Los Angeles (CNN) -- As the manhunt for the renegade ex-cop accused of killing three people in a revenge plot targeting the Los Angeles Police Department enters its second week, the big question facing authorities is: Where is Christopher Jordan Dorner?

The search, considered one of the largest in the history of Southern California, has taken authorities from Orange County to the border of Mexico, from Los Angeles to the Big Bear Lake resort area of the San Bernardino Mountains.

Even so, a week after Dorner allegedly began targeting police officers and their families, putting the region on edge, there was no sign of the man on Monday.

The city of Los Angeles put up $1 million in reward money Sunday for help catching Dorner, an announcement that followed news that the LAPD was reopening the case that resulted in his termination.

Dorner accused his training officer of kicking a mentally ill man during an arrest in 2007. The LAPD ruled the complaint unfounded and booted Dorner off the force for filing a false complaint. He challenged his firing in court and lost.

In a manifesto released last week, Dorner blamed racism and corruption in the LAPD for his termination and vowed to wage "unconventional and asymmetrical warfare" against LAPD officers and their families. He called it a "last resort" to clear his name and strike back at a department he says mistreated him.

Authorities say Dorner began making good on his threats on February 4 when he allegedly killed Monica Quan and her fiance, Keith Lawrence, in an Irving parking lot, south of Los Angeles.

Quan is the daughter of a now-retired Los Angeles police officer, who represented Dorner during the disciplinary hearing that resulted in his firing. The officer was among dozens named in the manifesto.

The retired officer told investigators that he received a call from someone identifying himself as Dorner, who told him he "should have done a better job of protecting his daughter," according to a federal arrest warrant affidavit. Investigators traced the call to Vancouver, Washington, but based on the timing of other sightings, they don't believe Dorner was in Vancouver at the time, the affidavit states.

Days later, early Thursday morning, Dorner allegedly opened fire on two LAPD police officers, wounding one, in the suburban city of Corona.

Roughly 20 minutes later, Dorner allegedly fired on two officers in the nearby city of Riverside, killing one and wounding another.

Since then, the LAPD has provided more than 50 police officers and their families -- many of whom were named in the manifesto -- with security and surveillance details.

There has been speculation, based in part on an arrest warrant affidavit filed last week, that Dorner could have crossed state lines into Nevada or made his way to Mexico.

Federal authorities, meanwhile, were asking anyone across the country with information about Dorner or his whereabouts to contact their local FBI or U.S. Marshals Service.