Abstract
Recently,EdwardBurgerandhisco-authorsintroducedandstudied in Burgeretal.(2008) [3] a newclassofcontinuedfraction algorithms.Inparticulartheyshowedthatforeveryquadratic irrationalnumber x thereexistinfinitelymanyeventuallyperiodic N-expansionswithperiod-length1;seealsoKomatsu(2009) [10] forrelatedproperties.In2011,MaxwellAnselmandSteven Weintraubfurtherstudiedthepropertiesof N-expansionsin Anselm andWeintraub(2011) [2]. Oneniceresulttheyobtainedis that every x between0and N has uncountablymany N-expansions for eachinteger N 2. Inthispaperwewillreprovethisresultand fromthiswestudytheergodicpropertiesofvarioussubclassesof N-expansions.