11institutetext: Tulane University
11email: {swang47, ychung3, zding1<EMAIL_ADDRESS>
# From Majority to Minority: A Diffusion-based Augmentation for
Underrepresented Groups in Skin Lesion Analysis

Janet Wang  Yunsung Chung  Zhengming Ding  Jihun Hamm

###### Abstract

AI-based diagnoses have demonstrated dermatologist-level performance in
classifying skin cancer. However, such systems are prone to under-performing
when tested on data from minority groups that lack sufficient representation
in the training sets. Although data collection and annotation offer the best
means for promoting minority groups, these processes are costly and time-
consuming. Prior works have suggested that data from majority groups may serve
as a valuable information source to supplement the training of diagnosis tools
for minority groups. In this work, we propose an effective diffusion-based
augmentation framework that maximizes the use of rich information from
majority groups to benefit minority groups. Using groups with different skin
types as a case study, our results show that the proposed framework can
generate synthetic images that improve diagnostic results for the minority
groups, even when there is little or no reference data from these target
groups. The practical value of our work is evident in medical imaging
analysis, where under-diagnosis persists as a problem for certain groups due
to insufficient representation.

###### Keywords:

Skin Lesion Analysis Diffusion Models Data Augmentation.

##  1 Introduction

Current AI-assisted diagnostic systems demonstrate expert-level capability in
classifying skin cancers, often identified visually. [6, 16, 2]. These systems
can potentially contribute to teledermatology as diagnostic and decision-
support tools, enhancing diagnostic accessibility in rural areas [3]. However,
despite such success, recent studies have highlighted their susceptibility to
under-diagnosing minority groups, such as those with underrepresented skin
types, hindering their ability to generalize across different demographic
groups [4, 11]. Although the majority group contains rich lesion information,
directly training models for cross-color classification using this data is
challenging due to the domain gap caused by varying skin types [24]. Prior
research has suggested using synthetic images generated from majority groups
to supplement the training for minority groups [19].

Augmenting skin condition data with synthetic images has been explored, owing
to its potential to address common challenges for skin lesion analysis, such
as data privacy, imbalance, and scarcity. Notably, Generative Adversarial
Networks (GANs) [10] and Diffusion Models (DMs) [5] have emerged as leading
techniques for generating high-quality skin lesion images. While GANs have
successfully produced photorealistic synthetic images, their generation is
uncontrollable [9, 18]. On the other hand, DMs pre-trained on extensive web
data have enabled higher controllability over image generation through the
guidance of textual prompts, allowing for the creation of diverse and high-
fidelity images of target skin conditions and types.

Existing studies have used diffusion models to augment skin disease datasets
and improve diagnosis for minority skin types using two public datasets:
Diverse Dermatology Images (DDI) [4] and Fitzpatrick17k [11]. Each image in
these datasets is annotated with skin type labels based on the Fitzpatrick
scoring system [7]. In their work, [21] generated multiple synthetic images
for each real image using Stable Diffusion [20] and then trained the
classifier on a dataset including real and synthetic data. They find that
diffusion models can enhance classifier performance across skin types in
binary malignancy classification on the DDI dataset, though the number of real
images remains a key driver in performance. Additionally, [22] sampled a small
number of seed images with skin type at the end of the Fitzpatrick spectrum
(FST I-II and FST V-VI) and carefully cropped the disease pathology, before
generating synthetic data from the seed data with OpenAI DALL·E 2’s inpainting
feature. They conducted class-wise data augmentation by incorporating
synthetic images of the target condition and minority skin type into the real
training set. Other related studies have focused on internal datasets [1, 15].

Despite these advancements, the potential to leverage diffusion models’
knowledge about skin variation and the rich lesion information from majority
groups to benefit minority groups remains underexplored. In this work, we
propose a novel diffusion-based augmentation framework capable of learning
skin lesion concepts from majority groups and generating images to improve
classification performance for minority groups. Unlike current works that
assume the existence of data from minority groups, we hypothesize that the
information gained from majority groups and the diffusion model’s knowledge is
sufficient to generate useful synthetic data. We test our hypothesis in a
challenging multi-condition task. The framework is illustrated in Fig. 1. We
conduct our experiments on the Fitzpatrick17k dataset, which includes lesions
that are less familiar to diffusion models than common skin cancer. This
dataset has a skewed skin type distribution, with light skin types (FST I-II)
being significantly more common than dark skin types (FST V-VI), thus forming
majority and minority groups. Our investigation focuses on images from both
groups and is structured around three scenarios with increasing difficulty:
(i) the training source includes some data from both groups; (ii) there is
limited data from the minority group in the training source; and (iii) the
training source lacks data from the minority group. Our findings reveal:

  * •

Our proposed method effectively leverages lesion information from the majority
group to generate synthetic images that can improve classification results for
the minority group across all settings, even without reference data from the
minority group.

  * •

Using synthetic images generated by our method to train classifiers
consistently outperforms training with only real images across various
architectures. Further improvement is observed when combining real and
synthetic data.

  * •

Our method is sensitive to information from the minority group. A notable
improvement can be observed when even a few examples from the minority group
are added to the training set.

Figure 1: Outline of the proposed augmentation framework. The framework pairs
each training image with a textual prompt describing the condition as an input
to train a latent diffusion model. Embeddings associated with new lesion
concepts are found through Textual Inversion. Compact matrices $A$ and $B$ are
optimized with LoRA to facilitate training with the new embeddings. During
inference, the trained model produces synthetic images from the training data
that mainly feature the majority groups via image-to-image generation, thus
conditioned on visual cues of lesions from images and textual prompts
describing the target condition and group attributes.

##  2 Methods

###  2.1 Latent Diffusion Models

FST |  | Basal Cell
---
Carcinoma
Folliculitis |  | Nematode
---
Infection
| Neutrophilic
---
Dermatoses
| Prurigo
---
Nodularis
Psoriasis |  | Squamous Cell
---
Carcinoma
Total
I | 85 | 30 | 15 | 70 | 7 | 113 | 100 | 420
II | 156 | 97 | 56 | 115 | 28 | 232 | 180 | 864
V | 24 | 31 | 32 | 31 | 29 | 64 | 40 | 251
VI | 7 | 9 | 12 | 15 | 9 | 21 | 23 | 96
Total | 272 | 167 | 115 | 231 | 73 | 430 | 343 | 1631

Table 1: Sample distribution across skin conditions by Fitzpatrick Skin Type.

We implement our method using Latent Diffusion Models (LDMs) [20], a class of
Denoising Diffusion Probabilistic Models (DDPMs) [13] that operate in the
latent space of an autoencoder, to enable DDPM training with limited
computational resources. LDMs include two core components: a pre-trained
autoencoder and a diffusion model. In our study, the encoder of the
autoencoder $\mathcal{E}$ encodes skin lesion images $x\in\mathcal{D}_{x}$
into a latent representation $z=\mathcal{E}(x)$, while the decoder $D$ maps
the latent representations back to images, such that
$D\left(\mathcal{E}(x)\right)\approx x$. The diffusion model is trained to
generate representations conditioned on prompts describing skin disease and
skin type, within the learned latent space. Let $c_{\theta}(y)$ be a model
that maps a conditioning input $y$ into a vector. We then learn the
conditional LDM via

$L_{LDM}\coloneqq\mathbb{E}_{z\in\mathcal{E}(x),~{}y,~{}\epsilon\in\mathcal{N}(0,1),~{}t}\Big{[}\|\epsilon-\epsilon_{\theta}(z_{t},t,c_{\theta}(y))\|_{2}^{2}\Big{]}\,,$

(1)

where $t$ is the time step, $z_{t}$ is the latent noise at time $t$,
$\epsilon$ is the unscaled noise sample, and $\epsilon_{\theta}$ is the
denoising network.

###  2.2 Concept Discovery via Textual Inversion

Our proposed framework leverages Textual Inversion [8] to capture a unique
embedding that accurately represents the targeted skin lesion concept from
training data. Skin lesion images paired with a string containing a
placeholder word (e.g., ‘An image of $\\{S_{*}\\}$’) are used to guide the
learning of new lesion embedding for the generative model. In particular, The
optimal embedding $v_{*}$ for the lesion concept $S_{*}$ is derived by
minimizing the reconstruction loss

$v_{*}=\mathrm{argmin}_{v}\mathbb{E}_{z\sim\mathcal{E}(x),y,\epsilon\sim\mathcal{N}(0,1),t}\Big{[}\|\epsilon-\epsilon_{\theta}(z_{t},t,c_{\theta}(y))\|_{2}^{2}\Big{]}\,.$

(2)

This objective encourages the embedded representation to encapsulate new
concepts specialized to skin disease datasets by re-using the same training
scheme as the original LDM model while keeping both $c_{\theta}$ and
$\epsilon_{\theta}$ fixed.

###  2.3 Fine-grained Detail Enhancement with LoRA

To enhance efficiency in fine-tuning LDM, we employ Low-Rank Adaptation (LoRA)
[14] in our framework, with the discovered tokens after textual inversion.
This fine-tuning strategy freezes the pre-trained model weights and introduces
two compact matrices $A$ and $B$, where $A\in\mathbb{R}^{n\times
r},B\in\mathbb{R}^{r\times n}$. The adaptation matrices $AB$ are integrated
into the attention layers to capture the fine visual details of the skin
lesion concept with target embedding $v_{*}$. The optimization is formulated
as

$L:=\mathbb{E}_{z\sim\mathcal{E}(x),y,\epsilon\sim\mathcal{N}(0,1),t}\Big{[}\|\epsilon-\epsilon_{\theta_{AB}}(z_{t},t,c_{\theta_{AB}}(v_{*}))\|_{2}^{2}\Big{]}\,.$

(3)

This step further strengthens the model to generate synthetic images that
reflect fine-grained features not initially present in the pre-trained model.

Architecture | Train | Accuracy | Precision | Recall | F1 Score
---|---|---|---|---|---
VGG-16 | real | 70.24 $\pm$ 0.12 | 72.49 $\pm$ 0.37 | 69.58 $\pm$ 0.30 | 70.48 $\pm$ 0.13
syn | 75.00 $\pm$ 0.64 | 75.77 $\pm$ 0.39 | 73.17 $\pm$ 0.29 | 72.42 $\pm$ 0.61
real + syn | 77.98 $\pm$ 0.40 | 81.51 $\pm$ 0.25 | 78.87 $\pm$ 0.23 | 77.45 $\pm$ 0.29
ResNet-18 | real | 68.45 $\pm$ 0,42 | 69.42 $\pm$ 0.57 | 69.05 $\pm$ 0.42 | 67.94 $\pm$ 0.35
syn | 69.05 $\pm$ 0.36 | 69.57 $\pm$ 0.82 | 69.05 $\pm$ 0.42 | 68.02 $\pm$ 0.33
real + syn | 71.36 $\pm$ 0.69 | 69.02 $\pm$ 0.48 | 68.45 $\pm$ 0.40 | 67.56 $\pm$ 0.59
ViT-B-16 | real | 70.38 $\pm$ 0.42 | 73.72 $\pm$ 0.69 | 70.82 $\pm$ 0.39 | 70.61 $\pm$ 0.59
syn | 74.19 $\pm$ 0.37 | 77.89 $\pm$ 1.01 | 74.04 $\pm$ 0.85 | 73.58 $\pm$ 0.53
real + syn | 78.65 $\pm$ 0.53 | 81.57 $\pm$ 0.47 | 79.17 $\pm$ 0.84 | 78.24 $\pm$ 0.64

Table 2: This table presents the results when the training set includes some
(291 non-seed) images from the minority group. The test set comprises 56 seed
images of the minority group (FST V-VI), uniformly distributed across the 7
conditions. Here, “real” indicates that the classifier is trained solely on
real images, while “syn” means that it is trained exclusively on synthetic
images generated by our framework. Accordingly, “real+syn” means that the
subsequent classifier is trained on a combination of both.

##  3 Experiments

We conduct our experiments using the Fitzpatrick17k dataset, where each image
is annotated with a condition and a Fitzpatrick Skin Type (FST) label. In line
with [22], our analysis narrows down to a subset of the Fitzpatrick17k
dataset, encompassing 7 conditions (Table 1). These conditions were selected
because they represent the largest sample sizes at the ends of the Fitzpatrick
Skin Type (FST I-II or V-VI) spectrum. Unlike [22], our study excludes
intermediate skin types (FST III-IV), to explore the efficacy of our
diffusion-based augmentation in a more challenging and explainable way. For
each condition, we randomly sample 8 images from the lightest (FST I-II) and
darkest (FST V-VI) skin type groups, resulting in 56 seed images for each
group.

We independently examine three scenarios: (i) the training source includes
some images of both dark and light skin types, and the target features
uniformly distributed seed images (56 of one skin type) over the 7 conditions;
(ii) there are many light-skinned images and a few dark-skinned seed images in
the training set. In this setting, the target consists of 291 holdout images
of dark skin type; and (iii) the source lacks dark-skinned images entirely and
includes only light skin type. Likewise, the target comprises of 291 dark-
skinned images. In all the settings, we generate 5 synthetic images for each
real one in the source training set during inference, using the fine-tuned
model, as illustrated by Fig. 1.

In setting (i), to ensure a sufficient number of examples for both majority
and minority groups (lightest vs. darkest skin types), we designate the seed
images as the target and utilize all remaining images as sources for generator
and classifier training. Meanwhile, this setting also serves as the basis for
hyperparameter tuning of the diffusion model, with the chosen hyperparameters
being fixed for subsequent experiments. In setting (ii), we reversed the roles
of the seed and non-seed images, creating a training set composed of 56 dark-
skinned seed images with other light-skinned ones, and a test set of 291 dark-
skinned images. In the final setting, we removed the seed images from the
training set to evaluate whether our pipeline could still generate synthetic
images that enhance classification performance for the target minority groups,
even without any visual references from them. Note that the same set of images
is used for training the diffusion model and classifier in all settings.

Implementation Details In each setting, we sampled 5 groups of seed images and
repeated the experiment 5 times. We used the Stable Diffusion 2.1 base [20]
and the Diffusers library [17] for fine-tuning the diffusion model and
generating synthetic images. We used pre-trained VGG-16 [23], ResNet-18 [12],
and ViT-B-16 [25] as classifier backbones and trained each classifier for 30
epochs using the Adam optimizer with an initial learning rate of 1e-3. A
weight-based sampler and StepLR scheduler were applied. Experiments were
conducted on two NVIDIA GeForce RTX 3090s.

Architecture | Train | Accuracy | Precision | Recall | F1 Score
---|---|---|---|---|---
VGG-16 | real | 58.79 $\pm$ 0.10 | 58.90 $\pm$ 0.03 | 58.26 $\pm$ 0.05 | 56.98 $\pm$ 0.04
syn | 62.86 $\pm$ 0.15 | 61.49 $\pm$ 0.15 | 63.32 $\pm$ 0.13 | 61.57 $\pm$ 0.24
real + syn | 63.66 $\pm$ 0.11 | 62.80 $\pm$ 0.12 | 64.08 $\pm$ 0.08 | 62.72 $\pm$ 0.18
ResNet-18 | real | 50.31 $\pm$ 0.30 | 50.45 $\pm$ 0.44 | 51.27 $\pm$ 0.30 | 48.23 $\pm$ 0.32
syn | 56.36 $\pm$ 0.16 | 56.58 $\pm$ 0.13 | 59.78 $\pm$ 0.08 | 55.58 $\pm$ 0.10
real + syn | 61.33 $\pm$ 0.17 | 59.75 $\pm$ 0,12 | 62.52 $\pm$ 0.17 | 59.94 $\pm$ 0.17
ViT-B-16 | real | 62.03 $\pm$ 0.19 | 63.09 $\pm$ 0.13 | 62.02 $\pm$ 0.04 | 61.05 $\pm$ 0.04
syn | 68.83 $\pm$ 0.19 | 70.07 $\pm$ 0.03 | 68.22 $\pm$ 0.25 | 68.34 $\pm$ 0.19
real + syn | 71.20 $\pm$ 0.07 | 71.61 $\pm$ 0.19 | 71.53 $\pm$ 0.01 | 71.17 $\pm$ 0.13

Table 3: Classification results when the training set contains few reference
images from the minority group (56). The test set has 291 non-seed images of
the minority group (FST V-VI).

##  4 Results

To assess the efficacy of our augmentation framework across the three
settings, we train the classifier on data that includes real images only,
synthetic images only, or a combination of both, respectively. Our evaluation
is based on four metrics: accuracy, precision, recall, and F1. First, in the
setting with some images from both majority and minority groups in the
training set, we observe that synthetic data enhances performance across all
architectures (Table 2). Specifically, classifiers trained on synthetic images
consistently outperform those trained solely on real ones, and the combination
of both types of data for training yields further improvements.

This trend of consistent improvement is also evident in the more challenging
scenarios where there are little or no reference images from the minority
groups (Tables 3 and 4) in the training set. Notably, significant improvement
is observed when just a few reference images from the minority group are
available in the training set for image generation and classification. The
transformer-based classifier demonstrates a larger improvement gap over the
real image baseline than the CNN-based models. In the most challenging
setting, with no reference images from the minority group, the improvement
margin narrowed, suggesting our pipeline effectively maximizes the use of
limited information from the seed images of the minority groups in the
training process. Despite these challenges, the sustained improvements in the
third setting validate our framework’s effectiveness in transferring
information across groups. Examples of real and synthetic image pairs for each
condition are presented in Fig. 2. Qualitatively, the synthetic images
generated by our augmentation framework introduce more diversity to the
training sets, including variations in skin color and lesion patterns.

Architecture | Train | Accuracy | Precision | Recall | F1 Score
---|---|---|---|---|---
VGG-16 | real | 55.58 $\pm$ 0.10 | 54.60 $\pm$ 0.11 | 51.84 $\pm$ 0.29 | 51.97 $\pm$ 0.45
syn | 57.62 $\pm$ 0.09 | 56.62 $\pm$ 0.15 | 55.42 $\pm$ 0.20 | 55.36 $\pm$ 0.22
real + syn | 58.08 $\pm$ 0.08 | 57.36 $\pm$ 0.13 | 55.78 $\pm$ 0.17 | 55.85 $\pm$ 0.23
ResNet-18 | real | 49.42 $\pm$ 0.36 | 49.79 $\pm$ 0.44 | 48.30 $\pm$ 0.23 | 47.42 $\pm$ 0.30
syn | 53.47 $\pm$ 0.32 | 52.39 $\pm$ 0.30 | 53.79 $\pm$ 0.30 | 51.97 $\pm$ 0.29
real + syn | 56.50 $\pm$ 0.15 | 55.72 $\pm$ 0.15 | 55.10 $\pm$ 0.16 | 54.76 $\pm$ 0.18
ViT-B-16 | real | 57.66 $\pm$ 0.35 | 61.89 $\pm$ 0.68 | 55.45 $\pm$ 0.36 | 55.96 $\pm$ 0.51
syn | 58.94 $\pm$ 0.23 | 62.79 $\pm$ 0.20 | 55.62 $\pm$ 0.22 | 56.53 $\pm$ 0.18
real + syn | 60.96 $\pm$ 0.31 | 64.57 $\pm$ 0.01 | 57.51 $\pm$ 0.29 | 58.92 $\pm$ 0.31

Table 4: Classification results when no image from the minority group is in
the training set. The test set has 291 non-seed images of the minority group
(FST V-VI).

To further investigate our framework’s generation capabilities, we conduct an
ablation study comparing our framework with various generation strategies.
This study focuses on the first setting, where the test set consists of light-
or dark-skinned seed images (56 for each type). We first examine the Stable
Diffusion’s vanilla text-to-image and image-to-image pipelines to generate
synthetic images. Next, we leverage Textual Inversion to learn the lesion
embeddings and generate synthetic images from these embeddings, with text-to-
image and image-to-image pipelines. Since image-to-image outperforms text-to-
image in both vanilla SD and Textual Inversion generation, we focus on image-
to-image generation after fine-tuning the diffusion model using LoRA, to
investigate if optimizing the diffusion model can benefit the generation even
more. We train a VGG-16 using only the synthetic images and then compare these
generation strategies with ours, as shown in Table 5.

Overall, training classifiers on synthetic images generated by text-to-image
models proves less effective than employing image-to-image techniques,
underscoring the importance of visual cues in augmenting skin lesion
classification. Additionally, using off-the-shelf models for image generation
yields less improvement than utilizing training strategies such as Textual
Inversion and LoRA, regardless of whether the target is a minority or majority
group. Finally, the combination of Textual Inversion and LoRA results in the
highest accuracy, thereby validating the practicality of our design which
integrates these two strategies. This improvement can be explained by the
model’s enhanced ability to associate the fine visual cues of the lesion with
the learned textual tokens.

Since a direct comparison with existing related works is challenging due to
the uncertain use of data, this ablation study can serve as an indirect
comparison. As introduced previously, related works either leveraged off-the-
shelf diffusion models such as DALL·E or fine-tuned a Stable Diffusion model
for text-to-image generation. Our results demonstrate that utilizing the dual
guidance of visual cues and text via fine-tuning diffusion models can maximize
the potential of diffusion-based augmentation to enhance the diagnosis for
minority groups.

Figure 2: Examples of real and synthetic image pairs. Synthetic images were
generated by a model trained on light-skinned images only, using prompts
conditioned on dark skin types.

| txt2img | img2img
---|---|---
2-7[1pt/1pt] target | vanilla | ti | vanilla | ti | lora | ti+lora
light (56) | 18.80 | 35.36 | 48.21 | 46.43 | 52.00 | 53.57
dark (56) | 21.22 | 44.64 | 69.64 | 71.43 | 73.21 | 79.57

Table 5: Classification accuracy with various generation strategies for the
first setting. Here, “vanilla” stands for Stable Diffusion’s original text-to-
image and image-to-image pipelines, “ti” for Textual Inversion, and “lora” for
LoRA.

##  5 Conclusion

In this work, we present an effective diffusion-based augmentation framework
that consistently improves classification results for the minority group, even
when training the classifier exclusively with synthetic images. This
improvement is observed regardless of the availability of reference data from
the minority group in the training set. The ablation study also validates that
our framework’s dual-guidance generation approach successfully learns novel
lesion concepts previously unfamiliar to the diffusion models. A practical
takeaway from this study is that, even in cases of extreme data scarcity,
existing data and diffusion models can still provide valuable insights,
maximizing information usage and achieving better performance. In the future,
we plan to apply this technique to other medical datasets characterized by
significant differences in group sizes. Additionally, as we used all synthetic
images generated for each setting without any filtering mechanism, we also aim
to investigate which types of synthetic data are useful for lesion diagnosis
and how to generate them.

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# Distribution of periodic orbits in the homology group of a knot complement

Solly Coles  Solly Coles, Department of Mathematics, Northwestern University,
Evanston, IL 60208, USA<EMAIL_ADDRESS>and  Richard Sharp
Richard Sharp, Mathematics Institute, University of Warwick, Coventry CV4 7AL,
UK<EMAIL_ADDRESS>
###### Abstract.

Consider a transitive Anosov flow on a closed $3$-manifold. After removing a
finite set of null-homologous periodic orbits, we study the distribution of
the remaining periodic orbits in the homology of the knot complement.

##  1\. Introduction

In a recent paper [19], McMullen discussed the way in which periodic orbits
$\mathscr{P}$ of the geodesic flow on the unit-tangent bundle of a compact
negatively curved surface (and more general Anosov flows on 3-manifolds)
behave like prime numbers, in the sense that they obey a _Chebotarev law_.
More precisely, if $M_{N}$ is the knot complement obtained by removing $N$
periodic orbits $\\{K_{1},\ldots,K_{N}\\}$ and $G$ is a finite quotient of
$\pi_{1}(M_{N})$ then each of the remaining periodic orbits
$\gamma\in\mathscr{P}_{N}:=\mathscr{P}\setminus\\{K_{1},\ldots,K_{N}\\}$
induces a well-defined conjugacy class $\langle\gamma\rangle_{G}$ in $G$ and
these conjugacy classes are equidistributed in the sense that, writing
$\ell(\gamma)$ for the least period of $\gamma$, for every conjugacy class
$C\subset G$ we have

$\lim_{T\to\infty}\frac{|\\{\gamma\in\mathscr{P}_{N}\hbox{ : }\ell(\gamma)\leq
T\,\langle\gamma\rangle_{G}=C\\}|}{|\\{\gamma\in\mathscr{P}_{N}\hbox{ :
}\ell(\gamma)\leq T\\}|}=\frac{|C|}{|G|},$

where $|A|$ denotes the cardinality of $A$. (See also Ueki [29] for results on
modular knots.) The aim of this paper is to provide precise asymptotic
information about how the periodic orbits in $\mathscr{P}_{N}$ link with the
removed orbits $\\{K_{1},\ldots,K_{N}\\}$. This may be viewed as a question
about how the orbits are distributed in the homology of the knot complement
$H_{1}(M_{N},\mathbb{Z})$. The distribution of periodic orbits in the first
homology group of $M$ is well understood ([1], [3], [12], [13], [15], [27])
and the above problem may be viewed as a generalisation of this theory. Many
of the ideas introduced by McMullen will be invaluable to our analysis.

We shall begin by recalling some classical distribution results for periodic
orbits. Let $M$ be a smooth closed (compact and without boundary) oriented
Riemannian 3-manifold and let $X^{t}:M\to M$ be a weak-mixing transitive
Anosov flow. This flow has a countably infinite set $\mathscr{P}$ of prime
periodic orbits: we denote a typical such orbits by $\gamma$ and its least
period by $\ell(\gamma)$. The following asymptotic formula is due to Margulis
[17], [18]:

$|\\{\gamma\in\mathscr{P}\hbox{ : }\ell(\gamma)\leq
T\\}|\sim\frac{e^{h(X)T}}{h(X)T},\quad\mbox{as }T\to\infty,$

where $h(X)$ is the topological entropy of $X$.

We may also consider how the periodic orbits are distributed with respect to
the homology of $M$. Let $b=b_{1}(M)$ denote the first Betti number of $M$,
which we shall assume to be at least $1$. For $\gamma\in\mathcal{P}$, we shall
write $[\gamma]$ for the torsion-free part of the homology class of $\gamma$
in $H_{1}(M,\mathbb{Z})$; we may identify this with an element of
$\mathbb{Z}^{b}$. Write $\mathscr{C}$ for the closure of the set of normalised
homology classes,

(1.1) $\mathscr{C}=\overline{\left\\{\frac{[\gamma]}{\ell(\gamma)}\hbox{ :
}\gamma\in\mathscr{P}\right\\}}\subset\mathbb{R}^{b}.$

This is a compact convex set with non-empty interior. We say that $X^{t}$ is
homologically full if the interior of $\mathscr{C}$ contains zero. Assuming
for the moment that $X$ is homologically full, write
$\mathscr{P}_{0}=\\{\gamma\in\mathscr{P}\hbox{ : }[\gamma]=0\\}$. Then there
exists $0<h^{*}\leq h(X)$ and $C>0$ such that

$|\\{\gamma\in\mathscr{P}_{0}\hbox{ : }\ell(\gamma)\leq T\\}|\sim
C\frac{e^{h^{*}T}}{T^{1+b/2}},\quad\mbox{as }T\to\infty,$

and a similar formula for $[\gamma]=\alpha$ for any fixed $\alpha\in
H_{1}(M,\mathbb{Z})$ (though the constant $C$ may change) [27].

Dropping the assumption that $X$ is homologically full, more general results
hold where, instead of restricting $\gamma$ to be null-homologous (or to lie
in a fixed homology class), we consider $\gamma$ such that $[\gamma]$ is
approximately $T\rho$, where $\rho\in\mathrm{int}(\mathscr{C})$ (Lalley [15]
for geodesic flows over surfaces, Babillot and Ledrappier [3] for the general
case). The right hand side then takes the form
$C_{\rho}(T)e^{H(\rho)T}T^{-(1+b/2)},$ where
$H:\mathrm{int}(\mathscr{C})\to\mathbb{R}^{+}$ is a natural entropy function
and $C_{\rho}(T)$ an explicit continuous function which is bounded above and
bounded below away from zero.

The object of this paper will be to consider analogous problems for the knot
complements introduced above, inspired by the approach of McMullen [19]. Let
$K_{1},\ldots,K_{n}\in\mathscr{P}_{0}$ be a finite set of (integral) null-
homologous periodic orbits, which we think of as knots in $M$, and consider
the knot complement $M_{N}:=M\setminus\\{K_{1},\ldots,K_{N}\\}$. Then, for any
$\gamma\in\mathscr{P}_{N}:=\mathscr{P}\setminus\\{K_{1},\ldots,K_{N}\\}$, we
can consider the torsion-free part of its homology class $[\gamma]_{N}\in
H_{1}(M_{N},\mathbb{Z})/\mathrm{torsion}$. Let

$\mathscr{C}_{N}=\overline{\left\\{\frac{[\gamma]_{N}}{\ell(\gamma)}\hbox{ :
}\gamma\in\mathscr{P}_{N}\right\\}}\subset
H_{1}(M_{N},\mathbb{R})\cong\mathbb{R}^{b+N}.$

Write

$\pi_{N}(T,\lfloor\rho T\rfloor):=|\\{\gamma\in\mathscr{P}_{N}\hbox{ :
}\ell(\gamma)\leq T,\ [\gamma]_{N}=\lfloor\rho T\rfloor\\}|,$

where $\lfloor\cdot\rfloor$ is defined by choosing a fundamental domain
$\mathfrak{F}_{N}$ for $H_{1}(M_{N},\mathbb{Z})$ as a lattice in
$H_{1}(M_{N},\mathbb{R})$ and setting $\lfloor\rho\rfloor$ to be the unique
element of $H_{1}(M_{N},\mathbb{Z})/\mathrm{torsion}$ satisfying
$\rho-\lfloor\rho\rfloor\in\mathfrak{F}_{N}$. Here is a sample theorem.

######  Theorem 1.1.

Let $M$ be a smooth closed oriented Riemannian $3$-manifold and let
$X^{t}:M\to M$ be a transitive Anosov flow which mixes exponentially fast.
There is a strictly positive real analytic function
$\mathfrak{h}_{N}:\mathrm{int}(\mathscr{C}_{N})\to\mathbb{R}^{+}$ such that,
for each $\rho\in\mathrm{int}(\mathscr{C}_{N})$, we have

$\pi_{N}(T,\lfloor\rho T\rfloor)\sim
c_{\rho}(T)\frac{e^{\mathfrak{h}_{N}(\rho)T}}{T^{1+(b+N)/2}},$

as $T\to\infty$, where $c_{\rho}(T)$ is bounded above and bounded below away
from zero.

######  Remark 1.2.

In fact, the result holds under a weaker hypothesis than exponential mixing.
See section 2 below.

We will now outline the content of the rest of the paper. In section 2, we
define Anosov flows and introduce the assumptions for our main result. In
section 3, we discuss the homology of a link complement, when the link is
given by finitely many null-homologous periodic orbits in $M$. In particular
we show that the first homology group is generated by the homology classes of
the remaining periodic orbits. In section 4, we see how the homology of orbits
is encoded by the modelling of our Anosov flow as a symbolic suspension flow.
In section 5, we describe some relevant tools from the thermodynamic formalism
for suspension flows. In section 6 we state our main results and show how they
follow from the main result of [3].

##  2\. Anosov flows

Let $M$ be a smooth, connected, oriented, closed (compact and without
boundary) Riemannian 3-manifold and let $X^{t}:M\to M$ be a $C^{1}$ flow
generated by the vector field $X$. We will assume $X^{t}$ is Anosov, meaning
that the tangent bundle has a continuous $DX^{t}$-invariant splitting
$TM=E^{0}\oplus E^{s}\oplus E^{u}$, where $E^{0}$ is the one-dimensional
bundle spanned by $X$ and where there exist constants $C,\lambda>0$ such that

  1. (i)

$\|DX^{t}v\|\leq Ce^{-\lambda t}\|v\|$, for all $v\in E^{s}$ and $t>0$;

  2. (ii)

$\|DX^{-t}v\|\leq Ce^{-\lambda t}\|v\|$, for all $v\in E^{u}$ and $t>0$.

This class of flows was introduced by Anosov [2]; for a good modern reference
see [10]. In addition, we assume that $X^{t}:M\to M$ is topologically
transitive, i.e. that there is a dense orbit.

We say that the flow is topologically weak-mixing if the equation $\psi\circ
X^{t}=e^{iat}\psi$, for $\psi\in C(M,\mathbb{C})$ and $a\in\mathbb{R}$, has
only the trivial solution where $\psi$ is constant and $a=0$. We say it is
topologically mixing if for all non-empty open sets $U,V\subset M$, there
exists $t_{0}$ such that $X^{t}(U)\cap V\neq\varnothing$ for all $t\geq
t_{0}$. For Anosov flows (and more general hyperbolic flows) these conditions
are equivalent and are also equivalent to the flow being mixing with respect
to the equilibrium state $\mu$ of any Hölder continuous function, i.e. that

$\rho_{F,G}(t):=\int F\circ X^{t}\,G\,d\mu-\int F\,d\mu\int G\,d\mu\to 0,$

as $t\to\infty$, for all $F,G\in L^{2}(\mu)$. A classical result of Plante
[23] is that a transitive Anosov flow fails to be topologically weak-mixing if
and only if it is the constant time suspension of an Anosov diffeomorphism.

An Anosov flow has a countably infinite set $\mathscr{P}$ of prime periodic
orbits. For $\gamma\in\mathscr{P}$, $\ell(\gamma)$ denotes the least period of
$\gamma$. We have that the flow fails to be mixing if and only if
$\\{\ell(\gamma)\hbox{ : }\gamma\in\mathcal{P}\\}$ is contained in a discrete
subgroup of $\mathbb{R}$. If $X^{t}:M\to M$ is mixing then
$|\\{\gamma\in\mathscr{P}\hbox{ : }\ell(\gamma)\leq T\\}|\sim e^{hT}/hT$,
where $h$ is the topological entropy [17],[18],[20].

A transitive Anosov flow is (bounded-to-one) semiconjugate to a suspension
flow over a subshift of finite type. (This will be discussed in greater detail
in section 4.) We shall consider flows such that the roof function cannot be
chosen to be locally constant; we call such flows not locally constant. This
is an open dense condition which is implied by the flow having good
asymptotics in the sense defined in [9]. (We omit the rather complicated
definition.) In particular, this holds if the flow is exponentially mixing,
i.e. that $\rho_{F,G}(t)$ converges to zero exponentially fast for the
equilibrium state of any Hölder continuous function and all sufficiently
regular $F$ and $G$. The class of exponentially mixing transitive Anosov flows
includes geodesic flows over compact surfaces of (not necessarily constant)
negative curvature [7]. In fact, Tsujii and Zhang have shown that any smooth
mixing Anosov flow on a closed $3$-manifold is exponentially mixing [28].

##  3\. Knot complements

We now wish to consider $M$ with a finite number of null-homologous periodic
orbits removed. (For convenience, we suppose that they are integrally null-
homologous, so we end up with integral linking numbers.) Let
$K_{1},\ldots,K_{N}$ be null-homologous knots in $M$ (for the moment, they do
not need to be periodic orbits). We need to understand the homology of the
complement $M_{N}=M\setminus\bigcup_{i=1}^{N}K_{i}$. We will denote the
homology class of $\gamma$ in $H_{1}(M_{N},\mathbb{R})$ by $[\gamma]_{N}$.

For each $i=1,\ldots,N$, replace $K_{i}$ with a tubular neighbourhood
$\mathscr{N}(K_{i})$ (an embedded solid torus), and (slightly abusing
notation) let

$M_{N}:=M\setminus\bigcup_{i=1}^{N}\mathrm{int}(\mathscr{N}(K_{i})).$

For each $i=1,\ldots,N$, let $\mathfrak{m}_{i}$ be a meridian in
$\partial\mathscr{N}(K_{i})$.

######  Lemma 3.1.

Let $K_{1},\ldots,K_{N}$ be null-homologous knots in $M$. Then

$H_{1}(M_{N},\mathbb{R})\cong H_{1}(M,\mathbb{R})\oplus\mathbb{R}^{N}.$

In particular, the first Betti number of $M_{N}$ is $b+N$. Furthermore, the
classes $[\mathfrak{m}_{1}]_{N},\ldots,[\mathfrak{m}_{N}]_{N}$ are linearly
independent in $H_{1}(M_{N},\mathbb{R})$.

###### Proof.

The first statement follows from the homology long exact sequence

$\displaystyle\longrightarrow H_{2}(M_{N},\partial M_{N})\longrightarrow
H_{1}(\partial M_{N})\longrightarrow H_{1}(M_{N})\longrightarrow
H_{1}(M_{N},\partial M_{N})\longrightarrow$

and the ‘half lives, half dies’ principle (Lemma 3.5 of [11]), which tells us
that the dimension of the image of the boundary homomorphism
$\partial:H_{2}(M_{N},\partial M_{N})\to H_{1}(\partial M_{N})$ is equal to
$N$.

For the second statement, we are indebted to a Mathematics Stack Exchange post
by Kyle Miller. By construction, the longitude $\mathfrak{l}_{i}$ is null-
homologous in $M\setminus\mathrm{int}(\mathscr{N}(K_{i}))$ and so we can
choose an embedded surface $S_{i}\subset
M\setminus\mathrm{int}(\mathscr{N}(K_{i}))$ with boundary $\mathfrak{l}_{i}$.
We may assume that $S_{i}$ intersects the other knots transversally and so we
obtain an embedded surface $S_{i}^{\prime}=S_{i}\cap M_{N}$ that represents
$[\mathfrak{l}_{i}]$ as a linear combination of the homology classes of the
meridians $[\mathfrak{m}_{1}]_{N},\ldots,[\mathfrak{m}_{N}]_{N}$. Since
$[\mathfrak{m}_{1}]_{N},\ldots,[\mathfrak{m}_{N}]_{N},[\mathfrak{l}_{1}]_{N},\ldots,[\mathfrak{l}_{N}]_{N}$
is a basis for $H_{1}(\partial M_{N},\mathbb{R})$,
$[\mathfrak{m}_{1}]_{N},\ldots,[\mathfrak{m}_{N}]_{N}$ span the image of the
inclusion homomorphism $i_{*}:H_{1}(\partial M_{N},\mathbb{R})\to
H_{1}(M_{N},\mathbb{R})$, and $\dim\ker i_{*}=N$, it follows that
$[\mathfrak{m}_{1}]_{N},\ldots,[\mathfrak{m}_{N}]_{N}$ are linearly
independent in $H_{1}(M_{N},\mathbb{R})$. ∎

We will need the following result. (The analogous statement for
$H_{1}(N,\mathbb{Z})$ may be found in [21].)

######  Proposition 3.2.

The torsion-free part of the group $H_{1}(M_{N},\mathbb{Z})$ is generated by
$\\{[\gamma]_{N}\hbox{ : }\gamma\in\mathscr{P}_{N}\\}$.

###### Proof.

This is a corollary of the main result in [19], where an analogue of the
Chebotarev density theorem is proved for weak-mixing Anosov flows. More
precisely, for a finite group $G$, and a surjective homomorphism
$\phi:\pi_{1}(M_{N})\to G$, given any conjugacy class $C$ in $G$ we have

(3.1) $\lim_{T\to\infty}\frac{|\\{\gamma\in\mathscr{P}_{N}\hbox{ :
}\ell(\gamma)\leq T,\,\phi(\gamma)\in
C\\}|}{|\\{\gamma\in\mathscr{P}_{N}\hbox{ : }\ell(\gamma)\leq
T\\}|}=\frac{|C|}{|G|}.$

If $H_{1}(M_{N},\mathbb{Z})/\mathrm{torsion}$ is not generated by the orbits
in $\mathscr{P}_{N}$ then there is some proper cofinite subgroup $H\leq
H_{1}(M_{N},\mathbb{Z}),$ such that

$\\{[\gamma]_{N}\hbox{ : }\gamma\in\mathscr{P}_{N}\\}\subset H.$

If we set $G=H_{1}(M_{N},\mathbb{Z})/H$ then we have a quotient homomorphism
$\phi:\pi_{1}(M_{N})\to G$ and the formula (3.1) is contradicted, since all
orbits in $\mathscr{P}_{N}$ satisfy $\phi(\gamma)=0$. ∎

##  4\. Symbolic dynamics

In this section, we discuss how to model our flows by symbolic systems, which
(as we will see in the next section) will also keep track of the homology and
linking numbers of periodic orbits. This is classical material, due
independently to Bowen and Ratner. We begin by defining subshifts of finite
type.

###  4.1. Subshifts of finite type and coding

Let $k\geq 2$, and let $\Gamma$ be a directed graph on $k$ vertices
$V(\Gamma)$, labelled $1,\ldots,k$, and with directed edges $E(\Gamma)\subset
V(\Gamma)\times V(\Gamma)$. (In particular, for each ordered pair of vertices
$(i,j)$ there is at most one directed edge from $i$ to $j$.) Define

$\Sigma(\Gamma):=\\{(x_{n})_{n\in\mathbb{Z}}\in
V(\Gamma)^{\mathbb{Z}}:(x_{n},x_{n+1})\in E(\Gamma)\text{ for all }n\\},$

i.e. the set of bi-infinite walks on $\Gamma.$ The shift map
$\sigma:\Sigma(\Gamma)\to\Sigma(\Gamma)$ is defined by
$\sigma((x_{n})_{n\in\mathbb{Z}})=(x_{n+1})_{n\in\mathbb{Z}}$.

We give $V(\Gamma)$ the discrete topology, $V(\Gamma)^{\mathbb{Z}}$ the
product topology, and $\Sigma(\Gamma)$ the subspace topology. The following
metrics are compatible with this topology. Fix $0<\theta<1$. For $x,y\in X$
set $d_{\theta}(x,y)=\theta^{n(x,y)}$, where

$n(x,y)=\sup\\{n\in\mathbb{Z}^{+}\hbox{ : }x_{i}=y_{i}\ \forall|i|<n\\}.$

With this metric in place, let $F_{\theta}$ denote the set of real-valued
$d_{\theta}$-Lipschitz functions on $\Sigma(\Gamma)$.

We will say that $\Gamma$ is aperiodic if there exists $N\geq 1$ such that,
for each ordered pair of vertices $(i,j)$, there is a path of length $N$ from
$i$ to $j$. This condition is equivalent to
$\sigma:\Sigma(\Gamma)\to\Sigma(\Gamma)$ being topologically mixing (i.e. that
there exists $N\geq 1$ such that for all non-empty open sets
$U,V\subset\Sigma(\Gamma)$, we have $\sigma^{n}(U)\cap V\neq\varnothing$ for
all $n\geq N$).

Given a strictly positive function $r\in F_{\theta}$, we can suspend
$\Sigma(\Gamma)$ with $r$ as roof function, to obtain the space

$\Sigma(\Gamma,r)=\\{(x,t)\in\Sigma(\Gamma)\times\mathbb{R}\hbox{ : }0\leq
t\leq r(x)\\}/(x,r(x))\sim(\sigma(x),0).$

The suspension flow
$\sigma^{r}_{t}:\Sigma(\Gamma,r)\rightarrow\Sigma(\Gamma,r)$ is defined by
$\sigma^{r}_{t}[x,s]=[x,s+t],$ where $[x,s]$ is the equivalence class of
$(x,s)$ in $\Sigma(\Gamma,r)$. According to the classic results of Bowen and
Ratner from the 1970s, Anosovs flows (and more general hyperbolic flows) may
be modelled by suspension flows of this type.

######  Theorem 4.1 (Bowen [4], Ratner [24]).

Let $X^{t}:M\rightarrow M$ be a topologically weak-mixing transitive Anosov
flow. Then there exists a suspension flow $(\sigma^{r},\Sigma(\Gamma,r))$,
over a topologically mixing $(\sigma,\Sigma(\Gamma))$, and a continuous
surjection $\pi:~{}\Sigma(\Gamma,r)\rightarrow M$ such that

  1. _(i)_

For all $t\in\mathbb{R}$, $\pi\circ\sigma^{r}_{t}=X^{t}\circ\pi$;

  2. _(ii)_

There exists $N\in\mathbb{N}$ such that $\\#\pi^{-1}(y)\leq N$ for all $y\in
M$;

  3. _(iii)_

$\sigma^{r}$ is topologically weak-mixing and transitive.

The map $\pi$ is constructed as follows. Using the structure of the stable and
unstable manifolds of $X$, one can construct a Markov section. In particular,
this gives a finite set $\\{R_{1},\ldots R_{k}\\}$ of codimension one cross-
sections to the flow (or rectangles), such that every orbit of $X^{t}$
intersects $R=\bigcup_{i=1}^{k}R_{i}$ infinitely often in the past and future,
and these intersections are transversal. Letting $\tau:R\to R$ be the first
return map to $R$ (or Poincaré map), define a directed graph $\Gamma$ on $k$
vertices $\\{1,2,\ldots,k\\}$ as follows. First, denote by $ij$ the directed
edge from vertex $i$ to vertex $j$. Include $ij$ in the directed edge set
$E(\Gamma)$ whenever $R_{i}\cap\tau^{-1}(R_{j})\neq\varnothing$. This yields a
connected directed graph, with at most two edges (of opposite direction)
between two vertices. The rectangles are chosen with the properties that there
exists $T>0$ such that $M\subset X^{[0,T]}R$, and that , for each
$x\in\Sigma(\Gamma)$, $\bigcap_{n\in\mathbb{Z}}\tau^{n}(R_{x_{n}})$ contains
exactly one point. With this, we can define $\pi^{\prime}:\Sigma(\Gamma)\to R$
by $\pi^{\prime}(x)\in\bigcap_{n\in\mathbb{Z}}\tau^{n}(R_{x_{n}}).$ Setting
$r:R\to\mathbb{R}^{+}$ to be the first return time to $R$, we can extend
$\pi^{\prime}$ to a map $\pi:\Sigma(\Gamma,r)\to M$, by
$\pi[x,s]=X^{s}(\pi^{\prime}(x)).$

###  4.2. Homology of periodic orbits via symbolic dynamics

We need to understand how periodic orbits for $X^{t}$ are encoded by $\pi$ and
how homology and linking may be captured by the symbolic model.

Given $1\leq i,j\leq k$, let $E_{ij}$ be the set of points on flow lines going
from $\text{int}(R_{i})$ to $\text{int}(R_{j}),$ and let

$U=\left(\bigcup_{i=1}^{k}\text{int}(R_{i})\right)\cup\left(\bigcup_{1\leq
i,j\leq k}E_{ij}\right).$

Let $\mathscr{P}_{\sigma^{r}}$ denote the set of prime periodic orbits for
$\sigma^{r}$. Given a periodic orbit $\gamma\in\mathscr{P}$, there is a
periodic orbit $\eta\in\mathscr{P}_{\sigma^{r}}$, with
$\ell(\gamma)=\ell(\eta)$, such that $\gamma=\pi(\eta)$. Furthermore, $\eta$
is unique as long as $\gamma\subset U$, i.e. $\gamma$ does not pass through
the edge of some rectangle. In the particular case of a 3-manifold, uniqueness
holds except for a finite number of orbits. (In higher dimensions, there many
be infinitely many non-uniquely coded orbits but they exhibit a slower rate of
exponential growth.) Thus $\pi$ induces a one-to-one correspondence for
periodic orbits in $\mathscr{P}$ and $\mathscr{P}_{\sigma^{r}}$, up to
removing finitely many.

An additional feature of the symbolic dynamics is that, as argued in [19], we
can arrange for the removed periodic orbits $K_{1},\ldots,K_{N}$ to sit in the
boundary of the sections. Indeed, suppose that, for some $1\leq n\leq N$,
$x\in K_{n}\cap\hbox{int}(R_{i})$ (by transversality, there can only be
finitely many such $x$). Then we may split $R_{i}$ into two pieces $R_{i,1}$
and $R_{i,2}$ at the point $x$, so that

$\\{R_{1},\ldots,R_{i-1},R_{i,1},R_{i,2},R_{i+1},\ldots,R_{k}\\}$

still forms a Markov section with the same properties as above. With this in
mind, we will henceforth assume that $K_{1},\ldots,K_{N}$ are in the boundary
of $R$.

Let $\bar{\Gamma}$ be the graph obtained by removing the direction from edges
of $\Gamma$, but retaining any multiple edges between vertices. (So if $ij$
and $ji$ are both directed edges then $\bar{\Gamma}$ has two undirected edges
between $i$ and $j$.) One can then choose a natural embedding
$\iota:\bar{\Gamma}\hookrightarrow U\subset M_{N}$. The following result
appears as Lemma 5.1 of [19].

######  Proposition 4.2.

The embedding $\iota:\bar{\Gamma}\to U$ induces a surjective homomorphism
$\iota_{*}:\pi_{1}(\bar{\Gamma})\to\pi_{1}(M_{N}).$

We now describe how the homology of periodic orbits can be encoded using the
symbolic dynamics. For each $i\in\\{1,\ldots,k\\}$, fix a path $p(1,i)$ from
$1$ to $i$ in $\bar{\Gamma}$, noting that paths in $\bar{\Gamma}$ need not
follow the directions of edges of $\Gamma$. That such paths exist for all $i$
is a consequence of transitivity of $\sigma.$ Let $p(i,1)$ be the path from
$i$ to $1$ obtained by following $p(1,i)$ backwards. For any vertex $j$ which
satisfies that $ij\in E(\Gamma)$, let $e(i,j)$ be the corresponding edge
between $i,j$ in $\bar{\Gamma}$. Then we can form a loop $K(i,j)$ in
$\bar{\Gamma}$ given by

$K(i,j)=1\xrightarrow{p(1,i)}i\xrightarrow{e(i,j)}j\xrightarrow{p(j,1)}1,$

which has a homotopy class $\langle K(i,j)\rangle\in\pi_{1}(\bar{\Gamma}).$
From Proposition 4.2, we have a surjective homomorphism
$\iota_{*}:\pi_{1}(\bar{\Gamma})\to\pi_{1}(M_{N}).$ We also have the Hurewicz
homomorphism

$q:\pi_{1}(M_{N})\to H_{1}(M_{N},\mathbb{Z})$

and the projection homomorphism

$\mathrm{proj}:H_{1}(M_{N},\mathbb{Z})\to
H_{1}(M_{N},\mathbb{Z})/\mathrm{torsion}\cong\mathbb{Z}^{b+N}.$

Let $\rho=\mathrm{proj}\circ
q\circ\iota_{*}:\pi_{1}(\bar{\Gamma})\to\mathbb{Z}^{b+N},$ and define
$f:\Sigma(\Gamma)\to\mathbb{Z}^{b+N}$ by

(4.1) $f(x)=\rho([K(x_{0},x_{1})]).$

Clearly, $f$ is locally constant. Further, as $\rho$ is a homomorphism, we
have that for a periodic point $x=\overline{x_{0}x_{1}\ldots x_{n-1}},$ the
Birkhoff sum

$\displaystyle f^{n}(x):$ $\displaystyle=f(x)+f(\sigma
x)+\ldots+f(\sigma^{n-1}x)$
$\displaystyle=\rho([K(x_{0},x_{1})]_{N}[K(x_{1},x_{2})]_{N}\cdots[K(x_{n-2},x_{n-1})]_{N}[K(x_{n-1},x_{0})]_{N})$
$\displaystyle=\rho([c(x_{0},x_{1},\ldots,x_{n-1},x_{0})]_{N}),$

where $c(x_{0},x_{1},\ldots,x_{n-1},x_{0})$ is the cycle

$x_{0}\xrightarrow{e(x_{0},x_{1})}x_{1}\xrightarrow{e(x_{1},x_{2})}x_{2}\to\ldots\to
x_{n-2}\xrightarrow{e(x_{n-2},x_{n-1})}x_{n-1}\xrightarrow{e(x_{n-1},x_{0})}x_{0}.$

This construction leads to the following.

######  Lemma 4.3.

Let $\eta\in\mathscr{P}_{\sigma^{r}},$ and $x\in\Sigma(\Gamma)$ the
corresponding periodic point for $\sigma$, with period $n\in\mathbb{N}.$ Then
$\ell(\pi(\eta))=\ell(\eta)=r^{n}(x),$ and $f^{n}(x)=[\pi(\eta)]_{N}$.

##  5\. Thermodynamic formalism for suspension flows

To formulate and prove our results, we will need to use the machinery of
thermodynamic formalism. Since we need to deal with linking numbers and
homology of the knot complement $M_{N}$, we will not be able to use
thermodynamic functions defined with respect to the Anosov flow $X^{t}:M\to
M$. Rather, we will work with the shift
$\sigma:\Sigma(\Gamma)\to\Sigma(\Gamma)$ and suspension flow
$\sigma_{t}^{r}:\Sigma(\Gamma,r)\to\Sigma(\Gamma,r)$.

Let $\mathscr{M}_{\sigma}$ denote the space of $\sigma$-invariant Borel
probability measures on $\Sigma(\Gamma)$. For a continuous function
$\varphi:\Sigma(\Gamma)\to\mathbb{R}$, we define its pressure (with respect to
$\sigma$) $P_{\sigma}(\varphi)$ by

$P_{\sigma}(\varphi)=\sup_{m\in\mathscr{M}_{\sigma}}\left\\{h_{m}(\sigma)+\int\varphi\,dm\right\\},$

where $h_{m}(\sigma)$ is the measure theoretic entropy. If $\varphi$ is Hölder
continuous then the supremum is attained at a unique measure $m_{\varphi}$,
which is called the equilibrium state for $\varphi$; this measure is ergodic
and fully supported.

Now let $\mathscr{M}_{\sigma^{r}}$ denote the space of $\sigma^{r}$-invariant
Borel probability measures on $\Sigma(\Gamma,r)$. There is a natural bijection
between $\mathscr{M}_{\sigma}$ and $\mathscr{M}_{\sigma^{r}}$ given by
$m\mapsto\widetilde{m}$, where $\widetilde{m}$ is given locally by

$\frac{m\times\mathrm{Leb}}{\int r\,dm}.$

For a continuous function $\Phi:\Sigma(\Gamma,r)\to\mathbb{R}$, we define its
pressure (with respect to $\sigma^{r}$) $P_{\sigma^{r}}(\Phi)$ by

(5.1)
$P_{\sigma^{r}}(\Phi)=\sup_{\mu\in\mathscr{M}_{\sigma^{r}}}\left\\{h_{\mu}(\sigma^{r})+\int\Phi\,d\mu\right\\}.$

One can put a metric on $\Sigma(\Gamma,r)$ [5] and consider Hölder continuous
functions with respect to this metric, but for our purposes it is sufficient
to consider functions satisfying the following (strictly weaker) condition. We
say that $\Phi$ is fibre-Hölder if $\varphi:\Sigma(\Gamma)\to\mathbb{R}$,
defined by

(5.2) $\varphi(x)=\int_{0}^{r(x)}\Phi(x,t)\,dt,$

is Hölder continuous. If $\Phi$ is fibre-Hölder then the supremum in (5.1) is
attained at a unique measure $\mu_{\Phi}$, which is called the equilibrium
state for $\Phi$; this measure is ergodic and fully supported. Furthermore,
$\mu_{\Phi}$ is given locally by the product

$\frac{m_{-P(\Phi)r+\varphi}\times\mathrm{Leb}}{\int
r\,dm_{-P(\Phi)r+\varphi}},$

where $\varphi$ is the function in (5.2).

It will also be convenient to pass from functions on $\Sigma(\Gamma)$ to
functions on $\Sigma(\Gamma,r)$, so we ‘lift’ functions in the following way.
Choose a smooth function $\kappa:[0,1]\to\mathbb{R}^{+}$ satisfying
$\kappa(0)=\kappa(1)=0$ and $\int_{0}^{1}\kappa(t)\,dt=1$. Let
$\varphi:\Sigma(\Gamma)\to\mathbb{R}$ be Hölder continuous and define
$\widetilde{\varphi}:\Sigma(\Gamma,r)\to\mathbb{R}$ by

$\widetilde{\varphi}(x,t)=\frac{\varphi(x)}{r(x)}\kappa\left(\frac{t}{r(x)}\right).$

Then

$\varphi(x)=\int_{0}^{r(x)}\widetilde{\varphi}(x,t)\,dt$

and $\widetilde{\varphi}$ is automatically fibre-Hölder. Furthermore, for a
periodic orbit $\eta$ of $\sigma^{r}$, corresponding to a point
$x\in\Sigma(\Gamma)$ of least period $n$,

$\frac{\varphi^{n}(x)}{r^{n}(x)}=\frac{1}{\ell(\eta)}\int_{\eta}\widetilde{\varphi},$

where
$\int_{\eta}\tilde{\varphi}=\int_{0}^{\ell(\eta)}\tilde{\varphi}(\sigma^{r}_{t}x_{\eta})\,dt$,
for $x_{\eta}\in\eta.$ This integral defines a probability measure,
$\mu_{\eta}$, by

$\int\Phi\,d\mu_{\eta}=\frac{1}{\ell(\eta)}\int_{\eta}\Phi.$

We now wish to consider pressure functions of a particular form. Fix a fibre-
Hölder function $F:\Sigma(\Gamma,r)\to\mathbb{R}^{d}$ (for some $d\geq 1$) and
let $f:\Sigma(\Gamma)\to\mathbb{R}^{d}$ be defined by the higher dimensional
analogue of (5.2), i.e.

$f(x)=\int_{0}^{r(x)}F(x,t)\,dt.$

We then have a set $\mathscr{C}_{F}\subset\mathbb{R}^{d}$ defined in the
following equivalent ways:

$\displaystyle\mathscr{C}_{F}$
$\displaystyle=\overline{\left\\{\frac{1}{\ell(\eta)}\int_{\eta}F\hbox{ :
}\eta\in\mathscr{P}_{\sigma^{r}}\right\\}}=\left\\{\int Fd\mu\hbox{ :
}\mu\in\mathscr{M}_{\sigma^{r}}\right\\}$ $\displaystyle=\left\\{\frac{\int
f\,dm}{\int r\,dm}\hbox{ : }m\in\mathscr{M}_{\sigma}\right\\}.$

We write $F=(F_{1},\ldots,F_{d})$ and, for
$u=(u_{1},\ldots,u_{d})\in\mathbb{R}^{d}$, write $\langle
u,F\rangle=u_{1}F_{1}+\cdots+u_{d}F_{d}$. We then define the pressure function
$\mathfrak{p}_{F}:\mathbb{R}^{d}\to\mathbb{R}$ by

$\mathfrak{p}_{F}(u)=P_{\sigma^{r}}(\langle u,F\rangle).$

This is real analytic and convex and satisfies

$\nabla\mathfrak{p}_{F}(u)=\int F\,d\mu_{\langle u,F\rangle}.$

We have the following standard result (see [14], for example).

######  Lemma 5.1.

The following are equivalent.

  1. (i)

For all $u\in\mathbb{R}^{d}\setminus\\{0\\}$, the function $\langle
u,F\rangle:\Sigma(\Gamma,r)\to\mathbb{R}$ is not cohomologous to a constant.

  2. (ii)

The pressure function $\mathfrak{p}_{F}$ is strictly convex.

  3. (iii)

$\mathrm{int}(\mathscr{C}_{F})\neq\varnothing$.

Assuming that any (and hence all) of the statements in Lemma 5.1 hold, we
define an entropy function
$\mathfrak{h}_{F}:\mathrm{int}(\mathscr{C}_{F})\to\mathbb{R}$ by

$\mathfrak{h}_{F}(\rho)=\sup\bigg{\\{}h_{\mu}(\sigma)\hbox{ :
}\mu\in\mathscr{M}_{\sigma^{r}},\,\int F\,d\mu=\rho\bigg{\\}}.$

This leads to the following lemma.

######  Lemma 5.2.

The map $u\mapsto\nabla\mathfrak{p}_{F}(u)$ is a smooth diffeomorphism between
$\mathbb{R}^{d}$ and $\mathrm{int}(\mathscr{C}_{F})$. Furthermore,
$\mathfrak{h}_{F}$ is a smooth function on $\mathrm{int}(\mathscr{C}_{F})$ and
$(\nabla\mathfrak{p}_{F})^{-1}=-\nabla\mathfrak{h}_{F}$.

###### Proof.

This essentially follows from Theorem 26.5 of [25]. Since

$\mathfrak{p}_{F}(u)=h_{\mu_{\langle u,F\rangle}}(\sigma^{r})+\left\langle
u,\int F\,d\mu_{\langle u,F\rangle}\right\rangle$

for a unique measure $\mu_{\langle u,F\rangle}\in\mathscr{M}_{\sigma^{r}}$, we
have

$\mathfrak{p}_{F}(u)=\sup_{z\in\mathscr{C}_{F}}\Big{(}\mathfrak{h}_{F}(z)+\langle
u,z\rangle\Big{)},$

with the supremum attained uniquely at $z=\int F\,d\mu_{\langle u,F\rangle}$.
This shows that the functions $\mathfrak{p}_{F}$ and $-\mathfrak{h}_{F}$ are
Legendre duals. Theorem 26.5 of [25] then gives that $\nabla\mathfrak{p}_{F}$
is a diffeomorphism onto its image with inverse $-\nabla\mathfrak{h}_{F}$. To
complete the proof, it only remains to show that
$\nabla\mathfrak{p}_{F}(\mathbb{R}^{d})=\mathrm{int}(\mathscr{C}_{F})$.

We now use an argument adapted from the proof of Lemma 7 in [16]. Let
$z\in\mathrm{int}(\mathscr{C}_{F})$ and choose $\epsilon>0$ such that
$B(z,2\epsilon)\subset\mathscr{C}_{F}$. Then, for $u\neq 0$,
$z+(u/\|u\|)\epsilon\in\mathscr{C}_{F}$. Thus

$\langle u,z\rangle+\|u\|\epsilon\leq\sup_{w\in\mathscr{C}_{F}}\langle
u,w\rangle=\sup\left\\{\left\langle u,\int F\,d\mu\right\rangle\hbox{ :
}\mu\in\mathscr{M}(\sigma^{r})\right\\}\leq\mathfrak{p}_{F}(u).$

Let $e_{z}:\mathbb{R}^{d}\to\mathbb{R}$ be defined by
$e_{z}(u)=\mathfrak{p}_{F}(u)-\langle u,z\rangle$. Then, by the above
inequality, $e_{z}(u)\geq 0$ for all $u\in\mathbb{R}^{d}$. Thus $e_{z}$ has a
finite minimum attained at a unique $u(z)\in\mathbb{R}^{d}$. Hence $\nabla
e_{z}(u(z))=0$, which is exactly $\nabla\mathfrak{p}_{F}(u(z))=z$. ∎

We will continue to use the notation of this proof: given
$\rho\in\mathrm{int}(\mathscr{C}_{F})$, let $u(\rho)\in\mathbb{R}^{d}$ be
defined by $\nabla\mathfrak{p}_{F}(u(\rho))=\rho$. Furthermore, we will write
$\mu_{\rho}\in\mathscr{M}_{\sigma^{r}}$ for the equilibrium state for $\langle
u(\rho),F\rangle$.

##  6\. Counting results

The main results of the paper will follow from counting results for periodic
orbits subject to constraints initiated by Lalley [14] (see also Sharp [26])
and given in streamlined form by Babillot and Ledrappier [3]. The results in
[3] are stated for hyperbolic flows but the machinery all works for suspension
flows.

We make the following definitions. Let $\mathscr{A}_{F}$ be the closed
subgroup of $\mathbb{R}^{d}$ generated by

$\left\\{\int_{\eta}F\hbox{ : }\eta\in\mathscr{P}_{\sigma^{r}}\right\\}$

and let $\widetilde{\mathscr{A}_{F}}$ be the closed subgroup of
$\mathbb{R}^{d+1}$ generated by

$\left\\{\left(\ell(\eta),\int_{\eta}F\right)\hbox{ :
}\eta\in\mathscr{P}_{\sigma^{r}}\right\\}.$

We choose a fundamental domain $\mathfrak{F}$ for $\mathscr{A}_{F}$ and, for
$\rho\in\mathbb{R}^{d}$, we define $\lfloor\rho\rfloor\in\mathscr{A}_{F}$ by
$\rho-\lfloor\rho\rfloor\in\mathfrak{F}$.

Let $g_{0}:\mathbb{R}\to\mathbb{R}$ be a continuous function with compact
support and let $g:\mathscr{A}_{F}\to\mathbb{R}$ be finitely supported.
Following [3], we will study the functional $N_{T}^{\rho}$ defined by

$N_{T}^{\rho}(g_{0}\otimes
g)=\sum_{\eta\in\mathscr{P}_{\sigma^{r}}}g_{0}(\ell(\eta)-T)g\left(\int_{\eta}F-\lfloor
T\rho\rfloor\right).$

The following is a special case of Theorem 1.2 in [3].

######  Proposition 6.1.

Suppose that $\widetilde{\mathscr{A}_{F}}=\mathbb{R}\times\mathscr{A}_{F}$.
Then, for all $\rho\in\mathrm{int}(\mathscr{C}_{F})$, we have

$N_{T}^{\rho}(g_{0}\otimes
g)\sim\frac{\sqrt{|\det\nabla^{2}\mathfrak{h}_{F}(\rho)|}}{(2\pi)^{d/2}}a_{\rho}(g_{0},g)\frac{e^{\mathfrak{h}_{F}(\rho)T+\langle
u(\rho),T\rho-\lfloor T\rho\rfloor\rangle}}{T^{1+d/2}},$

where

$a_{\rho}(g_{0},g)=\int_{\mathbb{R}}e^{\mathfrak{p}_{F}(u(\rho))x}g_{0}(x)\,dx\sum_{y\in\mathscr{A}_{F}}e^{-\langle
u(\rho),y\rangle}g(y),$

as $T\to\infty$.

We shall apply this result when $d=b+N$ and $F=\widetilde{f}$, where
$f:\Sigma(\Gamma)\to\mathbb{Z}^{b+N}$ is the locally constant function defined
in (4.1). Hence, by Proposition 3.2, $\mathscr{A}_{F}=\mathbb{Z}^{d}$.
However, there are some additional complications that we need to account for.
First, we note that

$\sum_{\gamma\in\mathscr{P}_{N}}g_{0}(\ell(\gamma)-T)g\left([\gamma]_{N}-\lfloor
T\rho\rfloor\right)=N_{T}^{\rho}(g_{0}\otimes g)+O(1),$

so the removed orbits do not cause a problem. Second, we need to show that
$\widetilde{\mathscr{A}_{F}}=\mathbb{R}\times\mathscr{A}_{F}$. We do this in
the next lemma.

######  Lemma 6.2.

Suppose that $X^{t}$ is not locally constant. Then, for $F$ as in case (i) and
case (ii) above, we have

$\widetilde{\mathscr{A}_{F}}=\mathbb{R}\times\mathscr{A}_{F}.$

###### Proof.

Clearly, $\widetilde{\mathscr{A}_{F}}$ is a closed subgroup of
$\mathbb{R}\times\mathscr{A}_{F}=\mathbb{R}\times\mathbb{Z}^{d}$. Therefore,
it is sufficient to show that if $\chi$ is a character of
$\mathbb{R}\times\mathbb{Z}^{d}$ which is trivial on
$\widetilde{\mathscr{A}_{F}}$ then $\chi$ is trivial on
$\mathbb{R}\times\mathbb{Z}^{d}$.

The characters of $\mathbb{R}\times\mathbb{Z}^{d}$ taken the form
$\chi_{t,u}(x,y)=e^{2\pi i(tx+\langle u,y\rangle)}$, with
$(t,u)\in\mathbb{R}\times\mathbb{R}^{d}/\mathbb{Z}^{d}$. If $\chi_{t,u}$ is
trivial on $\widetilde{\mathscr{A}_{F}}$ then

(6.1) $\chi_{t,u}\left(\ell(\eta),\int_{\eta}F\right)=\exp 2\pi
i\left(t\ell(\eta)+\int_{\eta}\langle
u,F\rangle\right)=1\quad\forall\eta\in\mathscr{P}_{\sigma^{r}}$

Passing to the shift map, this may be rewritten in the form

(6.2) $\exp 2\pi i(tr^{n}(x)+\langle u,f^{n}(x)\rangle)=1$

whenever $\sigma^{n}x=x$ and $n\geq 1$. By Proposition 5.2 of [22], equation
(6.2) may itself be rewritten as

(6.3) $2\pi tr+2\pi\langle u,f\rangle+\psi\circ\sigma-\psi=\varphi,$

where $\psi\in C(\Sigma(\Gamma),\mathbb{R})$ and $\varphi\in
C(\Sigma(\Gamma),2\pi\mathbb{Z})$. In particular, if (6.3) holds then $2\pi
tr$ is cohomologous to a locally constant function. Applying the not locally
constant condition, this forces $t=0$.

Substituting $t=0$ into (6.1),we have

(6.4) $\exp 2\pi
i\bigg{\langle}u,\int_{\eta}F\bigg{\rangle}=1\quad\forall\eta\in\mathscr{P}_{\sigma^{r}}.$

Since $\\{[\gamma]_{N}\hbox{ : }\gamma\in\mathscr{P}\\}$ generates the torsion
free part of $H_{1}(M_{N},\mathbb{Z})$, we have that
$u=0\in\mathbb{R}^{d}/\mathbb{Z}^{d}$. ∎

Thus we have shown that Proposition 6.1 may be applied in our cases. We may
pass from $N_{T}^{\rho}(g_{0}\otimes g)$ to a counting function by choosing
$g_{0}$ to approximate the indicator function of the interval $(-\delta,0]$,
say, and choosing $g$ to be the indicator function of an element of
$\mathscr{A}_{F}$. We also use the following notation. Let
$C_{b}(\mathbb{R}^{+})$ denote the set set of bounded continuous functions
from $\mathbb{R}^{+}$ to $\mathbb{R}$ and let

$C_{b,+}(\mathbb{R}^{+})=\left\\{f\in C_{b}(\mathbb{R}^{+})\hbox{ :
}\inf_{T>0}f(T)>0\right\\}.$

Here are our main results.

######  Theorem 6.3.

Let $M$ be a smooth closed oriented Riemannian $3$-manifold and let
$X^{t}:M\to M$ be a transitive Anosov flow which is not locally constant.
There is a strictly positive real analytic function
$\mathfrak{h}_{N}:\mathrm{int}(\mathscr{C}_{N})\to\mathbb{R}^{+}$ such that,
for each $\rho\in\mathrm{int}(\mathscr{C}_{N})$ and $\alpha\in
H_{1}(M_{N},\mathbb{Z})/\mathrm{torsion}$, we have

$\displaystyle|\\{\gamma\in\mathscr{P}_{N}\hbox{ : }T-\delta<\ell(\gamma)\leq
T,\ [\gamma]_{N}=\lfloor T\rho\rfloor+\alpha\\}|\sim
c_{\rho,\alpha}(T)\frac{e^{\mathfrak{h}_{N}(\rho)T}}{T^{1+(b+N)/2}},$

as $T\to\infty$, where $c_{\rho,\alpha}\in C_{b,+}(\mathbb{R}^{+})$.

If $0\in\mathrm{int}(\mathscr{C}_{N})$ then the next corollary gives a result
for periodic orbits with fixed homology class in $H_{1}(M_{N},\mathbb{Z})$.
However, by work of Dehornoy, there are examples of geodesic flows where this
condition fails, since all linking numbers of pairs of periodic orbits are
negative [6].

######  Corollary 6.4.

If $0\in\mathrm{int}(\mathscr{C}_{N})$ then, for every $\alpha\in
H_{1}(M_{N},\mathbb{Z})/\mathrm{torsion}$,

$|\\{\gamma\in\mathscr{P}_{N}\hbox{ : }\ell(\gamma)\leq T,\
[\gamma]_{N}=\alpha\\}|\sim
c(m)\frac{e^{\mathfrak{h}_{N}(0)T}}{T^{1+(b+N)/2}},$

as $T\to\infty$, for some $c(m)>0$.

######  Remark 6.5.

One may replace the condition $T-\delta<\ell(\gamma)\leq T$ with
$\ell(\gamma)\leq T$ by a simple argument. We can describe the functions
$c_{\rho,\alpha}$ and constants $c(\alpha)$ more explicitly; see [3], [14],
[15], [26], [27]. If $\gamma$ is null-homologous in $M$ then $[\gamma]_{N}$
allows us to read off the linking numbers of $\gamma$ with $K_{i}$,
$i=1,\ldots,N$.

Following the arguments of [3], we can show that the periodic orbits
considered in Theorem 6.3 become equidistributed with respect to
$\pi_{*}(\mu_{\rho})$, the projection of $\mu_{\rho}$ to $M$, as $T\to\infty$
(where $\mu_{\rho}$ is the equilibrium state defined at the end of section 5).
We have the following result.

######  Theorem 6.6.

Let $M$ be a smooth closed oriented Riemannian $3$-manifold and let
$X^{t}:M\to M$ be a transitive Anosov flow which is not locally constant. For
each $\rho\in\mathrm{int}(\mathscr{C}_{N})$ and $\alpha\in
H_{1}(M_{N},\mathbb{Z})/\mathrm{torsion}$, and for every continuous function
$\varphi:M\to\mathbb{R}$, we have

$\displaystyle\lim_{T\to\infty}\frac{\displaystyle\sum_{\gamma\in\mathscr{P}_{N}\hbox{
: }T-\delta<\ell(\gamma)\leq T,\ [\gamma]_{N}=\lfloor
T\rho\rfloor+\alpha}\frac{1}{\ell(\gamma)}\int_{\gamma}\varphi}{|\\{\gamma\in\mathscr{P}_{N}\hbox{
: }T-\delta<\ell(\gamma)\leq T,\ [\gamma]_{N}=\lfloor
T\rho\rfloor+\alpha\\}|}=\int\varphi\,d\pi_{*}(\mu_{\rho}).$

We end with a result for amenable covers of $M_{N}$. Let $G$ be any quotient
group of $\pi_{1}(M_{N})$, with identity element $1_{G}$. Then each periodic
orbit $\gamma\in\mathscr{P}_{N}$ defines a conjugacy class
$\langle\gamma\rangle_{G}$ in $G$. (The case where $G$ is finite was
considered at the start of the paper.) Suppose that $G$ sits above
$H_{1}(M_{N},\mathbb{Z})$, i.e. that $G=\pi_{1}(M_{N})/\Lambda$, where
$\Lambda$ is a normal subgroup of the commutator
$[\pi_{1}(M_{N}),\pi_{1}(M_{N})]$. Supposing that
$0\in\mathrm{int}(\mathscr{C}_{N})$, the exponential growth rate of periodic
orbits for which $\langle\gamma\rangle_{G}$ trivial is at most
$\mathfrak{h}_{N}(0)$. The results of [8] may be adapted to show that equality
holds if and only if $G$ is amenable. More precisely, we have the following
theorem.

######  Theorem 6.7.

Let $M$ be a smooth closed oriented Riemannian $3$-manifold and let
$X^{t}:M\to M$ be a transitive Anosov flow which is not locally constant and
such that $0\in\mathrm{int}(\mathscr{C}_{N})$. Let $G=\pi_{1}(M_{N})/\Lambda$,
where $\Lambda$ is a normal subgroup of $[\pi_{1}(M_{N}),\pi_{1}(M_{N})]$.
Then

$\lim_{T\to\infty}\frac{1}{T}\log|\\{\gamma\in\mathscr{P}_{N}\hbox{ :
}T-\delta<\ell(\gamma)\leq T,\
\langle\gamma\rangle_{G}=\\{1_{G}\\}\\}|\leq\mathfrak{h}_{N}(0).$

with equality if and only if $G$ is amenable.

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