OFFICIALS in Tehran are not shy about their aim of spreading influence abroad, nor of their apparent success. Even as the efforts of the West and its Sunni Arab allies look distinctly half-hearted, notably in their fight against Islamic State (IS), Tehran can claim, with only a pinch of hubris, to run three Arab capitals: Baghdad, Damascus and Beirut.

This week it may have added a fourth: Sana’a, Yemen’s capital, where on January 20th Shia Houthi rebels took over the presidential palace (see article). American and Saudi officials believe the rebel militia is backed by the Iranians, although they deny it in public (and boast of it in private).

The takeover in Yemen came soon after an Israeli drone strike exposed Iranian meddling in another part of the Middle East. The cross-border attack on the Syrian side of the Golan Heights killed six fighters from Hizbullah, the Iran-backed Shia party-cum-militia in Lebanon, days after the group denied that it was in the area. More surprisingly it also killed Mohammad Ali Allahdadi, an Iranian general. His presence suggests that Iran was trying to establish a presence in an area that has fallen out of Syrian government control and into the hands of rebels with whom Israel appears to be on friendly-enough terms. To its critics, Iran alarmingly holds sway from the Mediterranean Sea to the Fertile Crescent and the Gulf of Aden.

Iran’s biggest gains were handed to it by America when, after the terrorist attacks of September 11th 2001, it removed hostile regimes in Iraq and Afghanistan. Iran has continued to extend its influence, even after a wave of Sunni uprisings that started in 2011 seemed likely to weaken the Shia regime’s pull. The Quds Force, the foreign wing of Iran’s Revolutionary Guards, has exploited the region’s instability. Its tactics include assassinations and bombings overseas, and supplying arms and training to militias deemed helpful to its interests. “The Iranians are experts at taking advantage of chaos,” says Shimon Shapira, a retired military man now at the Jerusalem Centre for Public Affairs, a think-tank.

The rise of IS, a Sunni jihadist movement straddling Iraq and Syria, has only strengthened Iran. Nuri al-Maliki, Iraq’s former prime minister and an Iranian ally, was ousted in August, and American forces rode back into Iraq to rescue it from IS. But Iran has won much of the kudos. Iranian officials boast of being the ground force for America’s air strikes. Politicians—Iranian and Iraqi—talk of Baghdad being intact, and the government in place, thanks only to a ring of defences set up around the city by Qassem Suleimani, the head of the Quds Force; some call him “Supermani”.

Similarly Syria has fallen ever more under Iran’s spell. Where Hafez al-Assad, father of the current president, kept Iran at arms length, his son “sold Syria to the Iranians,” says a defected general. Mr Assad relies on Tehran for cash, advice and training for its paramilitary fighters. In Lebanon, Hizbullah’s military force rivals that of the country’s army, and it has maintained a tenuous military balance with Israel.

Not waving but drowning

Iran’s rising influence is a concern for Saudi Arabia and America. The Saudis are rich but mostly ineffectual and the Americans are reticent and often unwelcome outsiders. Whereas the Saudis are building a 600-mile-long wall along its border with Iraq to keep out IS militants, Iran has waded into Iraq to support Shia militias.

Yet Iran’s reach has limits. One motivation for Israel’s attack may have been to expose Iran’s covert activities at a time when America looks close to striking a deal with the country over its nuclear programme. In what was likely an effort to calm tempers, Israeli officials later said they did not know the Iranian general was present, but Hizbullah cadres in southern Lebanon threatened retaliation, possibly with guerrilla attacks over the border with Israel.

Iran, however, would probably not be able to sustain a significant escalation or open confrontation. Propping up the Syrian regime is reckoned to have cost billions of dollars that it can ill afford, now that oil prices have fallen and Western sanctions bite hard. Oil exports are down to less than half the pre-sanctions level of 2.5m barrels per day. Some in Lebanon say Iranian largesse to Hizbullah has been trimmed.

The growth of IS just over the border in Iraq has Iranian officials worried. Yet Sunni extremism is, in part, a consequence of Iran’s own policies. Mr Assad deliberately sought to destroy moderate rebels and stoke extremism to present himself as the only bulwark against jihadists. Disgust at the Syrian regime’s brutality has led Hamas, the Palestinian Islamist movement, to distance itself from its Iranian backers.

Moreover, Iran’s Shia allies are deepening sectarianism. Abuses by Iranian-backed Shia militias in Iraq are undermining attempts by the Baghdad government to woo the Sunni tribes whose support is needed to counter IS. And Hizbullah has lost favour around the Muslim world by moving away from its supposed raison d’être—resistance to Israel—and fighting instead to prop up Mr Assad.

Its meddling also undermines the overtures by Iran’s president, Hassan Rohani, and his foreign minister, Mohammad Javad Zarif, who want Iran to be accepted as a normal state—starting with a deal with America and other Western powers over its nuclear programme. Some in the West think a nuclear deal could be part of a grand bargain to stabilise the Middle East. But more likely Iran would simply pocket a deal on its nuclear programme and continue its current policy in the region. Indeed it is already hinting at doing so.

Some Iranian officials, who think Saudi Arabia has refused to cut its oil production in an attempt to weaken Iran, talk about making mischief there by stirring up the oppressed Shia minority in the east. “It’s not that we want to make trouble there,” says a political adviser to Ayatollah Ali Khamenei, Iran’s supreme leader. “But we have sent a message that if we wanted we could have the same benefits in the Gulf as we did in Iraq and Afghanistan.”