Pope Francis arrives in New York next week on a wave of good will, with favorability ratings among both American Catholics and the population in general that would be the envy of any U.S. president.

But a new survey of American Catholics by the Pew Research Center shows many U.S. adults who were raised Catholic have since left the church, and many of those who remain disagree with church leaders on a number of important issues.

The Pew survey finds that substantial majorities of American Catholics are remarkably accepting of family and living arrangements long frowned upon by the church. For instance, more than 8 in 10 Catholics say it is "acceptable" for unmarried couples have children. Even gay and lesbian couples are seen by most Catholics (66 percent) as acceptable parents.

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Catholics who say they attend Mass every week tend to be slightly less accepting of these nontraditional family arrangements. But even among these regular churchgoers, majorities are open to divorced parents as well as cohabiting and even same-sex couples raising children.

This high level of acceptance may be due in part to the fact that many Catholics, like Americans of all stripes, have themselves experienced different types of family and living arrangements. More than 4 in 10 (44 percent) say they have lived with a romantic partner outside of marriage.

Most American Catholics say the church should allow unwed couples who live together (61 percent) or those who have divorced and remarried without an annulment (62 percent) to receive Communion, and substantial majorities also disagree with the church's teachings in other areas. Sixty-two percent in the Pew survey say priests should be allowed to marry and 59 percent say the church should ordain women. Seventy-six percent say the church should lift its prohibition on the use of artificial birth control, and 46 percent say the church should recognize the marriages of gay and lesbian couples.

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Different Catholics

The pope also will find a church that has been shrinking as a share of the U.S. population. Indeed, in a 2014 Pew poll, 21 percent of U.S. adults said their religion is Roman Catholic. This is down from 24 percent in a 2007 Pew survey.

Many who identify themselves as Catholic remain loyal to the church. Seven in 10 say that, no matter what, they could not imagine leaving the Catholic Church. In addition, 4 in 10 say they attend Mass weekly or more and a whopping 95 percent say they go to Mass at least sometimes.

U.S. Catholics also see Pope Francis in a positive light, with 86 percent in the new Pew poll saying they view him favorably and 69 percent saying he is changing the church for the better.

At the same time, 52 percent all adults who were raised Catholic have left the church at some point in their lives. While some have later returned (11 percent of all those raised Catholic), 41 percent in the survey said they have not.

Further, most of those who were raised Catholic but have left the church and not returned (about two-thirds) no longer identify with Catholicism. Only about 8 percent of these ex-Catholics say they could see themselves returning to the church at some point.

Others still identify with Catholicism in one way or another even though it is not their primary religious affiliation. These so-called "cultural Catholics" -- who make up 9 percent of all U.S. adults -- either belong to another faith tradition (often Protestantism) or have no religious affiliation, but also say they consider themselves at least partially Catholic.

A diverse Church

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Francis' travels in the United States will be entirely in the Northeast -- in New York, Philadelphia and Washington, D.C. While the Northeast is still more heavily Catholic than the nation as a whole, there has been a shift in the Catholic population in recent years from the Northeast and Midwest to the South and West.

This regional shift comes at a time when the church has become more Hispanic. Indeed, according to a 2014 Pew poll detailing the nation's religious landscape, 34 percent of American Catholics are Hispanic, an increase of 5 percentage points from a similar 2007 poll.

Even with this regional shift, New York remains home to one of the largest shares of Catholics among the most populous U.S. metropolitan areas. A third of residents of the New York area are Catholic (33 percent), higher than Philadelphia's 26 percent and Washington's 19 percent.

David Masci is a senior writer and editor at Pew Research Center, where he specializes church-state issues, culture war issues, and religion and science.