A G4 CREW ENRTE TO HOU BELIEVE A TURN FOR TFC MAY HAVE CAUSED THEIR FLT TO ENTER MEXICAN AIRSPACE.

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FLT WAS BEING OPERATED UNDER IFR FROM CRQ TO HOU AS A PVT FLT. DURING ON-COURSE STEP CLB WE WERE CLRED DIRECT TO IPL BY ZLA. SHORTLY AFTER THAT THE ZLA CTLR INSTRUCTED US TO TURN 20 DEGS R FOR TFC. WE COMPLIED WITH THE INSTRUCTION. A COUPLE OF MINS LATER THE CTLR RECLRED US DIRECT IPL. AS THE ACFT BANKED L I LOOKED OUT MY WINDOW AND NOTICED THAT WE WERE VERY CLOSE TO THE US/MEXICO BORDER. IT EVEN OCCURRED TO ME THAT WE MIGHT HAVE CROSSED IT AS A RESULT OF COMPLIANCE WITH THE VECTOR CLRNC. AT THIS TIME THE CTLR QUERIED US AS TO OUR HDG. WE RESPONDED BY STATING THAT WE WERE IN THE L TURN TOWARDS IPL AS PREVIOUSLY CLRED. THE CTLR THEN ISSUED A HDG OF 060 DEGS; NO REASON FOR THE HDG WAS ISSUED AS PART OF THAT CLRNC. WE ACKNOWLEDGED AND COMPLIED WITH THE CLRNC. BEFORE WE COULD COMPLETE THE TURN THE CTLR AMENDED THE HDG TO 050 DEGS. AGAIN WE ACKNOWLEDGED THE CLRNC AND COMPLIED WITH IT. A FEW MINS LATER (PERHAPS 5 MINS) THE CTLR CAME BACK ON AND ASKED US TO RECYCLE OUR XPONDER; CLAIMING THAT HE WASN'T RECEIVING IT RELIABLY. WE ACKNOWLEDGED THE REQUEST AND COMPLIED BY RECYCLING THE XPONDER. I GAVE IT A MIN OR SO AND THEN INQUIRED AS TO THE OP OF THE XPONDER. THE CTLR REPLIED THAT HE WAS RECEIVING IT PROPERLY. WITHIN 5 MINS THE CTLR WE HAD BEEN DEALING WITH WAS RELIEVED AND A NEW VOICE CAME OVER THE RADIO. IT OCCURRED TO BOTH THE FO AND ME THAT WE HAD PERHAPS INDEED TRAVELED INTO MEXICAN AIRSPACE AS A RESULT OF THE VECTOR THE CTLR HAD ISSUED AND THAT HE MAY NOT HAVE HAD AUTH TO PUT US THERE. WE SURMISED THAT THE INQUIRIES ABOUT OUR HDG AND ABOUT RECYCLING THE XPONDER SEEMED TO SUPPORT THE NOTION OF A CTLR ATTEMPTING TO COVER SOME TRACKS AND XFER RESPONSIBILITY FOR THE TRANSGRESSION EITHER TO US (SAYING WE WERE NOT FLYING THE HDG HE INSTRUCTED US TO FLY IN RESPONSE TO OUR HDG RPT AS REQUESTED); OR TO DIFFICULTIES WITH HIS EQUIP OR OURS (XPONDER/RADAR DIFFICULTIES).