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eric.paulk
Eric Paulk
Eric Paulk is a lawyer and advocate working on issues impacting LGBTQ communities and communities of people living with and vulnerable to HIV acquisition. He serves as Deputy Director at Georgia Equality where he oversees all day-to-day operations, developing and managing local, state, and federal policy and advocacy activities that advance equity and fairness for LGBTQ communities and improves public and private responses to the HIV epidemic. Prior to joining Georgia Equality, Eric served as the Tyron Garner Fellow at Lambda Legal, the oldest and largest organization dedicated to advancing the civil rights of lesbians, gay men, bisexuals, transgender people and individuals living with HIV. At Lambda Legal, Paulk's work focused on HIV law and policy initiatives with an emphasis on legal issues disproportionately impacting Black LGBTQ communities. Eric has spoken at leading law schools and national advocacy conferences on HIV criminal justice reform, the impact of the school-to-prison pipeline on Black queer youth, and the intersections of race, poverty, HIV, and LGBTQ issues. He serves on the board of the Atlanta Harm Reduction Coalition, the Equality Federation, and the ProGeorgia Civic Engagement Table, a non-profit coalition of more than 30 voter engagement groups throughout the State of Georgia working to ensure access to the vote for historically marginalized communities. He is a former New Leadership Council Fellow and former Victory Institute Victory Empowerment Fellow. Most recently, Eric was named as one of the 100 most influential LGBTQ+ Georgians by the Out Georgia Business Alliance and as one of state's top 40 leaders under 40 by Georgia Trend Magazine. Mr. Paulk is a graduate of Morehouse College and received his law degree from Pace University Law School. Eric is currently serving an 18-month term as a Soros Justice Fellow where he is focused on creating a network of Black HIV Movement Lawyers to support and advocate for communities of people living with HIV.


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Category: New Inventions
Engineering, Inventions, New Inventions, Science & Technology News
The first camera for blind people
2C3D is a camera that enables the blinds to see. The camera, is a development and design of a tactile camera concept for the vision impaired. The camera creates 3D photos and videos and has a 3D screen. The screen, inspired by "Pin Toy," is built by numerous 3D pixels that shift depending on the photo to forms the 3D shot on the screen surface (giving the term "touch screen" a new and more literal interpretation).
The user can touch the screen while photographing and feel what the camera is seeing, in real time. When the users like what they feels, they can click and save the photo. The saved 3D file can be felt again later. The 2C3D performs as a camera for blind and as physical-digital photo album.
Continue reading… "The first camera for blind people"
Engineering, New Inventions, new technology, Science & Technology News, Transportation
Smart concrete could pave the way for high-tech , cost-effective roads
Every day, Americans travel on roads, bridges, and highways without considering the safety or reliability of these structures. Yet much of the transportation infrastructure in the US is outdated, deteriorating, and badly in need of repair.
Of the 614,387 bridges in the US, for example, 39 percent are older than their designed lifetimes, while nearly 10 percent are structurally deficient, meaning they could begin to break down faster or, worse, be vulnerable to catastrophic failure.
The cost to repair and improve nationwide transportation infrastructure ranges from nearly US$190 billion to almost $1 trillion. Repairing US infrastructure costs individual households, on average, about $3,400 every year. Traffic congestion alone is estimated to cost the average driver $1,400 in fuel and time spent commuting, a nationwide tally of more than $160 billion per year.
I am a professor in the Lyles School of Civil Engineering and the director of the Center for Intelligent Infrastructures at Purdue University. My co-author, Vishal Saravade, is part of my team at the Sustainable Materials and Renewable Technology (SMART) Lab. The SMART Lab researches and develops new technologies to make American infrastructure "intelligent," safer, and more cost-effective. These new systems self-monitor the condition of roads and bridges quickly and accurately and can, sometimes, even repair themselves.
Continue reading… "Smart concrete could pave the way for high-tech , cost-effective roads"
Healthcare, Inventions, Medicine, New Inventions, Science & Technology News
Medical innovations that will revolutionize the future of you healthcare
2020 has ingrained in me an age-old adage my mom loves to quote – health is wealth. Focus on our healthcare and the strain on our healthcare system has increased exponentially this year. While the world altogether has jumped up to help improve our healthcare systems, what can truly help is improved preventive methods, devices that help the patients monitor their health from home as well as to stay in touch with their doctors virtually while providing accurate data. The best example of the data's impact is how an Apple Watch helped saved a man's life by detecting problems with his heartbeat – and this is just the beginning. The products here show the best of healthcare we can provide to make this world a better place!
Continue reading… "Medical innovations that will revolutionize the future of you healthcare"
Engineering, Nanotechnology, New Inventions, Science & Technology News
Researchers create a single-molecule switch
A team of researchers has demonstrated for the first time a single-molecule electret—a device that could be one of the keys to molecular computers.
Smaller electronics are crucial to developing more advanced computers and other devices. This has led to a push in the field toward finding a way to replace silicon chips with molecules, an effort that includes creating single-molecule electret—a switching device that could serve as a platform for extremely small non-volatile storage devices. Because it seemed that such a device would be so unstable, however, many in the field wondered whether one could ever exist.
Along with colleagues at Nanjing University, Renmin University, Xiamen University, and Rensselaer Polytechnic Institute, Mark Reed, the Harold Hodgkinson Professor of Electrical Engineering & Applied Physics demonstrated a single-molecule electret with a functional memory. The results were published Oct. 12 in Nature Nanotechnology.
Continue reading… "Researchers create a single-molecule switch"
2021 Predictions, Engineering, mobile technology, New Inventions, Science & Technology News, Shopping
Amazon wants you to yell at your TV to buy things
Alphabet is wrangling mosquitoes, Apple's bendy phone and other patents from Big Tech.
Another week in lockdown has passed, and while the present might still feel quite uncertain, the future looks as zany as ever, at least as far as patents go. Alphabet is trying to trap mosquitoes, Amazon wants you to buy stuff off of your TV screen, Apple is getting in on the flexible phone trend, and Microsoft is trying to figure out your heart health from your camera.
And remember: The big tech companies file all kinds of crazy patents for things, and though most never amount to anything, some end up defining the future.
Continue reading… "Amazon wants you to yell at your TV to buy things"
Automation, New Inventions, Robots, Science & Technology News
LA debuts first firefighting robot in the country, deploys it in downtown blaze
The Thermite RS3 is capable of spraying 2,500 gallons per minute
A firefighting robot got its first major test Tuesday in Los Angeles when it was put to use for the first time in the United States to battle a major blaze.
The Los Angeles Fire Department said the Thermite RS3 robot was supposed to have its official public introduction in the afternoon but got called into duty a few hours early due to a blaze downtown.
"It had already gotten dirty at an early morning major emergency commercial structure fire that morning – proving its value from the start," the department said.
Continue reading… "LA debuts first firefighting robot in the country, deploys it in downtown blaze"
New Inventions, Science & Technology News, Space, Space Flight
Chitin could be used to build tools and habitats on Mars, study finds
The manufacturing process would require minimal energy and no specialized equipment.
Scientists mixed chitin—an organic polymer found in abundance in arthropods, as well as fish scales—with a mineral that mimics the properties of Martian soil to create a viable new material for building tools and shelters on Mars.
Space aficionados who dream of one day colonizing Mars must grapple with the stark reality of the planet's limited natural resources, particularly when it comes to building materials. A team of scientists from the Singapore University of Technology and Design discovered that, using simple chemistry, the organic polymer chitin—contained in the exoskeletons of insects and crustaceans—can easily be transformed into a viable building material for basic tools and habitats. This would require minimal energy and no need for transporting specialized equipment. The scientists described their experiments in a recent paper published in the journal PLOS ONE.
Continue reading… "Chitin could be used to build tools and habitats on Mars, study finds"
drones, Engineering, New Inventions, Robots, Science & Technology News
Insect-inspired robots that can jump, fly and climb are almost here
Roboticists who designed these robots, called Tribots, took inspiration from the real-life trap-jaw ant's locomotion strategies.
Did you envision a giant machine assembling cars, Data from "Star Trek," C-3PO from "Star Wars" or "The Terminator"? Most of us would probably think of something massive — or at least human size.
But a whole arm of robotics is focusing on bug-size 'bots (and smaller).
It's not just the size of tiny insects that are inspiring roboticists; it's also the many complex tasks and physical feats that comprise the everyday lives of many fleas, flies and other six-legged creatures.
Continue reading… "Insect-inspired robots that can jump, fly and climb are almost here"
Breakthrough Thinking, Internet, Microsoft, New Inventions, Science & Technology News
Microsoft's underwater server experiment resurfaces after two years
Turns out, dunking data centers is a great idea
Back in 2018, Microsoft sunk an entire data center to the bottom of the Scottish sea, plunging 864 servers and 27.6 petabytes of storage 117 feet deep in the ocean. Today, the company has reported that its latest experiment was a success, revealing findings that show that the idea of an underwater data center is actually a pretty good one.
On the surface, throwing an entire data center to the bottom of the ocean may seem strange, but Microsoft's Project Natick team hypothesized that placing would result in more reliable and energy-efficient data centers.
Continue reading… "Microsoft's underwater server experiment resurfaces after two years"
Inventions, New Inventions, New Viewpoints, Science & Technology News
Why Tokyo's new transparent public restrooms are a stroke of genius
Tokyo transparent bathroom
At first, it's hard to fathom how a public restroom with transparent walls could possibly help ease toilet anxiety — but a counterintuitive design by one of Japan's most innovative architects aims to do just that.
Around the world, public toilets get a foul rap. Even in Japan, where restrooms have a higher standard of hygiene than in much of the rest of the world, residents harbor a fear that public toilets are dark, dirty, smelly and scary.
To cure the public's phobia, the non-profit Nippon Foundation launched "The Tokyo Toilet Project," tasking 16 well-known architects to renovate 17 public toilets located in the public parks of Shibuya, one of the busiest commercial areas of Tokyo.
Continue reading… "Why Tokyo's new transparent public restrooms are a stroke of genius"
Inventions, Mass Transit, New Inventions, Science & Technology News, Transportation
Indian Railways to soon introduce these 20 in-house Made-in-India innovations
On the list is also a system to monitor real-time CCTV footage on board trains
The Western Railway has developed natural water coolers, with zero electric consumption, each at a cost of ₹1.25 lakh.
The North Central Railway has developed a vehicular system for ultrasonic flaw detection of rails.
The Railway Board has decided to implement 20 innovations by its employees to make train journeys safer and improve passenger comfort – such as a bell warning to alert travellers minutes before a train departs, real-time CCTV monitoring inside coaches, printing of unreserved tickets through mobile applications – on a mass scale.
While most of the 20 innovations are aimed at technical improvements to boost safety, some of the innovations are also directed at passenger comfort.
Continue reading… "Indian Railways to soon introduce these 20 in-house Made-in-India innovations"
Automation, New Inventions, Robotics, Science & Technology News
This wearable robotic arm can hold tools, pick fruit, and punch through walls
Doc Ock, is that you?
We've always had a soft spot for supernumerary robotic limbs here at The Verge, but this latest example of the genre is one of the most impressive we've seen to date. Designed by researchers at the Université de Sherbrooke in Canada, it's a hydraulic arm that sits on the wearer's hip and uses a three-fingered manipulator to carry out a range of tasks.
Continue reading… "This wearable robotic arm can hold tools, pick fruit, and punch through walls"

redpajama_set_name: RedPajamaCommonCrawl


The Bosch Tile Laser Square Laser Level features 2 lasers that can project up to 30 ft. to ensure sufficient space for markings. Integrated vials provide additional accuracy when attempting to achieve level surfaces. Batteries are included to let you get working right away.
Projects a horizontal and vertical laser chalk line on any surface up to 30 ft.
The Bosch GOL 26CK automatic optical level kit features a self-levelling compensator, both horizontal/vertical cross hairs and Stadia lines, 26x magnification lens and large aperture for clear image. Range: 330 ft., accuracy: up to 1/16 In. @ 100 ft., includes: hard case, lens cap, plumb bob, adjustment tools, aluminum tripod, aluminum levelling rod.
Tile Laser Square Laser Level The Bosch Tile Laser Square Laser Level features 2 lasers that can project up to 30 ft. to ensure sufficient space for markings. Integrated vials provide additional accuracy when attempting to achieve level surfaces. Batteries are included to let you get working right away.
Tile Laser Square Laser Level is rated 4.2 out of 5 by 83.
Rated 1 out of 5 by HomeDepotCustomer from Would not project line more than ten feet. ... Would not project line more than ten feet.
Rated 4 out of 5 by gscaleut from We bought it to give straight lines on the floor to lay tile on - works great for that since you ... We bought it to give straight lines on the floor to lay tile on - works great for that since you cover chalk lines with thin-set when you lay it down.
Rated 5 out of 5 by 6969rabbit from it worked great. it worked great.
Rated 5 out of 5 by Morris from Nice laser level I haven't used it, but seems very... Nice laser level I haven't used it, but seems very light weight and the laser is very bright which is good.
Rated 5 out of 5 by Docs from Does a nice job and is extremely handy when tiling alone. Does a nice job and is extremely handy when tiling alone.
Rated 3 out of 5 by Ru from Not bright enough. Not bright enough.
Rated 5 out of 5 by Tracy from I could have used this tool years ago. ... I could have used this tool years ago. Quickly I laid out a basement studio (walls, closet and a bath) by getting parallel and 90 degree angles off a desired wall. I will be using it to lay out tile next. The red laser is usable for many feet in two directions. Can't speak to longevity yet.
Rated 5 out of 5 by DK from The square is ready out of the box. Works... The square is ready out of the box. Works well as a right angle for laying out tile or other types of flooring.


After Nollywood actor and producer, Olayinka Solomon, shared a picture of her new house on her social media page, it attracted many comments, with some people questioning her source of income. In an interview with Saturday Beats, the actor noted that she only shared the picture of her house to appreciate God and not to attract undue attention to herself.
"I have been a landlady for a long time; I have some properties in my name. I actually bought this house; it was even a friend of mine that uploaded the picture on my behalf initially. I didn't mean to show off. In the entertainment industry, it is not easy to hide your accomplishments.
"I was surprised when I woke up the next day and noticed that I had been tagged on so many posts. It was the next morning I even posted the picture of the house on my platform. I live a very private life.
"I went through many things in the early stage of my career; I uploaded the picture to thank God and not to impress anybody. Everybody knows that acting cannot give me that kind of money to build a house; God is the source of my income," she told Saturday Beats.
Solomon has been absent from the movie scene for some years, and in the interview with one of our correspondents, she revealed that she had to leave the world of entertainment for a while because it wasn't bringing her money.
"I had to leave the industry for about eight years because it wasn't fetching me money, I was just focused on my business, the love for movies brought me back on set in 2018. I rarely act these days; I do more of movie productions now. In the early stage of my career, I faced some challenges, as I wasn't free with some of my senior colleagues at the time.
"My most challenging role was when I had to play triple roles at the same time. I am always on my own; I don't have friends or enemies in the industry. I would advise anyone that wants to go into the movie industry to stay focused and most importantly, have another source of income apart from acting. The industry is not easy for up-and-coming actors. As an up-and-coming actor, it doesn't really bring money. There are some movie productions that are not viable," she said.


Trends come and go, but the dad hat is a timeless style. Built with 100% cotton and the "Ring" graphic is directly embroidered onto the front panels and #GETYOKEDUP in the back. The strap-back closure makes this hat one size fits most.


Water logging has been disrupting livelihoods of about one million people in Bangladesh during past two decades. South-west Bangladesh is prone to water logging due to the vulnerable geographical settings, mismanagement and climate change. The worst hydro-geophysical vulnerability has been found in Keshabpur Thana of Jessore District where most of the land is waterlogged for over seven years. This research tried to support the policy making process by identifying people's social vulnerability and unique coping strategies under such adverse environmental condition. Necessary data were collected through questionnaire survey and Participatory Vulnerability Assessment (PVA) which includes Focus Group Discussions (FGD), Key Informants' Interview (KII) and case studies. About 270 households of the study area were surveyed. The target groups of this research were poor farmers and poor female headed households of four unions of Keshabpur Thana. Agricultural livelihood, housing, food security, health and education are found to be some socially vulnerable areas in Keshabpur under water logging. Major findings revealed the constraints towards reducing people's vulnerability such as climate change, poor coordination among stakeholders, limited institutional initiative of coping, etc. Accordingly researchers tried to put some necessary recommendations such as facilitating drainage through a coordinated approach among stakeholders, promoting immediate agricultural and institutional coping, construction of school buildings in higher grounds with access to higher roads, etc.


Avanity BROOKS-VS60-WT Brooks 60-in. Double Bathroom Vanity Granite Black Top.
Overall Dimensions: 61W x 22D x 35H in.. Contemporary-style double Bathroom vanity. Built using engineered wood and hardwood solids. Dual undermount china sinks included. Choose from several top options; faucets not included. Pre-drilled for dual 3-hole faucet assemblies. The Avanity BROOKS-VS60-WT Brooks 60 in. Double Bathroom Vanity – White provides your master-bathroom with a dual-basin, freestanding vanity that suits modern decors with panache. You'll find its wide frame complete with four soft-closing drawers and an open bottom shelf that runs the width of the piece. The vanity is constructed using medium-density fiberboard and fine poplar hardwood solids, finished in bright white and outfitted with brushed-nickel finished metal hardware. A solid stone top is provided, complemented by a matching backsplash. Two round vitreous china sinks are included with the set. A white marble top, a beige marble top, and a black granite top are yours to choose from (subject to availability). Each top comes pre-drilled for two widespread, 3-hole faucet assemblies (faucets no included).About Avanity CorporationAvanity's goal has always been to provide the public with the best products possible at the fairest prices. To this end, their customer service style is about listening to their customer, not just hearing them. Avanity is confident in all of their products and provides a one-year manufacturer's warranty to prove it. Avanity also takes note of increasing market trends to stay ahead of the game and provide the most cutting-edge products available. Color: Granite Black Top.
ACF by Nameeks ACF C02-W Cubical 22-in. Single Bathroom Vanity Set.
Modern wall-mounted vanity set. Engineered wood cabinet with metal accents. Wenge finish with chromed accents. 2 soft-close vanity doors. Choice of faucet drill holes. Rectangular white ceramic sink bowl. Includes vanity, sink, mirror, and light. Faucet not included. Make the most of your space with the stylish ACF by Nameeks ACF C02-W Cubical 22-in. Single Bathroom Vanity Set - Wenge. Modern and practical, this four-piece set comes complete with a two-door vanity cabinet, fitted ceramic sink, vanity mirror, and vanity light. It comes in a dark Wenge finish with chromed accents for a polished look. The vanity is wall-mounted for a free-floating design aesthetic. Bold cabinet pulls open the two soft-close doors to show plenty of storage space inside. The mirror and overhead light make this set complete.Component Dimensions:Vanity dimensions: 21.9W x 17.5D x 21.8H in.Mirror dimensions: 28.3W x 23.6H in.About NameeksFounded with the simple belief that the bath is the defining room of a Household, Nameeks strives to design a bath that shines with unique and creative qualities. Distributing only the finest European bathroom fixtures, Nameeks is a leading designer, developer, and marketer of innovative home products. In cooperation with top European manufacturers, their choice of designs has become extremely diversified. Their experience in the plumbing industry spans 30 years, and is now distributing their products throughout the world today. Dedicated to providing new trends and innovative bathroom products they offer their customers with long-term value in every product they purchase. In search of excellence, Nameeks will always be interested in two things: the quality of each product and the service provided to each customer.
Wyndham Collection WCV171736SCDCMUNRMED - 36" Single Bathroom Vanity, White Carrera Marble Countertop, Undermount Round Sink, 24" Medicine Cabinet.
Wyndham Collection WCS141460SESCMUNSM58 - 60" Single Bathroom Vanity, White Carrera Marble Countertop, Undermount Square Sink, & 58" Mirror.
Wyndham Collection WCS141472DESCMUNOM70 - 72" Double Bathroom Vanity, White Carrera Marble Countertop, Undermount Oval Sinks, & 70" Mirror.
Wyndham Collection WCS141460DESCMUNSM24 - 60" Double Bathroom Vanity, White Carrera Marble Countertop, Undermount Square Sinks, & 24" Mirrors.
Kohler K-72762-9M-VNT - 1.5 GPM Single Handle Bathroom Sink Faucet.
Wyndham Collection WCV161672DCDCXSXXM70 - 72" Double Bathroom Vanity, No Countertop, No Sinks, & 70" Mirror.
Wyndham Collection WCV161660SCDCXSXXM56 - 60" Single Bathroom Vanity, No Countertop, No Sink, & 56" Mirror.
Wyndham Collection WCS211172DESCXSXXM24 - 72" Double Bathroom Vanity, No Countertop, No sinks, & 24" Mirrors.


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NURSERY RHYME STORIES
Charlie Barker takes you through a regression of sorts in the sideshow today. Well, at the very least, a nostalgic look at the toddler years. There are nursery rhymes to be told, come in, come in!
You know that rhyme about the worms devouring your corpse after death? Ahh, what child doesn't hear that poem and smile. Well, it has a couple more stanzas that help keep you out of harm's way in the cemetery. Listen to HOLD YOUR BREATH and follow along. Second, it's the tale of a doll with a special tune to help you head off to dreamland. It's THE SLEEPYTIME SONG. Then, it's a classic nursery rhyme that had its time. But now, it's time to resurrect that old diddy in ONE IS FOR SORROW.
It's nice to feel like a kid again, ain't it? Next week we grow up a bit and strap on the ole feedback for some more tasty treats. Come back and Charlie Barker will give you something to chew on.
children's rhymes creepy poems hearse song horror stories nursery rhymes


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Financial arrangements for health systems in low-income countries
What is the aim of this overview?
The aim of this Cochrane Overview is to provide a broad summary of what is known about the effects of financial arrangements for health systems in low-income countries.
This overview is based on 15 systematic reviews. Each of these systematic reviews searched for studies that evaluated different types of financial arrangements within the scope of the review question. The reviews included a total of 276 studies.
This overview is one of a series of four Cochrane Overviews that evaluate different health system arrangements.
What are the effects of different ways of collecting funds to pay for health services?
Two reviews looked for studies that addressed this question and found the following.
- The effects of changes in user fees on utilisation and equity are uncertain (very low-certainty evidence).
- It is uncertain whether aid delivered under Paris Principles (ownership, alignment, harmonisation, managing for results, and mutual accountability) improves health compared to aid delivered without conforming to those principles (very low-certainty evidence).
What are the effects of different types of insurance schemes?
One systematic review looked for studies that addressed this question and found the following.
- Community-based health insurance may increase people's use of services (low-certainty evidence), but the effects on people's health are uncertain. It is uncertain whether social health insurance increases people's use of services (very low-certainty evidence).
What are the effects of different ways of paying for health services?
- It is uncertain whether increasing salaries of public sector healthcare workers improves the quantity or quality of their work.
What are the effects of different types of financial incentives for recipients of care?
Six systematic reviews looked for studies that addressed this question and found the following.
- Giving healthcare recipients incentives may improve their adherence to long-term treatments (low-certainty evidence), but it is uncertain whether they improve people's health.
- Giving healthcare recipients one-time incentives probably leads more people to return to start or continue treatment for tuberculosis (moderate-certainty evidence). The certainty of the evidence for other types of recipient incentives for tuberculosis is low or very low.
- Conditional cash transfer programmes (giving money to recipients of care on the condition that they take a specified action to improve their health) probably increase people's use of services (moderate-certainty evidence), but have mixed effect on people's health.
- Vouchers may improve people's use of health services (low-certainty evidence) but have mixed effects on people's health (low-certainty evidence).
- A combination of a ceiling and co-insurance probably slightly decreases the overall use of medicines (moderate-certainty evidence) and may increase health service utilisation (low-certainty evidence). The certainty of the evidence for the effects of other combinations of caps, co-insurance, co-payments, and ceilings is low or very low.
- Limits on how much insurers pay for different groups of drugs (reference pricing, maximum pricing, and index pricing) have mixed effects on drug expenditures by patients and insurers as well as the use of brand and generic drugs.
What are the effects of different types of financial incentives for health workers?
Five systematic reviews looked for studies that addressed this question and found the following.
- We are uncertain whether pay-for-performance improves health worker performance, people's use of services, people's health, or resource use in low-income countries (very low-certainty evidence).
- We are uncertain whether financial incentives for health workers improve the quality of care provided by primary care physicians or outpatient referrals from primary to secondary care (very low-certainty evidence).
- There is no rigorous research evaluating incentives (e.g. bursaries or scholarships linked to future practice location, rural allowances) for recruiting health workers to serve in remote areas. It is uncertain whether giving health workers incentives lead more of them to stay in underserved areas (very low-certainty evidence).
- No studies assessed the effects of financial interventions on the movement of health workers between public and private organisations in low- and middle-income countries.
How up to date is this overview?
The overview authors searched for systematic reviews published up to 17 December 2016.
Authors' conclusions:
Research based on sound systematic review methods has evaluated numerous financial arrangements relevant to low-income countries, targeting different levels of the health systems and assessing diverse outcomes. However, included reviews rarely reported social outcomes, resource use, equity impacts, or undesirable effects. We also identified gaps in primary research because of uncertainty about applicability of the evidence to low-income countries. Financial arrangements for which the effects are uncertain include external funding (aid), caps and co-payments, pay-for-performance, and provider incentives. Further studies evaluating the effects of these arrangements are needed in low-income countries. Systematic reviews should include all outcomes that are relevant to decision-makers and to people affected by changes in financial arrangements.
Read the full abstract...
One target of the Sustainable Development Goals is to achieve "universal health coverage, including financial risk protection, access to quality essential health-care services and access to safe, effective, quality and affordable essential medicines and vaccines for all". A fundamental concern of governments in striving for this goal is how to finance such a health system. This concern is very relevant for low-income countries.
To provide an overview of the evidence from up-to-date systematic reviews about the effects of financial arrangements for health systems in low-income countries. Secondary objectives include identifying needs and priorities for future evaluations and systematic reviews on financial arrangements, and informing refinements in the framework for financial arrangements presented in the overview.
We searched Health Systems Evidence in November 2010 and PDQ-Evidence up to 17 December 2016 for systematic reviews. We did not apply any date, language, or publication status limitations in the searches. We included well-conducted systematic reviews of studies that assessed the effects of financial arrangements on patient outcomes (health and health behaviours), the quality or utilisation of healthcare services, resource use, healthcare provider outcomes (such as sick leave), or social outcomes (such as poverty, employment, or financial burden of patients, e.g. out-of-pocket payment, catastrophic disease expenditure) and that were published after April 2005. We excluded reviews with limitations important enough to compromise the reliability of the findings. Two overview authors independently screened reviews, extracted data, and assessed the certainty of evidence using GRADE. We prepared SUPPORT Summaries for eligible reviews, including key messages, 'Summary of findings' tables (using GRADE to assess the certainty of the evidence), and assessments of the relevance of findings to low-income countries.
We identified 7272 reviews and included 15 in this overview, on: collection of funds (2 reviews), insurance schemes (1 review), purchasing of services (1 review), recipient incentives (6 reviews), and provider incentives (5 reviews). The reviews were published between 2008 and 2015; focused on 13 subcategories; and reported results from 276 studies: 115 (42%) randomised trials, 11 (4%) non-randomised trials, 23 (8%) controlled before-after studies, 51 (19%) interrupted time series, 9 (3%) repeated measures, and 67 (24%) other non-randomised studies. Forty-three per cent (119/276) of the studies included in the reviews took place in low- and middle-income countries.
Collection of funds: the effects of changes in user fees on utilisation and equity are uncertain (very low-certainty evidence). It is also uncertain whether aid delivered under the Paris Principles (ownership, alignment, harmonisation, managing for results, and mutual accountability) improves health outcomes compared to aid delivered without conforming to those principles (very low-certainty evidence).
Insurance schemes: community-based health insurance may increase service utilisation (low-certainty evidence), but the effects on health outcomes are uncertain (very low-certainty evidence). It is uncertain whether social health insurance improves utilisation of health services or health outcomes (very low-certainty evidence).
Purchasing of services: it is uncertain whether increasing salaries of public sector healthcare workers improves the quantity or quality of their work (very low-certainty evidence).
Recipient incentives: recipient incentives may improve adherence to long-term treatments (low-certainty evidence), but it is uncertain whether they improve patient outcomes. One-time recipient incentives probably improve patient return for start or continuation of treatment (moderate-certainty evidence) and may improve return for tuberculosis test readings (low-certainty evidence). However, incentives may not improve completion of tuberculosis prophylaxis, and it is uncertain whether they improve completion of treatment for active tuberculosis. Conditional cash transfer programmes probably lead to an increase in service utilisation (moderate-certainty evidence), but their effects on health outcomes are uncertain. Vouchers may improve health service utilisation (low-certainty evidence), but the effects on health outcomes are uncertain (very low-certainty evidence). Introducing a restrictive cap may decrease use of medicines for symptomatic conditions and overall use of medicines, may decrease insurers' expenditures on medicines (low-certainty evidence), and has uncertain effects on emergency department use, hospitalisations, and use of outpatient care (very low-certainty evidence). Reference pricing, maximum pricing, and index pricing for drugs have mixed effects on drug expenditures by patients and insurers as well as the use of brand and generic drugs.
Provider incentives: the effects of provider incentives are uncertain (very low-certainty evidence), including: the effects of provider incentives on the quality of care provided by primary care physicians or outpatient referrals from primary to secondary care, incentives for recruiting and retaining health professionals to serve in remote areas, and the effects of pay-for-performance on provider performance, the utilisation of services, patient outcomes, or resource use in low-income countries.
The impact of user fees on access to health services in low- and middle-income countries
The effect of direct payment policies on people's use of medicines
Effects of governance arrangements for health systems in low-income countries
The impact of conditional cash transfers on health outcomes and use of health services in low and middle income countries
Paying for performance to improve the delivery of health interventions in low- and middle-income countries
Effective practice & health systems > Financial arrangements
Wiysonge CS, Paulsen E, Lewin S, Ciapponi A, Herrera CA, Opiyo N, Pantoja T, Rada G, Oxman AD
Wiysonge CS, Paulsen E, Lewin S, Ciapponi A, Herrera CA, Opiyo N, Pantoja T, Rada G, Oxman AD. Financial arrangements for health systems in low-income countries: an overview of systematic reviews. Cochrane Database of Systematic Reviews 2017, Issue 9. Art. No.: CD011084. DOI: 10.1002/14651858.CD011084.pub2


text:
METRICH executes 3 warrants
Nov 25, 2014 | METRICH Enforcement Unit
Seneca County's METRICH unit executed three drug-related search warrants Monday.
The first two were executed simultaneously at 295 Jefferson St. and 290 1/2 Jefferson St. in Tiffin Monday morning. The third was at 346 N. Main St., Apt. 6, in Fostoria Monday evening.
Nicholas O. Webb, 31, formerly of Toledo, was arrested and charged with trafficking in cocaine, a fourth- degree felony, during execution of the Jefferson Street search warrants.
Officer Steven Beutler of Tiffin Police Department (left) escorts Nicholas O. Webb to a police cruiser after two drug-related search warrants were executed on Jefferson Street Monday morning.
Additional charges of trafficking prescription pills, trafficking cocaine, possession of cocaine, possession of marijuana, possession of criminal tools and permitting drug abuse are pending against Webb upon conclusion of the drug investigation and the return of laboratory analysis of substances discovered in one of the houses when the warrants were executed, according to a METRICH release.
Unit Coordinator Chuck Boyer said Webb, who was taken to Seneca County Jail, allegedly was selling cocaine from both residences.
The search warrants, which were signed by Seneca County Juvenile and Probate Court Judge Jay Meyer, yielded money, suspected marijuana, suspected cocaine, drug paraphernalia, criminal tools and evidence indicative of drug trafficking at 295 Jefferson St., according to the release. Webb and Krista Zeigler, 29, live at the residence.
Nothing illegal was discovered at 290 1/2 Jefferson St., the home of Lindy Warwick, 29, according to the release.
A grand jury is to decide whether Zeigler and Warwick will be charged with permitting drug abuse, Boyer said.
Additional officers from Tiffin Police Department, the department's special response team and its canine unit assisted METRICH with the search warrants. The operation started about 6 a.m., according to a Tiffin Police Department report.
Law enforcement executed a drug-related search warrant at 346 N. Main St., Apt. 6, in Fostoria Monday evening.
Boyer said Fostoria Police Department's uniformed officers responded to a domestic call and saw suspected drug-related items in plain view. They contacted the drug task force, he said.
Details about the search warrant in Fostoria were unavailable early this morning. Originally published in the Advertiser Tribune on November 25, 2014.


text:
Home All issues A&A, 538 (2012) A61 Article references
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Cited article:
Extranuclear Hα-emitting complexes in low-z (U)LIRGs: precursors of tidal dwarf galaxies?
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text:
[Eye Plus] Banpo Hangang Park brings colors to city at night
By Yim Hyun-su
Hangang Park is a crucial part of the recreational life of many of us living in Seoul, offering rare public open space for joggers, cyclists and couples on dates. When the sun goes down and evening approaches, the Banpo part of the park lights up in vivid colors that cast shimmering reflections on the Han River.
Located at Banpo-dong in Seocho district, Banpo Hangang Park boasts three buildings that radiates in changing colors. Known together as "sebit," which literally translates to "three colors" in Korean, the architectures are located on Sebitseom, an artificial island.
In the middle of Banpo Bridge and Dongjak Bridge, the park is also famous for the Moonlight Rainbow Fountain which makes for a great night-view spot.
Banpo Bridge is also known as "Jamsoogyo," meaning submerged bridge. The structure goes below the river's water level for an average of 13 days a year during the monsoon season.
Photos by Park Hyun-koo
Written by Yim Hyun-su
By Yim Hyun-su (hyunsu@heraldcorp.com)
[Today's K-pop] NCT launches music project led by Mark
Mnet's 'I Can See Your Voice' aims to become long-running music show
Kyobo Life asks US accounting oversight board to punish Deloitte Anjin accountants


text:
\section{Introduction}

A diffeomorphism is called partially hyperbolic if the tangent bundle admits an invariant decomposition $TM = E^s \oplus E^c \oplus E^u$, such that all unit vectors $v^{\sigma} \in E^{\sigma}_x, \sigma \in \{s, c, u\}$ for all $x \in M$ satisfy:
$$
  \|D_xf v^s \| < \|D_xf v^c\| < \| D_xf v^u\|
$$
and moreover $ \|D_f| E^s\| < 1$ and $\|D f^{-1} | E^{u}\| < 1.$ We call $f$  {\bf absolute} partially hyperbolic if it is partially hyperbolic and for any $x, y, z \in M$
$$
 \|D_xf v^s \| < \|D_yf v^c\| < \| D_zf v^u\|
$$
where $v^s, v^c, v^u$ belong respectively to $E_x^s, E^c_y$ and $E^u_z.$
In this paper when we require partial hyperbolicity, we mean absolute partial hyperbolicity.

The subbundles $E^s$ and $E^u$ integrate into $f$-invariant foliations, respectively the stable foliation, $\mathcal F^s$, and the unstable foliation, $\mathcal F^u$. These foliations have  a nice property called absolute continuity. Among different definitions for absolute continuity of foliations we choose the following one which is suitable for our purpose: A set of full volume measure on $M$ must intersect almost every leaf of $\mathcal F^s$ (or $\mathcal F^u$) in a set of  full Lebesgue measure of the leaf. Although the absolute continuity of $\mathcal F^s$ and $\mathcal F^u$ are mandatory for a (general)  $C^2$ partially hyperbolic diffeomorphism, this is not the case for the center
foliation $\mathcal F^c$ (it is not even true that there will exist such a foliation, but by \cite{BBI09} for all absolute partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ the center foliation exists). The center foliation might not be absolutely continuous, at least this is, in general, expected to happen for diffeomorphisms which preserves volume (see \cite{SX}, \cite{HP},  \cite{Gogolev1}). For many examples (some of them described below) the center foliation has atomic disintegration. In all such examples there exists a set $A \subset M$ of full volume measure, such that the intersection of $A$ with every leaf of $\mathcal F^c$ are $k$ points, where $k$ is a natural number independent of the leaf (in the ergodic context). In principle for a general partially hyperbolic difeomorphism the geometric structure of the support of disintegration measures is not clear. We do not have examples of atomic disintegration with infinitely many atoms.

 There exist essentially three known category of partially hyperbolic diffeomorphisms on  three-dimensional manifolds (see conjecture of Pujals in \cite{BW05}). We deal here with the so called Derived from Anosov (DA) diffeomorphisms. By a DA diffeomorphism we mean a partially hyperbolic diffeomorphim $f:\mathbb T^3 \rightarrow \mathbb T^3$ such that its linearization (see \ref{prepartial} in Preliminaries) is an Anosov diffeomorphism. Observe that by \cite{Htese} the linearization of $f$ is also partially hyperbolic in the sense that it admits three invariant sub-bundles.   The other two classes of partially hyperbolic diffeomorphisms are the skew-product type and perturbations of time-one of Anosov flows (see as well Hammerlindl-Potrie \cite{HP13} for a discussion and new results).

For the perturbation of a time-one map of the geodesic flow for a closed negatively curved surfaces (which is an Anosov flow), it was shown by A. Avila, M. Viana and A. Wilkinson \cite{AVW} that $\mathcal F^c$ has atomic disintegration or it is absolutely continuous. For a large class of skew-product diffeomorphisms, they announced that  they can prove an analogous result.

It is interesting to emphasize that (conservative) Derived from Anosov (DA) diffeomorphims on $\mathbb T^3$ show a feature that is not, so far, shared with any other known partially hyperbolic diffeomorphims on dimension three, it admits all three disintegration of volume on the center leaf, namely: Lebesgue, atomic, and, by a recent result of R. Var\~ao \cite{Va}, they can also have a disintegration which is neither Lebesgue nor atomic.

 More precisely, R. Var\~ao \cite{Va} showed that there exist Anosov diffeomorphisms with non-absolutely continuous center foliation which does not have atomic disintegration. Here we show a new behavior for DA diffeomorphisms on $\mathbb T^3$, and that is the existence of atomic disintegration (Theorem \ref{theorem:existatom}). This behavior can be verified for an open class of diffeomorphisms found by  Ponce-Tahzibi in \cite{PT}. In fact we prove  (Theorem \ref{theorem:oneatom}): if the disintegration of Lebesgue measure is atomic, then it is in fact mono-atomic, i.e there is just one atom per leaf. We should mention that the most important part of our proof is to conclude finiteness of atoms in the case of atomic disintegration. However, for general partially hyperbolic diffeomorphisms finiteness does not imply mono-atomic disintegration. See \cite{RW} and our discussion after Theorem A below for a contrast with the skew product case.\\

\textbf{Theorem \ref{theorem:oneatom}}  \textit{Let $f$ be a volume preserving accessible DA diffeomorphism on $\mathbb T^3.$ If volume has atomic disintegration on center leaves, then it has one atom per leaf.}\\

\begin{remark} \label{remark: nonnull} The accessibility hypothesis for partially hyperbolic diffeomorphisms with one dimensional center bundle implies  that $f^n$ is ergodic for all $n \geq 1.$ (see \cite{HHU1} or \cite{BW}.) This is the unique place that we use accessibility, so that the accessibility hypothesis can be substituted by any other  hypothesis that provides the ergodicity of all $f^n$. For instance, by Hammerlindl-Ures \cite{HU} we know that if the center exponent of $f$ is not zero almost everywhere (consequently the center exponent of $f^n$ is not zero almost everywhere, for all $n$) then it is ergodic (consequently $f^n$ is ergodic for all $n$). This observation will be important in the proof of Theorem B.
\end{remark}

 The conclusion about the number of atoms is not true for skew-products in general. Let $B := A \times Id$ where
$ A:= \left( \begin{array}{cc}
2 & 1 \\
1 & 1 \\ \end{array} \right)$. Then arbitrarily close to $B : \mathbb{T}^3 \rightarrow \mathbb{T}^3 $ there is an open set of partially hyperbolic diffeomorphisms $g$ such that $g$ is ergodic and there is an equivariant fibration $\pi: \mathbb{T}^3 \rightarrow \mathbb{T}^2$ such that the fibers are circles, $\pi \circ g = B \circ \pi$. And $g$ has positive central Lyapunov exponent, hence the central foliation is not absolutely continuous. In Ruelle and Wilkinson's paper \cite{RW}, we see that there exist $S \subset \mathbb{T}^ 3$ of full volume and $k \in \mathbb{N}$ such that $S$ meets every leaf in exactly $k$ points. In Shub and Wilkinson's example \cite{SW00} the fibers of the fibration are invariant under the action of a finite non-trivial group  and consequently in their example the number of atoms cannot be one.

In the next theorem we introduce a class of derived from Anosov diffeomorphisms where Theorem A can be applied. The work of Ponce, Tahzibi \cite{PT} guaranties that there exist DA diffeomorphisms satisfying these conditions.\\

\textbf{Theorem \ref{theorem:existatom}}
\textit{Let $f: \mathbb T^3 \rightarrow \mathbb T^3$ be a volume preserving, DA diffeomorphism. Suppose that its linearization $A$ has the splitting $T_AM = E^{su} \oplus E^{wu} \oplus E^s$ (su and wu represent strong unstable and weak unstable bundles.) If $f$ has $\lambda ^{c}(x) <0$ for Lebesgue almost every point $x \in \mathbb T^3$, then volume has atomic disintegration on $\mathcal F^{c}_f$, in fact the disintegrated measures  have one atom per center leaf.}\\

\begin{remark}\label{remark:proof.theorem}
The key point in the proof of Theorem \ref{theorem:existatom} is  show that we get atomic disintegration. Then, we use Theorem \ref{theorem:oneatom} (because of Remark \ref{remark: nonnull}) to obtain one atom per leaf.
\end{remark}

We look at the linearization of $f$, the linear Anosov $A$, as a partially hyperbolic for which the center leaf $\mathcal F^c_A$ is the expanding leaf $\mathcal F^{wu}_A$.

We mention that the examples of non-absolutely continuous weak foliation of Anosov diffeomorphisms was known by Saghin-Xia \cite{SX} and Baraviera-Bonatti \cite{BB} near to geodesic flows,  and by A.Gogolev near to hyperbolic automorphisms of the $3$-torus \cite{Gogolev1}. However, we are introducing examples of non-Anosov diffeomorphisms with non-absolutely continuous central foliation and prove atomic disintegration. It is not known whether the disintegration of the Lebesgue measure can be atomic in the case of Anosov diffeomorphisms.

\subsection{Maximal entropy measures and Lyapunov exponents}\label{subsec.max.entropy}

Lyapunov exponents are celebrated constants which are related to the entropy of invariant measures. In this paper  we denote by $\lambda^u(f), \lambda^c(f)$ and $\lambda^s(f)$ the Lyapunov exponents of Lebesgue measure (ergodic case).  When we are referring to the Lyapunov exponents of any other (ergodic) invariant measure $\mu$ we use the subscript $\lambda^{*}_{\mu}.$
The relationship between the Lyapunov exponents of a partially hyperbolic diffeomorphism and its linearization is an interesting issue. In \cite{MT}, the authors among other results proved a folkloric semi-rigidity property of unstable and stable Lyapunov exponents of  (absolute) partially hyperbolic diffeomorphisms of $\mathbb{T}^3:$

\begin{theorem}
Let $f$ be a $C^2$ conservative partially hyperbolic diffeomorphism on the $3-$torus and $A$ its linearization then

$$ \lambda^u(f,x) \leq \lambda^u(A) \; \mbox{and} \;\; \lambda^s(f,x) \geq \lambda^s(A)\; \mbox{for Lebesgue a.e. } \; x \in \mathbb{T}^3.$$
\end{theorem}

The above theorem relates the extremal Lyapunov exponents of a non-linear and linear partially hyperbolic diffeomorphism. For the central Lyapunov exponent we do not expect such behavior. However, it is a problem in \cite{MT}:

\begin{problem} In the context of the above theorem,
suppose that $\lambda^{c}(f) > 0$ and $\mathcal{F}^c$ is (upper leafwise) absolutely continuous. Is it true that $\lambda^{c}(f) \leq \lambda^{c}(A)$ ?
\end{problem}

The answer to the above problem is positive when $f$ is an Anosov diffeomorphism (see \cite{MT, Va}).

We emphasize that the above relations between Lyapunov exponents are valid for the volume measure. For other natural invariant measures the scenario can be different. Let $\mu_f$ be the unique maximizing measure for a partially hyperbolic diffeomorphism $f$ homotopic to Anosov $A.$ In \cite{Ures},  R. Ures  proved that if $\lambda^ c(A) > 0$, i.e $A$ has weak unstable sub-bundle then $\lambda^ c_{\mu_f} \geq \lambda^ c(A).$

In the following theorem we see that if the central Lyapunov exponent for Lebesgue measure of $f$ is strictly smaller than the central exponent of the linearization $A,$ then either the central or the unstable exponent of the maximizing measure of $f$ is strictly larger than the corresponding exponent of $A$.

Before stating our last theorem we give one more definition. A measure is called u-Gibbs if the disintegration subordinated to the unstable foliation (corresponding to all positive Lyapunov exponents) gives conditional measures equivalent to the Lebesgue on the leaf. These measures play an important role in dynamical systems, \cite{LY1}.\\

\textbf{Theorem \ref{ugibbs}}
\textit{Let $f : \mathbb T^3 \rightarrow \mathbb T^3$ be a DA diffeomorphism with $A: \mathbb T^3 \rightarrow \mathbb T^3$ its linearization. Assume that $\lambda^c(A)>0$ and $\lambda^c(f) < \lambda^c(A)$. Then $\mu_f$ is not $u$-Gibbs. Also, denoting by $\lambda^u_{\mu_f}, \lambda^c_{\mu_f}, \lambda^s_{\mu_f}$ the Lyapunov exponents of $f$ (for $\mu_f$ almost every point) we have
$$\lambda^c_{\mu_f} > \lambda^c(A) \text{ or } \lambda^u_{\mu_f} > \lambda^u(A) .$$}

We recall  the example of Ponce-Tahzibi in \cite{PT} where $f$ has negative central Lyapunov exponent and the linearization of $f$ has positive central Lyapunov exponent. The authors began with a linear Anosov diffeomorphism $A$ with splitting $E^{su} \oplus E^{wu} \oplus E^ {s}$ and after a modification they found $f$ partially hyperbolic ergodic volume preserving such that $\lambda^c(f) < 0.$ The point is that, in their construction $\lambda^u_{\mu_f} (f) =\lambda^u(A)$ and by the above theorem we conclude that $\lambda^c_{\mu_f} > \lambda^c(A).$ That is, although after perturbation the central Lyapunov exponent of Lebesgue measure drops, the central exponent of maximal entropy measure increases. We describe in some more details the construction of this example on \S \ref{subsec.pathological}.

\section{Preliminaries}\label{sec.preliminaries}

\subsection{Measurable partitions and disintegration of measures}\label{subsec.partition}

Let $(M, \mu, \mathcal B)$ be a probability space, where $M$ is a compact metric space, $\mu$ a probability measure and $\mathcal B$ the borelian $\sigma$-algebra.
Given a partition $\mathcal P$ of $M$ by measurable sets, we associate the probability space $(\mathcal P, \widetilde \mu, \widetilde{\mathcal B})$ by the following way. Let $\pi:M \rightarrow \mathcal P$ be the canonical projection, that is, $\pi$ associates a point $x$ of $M$ with the partition element of $\mathcal P$ that contains it. Then we define $\widetilde \mu := \pi_* \mu$ and $ \widetilde{\mathcal B}:= \pi_*\mathcal B$.

\begin{definition} \label{definition:conditionalmeasure}
 Given a partition $\mathcal P$. A family $\{\mu_P\}_{p \in \mathcal P} $ is a \textit{system of conditional measures} for $\mu$ (with respect to $\mathcal P$) if
\begin{itemize}
 \item[i)] given $\phi \in C^0(M)$, then $P \mapsto \int \phi \mu_P$ is measurable;
\item[ii)] $\mu_P(P)=1$ $\widetilde \mu$-a.e.;
\item[iii)] if $\phi \in C^0(M)$, then $\displaystyle{ \int_M \phi d\mu = \int_{\mathcal P}\left(\int_P \phi d\mu_P \right)d\widetilde \mu }$.
\end{itemize}
\end{definition}

When it is clear which partition we are referring to, we say that the family $\{\mu_P\}$ \textit{disintegrates} the measure $\mu$.

\begin{proposition}
 If $\{\mu_P\}$ and $\{\nu_P\}$ are conditional measures that disintegrate $\mu$, then $\mu_P = \nu_P$ $\widetilde \mu$-a.e.
\end{proposition}

\begin{corollary}
 If $T:M \rightarrow M$ preserves a probability $\mu$ and the partition $\mathcal P$, then  $T_*\mu_P = \mu_P$ $\widetilde \mu$-a.e.
\end{corollary}
\begin{proof}
 It follows from the fact that $\{T_*\mu_P\}_{P \in \mathcal P}$ is also a disintegration of $\mu$.
\end{proof}

\begin{definition}
We say that a partition $\mathcal P$ is measurable with respect to $\mu$ if there exist a measurable family $\{A_i\}_{i \in \mathbb N}$ and a measurable set $F$ of full measure such that
if $B \in \mathcal P$, then there exists a sequence $\{B_i\}$, where $B_i \in \{A_i, A_i^c \}$ such that $B \cap F = \bigcap_i B_i \cap F$.
\end{definition}
\begin{proposition} \label{prop:compact}
Let $(M,\mathcal B, \mu)$ a probability space where $M$ is a compact metric space and $\mathcal B$ is the Borel sigma-algebra. If $\mathcal P$ is a continuous foliation of $M$ by compact measurable sets, then $\mathcal P$ is a measurable partition.
\end{proposition}

\begin{theorem}[Rokhlin's disintegration] \label{teo:rokhlin}
 Let $\mathcal P$ be a measurable partition of a compact metric space $M$ and $\mu$ a borelian probability. Then there exists a disintegration by conditional measures for $\mu$.
\end{theorem}
In general the partition by the leaves of a foliation may be non-measurable. It is for instance the case for the stable and unstable foliations of a linear Anosov diffeomorphism. Therefore, by disintegration of a measure along the leaves of a foliation we mean the disintegration on compact foliated boxes. In principle, the conditional measures depend on the foliated boxes, however, two different foliated boxes induce proportional conditional measures. See \cite{AVW} for a discussion. We define absolute continuity of foliations as follows:

\begin{definition}
 We say that a foliation $\mathcal F$ is absolutely continuous if for any foliated box, the disintegration of volume on the segment leaves have conditional measures equivalent to the Lebesgue measure on the leaf.
\end{definition}

\begin{definition}
We say that a foliation $\mathcal F$ has atomic disintegration with respect to a measure $\mu$ if the conditional measures on any foliated box are a sum of Dirac measures. Note that this is equivalent to saying that there is a set of $\mu$-full measure that intersects each leaf on a discrete set.
\end{definition}

Although the disintegration of a measure along a general foliation is defined in compact foliated boxes, it makes sense to say that the foliation $\mathcal F$ has a quantity $k_0 \in \mathbb N$  atoms per leaf. The meaning of ``per leaf'' should always be understood as a generic leaf, i.e. almost every leaf. That means that there is a set $A$ of $\mu$-full measure which intersects a generic leaf on exactly $k_0$ points. Let's see that this implies atomic disintegration. Definition \ref{definition:conditionalmeasure} shows that it only make sense to talk about conditional measures from the generic point of view, hence when restricted to a foliated box $\mathfrak{ B}$, the set $A \cap \mathfrak{B}$ has $\mu$-full measure on $\mathfrak{B}$, therefore the support of the conditional measure disintegrated on $\mathfrak{B}$ must be contained on the set $A$. This implies atomic disintegration.

It is well worth to remark that  the weight of an atom for a conditional measure naturally depends on the foliated box, but a point $x$ is atom independent of the foliated box where we disintegrate a measure.

\subsection{ Preliminaries on Partial Hyperbolicity}\label{subsec.ph} \label{prepartial}

If $f: M \rightarrow M$ is a partially hyperbolic diffeomorphism with
$$T_xM = E^s(x) \oplus E^c(x) \oplus E^u(x)$$
the bundles $E^s$ and $E^u$ are tangent to invariant foliations $\mathcal{F}^s$ and $\mathcal{F}^u.$ $f$ is called {\bf accessible} if for any two points $x$ and $y$ there is a piecewise smooth curve connecting $x$ to $y$ and tangent to $E^s \cup E^u.$ If the central bundle is one dimensional and $f$ is volume preserving then accessibility of $f$ implies that all iterates $f^n$ are ergodic (\cite{BW}, \cite{HHU1}).

We define the Lyapunov exponents of $f$ by
$$\lambda^{\tau}(x) := \lim_{n\rightarrow \infty}\frac{1}{n} \log ||Df^n(x) \cdot v||$$

where $v  \in E^{\tau}$ and $\tau \in \{s,c,u\}$. If $f$ is ergodic, the stable, center and unstable Lyapunov exponents are constant almost everywhere. Otherwise they have no reason to be constant in the general case. \\

Any $A \in SL(3, \mathbb{Z})$ with at least one eigenvalue with norm larger than one, induces a linear partially hyperbolic diffeomorphism on $\mathbb{T}^3.$ Conversely for any partially hyperbolic diffeomorphism $f$ on $\mathbb{T}^3,$ there exist a unique linear diffeomorphism $A,$ such that $A$ induces the same automorphism as $f$ on the fundamental group $\pi_1(\mathbb{T}^3).$

Let $f: \mathbb{T}^3 \rightarrow \mathbb{T}^{3}$ be a partially hyperbolic diffeomorphism. Consider $f_* : \mathbb{Z}^3 \rightarrow \mathbb{Z}^3$ the action of $f$ on the fundamental group of $\mathbb{T}^3.$ $f_*$ can be extended to $\mathbb{R}^3$ and the extension is the lift of  a unique linear automorphism $A :  \mathbb{T}^{3} \rightarrow \mathbb{T}^{3}$ which is called the linearization of $f.$ It can be proved that $A$ is a partially hyperbolic automorphism of torus (\cite{BBI}). We will say that $f$ is {\bf derived from Anosov} (DA diffeomorphism) if  its linearization $f_*$ is an Anosov diffeomorphism.

Let $f$ be DA diffeomorphism defined  as above, then by \cite{Franks} we know that $f$ is semi-conjugated to its linearization by a function $h: \mathbb{T}^3 \rightarrow \mathbb{T}^3, h \circ f = A \circ h$. It follows from \cite{Ures} that $\mathcal{F}^c (A) = h(\mathcal{F}^c (f)).$ Moreover, there exists a constant $K  \in \mathbb{R}$ such that  if $\tilde{h} : \mathbb{R}^3 \rightarrow \mathbb{R}^3 $ denotes the lift of $h$ to $\mathbb{R}^3$ we have $\|\tilde h(x) - x\| \leq K$ for all $x \in \mathbb{R}^3.$

It is not difficult to see that in large scale $f$ and $A$ behaves similarly (see \cite{Htese}, lemma 3.6). More precisely, for each $k \in \mathbb{Z}$ and $C > 1$ there is an $M > 0 $ such that for all $x, y \in \mathbb{R}^3$,
\begin{equation}
\label{large-scale}
\|x - y\| > M \Rightarrow \frac{1}{C} < \frac{\|\tilde{f}^k(x) - \tilde{f}^k(y)\|}{\|A^k(x) - A^k(y)\|} < C.
\end{equation}
where $\tilde{f}: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ is the lift of $f$ to $\mathbb{R}^3.$

\begin{definition} \label{qi}
A foliation $\mathcal F$ defined on a manifold $M$  is quasi-isometric if the lift $\widetilde{ \mathcal F}$ of $\mathcal F$ to the universal cover of $M$ has the folowing property: There exist positive constant $Q$
such that for all $x, y$ in a common leaf of $\widetilde{ \mathcal F}$ we have
$$d_{\widetilde{\mathcal{F}}}(x, y) \leq Q || x - y||,$$
where $d_{\widetilde{\mathcal{F}}}$ denotes the riemannian metric on $\widetilde{\mathcal{F}}$ and $\|x-y\|$ is the distance on the ambient manifold of the foliation.
\end{definition}

For absolute partially hyperbolic diffeomorphisms on $\mathbb{T}^3$ the stable, unstable and central foliations are quasi isometric in the universal covering $\mathbb{R}^3$.

\subsection{``Pathological" example}\label{subsec.pathological}
As we remarked before, in theorem  \ref{theorem:existatom} one of the hypothesis is that the center Lyapunov exponent of the diffeomorphism $f$ and of its linearization $A$ have opposite sign. Since we have center leaf conjugacy between $f$ and $A$ and since in large scale the behavior of the center leafs  is similar (see (\ref{large-scale})), this hypothesis would imply that the asymptotic growth of the center leaves (which is a local issue) and global behavior of the center leafs of $f$ are opposite. In \cite{PT}, the authors constructed an example to show that this kind of phenomena occurs in an open set of partially hyperbolic dynamics and we briefly describe it here.

Start with the family of linear Anosov diffeomorphisms $f_k: \mathbb T^3 \rightarrow \mathbb T^3$ induced by the integer matrices:
$$A_k=\left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & 1 & -1 \\
-1 & -1 & k \end{array} \right), k \in \mathbb N. $$

This family of Anosov diffeomorphisms has two important characteristics that justify this choice.  Denote by $\lambda^s_k, \lambda^c_k, \lambda^u_k$ the three eigenvalues of $A_k$ with $\lambda^s_c < \lambda^c_k<\lambda^u_k$ and, for each $k$, denote by $E^s_k, E^c_k, E^u_k$ the stable, central and unstable fiber bundles with respect to $A_k$. Then an easy calculation shows that
$$\lambda^s_k \rightarrow 0, \lambda^c_k \rightarrow 1, \lambda^u_k \rightarrow \infty,$$
as $k \rightarrow \infty$.

Using a Baraviera-Bonatti \cite{BB} local perturbation method, for large $k$ the authors managed to construct small perturbation of $f_k$ and obtain partially hyperbolic diffeomorphisms $g_k$ such that the central Lyapunov exponent of $g_k$ is positive.

By taking the family $g_k^{-1}$ we obtain partially hyperbolic diffeomorphisms with negative center exponent and isotopic to  Anosov diffeomorphism with weak expanding subbundle. In fact any $f:=g_k^{-1}$ satisfy the desired properties.
By construction of the perturbation, it comes out that  the center-stable bundle of $f$ coincides with the sum of  stable and weak unstable bundle of Anosov linearizations $f_* = A$: $E^{cu}f (x) =  E_A^{wu}(x) \oplus E_A^{s}(x)$. Moreover, for any $x \in \mathbb{T}^3$ we have $\log J^{cs}f = \log J^sA + \log J^{wu}(A).$ As both $f$ and $A$ are volume preserving we conclude that $\displaystyle{\int}
\log J^u(f, x) d\mu = \lambda^u(A)$ and by ergodicity $\lambda_f^u(x) = \lambda^u(A)$ for $\mu$ almost every $x$ where $\mu$ is any invariant probability measure.

 This family of diffeomorphisms fulfills the hypothesis required in Theorems \ref{theorem:existatom} and \ref{ugibbs}.

\section{Proof of results}

\begin{maintheorema}\label{theorem:oneatom} Let $f$ be a volume preserving accessible DA diffeomorphism on $\mathbb T^3.$ If the volume has atomic disintegration on the center leaves, then it has one atom per leaf.
\end{maintheorema}
\begin{proof}
Let $h: \mathbb T^3 \rightarrow \mathbb T^3$ be the semi-conjugacy between $f$ and its linearization $A$, hence $ h\circ f = A \circ h$. We can assume that  $A$ has two eigenvalues larger than one, otherwise we work with $f^{-1}$.

 Let $\{R_i\}$ be a Markov partition for $A$, and define $\widetilde R_i := h^{-1}(R_i)$. We claim that
 \begin{equation} \label{interior}
 Vol \left( \bigcup_i int \;\widetilde R_i \right) =1.
 \end{equation}

Indeed, first look at the center direction of $A$. For simplicity we consider the center direction as a vertical foliation. This means that the rectangle $R_i$ has two types of boundaries, the one coming from the extremes of the center foliation and the lateral ones. We call $\partial_c R_{i}$ the boundary coming from these extremes of the center foliation, i.e
$$
 \partial_c R_i = \bigcup_{x \in R_i} \partial (\mathcal{F}^c_{x} \cap R_i).
$$ Since $h$ takes center leaves to center leaves, we conclude that the respective boundary for the $\widetilde R_i$ sets is $\partial_c \widetilde R_i = h^{-1}(\partial_c R_{i})$, and since $\bigcup_i \partial_c R_{i}$ is an $A$-invariant set, $\bigcup_i \partial_c \widetilde R_i $ is an $f$-invariant set. By ergodicity of $f$ it follows that $\bigcup_i \partial_c \widetilde R_i $ has zero or full measure. Since the volume of the interior cannot be zero, then the volume of $\bigcup_i \partial_c \widetilde R_i $ cannot be one. Therefore it has zero measure.

By (\ref{interior}) we can consider the partition $\widetilde {\mathcal P} = \{ \mathcal F^c_{R(x)} : x \in \widetilde R_i \text{ for some } i\}$ where $\mathcal F^c_{R(x)}$ denotes the connected component of $\mathcal F^c_f(x) \cap \widetilde R(x)$ which contains $x$. Thus we can consider the Rokhlin disintegration of volume on the partition $\widetilde {\mathcal P}$.  Denote this system of measures by $\{m_x\}$, so that each $m_x$ is supported in $\mathcal F^c_f(x)$.

\begin{lemma}
 There is a natural number  $\alpha_0 \in \mathbb N$, such that for almost every point, $\mathcal F^c_{R(x)}$ contains exactly $\alpha_0$ atoms.
\end{lemma}
\begin{proof}
The semi-conjugacy $h$ sends center leaves of $f$ to center leaves of $A$. Also, the points of the interior of the $\widetilde R_i$ satisfy that $f(\mathcal F^c_{R(x)}) \supset \mathcal F^c_{R(f(x))}$, which just comes from the Markov property of the rectangles $R_i$. This implies that
$$f_*m_x \leq m_{f(x)}$$
on $\mathcal F^c_{R(f(x))}$.
 Given any $\delta \geq 0$ consider the set $A_\delta =\{ x \in \mathbb{T}^3 \; | \; m_{x}(\{x\}) > \delta \}$, that is, the set of atoms with weight at least $\delta$.
 If $x\in A_{\delta}$ then
 $$\delta < m_x(\{x\}) =  f_{*}m_x(\{f(x)\}) \leq m_{f(x)} (\{f(x)\}).$$

 Thus $f(A_{\delta}) \subset A_{\delta}$, and by the ergodicity of $f$ we have that
$Vol(A_\delta)$ is  zero or one, for each $\delta \geq 0$. Note that $Vol(A_0)=1$ and $Vol(A_1)=0$. Let $\delta_0$ be the critical point for which $Vol(A_\delta)$ changes value, i.e, $\delta_0 = \sup \{\delta : Vol(A_{\delta}) = 1\}$. This means that all the atoms have weight $\delta_0$. Since $m_x$ is a probability we have an $\alpha_0:=1/\delta_0$ number of atoms as claimed.
\end{proof}

\begin{lemma}
There is a positive number $L_0$ such that on almost every center leaf there is a point $x$ such that $B_c(x,L_0)$ (ball centered on $x$ inside $\mathcal F^c(x)$ of size $L_0$) contains all the atoms of the leaf $\mathcal F^c_f(x)$.
 \end{lemma}
 \begin{proof}
  We divide the proof in two possible cases. The first one is the case where we have finite number of atoms on each center leaf and the second is that we have infinitely many atoms on each center leaf.\\

  {\noindent \bf Case one (Finite atoms):} Suppose we have a finite number of atoms on each center leaf. Given $L \in \mathbb R$, consider the following set
   $$B_L=\{x \; | \; B_c(x,L) \text{ contains all the atoms of } \mathcal F^c_x \}.$$
   For any large enough $L$ the set $B_L$ has positive volume. To prove that $Vol(B_L)=1$ for some large $L$, it is sufficient to prove that $f^{-n} (B_L) \subset B_L$ for some $n$. This is because, $f^{n}$ is ergodic. We do so by proving that $f^{-n} (B_c(x,L)) \subset B_c(f^{-n}(x), L)$, where $B_c(\xi,\rho)$ stands for the ball inside of the center leaf centered on $\xi$ and radius $\rho$ and $d_c$ is the distance on the center leaf coming from the metric when restricted to the center leaf.

   It is enough to prove that $f^{-n} (B_c(x,L)) \subset B_c(f^{-n}(x), L)$ on the lift, since the projection is locally an isometry. We now work on the lift, but we carry the same notation as it should not make any confusion. Let $K\in \mathbb R$ such that
   $$||h - Id||_{C^0} < K,$$

 And $Q$ coming from the quasi-isometry property of the center foliation.  Let $y, z \in B_c(x,L)$ be the extremals, i.e. $d_c(y,z)=2L$,

\begin{eqnarray*}
 d_c(f^{-n}(y), f^{-n}(z)) &\leq& Q \| f^{-n}(y) - f^{-n}(z) \|
 \leq Q ( \| hf^{-n}(y) - hf^{-n}(z)\|+ 2K) \\
				&=& Q ( \| A^{-n} h(y) - A^{-n}h(z)\| + 2K) \\
				&=& Q({{e^{-n\lambda^{wu}(A)}}} \|h(y) - h(z)\|  + 2K )\\
				&\leq& Q({{e^{-n\lambda^{wu}(A)}}}( \|y - z\| +2K) + 2K ) \\
				&\leq& Q({{e^{-n\lambda^{wu}(A)}}}( d_c(y,z) +2K) + 2K )				=Q({{e^{-n\lambda^{wu}(A)}}}( 2L +2K) + 2K )
 \end{eqnarray*}

  Since $K$ is fixed, first consider a large $L$ and then a large enough $n$ such that $$Q({{e^{-n\lambda^{wu}(A)}}}( 2L +2K) + 2K ) < 2L.$$

  For these choices of $n$ and $L$, we have $f^{-n} (B_c(x,L)) \subset B_c(f^{-n}(x), L)$.\\

  {\noindent \bf Case two (Infinite Atoms):} Suppose we have an infinitely many atoms on each center leaf.
   Let $\beta \in \mathbb R$ be a large number (for instance much bigger than $K Q$ where $K$ is the distance between $h$ and the identity map and $Q$ is the quasi isometric constant in the definition \ref{qi}). Since we have a finite number of $\widetilde R_i$, from the previous Lemma, we know that there is a number $\tau \in \mathbb R$ for which every center segment of size smaller then $\beta$ must contain at most $\tau$ atoms. But, since on each center leaf there are infinity many atoms, take a segment of leaf big enough so that it contains more then $\tau$ atoms. Iterate this segment backwards  and it will eventually be smaller than $\beta$ but containing more than $\tau$ atoms. Indeed,
   $$ h \circ f^{-n} = A^{-n} \circ h$$
   $$
   \|h (f^{-n}(x)) - h(f^{-n} (y)) \| = \| A^{-n} (h(x)) - A^{-n} (h(y))\| $$$$\leq e^{-n \lambda^{wu}(A)} \| h(x) -h(y) \|
   $$
  As $h$ is at a distance $K$ to identity we have
  $$
  \|f^{-n}(x) - f^{-n}(y)\| \leq e^{-n \lambda^{wu}(A)} \| h(x) -h(y) \| + K \leq \frac{\beta}{Q}
  $$
  So, finally by quasi isometric property
  $$
  d_c(f^{-n}(x), f^{-n}(y)) \leq \beta.
  $$

 The above contradiction implies that the number of atoms can not be infinite and now we proceed as in the previous case.

 \end{proof}

\begin{lemma}
The disintegration of the Lebesgue measure along the central leaves is mono-atomic, i.e there is just one atom per leaf.
\end{lemma}
 \begin{proof}
 We have a finite number of atoms on each center leaf and since the center foliation is an oriented foliation we may talk about the first atom. The set of first atom of all generic leaves is an invariant set with positive measure, therefore it has full measure. This means there is only one atom per center leaf, which concludes the proof.
 \end{proof}

  The above Lemma concludes the proof of the theorem.
\end{proof}

\begin{problem}
Is there any ergodic invariant measure $\mu$ with disintegration having more than one atom on leaves?
\end{problem}

We note once again that since the work of Ponce-Tahzibi \cite{PT} assures that the set of DA satisfying the hypothesis of the next theorem is non-empty, we prove that these diffeomorphisms have atomic disintegration.

\subsection{A Glimpse of Pesin Theory}
Before presenting the proof of Theorem $\ref{theorem:existatom}$, we recall some basic notions of Pesin theory. Let $f : \mathbb T^3 \rightarrow \mathbb T^3$ a partially hyperbolic diffeomorphism with splitting
$$T M = E^s \oplus E^c \oplus E^u.$$
Call $\Gamma$ the set of regular points of $f$, that is, the set of points $x \in \mathbb T^3$ for which the Lyapunov exponents are well defined. Then, for each $x \in \Gamma$ we define the Pesin-stable manifold of $f$ at $x$ as the set
$$W^s(x) = \left\{ y: \limsup_{n\rightarrow \infty} \frac{1}{n} d(f^n(x),f^n(y)) < 0 \right\}.$$

The Pesin-stable manifold is an immersed sub manifold of $\mathbb T^3$. Similarly we define the Pesin-unstable manifold at $x$, $W^u(x)$, using $f^{-1}$ instead of $f$ in the definition.

It is clear that for a partially hyperbolic diffeomorphism $W^s(x)$ contains the stable leaf $\mathcal{F}^s(x).$ In  Theorem \ref{theorem:existatom} we assume that the central Lyapunov exponent is negative and consequently the Pesin-stable manifolds are two dimensional. By $\mathcal W^c(x)$ we denote the intersection of the Pesin stable manifold  $W^s(x)$ of $x$ with the center leaf $\mathcal F_f^c(x)$ of $x$. These manifolds depends only measurable on the base point $x,$ as it is proved in the Pesin theory. However, there is a filtration of the set of regular points by Pesin blocks: $\Gamma = \bigcup_{l \in \mathbb{N}} \Gamma_{l}$ such that each $\Gamma_l$ is a closed (not necessarily invariant) subset and $x \rightarrow \mathcal W^c(x)$ varies continuously on each $\Gamma_l.$

\begin{maintheorema}\label{theorem:existatom}
Let $f: \mathbb T^3 \rightarrow \mathbb T^3$ be a volume preserving, DA diffeomorphism. Suppose its linearization $A$ has the splitting $T_AM = E^{su} \oplus E^{wu} \oplus E^s$ (su and wu represents strong unstable and weak unstable bundles.) If $f$ has $\lambda ^{c}(x) <0$ for Lebesgue almost every point $x \in \mathbb T^3$, then volume has atomic disintegration on $\mathcal F^{c}_f$, in fact it is one atom per center leaf.
\end{maintheorema}

\begin{proof}

To begin, we prove that the size of the weak stable manifolds $\mathcal{W}^c(x)$ is uniformly bounded for $x$ belonging to the regular set. In particular this enables us to prove that the partition (mod-$0$) by $\mathcal{W}^c(x)$ is a measurable partition.

\begin{lemma}
 The size of  $\{ \mathcal W^{c}(x) \}_{ \{x \; : \; \lambda^c(x) <0\} }$ is uniformly bounded for $x \in \Gamma$. More precisely, the image of  $ \mathcal W^{c}(x)$ by $h$ is a unique point.
\end{lemma}

\begin{proof}
Let $\tilde{f}:\mathbb R^3 \rightarrow \mathbb R^3$ and $\tilde{A}:\mathbb R^3 \rightarrow \mathbb R^3$ denote the lifts of $f$ and $A$ respectively and $\tilde{h}:\mathbb R^3 \rightarrow \mathbb R^3$ the lift of the semi-conjugacy $h$ between $f$ and $A$. Consider $\gamma \subset \widetilde{\mathcal W}^c(x)$, where $\widetilde{\mathcal W}^c(x)$ is the lift of $\mathcal W^c(x)$. Thus, $\gamma$ is inside the intersection of the center manifold of $\tilde{f}$ and the Pesin-stable manifold of $\tilde{f}$ passing through $x$.

Let us show that $\tilde{h}$ collapses $ \widetilde{\mathcal W}^{c}(x)$ to a unique point. If we prove that, it clearly comes out ( from the bounded distance of $h$ to identity that, the size of $ \mathcal W^{c}(x)$ is uniformly bounded. Suppose by contradiction that $h(\gamma)$ has more than one point. By semi-conjugacy $\tilde{h}(f^ n(\gamma)) = A^ n(\tilde{h}(\gamma)).$ As $\tilde{h}(\gamma)$ is a subset of weak unstable foliation of $\tilde{A}$ for large $n$ the size of $\tilde{A}^n(h(\gamma))$ is large. On the other hand, $\gamma$ is in the Pesin stable manifold of $f$ and consequently for large $n$, the size of $\tilde{f}^n(\gamma)$ is very small. As $\|\tilde{h} - id\| \leq K$ we conclude that for large $n$ the size of $\tilde{h}(\tilde{f}^n(\gamma))$ can not be very big. This contradiction completes the proof.
\begin{corollary}
The family $\{ \mathcal W^{c}(x) \}_{ \{x \; : \; \lambda^c(x) <0\} }$ forms a measurable partition.
\end{corollary}
This corollary uses the same idea of the proof of the Proposition \ref{prop:compact}. However, that proposition is proved for continuous foliations and we adapt the proof for the Pesin measurable lamination.

First of all we consider a new partition $\{ \overline{\mathcal W^c(x)} \}$ whose elements are the closure of the elements $\mathcal W^c(x)$, that is, $\overline{\mathcal W^c(x)}$ is a bounded length center segment with its extremum points. Since $h$ collapses the Pesin-stable manifolds of $f$ into points, two different elements $\mathcal W^c(x)$ and $\mathcal W^c(y)$ cannot have a common extrema, so that $\{\overline{\mathcal W^c(x)}\}$ is indeed a partition with compact elements. Let us prove that it is indeed a measurable partition.

Let $\{x_j\}_{j \in \mathbb{N}}$ be a countable dense set of $M = \mathbb{T}^3.$ For each $x_j$ and $k, l \in \mathbb{N}$ we define $C_l (x_j, k)$ to be the union of $\mathcal{W}^c(y),  y \in \Gamma_l$ such that $\mathcal{W}^c(y)$ intersects the closed ball $B(x_j, \frac{1}{k}).$ By continuity of $\mathcal{W}^c(.)$ on $\Gamma_l$ we conclude that $C_l (x_j, k)$ is closed and consequently measurable. Indeed, if $y_n \in C_l (x_j, k)$ converges to $y$ then $y \in \Gamma_l$ and moreover $\mathcal{W}^c(y)$ intersects the closure of $B(x_j, \frac{1}{k}).$

Now, we need to separate two weak stable manifolds by means of some $C_l (x_j, k).$ Taking two elements $\mathcal{W}^c(a)$ and  $\mathcal{W}^c(b)$ there exists $l \in \mathbb{N}$ such that $a , b \in \Gamma_l$ and it is enough to take small enough $k$ and some $x_j$ such that $C_l (x_j, k)$ contains $\mathcal{W}^c(a)$ and not $\mathcal{W}^c(b).$ It is easy to see that for each $x \in \Gamma$ we have $\mathcal{W}^c(x) = \bigcap_{k, l, j}C_l (x_j, k)^{*} $ where $C_l (x_j, k)^{*}$ is either $C_l (x_j, k)$ or $\mathbb{T}^3 \setminus C_l (x_j, k).$

Now, observe that if $y$ is an extremum point of $\mathcal W^c(x)$ then $y$ cannot have negative center Lyapunov exponent, otherwise it would be in the interior of $\mathcal W^c(x)$. So, the set of extremum points of the elements $\overline{W^c(x)}$ is inside the set of points with non negative center Lyapunov exponent, and therefore it has zero measure. Thus, removing such points, the measurability of the partition $\{\overline{\mathcal W^c(x)}\}$ implies the measurability of $\{\mathcal W^c(x)\}$, concluding the proof of the lemma.
\end{proof}

\begin{lemma}\label{lemma:gamma}
 There exist a set $\Lambda \subset \mathbb T^3$ and a real number $R > 0$,  such that $Vol (\Lambda) > 0.5$ and if $x \in \Lambda$, then $diam^c(\mathcal W^{c}(x)) > R$.
\end{lemma}
\begin{proof}
Comes from the fact that $\lim_{n \rightarrow \infty} Vol (\{x : diam^c (\mathcal W^c(x)) > 1/n\}) = 1$
\end{proof}

The next lemma is inspired on the work of Ruelle, Wilkinson \cite{RW}.

\begin{lemma}
Disintegration of volume on the measurable partition $ \{ \mathcal W^{c}(x) \}$ is atomic.
\end{lemma}

\begin{proof}
Let $\pi: \mathbb T^3 \rightarrow \mathbb T^3 / \{ \mathcal W^{c}(x) \}$ be the natural projection, $\nu := \pi_* Vol$ and $\Lambda$ as in Lemma \ref{lemma:gamma}. Consider $B= \pi (\Lambda)$ and $N$ be the  minimum number of balls with diameter $R/10$ needed to cover $\mathbb T^3$. We also denote $\{\eta_x\}$ the system of conditional measures of $Vol$ on the partition $\{\mathcal W^c(x)\}$ defined previously.
For $x \in \mathbb T^3$ define $$m(x) = inf \sum diam^c (U_i \cap \mathcal F^c_ f(x))$$
where the infimum is taken over all collections of closed balls $U_1, \ldots, U_k$ in $\mathbb T^3$ such that $k \leq N$ and $\eta_x (\cup_{i=1}^k U_j) \geq 0.5$.

We now define $$m=\mbox{ess} \sup_{x \in B \;}m(x).$$

We claim that $m=0$. Suppose, by contradiction that $m>0$. Then, given $\varepsilon>0$ there exist an integer $J$ such that
$$C \varepsilon^J N < m/2,$$ where $C>0$ is such that $d^c(f^i(y), f^i(z))\leq C \varepsilon ^i d^c(y,z)$ for all pair of points $y,z \in \Lambda$ with $z \in \mathcal W^c(y)$.

Let $\mathcal U$ be a cover of $\mathbb T^3$ by $N$ closed balls of diameter $R/10$. For $x \in \mathbb T^3$ such that $\pi(x)\in B$, let $U_1(x), \ldots, U_{k(x)}(x)$ such that $\eta_x(\cup_{i=1}^{k(x)}U_i(x)) \geq 0.5$. Note that

$$f^j_*\eta_x = \eta_{f^j(x)} \Rightarrow \eta_{f^j(x)}\left (\bigcup_{i=1}^{k(x)}f^j(U_i(x))\right) \geq 0.5. \; \forall i \in \mathbb N.$$

Also note that $ diam^c (f^j(U_i(x))) \leq \alpha \varepsilon^j.$

By Poincar\'e reccurence theorem, $y = f^{J_0}(x)$

\begin{eqnarray*}
 m(y) & \leq & \sum_{j=1}^{k(x)} diam (f^{J_0}(U_j(x))) \\
 &\leq & C k(x) \varepsilon^{J_0} \\
 & \leq & C N \alpha \varepsilon^{J_0} \\
 & < & m/2.
\end{eqnarray*}

Then, $m = ess \; sup_{x \in B}m(x) < m/2$, which is a contradiction with $m >0$.

Hence, $m=0$ implies that there is a sequence of closed balls with diameter going to zero and having measure greater then $0.5/N$. By Hammerlindl-Ures \cite{HU} we know that $f$ is ergodic, hence we have atomic disintegration.

\end{proof}

Once obtained atomicity we can apply Theorem \ref{theorem:oneatom} (see Remarks \ref{remark: nonnull} and \ref{remark:proof.theorem}) to get one atom per leaf.
\end{proof}

\section{Maximal entropy measure}

Given a volume preserving partially hyperbolic diffeomorphism $f: \mathbb T^3 \rightarrow \mathbb T^3$, we say that a measure $\mu$ is a {\bf maximizing entropy measure} if the metric entropy with respect to the measure is equal to the topological entropy of $f$, i.e,
$$h_{\mu}(f) = h_{top}(f).$$

Given a volume preserving partially hyperbolic diffeomorphism $f:\mathbb T^3 \rightarrow \mathbb T^3$ that is isotopic to a linear Anosov, the maximizing entropy measure is unique and is natural, in the sense that it is just the pull-back of the volume measure by the semi-conjugacy function $h$. More specifically, R. Ures showed in \cite{Ures}

\begin{theorem}
Let $f:\mathbb T^3 \rightarrow \mathbb T^3$ be an absolutely partially hyperbolic diffeomorphism homotopic to a hyperbolic linear automorphism $A:\mathbb T^3 \rightarrow \mathbb T^3$. Then, $f$ has a unique maximizing entropy measure $\mu_f$. More precisely, if $h$ is the semi-conjugacy between $f$ and $A$, i.e, $h \circ f = A \circ h$, then
$$m = h_{*}\mu_f$$

where $m$ denotes the Lebesgue measure on $\mathbb T^3$.
\end{theorem}

In the same article, R. Ures also showed that the center Lyapunov exponent for the maximizing measure entropy is greater or equal to the center exponent of the linear hyperbolic diffeomorphism. In the hypothesis of Theorem \ref{theorem:existatom} this fact contrasts with what happens to the center exponent with respect to the Lebesgue measure, and with this we prove in Theorem \ref{ugibbs} that $\mu_f$ cannot be $u$-gibbs.

\begin{theorem}[Theorem $5.1$ of \cite{Ures}] \label{theorem:raul}
Let $f:\mathbb T^3 \rightarrow \mathbb T^3$ be a $C^{1+\alpha}$ absolutely partially hyperbolic diffeomorphism homotopic to a hyperbolic linear automorphism $A$ with center Lyapunov exponent $\lambda^c_A > 0$. Let $\mu_f$ be the maximizing measure of $f$. Then, the center Lyapunov exponent of $\mu_f$, $\lambda^c_{\mu_f}$, satisfies
$$\lambda^c_{\mu_f} \geq \lambda^c_A.$$
\end{theorem}

To prove Theorem \ref{ugibbs} we need a celebrated result of Ledrappier-Young \cite{LY1}.

\begin{theorem}[Theorem $A$ of \cite{LY1}] \label{LY}
Let $f: M \rightarrow M$ be a $C^2$ diffeomorphism of a compact Riemannian manifold $M$ preserving a Borel probability measure $\mu$. Then $\mu$ has absolutely continuous conditional measure on unstable manifolds if, and only if,
$$h_{\mu}(f) = \int \sum_{i} \lambda^{+}_i(x) \dim E_i(x) d\mu(x)$$

where $a^{+}:= \max\{a,0\}$.
\end{theorem}

\begin{maintheorema}\label{ugibbs}
Let $f : \mathbb T^3 \rightarrow \mathbb T^3$ be a partially hyperbolic diffeomorphism with Anosov linearization $A: \mathbb T^3 \rightarrow \mathbb T^3$. Assume that $\lambda^c_A>0$ and $\lambda^c_f < \lambda^c_A$. Then $\mu_f$ is not $u$-gibbs. Also denoting by $\lambda^u_{\mu_f}, \lambda^c_{\mu_f}, \lambda^s_{\mu_f}$ the Lyapunov exponents of $f$ (for $\mu_f$ almost every point) then
$$\lambda^c_{\mu_f} > \lambda^c_A \text{ or } \lambda^u_{\mu_f} > \lambda^u_A .$$
\end{maintheorema}
\begin{proof}
By contradiction, assume that $\mu_f$ is $u$-Gibbs. Consider
$$A^{+}:=\{x\in M : \lambda^c(x) \text{ is defined and } \lambda^c(x) \geq \lambda^c_A \}.$$

By \ref{theorem:raul} we have that $\mu_f(A^{+}) = 1$. Since $\mu_f$ is $u$-gibbs, for some $x\in \mathbb T^3$ (actually for $\mu_f$ almost every $x$) the center unstable leaf $\mathcal F^{cu}(x)$ intersects the set $A^{+}$ in a set of positive leaf measure, that is,
$$m_{cu}(\mathcal F^{cu}(x) \cap A^{+}) >0.$$

Now, define $B=\mathcal F^{cu}(x) \cap A^{+}$ and set
$$C= \bigcup_{y \in B} \mathcal F^s.$$

Then, for all $y\in C$ we have that $\lambda^c(y) \geq \lambda^c_A$. Now by absolute continuity property, we get that
$$m(C)>0.$$

That is, we constructed a set of positive Lebesgue measure for which every point has center Lyapunov exponent bigger then $\lambda^c_A$ contradicting one of the hypothesis. So $\mu_f$ is not $u$-gibbs as we wanted to show.

Now, from Theorem \ref{LY}, given an invariant measure $\mu$ we can write
$$h_{\mu}(f) = \lambda^u_{\mu}\gamma_1 + \lambda^c_{\mu}\gamma_2$$

for some constants $0< \gamma_1,\gamma_2 \leq 1$ where
$$\gamma_1=\gamma_2=1 \Leftrightarrow \mu_f \text{ is }u-\text{gibbs}.$$

Also, we know that $h_{top}(f) = h_{top}(A)$. Then, since $\mu_f$ is the maximizing entropy measure it follows that
\begin{equation} \label{desig}
\lambda^u_{\mu_f}\gamma_1 + \lambda^c_{\mu_f}\gamma_2 = \lambda^u_A + \lambda^c_A.
\end{equation}

Since $\mu_f$ is not $u$-Gibbs, then $\gamma_1$ and $\gamma_2$ cannot be both $1$. So
$$\lambda^u_{\mu_f} + \lambda^c_{\mu_f}  > \lambda^u_{\mu_f}\gamma_1 + \lambda^c_{\mu_f} \gamma_2 = \lambda^u + \lambda^c.$$

So either $\lambda^c_{\mu}>\lambda^c$ or $\lambda^u_{\mu} > \lambda^u.$
\end{proof}

We end by analyzing the disintegration of the measure of maximal entropy.

\begin{maintheorema}\label{theorem:atomic.max.mesure}
Let $f:\mathbb T^3 \rightarrow \mathbb  T^3$ be a partially hyperbolic diffeomorphism with Anosov linearization. If the disintegration of the maximizing entropy measure $\mu_f$ on the central leaves is atomic then it has exactly one atom per leaf.
\end{maintheorema}
\begin{proof}
The proof is analogous to the proof of Theorem \ref{theorem:oneatom}, in this case we have to verify that
\begin{itemize}
\item $\mu_f$ is ergodic and

\item the $\mu_f(\cup_i \partial_c \widetilde R_i)=0$ (where $\partial_c \widetilde R_i$ as defined in the proof of Theorem \ref{theorem:oneatom}).
\end{itemize}

The first item was already observed by R. Ures in \cite{Ures}.
The second item comes from the definition of $\mu_f$. We know that the entropy maximizing measure $\mu_f$ is unique and that $h_{*}\mu_f = m$. Now, by the definition of the sets $\widetilde{R}_i = h^{-1}(R_i)$ we have that $\partial_c \widetilde{R}_i = h^{-1}(\partial_c R_i)$. Thus
$$\mu_f(\partial_c \widetilde{R}_i ) = \mu_f (h^{-1}(\partial_c R_i)) = h_{*}\mu_f(\partial_c R_i) = m(\partial_c R_i) =0.$$

The proof follows as in the proof of Theorem \ref{theorem:oneatom}.
\end{proof}

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
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Everyone knows timing is critical. However, it is as bad to be too early as too late. Business offers many areas in which moving too early can work against success. As a SCORE mentor, I have seen these at close hand.
Don't seek funding prematurely. You should start a business with enough money to get you through the first two years. You'll know how much if you've assembled a thorough, realistic business plan (you should never start without one). Two years will establish a business history you can present to investors, with cash flow and growth records that prove you have been successful. It would be prudent to delay your business start to save additional funds.
Don't rush to get a patent. Consider these points: First, a patent only has value if you have the money to fight for it in court. Few start-ups do. Pursuing the patent will distract you from what you really need to do: start producing the products your ideas enable. Balance the timing against the risk someone might independently develop (or copy) your idea. Focus on a patent only if you plan to license your technology. Early production might refine your patent, to strengthen its protection for your intellectual property.
Don't expand too rapidly. Businesses should grow, but each expansion stretches resources. Use your business plan to analyze and cost out what you will need. It is often productive to phase in product line growth in increments to allow support to grow at a similar, manageable rate. That lets your customers see a steady growth in your offerings without any interruption of your service.
Don't rush into an expensive ad program. Financial resources will be thin at the beginning. Don't waste them on something you don't need yet. You can generate tremendous exposure by aggressively using social media. And it's free! Repeat: FREE. Even though you feel time constrained, you actually have more time to spare than money. Twitter, Facebook and the like will not require massive time. Set up a strong, dynamic web page.
Some elements of your business should not be delayed. Never procrastinate. When you know something must be done, do it. Some delays can be painful, like failure to remit sales tax and employee payroll taxes on time. Prioritize and expedite.
Don't delay firing bad employees. Not all employees meet expectations. A constant complainer will eventually taint others. Shirkers are adept at off-loading tasks onto others. The victims could become complainers. The list grows: the chronically tardy, the frequently absent, the liquid lunch crowd and others are detrimental to your business. Strike fast. If you try to reform them, set a short deadline. If they're incorrigible, terminate them swiftly (after you have carefully documented their transgressions and your warnings. Don't explain why they're gone. Other employees will know without being told. Remember, your first responsibility is to your business, not to employees whose behavior could harm it.
Never hesitate to ask for help. You can't kick problems under the desk and hope they go away. If its beyond your skill set, find someone who can help. SCORE (part of SBA) has mentors who are ready to provide free counseling to small businesses who contact us. SBA offers inexpensive services through Small Business Development Centers at many community colleges. Don't fear to question a friend in a similar business. Good advice will let you attack problems as they arise, not when it's too late.
In most situations, listing the pluses and minuses of acting now or postponing will help you decide correctly. You can't run everything on the same time track. Prioritizing the timing of events is a critical business skill.