Dear if your only business is renting a home?
If you were a Florida LLC engaged in the same practice in New York, you need to register the foreign corporation. N.Y. BSC. LAW § 1301.
The Florida statute is similar to New York. See: http://www.flsenate.gov/Laws/Statutes/2012/608.501
You may be correct that mere ownership in the name of the New York LLC is not doing business as defined in the Florida statute;
"***608.501 Foreign limited liability company; authority to transact business required.—(2) The following activities, among others, do not constitute transacting business within the meaning of subsection (1):***(m) Owning, without more, real or personal property***", but you will not "win" your argument posting in the New York forum of AVVO, and to "win" in Florida, you will need an attorney.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.
You may want to repost your inquiry in the Florida section, as your question pertains directly to Florida law as opposed to NY.
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I suggest you ask a Florida attorney so try re-posting the question in the Florida Avvo section.
I am a former federal and State prosecutor and now handle criminal defense and personal injury/civil rights cases. Feel free to check out my web site and contact me at (212) 577-9797 or via email at Eric@RothsteinLawNY.com. I was named to the Super Lawyers list as one of the top attorneys in New York for 2012. No more than 5 percent of the lawyers in the state are selected by Super Lawyers. The above answer is for informational purposes only and not meant as legal advice.