Olivier Pourret
email: olivier.pourret@unilasalle.fr

David William Hedding
email: heddidw@unisa.ac.za

Dasapta Daniel Enrique Ibarra
email: dibarra@berkeley.edu

Erwin Irawan

Haiyan Liu

Jonathan Peter Tennant

Dasapta Erwin Irawan

International disparities in open access practices in the earth sciences

Keywords: article processing charges, geoscience, open science, predatory journals, preprints, repositories

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Scholarly journals are the primary vehicle for communicating research to other researchers and have traditionally been run by various societies and associations. [START_REF] Fyfe | Untangling academic publishing: a history of the relationship between commercial interests, academic prestige and the circulation of research[END_REF] Some journals have remained independent and continue to be run by scholarly communities, whereas most have been gradually acquired by commercial publishers. [START_REF] Larivière | The oligopoly of academic publishers in the digital era[END_REF] In choosing a journal for submitting their manuscripts, authors consider a number of factors: coverage by indexing services, readership, type of journal, the journal's Impact Factor (IF), language, type of article (a regular research article, a review article, a commentary, and so on), average time taken for peer review, reputation of the journal, and article-processing charges (APCs) or any other charges.

The digital age is also forcing scholarly publishing to undergo a major transformation. A decline in print versions, the high costs of journal subscriptions, and the increasing costs of publication are all spurring scientists to look for alternatives to the traditional scientific publishing. [START_REF] Tennant | Ten hot topics around scholarly publishing[END_REF][START_REF] Pourret | Open access publishing practice in geochemistry: overview of current state and look to the future[END_REF] In addition, according to STM (formerly the International Association of Scientific, Technical and Medical Publishers), [START_REF] Johnson | The STM report: an overview of scientific and scholarly publishing[END_REF] two-thirds of the scholarly literature produced in 2016 remains mostly inaccessible to the public because the work is hidden behind prohibitively expensive subscription paywalls. This constraint is driving the ever-increasing move to open access (OA), which, while generally slow, marks a significant shift in the financial models used by major publishers within a scientific, technical, and medical information publishing market that generated $25.7 billion in 2017. Open access has introduced greater diversity in publishing routes and highlighted major problems related to publishing ethics, such as copyright infringement and inappropriate expenditure of public funds. Ensuring that researchers as authors and their institutions do not have to pay even more to read and publish papers than they currently do has become a critically important part of the OA transition. [START_REF] Tennant | The academic, economic and societal impacts of open access: an evidence-based review[END_REF] Academic publishing through the OA route aims to make more scientific content accessible online and has been around in various forms for almost three decades. However, OA too often gets conflated with just one mechanism, namely the author-facing business model of APCs, whereby authors pay to cover the cost of publishing their papers, [START_REF] Pourret | Comments on "Factors affecting global flow of scientific knowledge in environmental sciences[END_REF] which puts the already disadvantaged academics to even greater disadvantage. Indeed, Pourret et al. [START_REF] Pourret | Comments on "Factors affecting global flow of scientific knowledge in environmental sciences[END_REF] highlight the different forms of OA, which are identified as gold, bronze, green, or diamond OA: neither green nor diamond OA involves APCs, and green OA corresponds to self-archiving by authors on their personal website or in an archive of near-final and peer-reviewed versions of their work. Trusted archives are usually preferable. Diamond OA denotes free (without any APCs) access to content from a journal's website. Gold OA involves APCs for immediate publishing access whereas bronze OA refers to free-to-read articles on the publisher's website but without any explicit open license, which potentially prohibits future re-use of the articles. These distinctions are critical, because APCs typically associated with OA publishing may put to disadvantage researchers from developing countries who do not have the funds to pay the APCs. [START_REF] Hedding | Comments on "Factors affecting global flow of scientific knowledge in environmental sciences[END_REF] Retraction Watch (https:// retractionwatch.com/2020/06/16/failure-fails-as-publisherprivileges-the-privileged/) recently highlighted the possibility that some publishers (MDPI, for example), by favouring the submissions of researchers from developed countries, privilege the privileged even more. This brief account of OA is important when one considers that the ultimate aim of publishing research is to disseminate information and describe advances in science for the good of society, especially in the increasingly important context of the United Nations Sustainable Development Goals, [START_REF] Blanc | Towards integration at last? The sustainable development goals as a network of targets[END_REF] and earth sciences -the focus of this paper -among other fields of science have been organized to solve local problems in dealing with the earth system.

It is against this background that we briefly discuss key differences in publication strategies between earth scientists from six countries across the world. More specifically, we discuss the differences in our experiences and understanding of access (OA versus paywall) and its scope, namely regional versus international.

Material and methods

Raw data were derived from two major indexing databases, namely Scopus and the Web of Science (WoS). Using both the databases, relevant information was extracted for 2018 and limited to affiliations from the six countries in which the authors of this paper work, namely China, France, Indonesia, South Africa, the United Kingdom, and the United States of America (Table 1). The total number of articles and that of OA articles covering the categories 'Geochemistry and Geophysics' and 'Geology' of the Web of Science and the 'Earth and Planetary Sciences' category from Scopus were counted. It should be noted that the same article can be counted more than once if the work was the result of international collaboration. We also selected the top ten earth-science journals (based on the number of articles published by them in the Scopus category 'Earth and Planetary Sciences') that also publish OA articles. For those ten journals, we counted the number of OA articles, calculated the proportion of OA articles to the total number of articles published, and also noted the status (fully OA or hybrid), the amount of APCs, the IF for 2019, and the name of the publisher (Table 2).

Results

In 2018, a total of 13,436 articles were published in the WoS category 'Geochemistry and Geophysics'; 30,189 in the category 'Geology'; and 106,246 in the Scopus category 'Earth and Planetary Sciences' . Of the total of all the three, only 24%-31% were OA, and the UK had the largest share (46%-54%) of the OA articles, much greater than that for any of the other five countries, with China accounting for the lowest share (18%-20%) (Table 1). Between the two databases, Scopus was ahead of WoS in terms of both absolute number and the proportion of OA articles.

Of the top ten journals that publish OA articles in earth sciences, six are fully OA and four are hybrid journals (Table 2). Fully OA journals are mostly published by either emerging publishers (MDPI 10 and Hindawi, for example) or long-time OA publishers (Copernicus Publications, which is the publishing arm of the European Geophysical Union, for example), and their APCs ranged from $1000 to $2200 and the journal IFs, from 1.298 to 5.414 (one of the journals was not indexed). The hybrid journals are published by the older or mainstream publishers including EDP Sciences, Wiley/AGU, Oxford/Royal Astronomical Society, and American Meteorological Society, and their APCs are much higher ($1100-$3035)-as are their journal IFs (4.580-6.209).

Discussion

Open-access policy

Most of the papers in earth sciences from China were earlier published in hybrid journals. [START_REF] Mallapaty | China bans cash rewards for publishing papers[END_REF] This trend can be attributed to a historical incentive for researchers to publish in top journals (that is, journals with high IFs and placed in the first quartile as categorized by the Chinese Academy of Sciences) or, in other words, those that publish the mostreliable work. [START_REF] Brembs | Prestigious science journals struggle to reach even average reliability[END_REF] In many cases, one of the easiest options for Chinese authors was to publish their research in a 'high-impact' predominantly English hybrid journal without paying the APCs and place their paper behind a paywall simply because they could not afford to pay the APCs. However, this policy changed in early 2020. China recently published a national-level policy to ban the use of journal-based metrics as assessment criteria for academic promotion and recruitment, which should in future give priority to innovation and significance of the representative achievements in solving practical problems. [START_REF] Mallapaty | China bans cash rewards for publishing papers[END_REF] Further, publication in a restrictive list of Chinese journals is being proposed as one of the prerequisites to apply for top national awards. A move away from high IF journals to Chinese journals could be a real game changer, because Chinese researchers are prolific in publishing. [START_REF] Tollefson | China declared world's largest producer of scientific articles[END_REF] This policy is seen as a responsible first step in reforming the evaluation of research in China and may encourage other nations to adopt similar policies. Specifically, the new policy tackles perverse incentives that drive the 'publish or perish' culture that might be encouraging questionable research practices. Owing to the drive to address (local) practical problems in this new policy and the need to target a specific audience, more research will probably be published in Chinese national journals (Acta Petrologica Sinica and Geology in China, for example), the majority of which levy APCs and are fully OA by default and continue to feed the common misconception that to comply with OA, authors need to pay APCs. Indeed, there are toll-access journals -journals that are not OA -that have page charges and there are OA journals without any page charges or APCs.

Pourret et al. [START_REF] Pourret | Open access publishing practice in geochemistry: overview of current state and look to the future[END_REF] maintain that publicly funded research in the United Kingdom has to be made available through OA in order to abide by the UK Research and Innovation Policy: UK research councils provide universities with a tranche of money specifically dedicated to cover the costs of gold OA publishing in the form of APCs. Each university then uses that tranche in whichever way it sees fit: some cover the cost of gold OA on a first-come, first-served basis, whereas others favour publications they believe will have a higher impact. Any publication not selected for gold OA (for example because it was not deemed impactful enough or because the money has run out) has to be published through the green OA route at no charge to its authors, and there is a general policy of self-archiving in order for works to be eligible for assessment in the UK's Research Excellence Framework. Some universities also have restrictions on publishing in hybrid journals because of their lower quality and relatively high costs. Currently the Joint Information Systems Committee (https://www.jisc.ac.uk) is negotiating national-level agreements with commercial publishers. These contracts involve donating millions of pounds of public money each year to sustain the dysfunctional commercial publishing sector while simultaneously neglecting to invest in open scholarly infrastructure and thus, although often termed 'transformative agreements' , it can be argued that a more accurate term could be 'stagnation agreements' . This situation is being replicated in many countries around the world as they try to realign themselves with recent changes implied by Plan S. [START_REF] Tennant | How can we achieve a fully open future?[END_REF] The movement around Plan S (https://www.coalition-s. org), a funder-led initiative launched in September 2018, aims to accelerate full transition towards OA. These initiatives have opened up discussions on journals' and research communities' abilities to correctly and sustainably shift towards a dominantly OA model. [START_REF] Pourret | Open access publishing practice in geochemistry: overview of current state and look to the future[END_REF] Leaders in the field of higher education in many countries including South Africa are looking to move to a European model. [START_REF] Paterson | New publishing model pins hopes on 'unity of purpose[END_REF] One major consequence of Plan S in the UK is an increase in the number of OA publications with a shift from hybrid journals to fully OA journals. [START_REF] Pourret | The growth of open access publishing in geochemistry[END_REF] Other countries such as Denmark and France (and most of the European countries) are having considerably more success by investing, through libraries, in green OA as part of their national policy. [START_REF] Pourret | Comments on "Factors affecting global flow of scientific knowledge in environmental sciences[END_REF] This policy means that, to the largest possible extent, researchers and their institutes ensure that a peerreviewed copy of a manuscript that is accepted for publication is uploaded to the appropriate institutional repositories whenever legally and technically possible. [START_REF] Pourret | Comments on "Factors affecting global flow of scientific knowledge in environmental sciences[END_REF] In Denmark, a green OA policy has existed since 2016. This policy, as described above, does not constrain researchers in their choice of a publishing channel, because virtually all journals allow such uploading or even deposit articles automatically in appropriate repositories, after an embargo period, often on behalf of authors. At the moment, about 45% of the annual research output of Denmark is being uploaded into its universities' repositories (https://www.oaindikator.dk/en/), a share comparable to that of France: 49% of the research publications in France in 2018 are available as OA (including 'green' OA, not considered in our evaluation; https://ministeresuprecherche. github.io/bso/). Furthermore, some libraries in Denmark have allocated specific funds for paying APCs, albeit requiring that the corresponding manuscripts be made available through institutional repositories as well for the publications to qualify as green OA in the sense of the national policy.

Indonesia recently became the world leader for publishing research through OA, thanks largely to the efforts to index its journals in the Crossref registry. [START_REF] Van Noorden | Indonesia tops open-access publishing charts[END_REF] However, this significant shift to OA scientific publishing is yet to change the way the Indonesian government evaluates staff performance and research impact. [START_REF] Irawan | Indonesia publishes the most open-access journals in the world: what it means for local research[END_REF] The newest regulation on staff promotion in Indonesia, released in January 2020, continues to favour metric-based measurement of research output by giving a maximum score of 40 to articles published in journals with higher IFs but only 25 to those published in local journals. Publishing in journals with high IFs or in journals in the first quartile of the Scimago list is mandatory for being promoted to a professor. Currently all major universities in Indonesia not only pay the APCs but also offer other incentives for publishing in the so-called reputable journals (first-quartile journals). The way the Indonesian government allocates funding to higher education has been distorted as more money flows into the endpoint of research and innovation rather than into constructing research infrastructure. In the long run, this policy does not create resilience in the local academic or research ecosystem. A similar phenomenon is seen in other nations such as India, a particularly relevant example because the country is planning to implement a one nation -one subscription plan. [START_REF] Irawan | India's plan to pay journal subscription fees for all its citizen may end up making science harder to access[END_REF] South Africa currently has no formal policies to publish OA. Although not linked directly to any policy, universities are known to pay APCs 20 but this monetary support is typically capped at less than $900. There is no stipulation that such support must be used for various types of OA or for publishing in hybrid journals, the only criterion being that the journal must be accredited by the South African Department of Higher Education, Science and Innovation. Unfortunately, some universities in South Africa pay a bonus to faculty members for publishing in the accredited journals (irrespective of whether they are predatory or OA). This incentive has resulted in some dubious publication practices. [START_REF] Hedding | Payouts push professors towards predatory journals[END_REF] In USA, OA policies in the earth sciences community seem to be mixed. Gold OA is covered by some universities, typically those with sufficient resources, or whenever grants specifically budget for it. Additionally, some institutions, in conjunction with their subscription plans, have negotiated discounted APCs with society journals such as those published by AGU. These arrangements are viewed as beneficial to publishers, institutions, and societies because the arrangements help to keep the subscriptions down. However, as evidenced by recent ongoing negotiations (since 2019) between Elsevier and the University of California system, such arrangements can be a sticking point and have resulted in the University of California unsubscribing Elsevier's journals and calling for faculty to withdraw from editorial boards of those journals. Major US societies such as AGU and the Geochemical Society host journals with hybrid OA options as well as full OA. Further, funding agencies such as the National Science Foundation require principal investigators to deposit papers arising from the work funded by its grants into a public-access repository (https://par.nsf.gov/).

Global inequalities

The philosophy dominated by APCs has created a complex system and a hierarchy of financial privileges around OA publishing. [START_REF] Pourret | The growth of open access publishing in geochemistry[END_REF] Gold OA is now mostly funded by institutions through 'read and publish' agreements or by direct support from agencies that fund research and in some cases by researchers themselves. The non-OA and green self-archiving routes are typically funded only by institutions and funding agencies (because there are no author-facing charges). If a researcher happens to be with an institution that can afford to pay both APCs and journal subscriptions, this does not seem to be a major hurdle: the cost is supported externally, and authors have no incentive to publish in less-expensive platforms that might be perceived as having a lower value. However, for other institutions (with lower budgets) and for researchers working in them, cost certainly remains an obstacle. Although many authors seem to equate OA with a specific form of business model (APC-driven gold OA), this is clearly erroneous, and the myth should be dispelled. In this situation, those researchers who can afford to publish in OA journals, particularly in those with a high IF and steep APCs, have an advantage over those who enjoy no such financial security and are restricted in their choice of journals owing to their inability to afford APCs. Given that it is now recognized that OA publishing tends to lead to increased 'impact' for researchers, [START_REF] Mckiernan | How open science helps researchers succeed[END_REF] the inherent bias of the current APC-based OA publishing perpetuates this inequality through the 'Matthew effect' (the rich get richer and the poor get poorer). The switch from pay-to-read to pay-to-publish has left essentially the same people behind, 8 with some academics not having enough purchasing power (individually or through their institutions) for either option. [START_REF] Ross-Hellauer | Open science: who is left behind? LSE Impact Blog[END_REF] Virtually everyone who might benefit from access to research has limited access to papers behind paywalls. Hedding 8 notes that for many countries it is extremely expensive for university libraries and non-academics to pay to access published scientific content, a problem even greater for financially poorer nations. Open access may conceptually address these shortcomings by providing greater access to readers, but that often simply shifts the financial burden to researchers. Moving towards OA creates inequalities between countries that have substantial financial resources and those that find it difficult to pay the often high APCs. That some countries have allowed the scholarly publishing system to essentially become a public financing machine for this inequity is a paradox and indicates horrendous mismanagement of relevant funding streams, failure to understand even basic market principles, and the diversion of public funds to protect the commercial sector. However, Indonesia has more than 1570 OA journals and ranks second, next only to the UK, in the Directory of Open Access Journals (DOAJ; https://doaj.org/). The majority, about 70%, of these journals do not levy APCs [START_REF] Irawan | Era baru publikasi di Indonesia: status jurnal open access di Directory of Open Access Journal (DOAJ)[END_REF] and are funded by local universities and research institutions and published locally. Journals listed in the DOAJ mostly publish English-language articles but only represent one-sixth of the Indonesian journals listed in GARUDA, the country's national database (http:// garuda.ristekdikti.go.id/), which currently indexes more than 1.1 million articles published in more than 9600 Indonesian journals maintained by more than 1600 publishers. Journals that charge only moderate APCs (Indonesian Journal on Geoscience, for example) are mostly the ones that are indexed by the indexing services and are considered to be of higher quality as a result. Regulations in Indonesia's higher education system give a higher score to articles published in journals and conference proceedings listed in Scopus, which lists only 47 Indonesian journals at present. However, even indexing services such as Scopus have been infiltrated by predatory journals, [START_REF] Hedding | Payouts push professors towards predatory journals[END_REF] which continue to threaten the credibility of current scholarship systems. Nevertheless, journals indexed by Scopus are now considered to be the elite journals in Indonesia. Either way, it is important to note the perplexing scenario in which the current Indonesian evaluation system seems explicitly designed to penalize Indonesian researchers who share and publish their work in Bahasa Indonesia (the Indonesian language) and Indonesian journals.

Access to global literature appears to be declining whereas it was expected to increase in the era of a globalizing world and the World Wide Web. For example, Hedding [START_REF] Hedding | Comments on "Factors affecting global flow of scientific knowledge in environmental sciences[END_REF] notes that researchers, particularly students and non-academics (including policymakers), in many poor countries from the Global South have increasingly limited access to papers behind paywalls. Thus, the ultimate goal of OA publishing should be to make research more accessible to researchers, students, and non-academics. [START_REF] Hedding | Comments on "Factors affecting global flow of scientific knowledge in environmental sciences[END_REF] Although from a different perspective, the push for the decolonization of research in South Africa has raised similar concerns. [START_REF] Breetzke | The changing and challenging research landscape in South Africa[END_REF] Nordling 26 explains that decolonization is a movement to eliminate, or at least to minimize, the disproportionate legacy of white European thought and culture in education (including research) but notes later that decolonization is not well defined in the natural sciences and its relevance is even contested. Nevertheless, some South African researchers bemoan the lack of credit for publishing in local African journals. [START_REF] North | Out of Africa: the underrepresentation of African authors in high-impact geoscience literature[END_REF] This is even truer of other African countries (Democratic Republic of Congo, for example). [START_REF] Sooryamoorthy | The production of science in Africa: an analysis of publications in the science disciplines, 2000-2015[END_REF] As highlighted for Indonesia and as OA voices from the Global South (especially Brazil and Mexico) have shown, green OA can be successful without capitulating to corporate publishers or expecting authors to pay high APCs. [START_REF] Scherlen | Building bridges for social justice in global publishing: seeking the Mexican perspective[END_REF] Although more recognition could be given to African researchers publishing in African journals, the potential threat of predatory journals to African research communities is relatively high. [START_REF] Mouton | The extent of South African authored articles in predatory journals[END_REF] Therefore, although African researchers (and other researchers from the least developed countries) need to publish locally, this should be done while maintaining quality-the same problem faced by much of the rest of the world. To compound the problem in the case of the earth sciences community in Africa, very few local journals focus on earth sciences, for example the Journal of African Earth Science published by Elsevier with virtually unaffordable OA options for African researchers. These inequalities have led to inequities. Indeed, according to the DOAJ, approximately 71% of fully OA journals do not levy APCs but that seems to apply to only a few journals in earth sciences (Geochemical Perspectives Letters and Volcanica, for example). The bias inherent within the current APC-based OA publishing perpetuates the inequality through the Matthew effect, ultimately reinforcing the journal-coupled prestige economy that currently governs our global research systems. [START_REF] Pourret | The growth of open access publishing in geochemistry[END_REF] We are well aware that our study has some limitations: our analysis was limited to only one year (given the dynamic nature of the data, especially on APCs); that we considered only six countries (although they contributed 64%-70% of the articles and 69%-79% of the OA articles); and that we chose only two databases (although the most important ones). However, we do maintain that our findings offer some valuable insights.

Conclusion

Although being mindful of the major disparities described above, the most important thing is to conduct research and disseminate its findings as widely as possible. We therefore call for greater unification of the global earth sciences community to focus on non-profit and community-driven solutions for OA publishing and open science (the EarthArXiv, for example). Indeed, the migration of EarthArXiv to new infrastructure as a result of an emerging collaboration with California Digital Library is a good opportunity to further highlight the need for not-for-profit and community-driven infrastructure for preprint repositories. It is time to return the sovereignty of research in earth sciences to those who perform it and to those who need it-as reflected by the earth sciences community, it is essential to continue the ongoing discussion on the 'bibliodiversity' manifesto.

Table 1 . Open-access publishing in earth sciences, by country

 1

				Web of Science category				Scopus category
		Geochemistry and Geophysics		Geology		Earth and Planetary Sciences
		Total	Number	Propor-	Total	Number	Propor-	Total	Number	Proportion
	Country	number of	of OA	tion of OA	number	of OA	tion of OA	number of	of OA	of OA articles
		articles	articles	articles (%)	of articles	articles	articles (%)	articles	articles	(%)
	China	3,492	651	19	7,277	1,295	18	30,877	6,321	20
	France	1,145	333	29	2,085	851	41	6,949	2,916	42
	Indonesia	26	6	23	173	91	53	561	254	45
	South Africa	134	26	19	415	81	20	1466	538	37
	UK	1,063	574	54	2,545	1,618	64	9,749	4,516	46
	USA	3,569	902	25	6,761	2,523	37	25,108	11,486	46
	Total	13,436	3,271	24	30,189	9,369	31	106,241	33,135

31

Note: All data as accessed on 26 Feb. 2020.

Table 2 . Top ten journals in earth sciences (by number of articles indexed in Scopus)

 2

		Open-access articles		Other information
		Number	%	Publishing model Article-processing	Impact Factor	Publisher
	Journal				charges ($)	for 2019
	Remote Sensing	1,963	100% Fully open access	CHF 2200	4.118	MDPI
	Astronomy and	1,805	99% Hybrid	3000	6.209	EDP Sciences
	Astrophysics
	Atmospheric Chemistry	941	100% Fully open access	1000	5.414	Copernicus/EGU
	and Physics
	Geophysical Research	700	48% Hybrid	2500	4.580	Wiley/AGU
	Letters
	Monthly Notices of the	573	17% Hybrid	3035	5.356	Oxford/Royal Astronomical
	Royal Astronomical						Society
	Society
	Minerals	556	100% Fully open access	CHF 1800	2.380	MDPI
	Shock and Vibration	504	100% Fully open access	2200	1.298	Hindawi
	Geosciences	450	100% Fully open access	CHF 1200	-	MDPI
	Journal of Climate	417	77% Hybrid	1100	5.707	American Meteorological
							Society
	Biogeosciences	416	100% Fully open access	1000	3.480	Copernicus/EGU
	Other journals	97,916	26% -
	Total	106,241	31% -

Acknowledgement

In memory of Jonathan P Tennant: You opened so much for so many. It is your time to have your way opened. Take some rest, Jon. You were too young to die; we miss you.

Funding

This research did not receive any funding.

Conflict of interest

The authors declare no conflict of interest.

Authorship contributions

OP prepared the manuscript with contributions from all co-authors.


Amira Mchaalia

Amira Mchaalia On

ON TOTALLY PERIODIC ω-LIMIT SETS FOR MONOTONE MAPS ON REGULAR CURVES

Keywords: 2000 Mathematics Subject Classification. 37B20, 37B45, 54H20 Minimal set, regular curve, ω-limit set, totally periodic, monotone map

published or not. The documents may come

Introduction

The structure of ω-limit sets plays an important role in studying dynamical system since it represents asymptotically the state of the orbit. A class of ω-limit sets composed of periodic points called totally periodic ω-limit is investigated. This notion was introduced by Marzougui and Naghmouchi in [START_REF] Marzougui | On totally periodic ω-limit sets[END_REF]. A compact metric space X is said to have the ω-FTP property if for each self continuous map f of X, every totally periodic ω-limit set of a point x ∈ X is finite. We mention that totally periodic ω-limit set appeared in the definition of ω-chaos, introduced by Li in [START_REF] Li | ω-chaos and topological entropy[END_REF]. It also appeared as ω-limit set of "spiral points" as proved in [2, Proposition 3.2] by Artigue. In zero-dimension, it was shown in [START_REF] Marzougui | On totally periodic ω-limit sets[END_REF] that, a zero dimensional compact metric space has the ω-FTP property. In one-dimension, it was shown in ( [START_REF] Abdelli | ω-limit sets for monotone local dendrite maps[END_REF]) that any local dendrite has the ω-FTP property when we restricted to monotone maps; that is any totally periodic ω-limit set of a monotone map on a local dendrite is finite. The later result does not hold when we remove the restriction of monotonicity; it was shown in [START_REF] Marzougui | On totally periodic ω-limit sets[END_REF] that, there is a continuous map on a dendrite having an infinite totally periodic ω-limit set. In [START_REF] Askri | On totally periodic ω-limit sets in regular continua[END_REF]. it was shown that a regular curve has the ω-FTP property when we restricted to one-to-one continuous maps. In this line, we generalize in the present paper the above results to monotone maps on regular curves.

Our main result is the following theorem.

Theorem 1.1. Let X be a regular curve and f : X → X be a monotone map. Then any totally periodic ω-limit set of f is finite.

In other words, a regular curve has the ω-FTP property when we restricted to monotone maps.

Notice that Theorem 1.1 cannot be extended to other classes of systems such as rational curves; for instance, it was constructed in [START_REF] Naghmouchi | Dynamics of monotone graph, dendrite and dendroid maps[END_REF]Theorem D] an example of a homeomorphism of a rational curve (in fact a dendroid) for which some points have infinite ω-limit sets consisting only of fixed points.

On the other hand, further results are obtained with regard to expansivity, we prove that any totally periodic ω-limit set for expansive continuous map on a compact metric space X is finite (see Proposition 5.2).

Preliminaries

Let X be a compact metric space with metric d and let f : X → X be a continuous map. Let Z, Z + and N be the sets of integers, non-negative integers and positive integers, respectively. For n ∈ Z + denote by f n the n-th iterate of f ; that is, f 0 = id X : identity of X and

f n = f • f n-1 if n ∈ N. A point x ∈ X is called periodic of period n ∈ N if f n (x) = x and f i (x) = x for 1 ≤ i ≤ n -1. If n = 1, x is a fixed point of f . For any x ∈ X, the subset Orb f (x) = {f n (x) : n ∈ Z + } is called the orbit of x (under f ). If x is a periodic point, the orbit of x is called periodic orbit.
We denote by P (f ) and Fix(f ) the set of periodic points and fixed points of f , respectively.

A subset A ⊂ X is called f -invariant if f (A) ⊂ A. It is called strongly f -invariant if f (A) = A.
We define the ω-limit set of a point x ∈ X to be the set

ω f (x) = {y ∈ X : ∃ n i ∈ N, n i → +∞, lim i→+∞ d(f n i (x), y) = 0} = ∩ n∈N {f k (x) : k ≥ n}.
It is non-empty, closed and strongly f -invariant set (cf. [4, Chapter IV, Lemma 2]). For any subset A of X, we denote by A, Bd(A) and int(A) the closure, boundary and interior of A, respectively. The symbol card(A) stands for the cardinality of A and diam(A) = sup

x,y∈A d(x, y) means the diameter of A.

We denote by B(x, ε) the open ball centered at x of radius ε.

Definition 2.1 ([8], p. 131). Let X, Y be two topological spaces. A continuous map f : X -→ Y is said to be monotone if for any connected subset C of Y , f -1 (C) is connected.
When f is closed and onto, Definition 2.1 is equivalent to that the preimage of any point by f is connected (cf. [START_REF] Kuratowski | Topology[END_REF], p. 131). Notice that f n is monotone for every n ∈ N when f itself is monotone.

In the definitions below, we use the terminology from Nadler [START_REF] Nadler | Continuum Theory: An Introduction[END_REF].

-A continuum is a compact connected metric space. -A space is called degenerate if it is a single point, otherwise it is called non-degenerate.

-A dendrite is a locally connected continuum which contains no circle.

-A local dendrite is a continuum such that every point of which has a dendrite neighborhood. A local dendrite is then a locally connected continuum containing only a finite number of circles (cf. [START_REF] Kuratowski | Topology[END_REF], Theorem 4, p. 303).

-A regular curve is a continuum every point of which x and each open neighborhood V of x in X, there exists an open neighborhood U of x in X such that U ⊂ V and the boundary set Bd(U ) of U is finite. Each regular curve is locally arcwise connected. Every sub-continuum of a regular curve is a regular curve (cf. [START_REF] Kuratowski | Topology[END_REF] and [START_REF] Nadler | Continuum Theory: An Introduction[END_REF]). Notice that every local dendrite is a regular curve (cf. [START_REF] Kuratowski | Topology[END_REF], p. 303). There exists a regular curve which does not a local dendrites for example the Sierpińsky triangle.

-A hereditarily locally connected is a continuum such that every sub-continuum of it is locally connected.

-A Peano continuum is a locally connected continuum. In particular, each regular curve is a Peano continuum.

-A continuum X is said to be a dendroid provided that X is an arcwise connected hereditarily unicoherent space; that is if X = A ∪ B and A, B are subcontinua of X, then A ∩ B is connected. Given a compact metric space X with metric d, we denote by 2 X (resp. C(X)) the set of all non-empty compact subsets (resp. continua) of X. We endow 2 X with the Hausdorff metric d H defined as follows: [START_REF] Nadler | Continuum Theory: An Introduction[END_REF], for more details). For any B ∈ C(X), we denote by mesh(B) = sup{diam(C) : C is a connected component of B}.

For A, B ∈ 2 X , d H (A, B) = max(sup a∈A d(a, B), sup b∈B d(b, A)), where d(x, M ) = inf y∈M d(x, y) for any x ∈ X and M ∈ 2 X . For A = {a}, we write simply, d H (A, B) = d H (a, B). It is well known that (2 X , d H ) is a compact metric space (see

Some results

Lemma 3.1. Let X be a compact metric space and f be a continuous self-mapping of X. Then for all n ∈ N, P (f n ) = P (f ). Lemma 3.2. Let X be a metric compact space and f a continuous self mapping of X. Then for all

x ∈ X, ω f (f i (x)) = ω f (x), for all i ∈ N. Lemma 3.3. ([4], p. 70) Let X be a metric compact space and f a continuous self mapping of X. Then for all N ∈ N and x ∈ X, ω f (x) = ∪ N -1 i=0 ω f N (x) (f i (x)). Lemma 3.4. ([11, Theorem 1.1])
Let X be a compact metric space, f be a continuous self-mapping of X and x ∈ X. If ω f (x) is totally periodic, then ω f (x) has finitely many connected components that form a periodic cycle.

Recall that a p-tuple

(A 0 , . . . , A p-1 ) of subsets of X is called a periodic cycle if f (A 0 ) = A 1 , f (A 1 ) = A 2 , . . . , f (A p-1 ) = A 0 . Proposition 3.5. ([9, Theorem 1.6]) Let (C n ) n∈N be a sequence of disjoint sub-continua of a regular curve X. Then lim n→+∞ diam(C n ) = 0.
Lemma 3.6. ([8, Theorem 4, p. 257]) Let X be a Peano continuum and ε > 0 be a given number. Then there exists η > 0 such that for every pair of points (x, y) such that d(x, y) < η, there exists an arc I joining x and y with diam(I) < ε.

Lemma 3.7. ([5, Lemma 4.2]) If X is hereditarily locally connected, then for any sequence (A n ) n≥0 in C(X) that converges to A ∈ C(X) (with respect to Hausdorff metric), we have lim n→+∞ mesh(A n \ A) = 0.
Lemma 3.8. Let X be a regular continuum.Then for each sub-continuum A of X, there is a decreasing sequence (U n ) n∈N of open connected neighborhoods of A with finite boundary such that lim

n→+∞ mesh(U n \ A) = 0. Proof. It is clear that A = n∈N B(A, 1 n ), where B(A, 1 n ) = {x ∈ X : d(x, A) < 1 n }.
Since A is compact and X is locally connected, there exists, for each

n ∈ N, an open connected neighborhood V n of A in B(A, 1 n ) with finite boundary such that A = n∈N V n . For each n ∈ N, set W n = n i=1 V i . Then (W n ) n∈N is
mesh(U n \ A) = 0. Lemma 3.9. Let F be a non degenerate closed subset of X. Then (1) η(F ) := inf a∈F d H (a, F ) > 0.
(2) For any a ∈ F , there is c ∈ F such that d(a, c) ≥ η(F ).

Proof. (1) Since d(a,F) = 0, for a ∈ F , so d H (a, F ) = sup b∈F d(a,b). There exists a 0 ∈ F and b 0 ∈ F such that inf a∈F d H (a, F ) = d H (a 0 , F ) = d(a 0 , b 0 ). If a 0 = b 0 , so d(a 0 , b) = 0, ∀b ∈ F . This implies that F = {a 0 }, a contradiction since F is non degenerate. We conclude that η(F ) = inf a∈F d H (a, F ) > 0.
(2) For any a ∈ F , d H (a, F ) ≥ η(F ) and so there is

c ∈ F such that d(a, c) = d H (a, F ) ≥ η(F ).
4. Proof of Theorem 1.1

Suppose the theorem does not hold. Then we could find x ∈ X such that ω f (x) totally periodic (that is ω f (x) ⊂ P (f ) ) and infinite.

Lemma 4.1. Let x ∈ X such that ω f (x) is totally periodic. Then, there are N ∈ N and 0 ≤ i ≤ N -1 such that:

(1) int ω f N (f i (x)) Fix(f N ) ∩ ω f N (f i (x)) = ∅.
(

) ω f N (f i (x)) is connected. Proof. (1) As ω f (x) is totally periodic, so ω f (x) = ∪ n∈N Fix(f n ) ∩ ω f (x) 2
. By the Baire property, there is N ∈ N such that

V := int ω f (x) Fix(f N ) ∩ ω f (x) = ∅.
As

ω f (x) = N -1 i=0 ω f N (f i (x))( Lemma 3.3), so there exists 0 ≤ i ≤ N -1 such that V ∩ ω f N (f i (x)) = ∅. As V ∩ ω f N (f i (x)) ⊂ Fix(f N ) ∩ ω f N (f i (x)), thus int ω f N (f i (x)) Fix(f N ) ∩ ω f N (f i (x)) = ∅.
(2) Set g = f N and y = f i (x). From (1), int ωg(y) Fix(g) ∩ ω g (y) = ∅. We have ω g (y) ⊂ ω f (y) = ω f (x) (Lemma 3.2). As P (f ) = P (g) (Lemma 3.1), so ω g (y) ⊂ P (g). Following Lemma 3.4,

ω g (y) = m-1 i=0 C i , for some m ∈ N,
where the (C i ) 0≤i≤m-1 are the connected components of ω g (y) with g(C i ) = C i+1 for every i ∈ {0, . . . , m -1}. As ω g (y) ∩ Fix(g) = ∅, so there is a ∈ ω g (y) ∩ Fix(g). Therefore a ∈ C i for every i ∈ {0, . . . , m -1}, which implies that ω g (y) = C 0 . In result ω g (y) is connected.

Set g = f N and y = f i (x). From Lemma 4.1, we have the following properties.

-ω g (y) ⊂ P (g).

-int ωg(y) Fix(g) ∩ ω g (y) = ∅.

-ω g (y) is a regular curve (since ω g (y) is a sub-continuum of X).

It follows that we can find an open connected set O ⊂ Fix(g) ∩ ω g (y). For the sequel, we let S a = ∪ such that V ⊂ O \ {a 1 , a 2 , . . . , a k }. This implies that for all a ∈ V , S a ⊂ U . The proof is complete.

> i ∈ N. As a, b ∈ Fix(g), a = g j-i (g i (c)) = g j-i (b) = b,
Step 1. For F = O, we denote by η := η(F ) > 0 be given as in Lemma 3.9. By Lemma 3.8, there exists a sequence of open connected neighborhoods (U n ) n∈N of ω g (y) in X with finite boundary such that mesh(U n \ ω g (y)) < η 2 . We let U := U N and set card(Bd(U)) = k. From Lemma 4.2, there exists an open connected subset V ⊂ O of ω g (y) such that for all a ∈ V , S a ⊂ U . Let a ∈ V . As {a} ω g (y), there is an open connected neighborhood W of a in X such that W ∩ ω g (y) ⊂ V and ω g (y) \ W = ∅. We can find two increasing sequences of integers (n i ) i∈N , (t i ) i∈N with the following properties:

(1)

g n i (y) ∈ W with lim i→+∞ g n i (y) = a, (2)
g n i +j (y) ∈ W for each j = 0, . . . , t i , with t i ≥ k and Since ω g (y) is infinite and ω g (y) ⊂ P (g), so g m (y) / ∈ ω g (y), for every m ∈ N k . Let I m , m ∈ N k , be the connected component of T m \ ω g (y) containing g m (y). Then one can write I m = [g m (y), a m ) with I m ∩ ω g (y) = {a m }. We see that a m ∈ W ∩ ω g (y) ⊂ V ⊂ O ⊂ Fix(g) ∩ ω g (y), for any m ∈ N k .

t i ≥ i, ∀ i ∈ N, (3)
g n i +t i +1 (y) ∈ X \ W and since b is a fixed point of g, lim i→+∞ g n i +t i -k (y) = b = lim i→+∞ g n i +t i +1 (y) ∈ W \ W. Observe that a ∈ W ∩ ω g (y) and b ∈ (W \ W ) ∩ ω g (y). So a = b. We let N k = ∪ i∈N {n i , . . . , n i + t i -k}. Since lim i→+∞ t i = +∞, N k is infinite. By definition, g m (y) ∈ W , for each m ∈ N k and lim m→+∞,m∈N k d(g m (y), W ∩ ω g (y)) = 0. For each m ∈ N k , there is b m ∈ W ∩ ω g (y) such that d(g m (y), W ∩ ω g (y)) = d(g m (y), b m ). Since W is a sub-continuum of X,
Claim 1. Under the notations above, there exists N ∈ N such that for all m ∈ N k , m ≥ N , the family

F m = {I m+j : 0 ≤ j ≤ k, m + j ∈ N k } is not pairwise disjoint.
Proof. Suppose that, there exists an increasing sequence of positive integers (m i ) i≥1 such that the family F m i is pairwise disjoint. By passing to a subsequence of (m i ) i≥1 , one can assume that (I m i ) i≥1 converges to I ∈ C(X) (in Hausdorff metric). As lim i→+∞ diam(T m i ) = 0, then lim i→+∞ diam(I m i ) = 0 = diam(I), so I is degenerate i.e. I = {c}. As

g m i (y) ∈ I m i ⊂ W and lim i→+∞ d H (c, I m i ) = 0, so lim i→+∞ g m i (y) = c ∈ W ∩ ω g (y). Since W ∩ ω g (y) ⊂ V ⊂ O ⊂ Fix(g), so for all 0 ≤ j ≤ k, lim i→+∞ g m i +j (y) = g j (c) = c.
C j = g -(mp-m) (I mp+j ), m p + j ∈ N k .
By monotonicity of g, C j is a sub-continuum of X. Then there is an arc J j = [g m+j (y), a m p+j ] ⊂ C j . Each J j meets O c in at least g m+j (y) and meets O c in at least a mp+j ∈ Fix(g) ∩ ω g (y). Let us show that J j \ ω g (y) = J j \ {a m p+j }:

It is clear that J j \ ω g (y) ⊂ J j \ {a m p+j }. Conversely, if z ∈ J j ∩ ω g (y), then g mp-m (z) ∈ I m p+j ∩ ω g (y) = {a m p+j }. As a m p+j ∈ Fix(g), then g mp-m (a m p+j ) = g mp-m (z). As ω g (y) ⊂ P (g), then the restriction g mp-m |ωg(y)
is a homeomorphism and therefore z = a m p+j . In result, J j \ {a m p+j } ⊂ J j \ ω g (y) and hence the equality holds. We conclude that J j \ {a m p+j } is connected and hence diam(J j ) > η 2 . Moreover we have that J j ⊂ U (since otherwise, J j \ {a m p+j } ⊂ U \ ω g (y) and so diam(

J j \ {a m p+j }) ≤ mesh(U \ ω g (y)) < η 2 . A contradiction). Therefore C j ∩ Bd(U ) = ∅, for each m p + j ∈ N k . Let us show that C j ∩ C j ∩ Bd(U ) = ∅ for all j = j . We have I mp+j ∩ I mp+j ⊂ {a m p+j } for j = j (since a m p+j = a m p+j ). Then C j ∩ C j = g -(mp-m) (I mp+j ∩ I mp+j ) ⊂ g -(mp-m) (a m p+j ) ⊂ S am p+j .
As a mp+j ∈ V , so by Lemma 4.2, S am p+j ⊂ U . Hence C j ∩ C j ⊂ U and so C j ∩ C j ∩ Bd(U ) = ∅. We conclude that Bd(U ) has at least k + 1 points, which contradicts Bd(U ) = k. This completes the proof of Claim 1.

Step 2. From Claim 1, there is some m

∈ N k with m + j 1 (m), m + j 2 (m) ∈ N k such that I m+j 1 (m) ∩ I m+j 2 (m) = ∅. Hence I m+j 1 (m) ∪ I m+j 2 (m)
is arcwise connected and so we can find an arc

P m = [g m+j 1 (m) (y), g m+j 2 (m) (y)] ⊂ I m+j 1 (m) ∪ I m+j 2 (m) .
We have that P m ∩ ω g (y) = ∅. Since P m ⊂ I m+j 1 (m) ∪ I m+j 2 (m) , we have lim

m→+∞,m∈N k diam(P m ) = 0. Claim 2. Let B m = g -j 1 (m) (P m ), m ∈ N k . Then
g m l +j 1 (m l ) (y) = c ∈ ω g (y)
. As P m l ∩ ω g (y) = ∅ for all l ∈ N, so c / ∈ P m l . Then we can construct a subsequence (P m lq ) q≥0,m lq ∈N k which is pairwise disjoint: indeed, as (P m l ) m l ∈N k converges to {c}, then for each neighborhood U 0 of c, there is n 0 ∈ N such that for all l ≥ n 0 , P m l ⊂ U 0 . Since c / ∈ P m l , for each m l ∈ N k , so we can find an open neighborhood U 1 ⊂ U 0 of c such that P mn 0 U 1 . Moreover, since (P m l ) m l ∈N k converges to {c}, so there is n 1 ∈ N such that for all l ≥ n 1 , P m l ∈ U 1 , we obtain P mn 0 ∩ P mn 1 = ∅. We continue in this fashion obtaining a subsequence (P m lq ) q≥0,m lq ∈N k which is pairwise disjoint. Since 0 ≤ j 1 (m lq ) ≤ k, we can assume that j 1 (m lq ) = j 0 , for all q ∈ N. Then the B m lq = g -j 0 (P m lq ), q ∈ N, are connected and pairwise disjoint sets. Hence, by Proposition 3.5, lim q→+∞,m lq ∈N k diam(B m lq ) = 0. This contradicts the fact that diam(B m l )> δ, for all l ∈ N, m l ∈ N k . This completes the proof.

End of the proof of Theorem 1.1. Recall that ω g (y) ⊂ P (g) and ω g (y) ∩ Fix(g) = ∅ is infinite. From the notation given in Step 1, we have that lim i→+∞ g n i (y) = a and lim i→+∞

g n i +t i -k (y) = b with a = b ∈ ω g (y) ∩ Fix(g).
From Claim 2, B m = g -j 1 (m) (P m ) is connected for each m ∈ N k and contains the points g m (y) and g m+j 2 (m)-j 1 (m) (y). Moreover, for each m ∈ N k , we have B m ∩ ω g (y) = ∅; otherwise, there is m ∈ N k such that z ∈ B m ∩ ω g (y) and so g j 2 . Since lim m→+∞ d (g m (y), ω g (y)) = 0, so there is m 0 ∈ N such that for all m ≥ m 0 , m ∈ N k , B m ⊂ U . Thus there exists q ∈ N such that for n q ≥ m 0 , g nq (y) ∈ W a and {g nq+tq-k (y), . . . , g nq+tq (y

)} ⊂ W b . For each m ∈ N k , denote by -s(m) = m + j 2 (m) -j 1 (m), where m + j 1 (m), m + j 2 (m) ∈ N k , -p nq = inf{i ∈ N : s i (n q ) ∈ {n q + t q -k + 1, . . . , n q + t q }}, -K q = B nq ∪ B s(nq) ∪ • • • ∪ B s pn q -1 (nq) .
It is easy to see that K q is connected. Since for all 0 ≤ i < p nq , s i (n q ) ∈ N k , so B s i (nq) ∩ ω g (y) = ∅ and then K q ∩ ω g (y) = ∅. Moreover as K q meets W a in at least g nq (y) and meets W b in at least g sp nq (y), thus diam(K q ) > d(a,b)

2 . Since for all n q ≥ m 0 , B nq ⊂ U , K q ⊂ U and

K q ∩ ω g (y) = ∅, thus K q ⊂ U \ ω g (y). Therefore diam(K q ) ≤ mesh(U \ ω g (y)) < d(a,b)
2 . A contradiction.

Expansivity and totally periodic ω-limit sets

In this section, we consider the notion of expansivity of continuous maps on compact metric space.

Definition 5.1. [14] A continuous map f : X -→ X on a compact metric space X is said to be expansive if there is real constant c > 0 such that for any x, y ∈ X and x = y, there is an integer n ∈ Z + such that d(f n (x), f n (y)) > c.

It is well known that the Cantor set and the torus T 2 admit expansive homeomorphisms [START_REF] William | A note on unstable homeomorphisms[END_REF]. Kawamura showed [START_REF] Kawamura | A direct proof that each Peano continuum with a free arc admits no expansive homeomorphisms[END_REF] that a Peano continuum which contains a free arc admits no expansive homeomorphisms. In particular, there is no expansive homeomorphism on an arc and a circle. In [START_REF] Kato | The nonexistence of expansive homeomorphisms on dendroids[END_REF], Kato showed that no dendroid admits an expansive homeomorphism. With respect to expansivity, we show the following.

Proposition 5.2. If f : X -→ X is an expansive continuous map on a compact metric pace X, then any totally periodic ω-limit set is finite.

Lemma 5.3. ([4, Lemma 4]) Let (X, f ) be a dynamical system and let

x ∈ X such that ω f (x) is infinite. If z is an isolated point in ω f (x), then it is not periodic.

Proof of Proposition 5.2. Let x ∈ X such that ω f (x) ⊂ P (f ) and assume that ω f (x) is infinite. We have that ω f (x) = ∪ n∈N Fix(f n ) ∩ ω f (x). Then we claim that there exists N ∈ N such that Fix(f N ) ∩ ω f (x) is infinite.

Otherwise, ω f (x) would be compact countable, so it contains an isolated point a and therefore by Lemma 5.3, a / ∈ P (f ). A contradiction. Now let (x n ) n∈N ⊂ Fix(f N ) ∩ ω f (x) be an infinite sequence which converges to some point y. From expansivity of f , there is a constant c > 0 such that for any n ∈ N, there is an integer k n such that d(f kn (x n ), f kn (x n+1 )) > c. One can write k n = s n N + r n with 0 ≤ r n < N . We may assume, by passing to a subsequence if necessary, that r n = r, for every n. It follows that d(f r (x n ), f r (x n+1 )) > c, for every n and so letting n → +∞, we obtain that d(f r (y), f r (y)) = 0 ≥ c. A contradiction.

In [START_REF] Artigue | Minimal expansive systems and spiral points[END_REF], Artigue introduced the notion of turning point for a continuous map f : X → X on a metric space (X, d) as follows: we say that x ∈ X is a spiral point of f if there is m ∈ N such that lim n→+∞ d(f n (x), f n+m (x)) = 0.

From [2, Proposition 3.2] and Proposition 5.2, we obtain: Corollary 5.4. If f : X → X is an expansive continuous map on a metric space (X, d) and x ∈ X is a spiral point of f , then ω f (x) is finite.

-

  An arc I (resp. a circle) is any space homeomorphic to the compact interval [0, 1] (resp. to the unit circle S 1 = {z ∈ C : |z| = 1}). For an arc I = [a, b], denote by γ(I) = {a, b} the set of its endpoints, [a, b) = I \ {b} and (a, b) = I \ γ(I) = I \ {a, b}.

i≥0gLemma 4 . 2 .

 42 -i (a), for any a ∈ O. Let U be an open neighborhood of ω g (y) in X with finite boundary. Then there exists an open connected subset V of ω g (y) such that V ⊂ O and S a ⊂ U , for all a ∈ V .Proof. First, the sets (S a ) a∈O are pairwise disjoint connected sets; for a = b ∈ O, we have S a ∩ S b = ∅; indeed otherwise, there is c ∈ S a ∩ S b , so g j (c) = a and g i (c) = b for some j

  so it is a Peano continuum. By Lemma 3.6, one can find an arc T m = [b m , g m (y)] ⊂ W such that lim m→+∞,m∈N k diam(T m ) = 0.

  In result, we have that lim i→+∞,m i +j∈N k I m i +j = {c} (in Hausdorff metric). Now apply Lemma 3.9 for c ∈ O: there is c ∈ O such that d(c, c ) ≥ η. Then we can find two open neighborhoods O c and O c of c and c in U , respectively such that d(O c , O c ) > η 2 . As c ∈ O ⊂ ω g (y) ∩ Fix(g), so there is m ≥ 1 such that {g m (y), g m+1 (y), . . . , g m+k (y)} ⊂ O c . Let m p > m such that ∪ mp+j∈N k I mp+j ⊂ O c and set

1

 1 (m) (z) ∈ P m ∩ ω g (y), this leads to a contradiction with P m ∩ ω g (y) = ∅. By Claim 2, lim m→+∞,m∈N k diam(B m ) = 0. So let W a , W b be two open neighborhoods of a and b, respectively such that d(W a , W b ) > d(a,b) 2. By Lemma 3.8, there exists a connected neighborhood U := U N of ω g (y) such that mesh(U \ ω g (y)) < d(a,b)

  a contradiction. By monotonicity of g, S a is connected. Set card(Bd(U )) = k. So there are at most k points a 1 , a 2 , . . . , a k ∈ O such that S a i ⊂ U for every i ∈ {1, . . . , k}: Otherwise, there are at least k + 1 points a 1 , a 2 , . . . , a k+1 ∈ O such that S a

i ⊂ U for all 1 ≤ i ≤ k + 1. As a i ∈ S a i ∩ U and S a i is connected, so S a i meets Bd(U ) for every 1 ≤ i ≤ k + 1. Hence card(Bd(U)) ≥ k + 1, a contradiction. As O \ {a 1 , a 2 , . . . , a k } is open in ω g (y)

and non-empty (since ω g (y) is infinite), so we can find a non-empty open connected subset V of ω g (y)

  ) > 0. Then there is δ > 0 and a sequence of positive integers (m l ) l ⊂ N k such that diam(B m l ) > δ, for all l ∈ N. Recall lim m→+∞, m∈N k diam(I m ) = 0, where I m = [g m (y), a m ) and lim l→+∞,m l ∈N k diam(P m l ) = 0. By passing to a subsequence if necessary, one can assume that (P m l ) m l ∈N k converges necessarily to a degenerate set, say {c}. Hence, lim

				we have
		lim m→+∞, m∈N k	diam(B m ) = 0.
	Proof. Suppose that	lim m→+∞, m∈N k	diam(B m

l→+∞, m l ∈N

Acknowledgements. This work was supported by the research unit: "Dynamical systems and their applications", (UR17ES21), Ministry of Higher Education and Scientific Research, Faculty of Science of Bizerte, Bizerte, Tunisia.


Bogdan Vrusias
email: b.vrusias@surrey.ac.uk

Dimitrios Makris
email: d.makris@kingston.ac.uk

Ademola Popoola
email: a.popoola@surrey.ac.uk

Graeme Jones
email: g.jones@kingston.ac.uk

A system for tracking and annotating illegally parked vehicles

The large and still increasing number of vehicles on the roads has imposed the need for continuously monitoring the traffic behaviour, especially when it comes to illegally parked vehicles that disturb the normal flow of traffic. This paper presents an automatic method for identifying such events in CCTV video, by first tracking all the related objects and then annotating the events with appropriate keywords for storing and retrieval purposes. The method proposed makes use of a combination of video object tracking algorithms and techniques for capturing knowledge in keyword-based ontology structures. Starting from lowlevel visual information extracted from each video frame, high-level semantics such as moving objects are identified and classified. A probabilistic model, which takes its inputs from the visual modules, is used for identifying illegally parked vehicles. Finally, the keyword ontology, constructed automatically from expert descriptions, is linked to objects identified in the video. The results of the conducted experiments are encouraging.

Introduction

There has been a significant shift of interest of researchers in the area of visual surveillance, to the topic of event detection. This is justified by the maturing of motion detection [START_REF] Stauffer | Adaptive Background Mixture Models for Real-Time Tracking[END_REF] and motion tracking methods [START_REF] Haritaoglu | W 4 : Real-Time Ssurveillance of People and their Activities[END_REF] [START_REF] Isard | CONDENSATION -conditional density propagation for visual tracking[END_REF][12], but is also underpinned by the CCTV users who require systems that are able to detect specific type of events. For instance, the iLids project [START_REF]Home Office Scientific Development Branch Imagery library for intelligent detection systems (i-LIDS)[END_REF] identified 4 types of events that are of particular interest for the UK surveillance market.

Event detection can be broadly categorised in two categories: First, abnormal event detection aims to specify unusual events that are not expected to occur in the scene and may need the attention of the CCTV operator. A common approach is to measure the likelihood of an event using an appropriate probabilistic model (e.g. HMM in [START_REF] Galata | Learning variablelength Markov models of behavior[END_REF]) and define the outliers of this model to be the abnormal events.

Second, user-defined event detection aims to recognise specific events of interests for a given scene (e.g. "abandoning bag" or "intruding sterile zone"). Therefore, user's knowledge event is incorporated into to event detection mechanism, such a rule-based system or a Bayesian network [START_REF] Hongeng | Video-based event recognition: activity representation and probabilistic recognition methods[END_REF].

However, user's knowledge needs to be encoded to the "system's language" and there is no space for users to communicate with the system using natural language. This may be important, because the events may described differently, based on the background, and experience of the CCTV operator.

Our work aims to bridge the gap between visual semantics that are detected by video processing algorithms and ontologies that are extracted by statistical natural language processing [START_REF] Vrusias | A Framework for Ontology Enriched Semantic Annotation of CCTV Video[END_REF] [START_REF] Hare | Bridging the semantic gap in multimedia information retrieval -top-down and bottomup approaches[END_REF]. Our contribution is a system that processes and annotates videos for indexing and retrieval purposes. We focus on the problem of illegal parking, as presented by the iLids project.

The structure of the paper is as follows; Section 2 discusses the video processing methods to extract video semantics. Section 3 presents how a CCTV ontology is constructed using statistical natural language processing. The problem of linking visual semantics to ontologies is addressed in section 4. Section 5 focuses on the case study of "illegal" parking and demonstrates the results of our system. Finally, the paper is concluded in section 6.

Video Semantics

We identify three types of important semantics for the interpretation of CCTV videos: Actors, actions and static scene features. Combinations of those semantics are sufficient to describe the type of events that interest CCTV operators.

However, extracting these visual semantics from raw video is not straightforward. In our work, we firstly estimate the blobs and the tracks of moving objects in the sequence, which then are used to estimate models of actors and areas. Finally, actor labels (pedestrian/vehicle) are attached to tracks and areas are marked according to their usage by the actors. Figure 1 shows the data hierarchy of this process.

Object Tracking

We apply a motion detection algorithm [START_REF] Renno | Evaluating Motion Detection Algorithms: Issues and Results[END_REF] that is able to deal with illumination changes and shadows to separate foreground moving objects from background in individual fames. Then we use a motion tracking algorithm [START_REF] Xu | Partial observation vs. blind tracking through occlusion[END_REF] to establish the temporal correspondence of these blobs, represented by a set of trajectories Since colour and geometric features are important for the extraction of visual semantics, we ensure their invariance in time and 3D space respectively by colour [START_REF] Renno | Application and Evaluation of Colour Constancy in Visual Surveillance[END_REF] and geometric calibration [START_REF] Renno | Learning Surveillance Tracking Models for the Self-Calibrated Ground Plane[END_REF]. Both types of calibration are automatic to avoid the practical issues of manual calibration, such as access to the physical place of the surveyed scene and the burden of human effort.

Object Recognition

Object Classification based on the geometric attributes of the detected blobs and the dynamics of their trajectories is used to distinguish between pedestrians and vehicles ( Our classification methodology uses an ensemble of classifiers that, currently, assigns objects to one of two classes: people & vehicles and, circumvents issues relating to feature selection [START_REF] Renno | Object Classification In Visual Surveillance Using Adaboost[END_REF]. Though primarily based upon Adaboost classifier, our methodology seeks to in-rich classifier performance through the determination of a metric's relative importance.

In order to store the output of the processed video, the information is encapsulated in predefined metadata files. The video metadata produced by the proposed method describes the tracked objects that can be captured in a video. The data stored for each object per frame includes: locations for bounding boxes, speed, bounding box size, object recognition label. Each object found in a video has its own XML file to and all of the separate XML files are summarized into a header XML file which contains a reference to all the individual files (Fig. 3).

Scene Semantics

To determine activity-based scene features, we exploit the observed motion of the actors. We assume that activity-based scene features influence and/or enforce specific types of behaviour. For instance, roads constrain vehicles to move along specific lanes in a particular direction; similarly pavements constraints the pedestrian motion; gates, doors and regions at the image boundaries are related to entrance/exit events where actors will appear or disappear. Therefore,

Pixels Blobs Trajectories

Actor labels Scene labels Action labels accumulation of observations of the same behaviour in a specific location provides a clue for the existence of a correspondent activity-based feature [START_REF] Makris | Learning Scene Semantics[END_REF].

Video Concept Relationship Structures

Most systems that capture knowledge lack of the ability to make this knowledge available to end users. Being able to annotate, index, and retrieve specific information without missing useful records is a challenging task. Video annotation in particular is one of the most challenging areas for information retrieval. To overcome this, a method for capturing the relationships between concepts used by experts in the process of video monitoring is described in this section. The method has three main stages: first, only the salient concept objects and actions are identified; second, the relationship between the concepts is determined and encoded in a domain specific ontology; and third, rules and reasoning are formed for the annotation purposes.

Identifying key objects and actions

The domain specific multimedia ontology is intended for documenting the semantics of video scenes that cannot easily be captured through video analysis techniques alone. Statistical text analysis is used to capture this ontology from expert descriptions of videos. Other existing ontology structures are used in order to complement the one built. The ontologies used here are acquired from two sources: the WordNet lexical database and the Police Information Technology Organisation (PITO), now the National Policing Improvement Agency (NPIA), terminology.

The components built and used in this process integrate with Sheffield's GATE system. These components build on existing GATE plug-ins from ANNIE, for preliminary NLP tasks of such as POS tagging and sentence splitting. OWL is used as the syntax for representing the ontology which captures the weight of the co-occurrence relationship between objects and actions concepts found in the videos [START_REF] Newbold | Lexical Ontology Extraction Using Terminology Analysis: Automating Video Annotation[END_REF]. The relationship weight is calculated using a function of frequency and distance between terms, with the purpose to later determine the strongest relationships between objects and actions. This information is the further used to associate the identified video objects with concepts in the ontology, for potential use in video search expansion.

The pipeline for these resources are (Figure 4): In order to identify objects and actions in the text descriptions of videos (transcripts), the GATE ANNIE tokeniser and Part of Speech (PoS) tagger are used to identify all the nouns and verbs, and to provide a basis for the identification of compound nouns according to specified patterns of PoS annotations. Some basic stemming does occur for the initial assessment of potential object and action concepts. At this point no frequency or other statistical information is considered.

The initial linguistic identification of concepts is supplemented by a statistical approach to identify other potential concepts that may have been incorrectly classified by the PoS tagger. Transcripts undergo statistical analysis to determine salience using frequency and weirdness information to provide statistical evidence for concepts and to act as confirmation for the linguistic extraction [START_REF] Newbold | Lexical Ontology Extraction Using Terminology Analysis: Automating Video Annotation[END_REF]. Results are classified as either objects or actions via the WordNet Verification (2.3) process. The British National Corpus (BNC) is then used as a reference corpus, and thresholds are adjusted by modifying parameters for the distributions. Similar approaches that process text, available with multimedia, and extract terms using combinations of word frequency, tf*idf and entropy, and application of stemming [START_REF] Jaimes | Semi-automatic, data-driven construction of multimedia ontologies[END_REF], where resulting keywords are used in the manual construction of an ontology.

Information and procedures defined by the National Policing Improvement Agency (NPIA), formerly known as the Police Information Technology Organisation (PITO), are also used as a filtering stage in the process of building the video ontology. PITO/NPIA specified a terminology for describing events or incidents. PITO provide data definitions which help sub-divide information into simpler data elements. The PITO/NPIA elements were transformed into an ontology by re-interpreting the category and sub-category information as classes and sub-classes. The PITO/NPIA ontology is used for filtering purposes when classifying objects and actions in the generated ontology. This information is used to associate keywords for more efficient relationship analysis in Event Mapping (Figure 4, 2.5).

Building a video ontology

Several knowledge structures for holding data have been used in the literature; the ontology data structure is used here as a way for storing and retrieving relationships between video concepts of objects and actions. Domain ontologies are useful for sharing knowledge between more than one application on the web, and they are structured in such a way that allows machine-based communication. They are often used in e-commerce to allow machine-based communication between buyer and seller. They are also used in some search engines to allow searching of words which are semantically similar but syntactically different therefore enhancing the search results, this method is sometimes known as query expansion, and is adopted for this paper too.

With objects and actions identified through the previous processes and filtering thereof, we analyze connections between the objects and actions to produce two types of event, 'action-object' or AO event and 'agent-action-recipient' or AAR events. An AO Event associates one object and one action. An AAR Event represents the relationship between two objects and the action that connects them. We calculate collocation distances between words connected in events to ascertain dominant patterns. The connections are used to create instances of the AO Event and AAR Event classes in the ontology.

The REVEAL project makes use of OWL DL and the RDF/XML syntax to produce a representation of the objects and actions contained within a surveillance video and the events linking them together.

Expanding the video ontology

The videos that will be searched by the system contain Objects, Actions, ActionEvents and ObjectEvents contained within the ontology. This ontology was produced from descriptions of videos given manually by several different CCTV experts watching them. The process of creating the ontology looks at the descriptions and locates object related and action related words, and the distance between them:

Object: a noun that relates to an object that was described in the video, such as 'person' or 'car'.

Action: a verb that relates to an action that was described in the video, such as 'cross' or 'drive'.

ActionEvent: this relates to an event described in the video in the form action_object, such as 'cross_person'.

ObjectEvent: this relates to an event described in the video in the form object_object_action, such as 'car_person_drive'.

An exemplar output of the system is shown here:

<rdf:Description rdf:about="car_stop_road"> <agent rdf:resource="car"/> <action rdf:resource="stop"/> <recepient rdf:resource="road"/> <freq>7</freq><dist>3.5</dist> </rdf:Description>

The query expansion that was performed uses the WordNet ontology and is created to supply synonyms, hyponyms and meronyms as the search criteria specified, and use these words along with the original criteria when searching the generated video ontology. Different levels of expansion for each of the search words, supplied by the user, are available, and also narrowing the expansion performed. Although this sometimes led to some inaccurate results being returned.

Linking video semantics to ontologies

The combination of different modalities of the same information can produce high-performance retrieval systems. The semantic gap, is the difference between ambiguous formulation of contextual knowledge in a powerful language e.g. natural language and its sound, reproducible and computational representation in a formal language e.g. programming language [START_REF] Vrusias | A Framework for Ontology Enriched Semantic Annotation of CCTV Video[END_REF] [START_REF] Hare | Bridging the semantic gap in multimedia information retrieval -top-down and bottomup approaches[END_REF].

A Bayesian Network is used in this study as a 'concept detector'. Features like Elapsed Time, Distance to Stop Zone, obtained from experts in a workshop conducted during the REVEAL project, were used to construct the Bayesian network. As suggested by [START_REF] Neil | Building large-scale Bayesian networks[END_REF], the network structure was designed by incorporating so-called synthetic nodes in order to reduce the number of parents of each child node. In order to reduce bias during probability elicitation from experts, we followed the method used in [START_REF] Van Der Gaag | Probabilities for a probabilistic network: a case study in oesophageal cancer[END_REF], where probabilities were obtained using text fragments and a scale with both numerical and verbal anchors.

The system relies on the use of keywords or tags generated from the low level object detector/recognition (section 2.2) and the concept detector (Bayesian network). However, two major problems are associated with "plain" keywords: synonymy and polyzemy, which affect the performance of semantic retrieval. The concept synonymy is used to describe the fact that there are many ways to refer to the same object, while polyzemy is referred to the fact that most words have more than one distinct meaning [START_REF] Yang | Learning the Semantics in Image Retrieval -A Natural Language Processing Approach[END_REF]. We have taken advantage of the lexical relationships -synonymy, hyponymy and hypernymyin WordNet to carry out tag expansion. The pipeline for these resources is shown in Figure 5.

Case Study: tracking "illegally" parked vehicles

The dataset was supplied by the REVEAL project and consisted of 12 videos, manual descriptions of those videos, an ontology describing the objects, actions and events occurring within all 12 videos and video metadata supplying the bounding box locations for objects identified within the video. The descriptions for the videos were produced by 6 different people each commenting on the same 12 videos. Videos are split into 20 second periods and the 6 descriptions, from the 6 different people, for that 20 second time period are merged into one file.

The videos are segmented into 20 second periods, each of which has its own description and for each query used in the evaluation the segments of video that are expected to be returned are marked manually. Then, using the expected results and the actual results from the system precision and recall are calculated. Due to the fact that the expected results rely on the integrity of the video descriptions given by people it is feasible that a description could be wrong, however it is assumed for this evaluation that the descriptions are reliable. The main focus is placed in identifying 'illegally' parked vehicles, so the clips are described in terms of the presence of 'interesting events', which are vehicles initially in motion that subsequently become stationary i.e. have a velocity = 0. Within the set of 'interesting' events, a subset of that, 'illegally' parked vehicles, is identified. Conditions for a vehicle to be classified as being 'illegally' parked, it had to be identified as (i) an object previously in motion (ii) classified as a 'vehicle' (iii) become stationary for more than one minute (as defined by the NPIA for the iLids dataset) and (iv) be parked in a 'no stop' location. A total of 41 interesting events were identified in the 12 clips used for this case study, with 18 of these events -almost half -further classified as illegally parked vehicles (Table 1). In each clip, the low level object detector automatically identify and classify moving objects, using initial tags: 'vehicles' and 'person', as described in section 2.2 (Figure 8). Since the aim of this case study is to annotate 'illegally' parked vehicles, we focus on objects classified as 'vehicles'. When a previously moving 'vehicle' object stops, the object is selected as 'interesting', and its properties, in relation to other objects present in the same timeframe, are collated. Features like Elapsed Time, Distance to Stop Zone, obtained from experts in a workshop conducted during the REVEAL project, were used to construct a Bayesian network. The data obtained from the video footage are used in the Bayesian network to classify the stopped vehicle as being 'illegally' parked i.e. 'abandoned' or not. Thus, the Bayesian network is used as a 'concept detector', the concept being 'abandoned' or 'illegally' parked vehicles. Once a 'vehicle' object is identified or labelled as 'abandoned' or illegally parked, we use the lexical relationships in WordNet to perform tag or label expansion in order to generate additional tags or labels. For example, the 'vehicle' tag generates hyponym expansions such as 'motor vehicle' and 'car'. The initial tag and associated expansions are then used to query the CCTV ontology, to extract existing, statistically significant relationships that are linked to these tags. The relationships between concepts determined through relationship extraction module are detailed in Table 2, showing the 'agent-actionrecipient' or AAR events.

The initial and expanded tags, as well as associated relationships are then used to 'annotate' the video footage (Figures 9-10

Conclusions

A system for automatically annotating "illegally" parked vehicles is described. The method combines video processing for identifying blobs and consequently conceptual entities in videos, together with statistical text analysis methods for building knowledge structures that capture relationships between video content concepts. In order to extract high level semantic annotations from the videos, the visual and text extracted semantics are combined via a Bayesian network classifier. The system successfully identified the "illegally" parked vehicles present in the videos. Although the case study focused on a particular scenario, the method followed can potentially be used to identify other scenarios such as people running, traffic congestions, overtaking, or unusual behaviour in general, which can be of great use in traffic control or public safety.

Figure 1 .

 1 Figure 1. Data hierarchy

  Fig 2).

Figure 2 .

 2 Figure 2. Pedestrians (P) and vehicles (V) attached to extracted blobs.

Figure 3 .

 3 Figure 3. An example of the header XML file and its relationship to an individual object XML file which describes a tracked object.

Figure 4 .

 4 Figure 4. Pipeline for the automatic generation of the term-based CCTV ontology [1].

Figure 5 :

 5 Figure 5: Pipeline for the automatic annotation of CCTV footage

Figure 6 :

 6 Figure 6: Scene from a video

Figure 7 :

 7 Figure 7: Two examples from six transcripts describing the same scene as featured in figure 6.

Figure 8 .

 8 Figure 8. Initial tags/labels from Low Level Object Detector.

.) Table 2 :Figure 9 .Figure 10

 2910 Figure 9. Exemplar annotations produced by each module of the system

Table 1 .

 1 'Interesting' events & 'illegally' parked vehicles in 12 clips

	Clip	'Interesting'	'Illegally' Parked
	No.	Event Count	Vehicle Count
	1	4	2
	2	3	1
	3	7	3
	4	7	3
	5	4	1
	6	1	1
	7	3	0
	8	2	1
	9	1	1
	10	2	0
	11	4	2
	12	3	3
	Totals	41	18

Acknowledgement

This work was supported EPSRC sponsored REVEAL project under Grant No.GR/S98450/01, GR/S98443/01 jointly undertaken by the University of Surrey and Kingston University. We are grateful to our project partners and expert end-users, John Armstrong from Surrey Police, the UK National Policing Improvement Agency (NPIA), and Home Office -Scientific Development Branch (HOSDB). Special thanks to Neil Newbold for building the domain ontology system and David Williams for building the video retrieval system.