp}$ are combinatorially equivalent.

Recall the minimally transverse condition means each blown annulus of
$\varphi_{i}^{\sharp}$ not contained in $\partial M$ is intersected by $S_{i}$
in a positive number of core curves.

######  Theorem 8.2.

Let $S_{1}$ and $S_{2}$ be two isotopic properly embedded surfaces minimally
transverse to an almost pseudo-Anosov flow $\varphi^{\sharp}$. Then any two
compatible lifts of isotopic components of $S_{1}$ and $S_{2}$ to
$\widetilde{M}$ have the same projection to the orbit space. Further, $S_{1}$
and $S_{2}$ are isotopic along flowlines.

The main tool for proving Theorem 8.1 and Theorem 8.2 is the structure theory
of shadows described in Section 7.

###  8.1. Shadows of isotopic surfaces are unique

To begin, we need to recall and extend the notation for flow spaces from
Section 3.5. We let $\mathcal{O}$ denote the flow space of the original flow
$\varphi$, lifted to the universal cover $\widetilde{M}$, and let
$\Theta:\widetilde{M}\to\mathcal{O}$ denote the quotient map.

In the case where $M$ has boundary, $\mathcal{O}$ will have lines that are the
images of the planar components of $\partial\widetilde{M}$. We let
$\mathcal{O}^{*}$ denote the image of $\mathcal{O}$ under the quotient map $p$
that collapses each blown annulus to a point, which simply collapses all
boundary lines The resulting space is not locally compact at the images of the
lines. If $M$ has no boundary, then $\mathcal{O}^{*}=\mathcal{O}$.

Suppose, as in the statement of Theorem 8.1, that $S_{1},S_{2}\subset M$ are
isotopic surfaces which are respectively minimally transverse to (generalized)
dynamic blowups $\varphi_{1}^{\sharp}$ and $\varphi_{2}^{\sharp}$ of
$\varphi$. We denote by $\mathcal{O}_{i}$ the flow space of
$\varphi_{i}^{\sharp}$, and by $p_{i}\colon\mathcal{O}_{i}\to\mathcal{O}^{*}$
the map collapsing all blown segments to points (note we are identifying the
result of this collapsing with $\mathcal{O}^{*}$ in the obvious way).

Since $\varphi_{1}^{\sharp}$ and $\varphi_{2}^{\sharp}$ are dynamic blowups of
$\varphi$, there are homeomorphisms

$g_{i}\colon M\smallsetminus(\text{blowup locus of }\varphi_{i}^{\sharp})\to
M\smallsetminus(\text{blowup locus of }\varphi)$

which are orbit equivalences from the restriction of $\varphi^{\sharp}_{i}$ to
that of $\varphi$, and homotopic to the identity through maps
$M\smallsetminus(\text{blowup locus of }\varphi_{i}^{\sharp})\to M$. The
following diagram describes some of the spaces and maps at play, but need not
commute, because $\varphi_{1}^{\sharp}$ and $\varphi_{2}^{\sharp}$ are
distinct flows.

(8.3)
${\widetilde{M}}$${\mathcal{O}_{1}}$${\mathcal{O}}$${\mathcal{O}_{2}}$${\mathcal{O}^{*}}$$\scriptstyle{\Theta_{1}}$$\scriptstyle{\Theta}$$\scriptstyle{\Theta_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{p}$$\scriptstyle{p_{2}}$

Now, we choose compatible lifts $\widetilde{S}_{1}$ and $\widetilde{S}_{2}$ of
components of $S_{1}$ and $S_{2}$. For brevity we denote
$\Theta_{i}(\widetilde{S}_{i})$ by $\Omega_{i}$. Let $\mathring{\mathcal{O}}$,
$\mathring{\mathcal{O}}_{i}$, $\mathring{\mathcal{O}}^{*}$ respectively denote
the spaces $\mathcal{O}$, $\mathcal{O}_{i}$, $\mathcal{O}^{*}$ minus all blown
segments and/or singular points (these are all canonically homeomorphic via
the maps in Equation 8.3). Applied to a set or map, the hovering circle
notation denotes restriction to the corresponding set. Here are some salient
properties of these objects:

  * •

Each
$\mathring{p}_{i}\colon\mathring{\mathcal{O}}_{i}\to\mathring{\mathcal{O}}^{*}$
is a homeomorphism, so it respects interiors and closures in its domain and
range.

  * •

Regularity of open sets (recall this means
$A=\operatorname{int}(\operatorname{cl}(A))$) is preserved upon passage to
open subspaces. Each $\Omega_{i}$ is regular in $\mathcal{O}_{i}$ by Corollary
7.8, so $\mathring{\Omega}_{i}$ is regular in $\mathring{\mathcal{O}}_{i}$.

######  Lemma 8.4.

With the above notation, $p_{1}(\Omega_{1})=p_{2}(\Omega_{2})$.

###### Proof.

We first consider the sets of regular periodic points $P_{1}$ in $\Omega_{1}$
and $P_{2}$ in $\Omega_{2}$, that is periodic points which correspond to
regular orbits of the flows. We will show that $p_{1}(P_{1})=p_{2}(P_{2})$ in
$\mathcal{O}^{*}$.

Let $y_{1}\in P_{1}$ be a regular point. Its preimage
$\widetilde{\gamma}_{1}=\Theta_{1}^{-1}(y_{1})$ projects to a periodic
$\varphi^{\sharp}_{1}$-orbit $\gamma_{1}$ in $M$. Let
$\gamma_{2}=g_{2}^{-1}(g_{1}(\gamma_{1}))$. This is a periodic orbit of
$\varphi^{\sharp}_{2}$, which is homotopic to $\gamma_{1}$ because $g_{1}$ and
$g_{2}$ are homotopic to the identity.

We can lift a homotopy from $\gamma_{1}$ to $\gamma_{2}$ to produce
$\widetilde{\gamma}_{2}$, a lift of $\gamma_{2}$ to $\widetilde{M}$. We know
that $\widetilde{\gamma}_{1}$ passes from the negative side of
$\widetilde{S}_{1}$ to the positive side, intersecting it once. We claim that
$\widetilde{\gamma}_{2}$ intersects $\widetilde{S}_{1}$ finitely many times,
starting on the negative side and ending on the positive side. This follows
from the fact that, because $\gamma_{1}$ intersects $S_{1}$ positively, the
distance from $\widetilde{\gamma}_{1}$ to $\widetilde{S}_{1}$ is a proper
function, and on the other hand the homotopy from $\widetilde{\gamma}_{1}$ to
$\widetilde{\gamma}_{2}$ has bounded tracks.

Similarly, since the lifted homotopy between $\widetilde{S}_{1}$ and
$\widetilde{S}_{2}$ has bounded tracks, we then have that
$\widetilde{\gamma}_{2}$ also passes from the negative side of
$\widetilde{S}_{2}$ to the positive side (note there can be at most one point
of intersection). Thus $\Theta_{2}(\widetilde{\gamma}_{2})$ is in $P_{2}$.

Now $\Theta_{1}(\widetilde{\gamma}_{1})$ and
$\Theta_{2}(\widetilde{\gamma}_{2})$ will have the same image in
$\mathcal{O}^{*}$ by construction. Because the roles of $S_{1}$ and $S_{2}$
are symmetric, we conclude that $P_{1}$ and $P_{2}$ have the same images in
$\mathcal{O}^{*}$. Note that $P_{i}\subset\mathring{\mathcal{O}}_{i}$, and
$P_{i}$ is dense in $\mathring{\Omega}_{i}$. Using this, and the properties
itemized before the lemma statement, we have (where interiors and closures are
taken in the _punctured_ spaces $\mathring{\mathcal{O}}_{i}$ and
$\mathring{\mathcal{O}}^{*}$):

$\displaystyle\mathring{p}_{1}(\mathring{\Omega}_{1})$
$\displaystyle=\mathring{p}_{1}(\operatorname{int}(\operatorname{cl}(\mathring{\Omega}_{1})))=\mathring{p}_{1}(\operatorname{int}(\operatorname{cl}(P_{1})))=\operatorname{int}(\operatorname{cl}(\mathring{p}_{1}(P_{1})))$
$\displaystyle=\operatorname{int}(\operatorname{cl}(\mathring{p}_{2}(P_{2})))=\mathring{p}_{2}(\operatorname{int}(\operatorname{cl}(P_{2})))=\mathring{p}_{2}(\operatorname{int}(\operatorname{cl}(\mathring{\Omega}_{1})))$
$\displaystyle=\mathring{p}_{2}(\mathring{\Omega}_{2}).$

The following lemma implies that $p_{1}(\Omega_{1})=p_{2}(\Omega_{2})$,
completing the proof. ∎

######  Lemma 8.5.

The set $p_{i}(\Omega_{i})$ is uniquely determined by
$\mathring{p}_{i}(\mathring{\Omega}_{i})$ in the following sense:

Let $q$ be a singular point of $\mathcal{O}^{*}$ that is contained in the
frontier of $\mathring{p}_{i}(\mathring{\Omega}_{i})\subset\mathcal{O}^{*}$.
Then $q\in p_{i}(\Omega_{i})$ unless $\mathring{p}_{i}(\mathring{\Omega}_{i})$
meets exactly two adjacent quadrants of $q$.

###### Proof.

We have $q\in p_{i}(\Omega_{i})$ if and only if $\Omega_{i}$ intersects the
tree of blown segments $p_{i}^{-1}(q)$, so this follows immediately from the
local structure described by Proposition 7.2. The excluded case is column (d)
of Figure 20. ∎

###  8.2. Uniqueness of combinatorial type

We can now prove that the isotopy class of a surface almost transverse to a
pseudo-Anosov flow determines a unique combinatorial type of minimal dynamic
blowup transverse to the surface.

###### Proof of Theorem 8.1.

Let $A$ be a blown annulus complex of $\varphi_{1}^{\sharp}$, and let $U$ be a
standard neighborhood for $A$. Let $D$ be a meridional disk or annulus of $U$,
depending on whether $U$ is solid or hollow. Then there are singular stable
and unstable foliations of $D$; as described in Section 3.2, the combinatorial
type of $\varphi_{1}^{\sharp}$ is determined by which quadrants of this
bifoliation meet along blown segments (the intersections of blown annuli with
$D$), as $A$ varies over all blown annulus complexes.

Suppose $Q_{1}$ and $Q_{2}$ are two quadrants meeting along a blown segment
$s$ corresponding to the immersed annulus $A_{s}\subset A$. By minimality,
there is an arc of $S_{1}\cap D$ lying in $Q_{1}\cup Q_{2}$ and intersecting
$s$ in a point. This point corresponds to a closed curve $\alpha\subset
S_{1}\cap A_{s}$. Let $S_{1}^{\alpha}$ be the component of $S_{1}$ containing
$\alpha$. Choose intersecting lifts $\widetilde{S}_{1}^{\alpha}$,
$\widetilde{A}_{s}$ to $\widetilde{M}$. Then $p_{1}(\Omega_{1})$ meets two
nonadjacent quadrants of $p_{1}(\widetilde{A}_{s})$.

Conversely if $p_{1}(\Omega_{1})$ meets two nonadjacent quadrants of a
singular point corresponding to a lift of the complex $A$ to $\widetilde{M}$,
then $\Omega_{1}$ meets their preimage quadrants in $\mathcal{O}_{1}$, and
Section 7.2 explains that this corresponds to a curve of intersection between
$A$ and a blown annulus, meaning that the two corresponding quadrants in $D$
must meet along a blown annulus.

Reasoning in this way for all blown annulus complexes, we see the
combinatorial type of $\varphi_{1}^{\sharp}$ is uniquely determined by
$p_{1}(\Omega_{1})$. The same is true for $\varphi_{2}^{\sharp}$. Lemma 8.4
finishes the proof of combinatorial equivalence. ∎

As a consequence, whether or not a dynamic blowup is necessary to achieve
transversality can be read off from the data of a relative veering
triangulation and a relatively carried surface:

######  Corollary 8.6.

Let $\varphi$ be a transitive pseudo-Anosov flow on $M$ and let $\tau$ be any
compatible veering triangulation. Then, up to isotopy, $S$ is (honestly)
transverse to $\varphi$ if and only if $S$ is carried by $\tau$ without
ladderpole annuli.

###### Proof.

If $S$ is carried by $\tau$ without ladderpole annuli, then the fact that
$\tau$ is positive transverse to $\varphi$ and the picture within each tube
show that $S$ is indeed transverse to $\varphi$.

Let $S$ be transverse to $\varphi$ and suppose that $S$ is isotopic to a
surface $S_{1}$ that is efficiently carried by a veering triangulation with
ladderpole annuli. Then $S_{1}$ is minimally transverse to a nontrivial
dynamic blowup $\varphi_{1}^{\sharp}$ of $\varphi$, contradicting Theorem 8.1.
∎

###  8.3. Uniqueness of transverse position

###### Proof of Theorem 8.2.

Recall the setting: we have two isotopic surfaces $S_{1}$ and $S_{2}$
minimally transverse to a single almost pseudo-Anosov flow $\varphi^{\sharp}$.

We first wish to show that compatible lifts of any pair of isotopic components
of $S_{1}$ and $S_{2}$ to $\widetilde{M}$ have the same projection to the
orbit space $\mathcal{O}$ of $\varphi^{\sharp}$; for notational simplicity we
assume that $S_{1}$ and $S_{2}$ are connected. Let $\widetilde{S}_{1}$ and
$\widetilde{S}_{2}$ be compatible lifts to $\widetilde{M}$. By Lemma 8.4,
their images contain the same regular points and intersect exactly the same
quadrants at each blowup tree in $\mathcal{O}$. Hence they are equal.

Now we show that $S_{1}$ and $S_{2}$ are isotopic along flow lines. The above
implies that compatible lifts of components of $S_{1}$ and $S_{2}$ are
sections of the bundle of lifted flow lines over their common shadow in
$\mathcal{O}$. The difference of these sections is a
$\pi_{1}(S_{i})$–invariant function, and by adding $t\in[0,1]$ times this
function to one section, and projecting to $M$ we obtain a homotopy through
transverse surfaces between the components.

In fact, we can see that this homotopy is an isotopy as follows: The preimage
$q^{-1}(S_{i})$ (where $q\colon\widetilde{M}\to M$ is the covering map)
divides $\widetilde{M}$ into regions, and its coorientation induces an
orientation on the dual tree $T_{i}$ of this subdivision. Because $S_{1}$ and
$S_{2}$ are isotopic, there is an isomorphism between these oriented trees
that preserves the labeling of the edges by conjugates of
$\pi_{1}(S_{1})=\pi_{1}(S_{2})$ in $\pi_{1}(M)$. Now a flow line $\ell$ in
$\widetilde{M}$, because it is positively transverse to $S_{i}$, determines a
subset in $T_{i}$ which is either an oriented line, ray, or segment or a
single vertex. Because compatible lifts of $S_{1}$ and $S_{2}$ have the same
shadows, the two lines are the same (up to the isomorphism). In other words,
the intersection points $\ell\cap q^{-1}(S_{1})$ and $\ell\cap q^{-1}(S_{2})$
appear in the same order along $\ell$, as labeled by their corresponding
subgroups (or shadows). The linear interpolation given above must therefore
preserve the embeddedness of the intersection points, and hence is an isotopy.
∎

###  8.4. An application: boundary leaves are periodic

The goal of this subsection is to prove the following proposition. The case
where $M$ is closed and $\varphi$ is pseudo-Anosov was proven by Fenley
[Fen99b].

######  Proposition 8.7.

Let $S$ be a connected surface transverse to a transitive almost pseudo-Anosov
flow $\varphi$. Let $\widetilde{S}$ be a lift of $S$ to $\widetilde{M}$, and
let $\Omega$ be its projection to the flow space. Then any component of
$\operatorname{fr}(\Omega)$ is periodic.

For the almost pseudo-Anosov flow $\varphi$ on $M$, let $\kappa$ be a finite
collection of orbits that kills its perfect fits (Section 3.5.1), and let
$\tau$ be the associated veering triangulation on the fully-punctured manifold
$M\smallsetminus\kappa_{s}$ (Theorem 3.4). Let $U$ be a standard neighborhood
of $\kappa_{s}$, so that $\tau$ is a veering triangulation of $M$ relative to
the tube system $U$.

We begin with the following additional setup. By Theorem 5.1, since $S$ is
transverse to $\varphi$, we can isotope $S$ so that it is relatively carried
by $\tau$. Further, we may choose an _efficient_ carried position. In such a
position, whenever $S$ meets a component of $U$, it intersects the
corresponding component of $\kappa_{s}$. Note that since $S$ is still
transverse to $\varphi$, we have not changed its shadow $\Omega$ by Theorem
8.2.

Let $\widetilde{S}$ be a lift of $S$ to $\widetilde{M}$. The surface
$\widetilde{S}$ traverses various sectors of the preimage of $\tau_{U}$ in
$\widetilde{M}$, and each sector of this branched surface is contained in a
face of the preimage $\widetilde{\tau}$ of $\tau$ in $\widetilde{M}$. We say
such a face is _traversed_ by $\widetilde{S}$ if it contains a sector in the
preimage of $\tau_{U}$ traversed by $\widetilde{S}$. Recall the definitions of
$\mathring{\mathcal{O}}$ and $\mathring{\Omega}$ from Section 8.1 (the circle
notation denotes removal of all non-regular points).

######  Lemma 8.8.

The punctured image $\mathring{\Omega}=\mathring{\Theta}(\widetilde{S})$ in
$\mathring{\mathcal{O}}$ is ideally triangulated by the images of the faces of
$\widetilde{\tau}$ that are traversed by $\widetilde{S}$.

###### Proof.

We begin by noting that the faces of $\widetilde{\tau}$ traversed by
$\widetilde{S}$ triangulate their image in $\mathring{\mathcal{O}}$. To see
this, note that the union of these faces in $\widetilde{M}$ determine a
properly embedded surface $\widetilde{S}_{\triangle}$ in
$\widetilde{M}\smallsetminus\widetilde{\kappa}_{s}$ whose image in
$M\smallsetminus\kappa_{s}$ is a surface carried by $\tau$. It follows that
the projection $\mathring{\Theta}$ restricted to $\widetilde{S}_{\triangle}$
is a homeomorphism onto its image in $\mathring{\mathcal{O}}$. We denote this
image by $\mathring{\Omega}_{\triangle}$. For details, see [LMT23b, Lemma
9.4]) where it is also established that
$\operatorname{fr}(\mathring{\Omega}_{\triangle})$ is a disjoint union of
vertical/horizontal leaves in $\mathring{\mathcal{O}}$.

We now show that if a face $f$ of $\widetilde{\tau}$ is traversed by
$\widetilde{S}$ then its image $\widehat{f}$ in $\mathring{\mathcal{O}}$ is
contained in $\mathring{\Omega}$. If $\widehat{f}$ meets $\mathring{\Omega}$
but is not contained in $\mathring{\Omega}$, then it is crossed by a component
$\ell$ of $\operatorname{fr}(\widetilde{S})$. By Proposition 7.7, $\ell$ is
stable/unstable slice leaf or slice ray. Hence there is a neighborhood
$\widehat{V}$ of an end of $\widehat{f}$ (i.e. a deleted neighborhood of a
vertex) whose preimage has closure $\widetilde{V}$ in $\widetilde{M}$ that is
a closed neighborhood of a component of $\widetilde{\kappa}_{s}$ that does not
meet $\widetilde{S}$. But as noted above, efficient position implies that
$\widetilde{S}$ does not meet the corresponding component of $\widetilde{U}$,
contradicting that $f$ is traversed by $\widetilde{S}$. This gives
$\mathring{\Omega}_{\triangle}\subset\mathring{\Omega}$.

We finish by demonstrating the reverse inclusion. Suppose
$p\in\mathring{\Omega}$. The orbit $\mathring{\Theta}^{-1}(p)$ passes through
$\widetilde{S}$. If it does so outside of $\widetilde{U}$ then clearly
$p\in\mathring{\Omega}_{\triangle}$, so suppose $\gamma_{p}$ passes through
$\widetilde{S}$ inside a component of $\widetilde{U}$ corresponding to an
ideal vertex $v$ of $\mathring{\mathcal{O}}$. There are two adjacent quadrants
of $v$ such that $p$ lies in the interior of their union. Moreover, since
$\widetilde{S}$ intersects the corresponding component of $U$, it traverses a
face $\widetilde{f}$ of $\widetilde{\tau}$ whose projection $\widehat{f}$ lies
in the interior of the two quadrants. By definition, $\widehat{f}$ lies in
$\mathring{\Omega}_{\triangle}$. Since
$\operatorname{fr}(\mathring{\Omega}_{\triangle})$ is a union of vertical and
horizontal leaves of $\mathring{\mathcal{O}}$, we have
$p\in\mathring{\Omega}_{\triangle}$. ∎

We now turn to the proof of the proposition.

###### Proof of Proposition 8.7.

Continuing with the discussion above, let $\lambda$ be a component of
$\operatorname{fr}(\Omega)$. Since every singular leaf is periodic, we may
assume that $\lambda$ is a regular stable or unstable leaf and contains no
points of $\widehat{\kappa}$. Without loss of generality, we assume $\lambda$
is unstable.

We recall the notion of an _upward flip move_ on a surface relatively carried
by $\tau^{(2)}_{U}$. These moves were essentially first considered in [MT17]
but the version we use here is from [Lan22]. If $S$ is carried with positive
weights on the two bottom faces of a tetrahedron , there is an isotopy
sweeping the corresponding portion of $S$ across  into a neighborhood of the
two top faces of . See Figure 22.

Figure 22. A picture of a flip move.

Note that $S$ is not a cross section to $\varphi$, since in that case we would
have $\Omega=\mathcal{O}$. By [Lan23, Proposition 4.5], we may perform only
finitely many upward flips before arriving at a relatively carried position
for $S$ in which no upward flips are possible. We fix this position of $S$ and
consider the associated triangulation of $\mathring{\Omega}$ in
$\mathring{\mathcal{O}}$ given by Lemma 8.8. By the definition of $B^{s}$, the
train track $B^{s}\cap S$ has no large branches at this point. As a
consequence there is a finite collection of simple closed curves in $S$
carried by $B^{s}\cap S$ which we will call _stable loops_. We observe that
any infinite train route in $B^{s}\cap S$ must eventually circle around one of
these stable loops forever.

Let $t_{1}$ be an ideal triangle of the ideal triangulation of
$\mathring{\Omega}$ which is near enough to $\lambda$ so that there exists a
leaf $\ell^{s}$ of the stable foliation passing through the interior of
$t_{1}$ and also intersecting $\lambda$. If we orient $\ell^{s}$ from $t_{1}$
toward $\lambda$, it determines an infinite path of ideal triangles
$(t_{1},t_{2},t_{3},\dots)$ in $\mathring{\Omega}$. If $e_{i}$ is the edge
shared by $t_{i}$ and $t_{i+1}$, we see that the sequence $(e_{n})$ limits on
$\lambda$. Indeed, this follows from the proof of [LMT23b, Lemma 9.4]. A
picture of this situation is shown in Figure 23.

Figure 23. Notation from the proof of Proposition 8.7.

Each $t_{i}$ has a “widest” edge, by which we mean an edge that intersects
every leaf of $\mathcal{O}^{s}$ that intersects $t_{i}$. The projection of
$\widetilde{S}\cap\widetilde{B}^{s}$ to $\mathcal{O}$ gives a train track in
$\Omega$ which is dual to our ideal triangulation, and endows each ideal
triangle in $t_{i}$ with a switch of a train track that points toward the
widest edge of $t_{i}$. In particular there is a train route from the widest
edge of $t_{i}$ to each of the other edges of $t_{i}$. Note that since
$\ell^{s}$ is stable, it passes through the widest edge of each $t_{i}$.
Therefore there is an infinite train route $\widehat{\rho}$ in the train track
$\widetilde{S}\cap\widetilde{B}^{s}$ passing through the triangles
$t_{1},t_{2},t_{3},\dots$ in that order. Let $\widetilde{\rho}$ be the lift of
$\widehat{\rho}$ to $\widetilde{S}\subset\widetilde{M}$, and let $\rho$ be the
projection of $\widetilde{\rho}$ to $S\subset M$. As observed above, the route
$\rho$ is eventually periodic, winding around a stable loop $\gamma\subset S$.
By truncating an initial subsequence of $(t_{n})$ and relabeling, we can
assume that $\rho$ is actually periodic and that $\widetilde{\rho}$ is
contained in a lift $\widetilde{\gamma}$ of $\gamma$. We orient $\gamma$ so
that the orientations of $\widetilde{\rho}$ and $\widetilde{\gamma}$ are
compatible.

Let $g$ be a deck transformation of $\widetilde{M}$ which corresponds to
$\gamma\in\pi_{1}(S)\leq\pi_{1}(M)$ and preserves $\widetilde{\gamma}$,
translating it in the positive direction. The action of $g$ on $\widehat{S}$
is cellular and in particular acts as translation on the (truncated) sequence
$(e_{n})$, sending each edge to one which is closer to $\lambda$. Since
$(e_{n})$ limits on $\lambda$ we conclude that $\lambda$ is preserved by the
action of $g$, hence is periodic. ∎

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