My ex-wife has totally broken her promise under the judgment authorizing our divorce agreement in the States,to provide a bank guarantee to ensure she resepcts my visitation rights with our children after she relocates to Israel with them. What legal action can I take in Israel against her?

Firstly, if you have not yet had he US judgment "mirrored" in Israel, then you should take action to do so, under the 1958 Recognition of Foreign Judgments Act. Once the settlement is recognized in Israel, it will then become enforceable, and you can take action to enforce it, both in relation to  your visitation rights, and to the financial guarantee. Another option, once the settlement agreement is recognized in Israel, is to file for contempt of court, of the Israeli judgment.

In June 2014 . representing the father, our law practice filed to enforce a "Mirror Order" obtained in Israel, of a South African settlement agreement, from 2013, which had been recognized by Tel Aviv Family Court ,concerning the relocation of a 5 year old boy, to Israel. The  mother had relocated to Israel, with the minor , as agreed, but had failed to provide the $100.000 bank guarantee which she had undertaken to produce. The proceedings, which also included an application for contempt of court ,ended with a compromise whereby the $100,000 was provided by a combination of a bank guarantee and a lien on a relative’s apartment in Israel, with the court exercising its discretion regarding costs, separately, which it did, ruling in favour of the father.