Featured Images Filtered in Home Page: how to get this link?

hi, i have a theme that display in home page liks to portfolio category and show the feature images of last created porfolio. if we click on category links the featured images are filtered so the home show only porfolio post of that category withoud recharge the page.

how is it possible to know how get the link at home with specific portfolio itemes? otherwise how get the link to Dynamic AJAX Galleries?