DRESDEN, Germany — As Germany woke up to the news of a third attack by an asylum seeker within a week on Monday, German authorities faced uncomfortable questions.

Worries about what might be next fueled accusations of mistakes and misjudgments by authorities and leading politicians. Last year's influx of more than 1 million refugees into the country has become an uncontrollable risk, critics said, as mental-health care for those newly arrived lags behind and as security services say it might be impossible to prevent future attacks by individuals.

Above all, it is the frequency of the attacks that has shocked Germans. When the Syrian refugee detonated his explosive device in the southern German city of Ansbach, where he lived, on Sunday evening — killing himself and injuring at least 15 bystanders — it was a terrible end to a disturbing week. On Saturday, a Syrian refugee killed a woman in western Germany with a machete. Last Monday, a teenage Afghan refugee hit and injured passengers on a train near Würzburg with an ax before he was shot by police.

Sunday's attacker had applied for asylum in 2014, but his request was rejected last year. Asylum rejections are not uncommon in Germany: Authorities decline more than half of such requests.

But despite that, thousands of rejected asylum seekers are officially "tolerated" in the country — meaning that they are not considered as refugees but will also not be deported instantly. Germany does not deport refugees who are facing health issues or who could be threatened in their countries of origin.

The Syrian who killed himself and injured others on Sunday was also allowed to stay in the country, although he was set to be deported to Bulgaria. It is unclear whether plans to deport him had only recently emerged or whether they had stalled.

It is also unclear why he was set to be deported to Bulgaria. German media outlets speculated Monday that the Syrian was accused of crimes there. But the fact that he was not deported might also point to the possibility that he entered the European Union through that country in Eastern Europe. Until the middle of last year, it was common practice for E.U. countries to deport those refugees whose applications had been rejected back to the E.U. member states through which they first entered the union, often to eastern Europe or Italy. The so-called Dublin agreement was controversial — especially in the south of Europe where most refugees enter the continent — and complaints led to a rethinking of that practice among some northern countries last year. German Chancellor Angela Merkel abandoned that rule in August for Syrians, saying they would not be deported if they reached Germany.

Regardless of the specifics of why the attacker was not deported, statistics show that the he was among many refugees who were allowed to stay in the region of Bavaria despite having their asylum requests rejected. Two of the three attacks launched in the past seven days occurred in Bavaria.

Bavaria is considered rather strict in deporting individuals who failed to receive a status as refugees. Whereas more than 45 percent of all rejected asylum seekers were still in the country more than one year after they were ordered to leave, the German average among all regions is 55 percent.

In the city of Bremen, more than 90 percent of those who were supposed to leave stayed.

Those "tolerated" refugees have little governmental support, although they face an uncertain future — conditions that might fuel radicalization, experts say.

"It could also be a mixed process: of post-traumatic stress disorder developed during the journey together with jihadi ideology, which would quickly turn depression, frustration, anger, sense of helplessness and guilt into violent action," said Daniel Koehler, director of the German Institute on Radicalization and De-Radicalization Studies and a fellow at George Washington University’s Program on Extremism.

German authorities lag behind in fighting radicalization, critics say. "For a period, German authorities struggled to provide shelter and food," Koehler said, referring to last year, when thousands of refugees arrived in the country every day. Extensive counter-radicalization programs emerged much later on and still lack coherence.

Koehler also questioned whether authorities had thoroughly checked all newly arrived refugees to the extent they said they have. "It is simply impossible for the German authorities to check with Syrian or Iraqi authorities if the claimed identity is correct or if there are any known criminal charges against that person," he said. "For German authorities, the life of a refugee ‘starts’ the moment he or she enters the country."

And even when authorities decide it is time for someone to leave, many of them still stay.

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